Research ArticleDirection of Arrival Estimation Based on the MultistageNested Wiener Filter

Xiaodong He and Bin Tang

School of Electronic Engineering University of Electronic Science and Technology of China Chengdu 611731 China

Correspondence should be addressed to Xiaodong He winter hehotmailcom

Received 23 October 2014 Revised 3 January 2015 Accepted 5 January 2015

Academic Editor Zujun Hou

Copyright copy 2015 X He and B Tang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel direction of arrival (DOA) estimation technique based on data level and order recursive Multistage Nested Wiener Filters(MSNWF) which is used in adaptive beamforming for subarray signal is proposed in this paper The two subarrays using the samearray geometry are used to form a signal whose phase relative to the reference signal is a function of theDOATheDOA is estimatedby calculating the phase-shift between the reference signal and its phase-shifted version The performance of this DOA estimationtechnique is significantly improved due to the application of order recursive MSNWF for the rejection of interference signals Thecomputation of the proposed method is simple and the number of detectable signal sources could exceed the number of antennaelements

1 Introduction

In the last two decades smart antenna has been widelyused in many applications such as radar sonar and wirelesscommunication systems [1] It is also utilized in tracking[2 3] localization [4 5] intelligent transportation [6] ultra-wideband wireless sensor networks [7] array calibration [8]scatter cluster model [9] and antijamming [10] For examplethe multiple input multiple output (MIMO) radar utilizesmultiple sensor array antennas to simultaneously transmitand receive diverse waveforms which estimates the signalparameters to locate and track the target [2] Distributedsensor networks have been used for enhancing signal to noiseratios for space-time localization and tracking of remoteobjects using phased array antennas [4] Radio frequencyidentification (RFID) is widely used for electronically identi-fying locating and tracking products animals and vehiclesas a very valuable business and technology tool [5] Vehicularad hoc networks (VANETs) could be a benefit to the trafficsafety and efficiency [6] The performance of array pro-cessing algorithms is improved by the sensor array locationerror calibration which made the algorithms insensitiveto the model uncertainties and deterministic signals withunknown waveforms [8] The performance of the wirelesscommunication system is evaluated based on scatter cluster

models by estimating the corresponding parameters [9]Array sensor and subarray adaptive beamforming techniquesobtain the best antijamming performance widely used inGNSS receivers [10] active radar and sonar [11] In thesesensor networks implication systems and scenarios directionof arrival (DOA) is an important parameter that is needed tobe estimated to determine the direction of the located andtracked target or the position of the sensor nodes

Considerable research efforts have been made in theDOA estimation and various array signal process techniquesfor DOA estimation have been proposed [12ndash18] The mostcommonly used DOA estimation techniques include (1)spectrum based methods such as Bartlett [14] and Capon[15] (2) subspace-based algorithm such as multiple signalclassification (MUSIC) [16] (3) parametric methods suchas estimation of signal parameters via rotational invariancetechnique (ESPRIT) [17] In Capon techniques the DOAs aredetermined by finding the directions in which their antennaresponse vectors lead to peaks in the spectrum formed bythe covariance matrix of the observation vectors Thus thecapacity of this DOA estimation technique is less than thenumber of antenna elements bounded by the covariancematrix of the observation vectors In MUSIC techniquesthe DOAs of target signals are determined by finding thedirections in which their antenna response vectors lead to

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 634637 11 pageshttpdxdoiorg1011552015634637

2 International Journal of Distributed Sensor Networks

peaks in the MUSIC spectrum formed by the eigenvectorsof the noise subspace Thus the capacity of this DOAestimation is equal to the rank of the reciprocal subspaceof the selected noise subspace and is also less than thenumber of the antenna elements In ESPRIT techniques twovirtual subarrays structures are proposed to obtain two signalsubspaces The eigenvectors of the relevant signal subspacesare rotated for the DOAs of the signals As a result thecapacity of DOA estimation using ESPRIT is bounded by thenumber of subarraysThe application of the above techniquesis limited to cases where the number of signal sourcesis less than that of antenna elements These techniquesrequire subspace estimation eigendecomposition and thecomputation of covariance matrix inversion which leadsto high computational complexity and they are therebylimited to the applications where fast DOA estimation isrequired Furthermore in the presence of interference thesetechniques need to estimate the DOAs of all the target signalsand interference which decreases the accuracy of DOAestimation

The application of adaptive beamforming in DOA esti-mation has become the research focus on interference exis-tence [18] In [18] Wang et al developed a new structureof DOA estimation based on subarray beamforming Thistechnique has clear advantage on the DOA estimation wheninterference exists but it still needs the computation ofmatrixinversionwhich is not easy to be applied to a practical systemBased on this structure a DOA estimation technique basedon Multistage Nested Wiener Filter (MSNWF) [19ndash25] isproposed in [26] In [26] Yu stated an original MSNWFalgorithm [27] to estimate the DOAs which used a filter andblocking matrix to avoid the calculation of covariance matrixinversion In this technique however it cannot calculatethe coefficients of Wiener filter in backward recursion ifthe forward recursion does not finish the calculation of thematch filter and blockingmatrix And themean squared error(MSE) can not be determined when adding a new stageexcept for the last stage

In this paper a data level order recursive MSNWFDOA estimation technique that uses a reference signal isproposed in detail This DOA estimation technique uses twosubarray adaptive beamformers based on the data level orderrecursive MSNWF to construct the same array geometry forforming the phase-shift and rejecting interference at sametime The DOAs of the target signals are estimated from thephase-shifts by using reference signal after the rejection ofinterference Therefore the performance of DOA estimationis significantly improved This technique can be widely usedfor the implementation of hardware systems such as wirelesscommunication system active radar sonar and space-timeadaptive process (STAP) systems [28 29]

The advantages of the data level order recursive MSNWFDOA estimation are as follows (1) Since the use of data levelorder recursive MSNWF in this DOA estimation techniquerealizes the subspace eigendecomposition computation ofinversion of covariance matrix becomes unnecessary andthus reduces the complexity of computation the data levelMSNWF DOA estimation technique can be easily applied inhardware platform (2) An orthogonal basis for the Krylov

subspace spanned cross correlation vector and covariancematrix improves the computational efficiency when calcu-lating the weight vector of the match filter And the orderrecursion could update the weight vector of the match filterand the MSE at new stage

The paper is organized as follows In Section 2 the signalmodel is described In Section 3 the structure of the data levelorder recursive MSNWF DOA estimation system MSNWFbased adaptive beamforming including data level recursionof match filters and order recursion and DOA calculationof the proposed method are presented Design examples andsimulation results are given in Section 5 and conclusions aredrawn in Section 6

2 Signal Model

Consider a uniform linear array (ULA) system that uses 119872elements with adjacent element spacing 119889 deployed at a basestation Assume that 119870 narrowband signals and 119875 unknowninterference sources are received at the ULA with differentDOAs 120579119896 119896 = 1 2 119870 + 119875

Using complex envelope representation the receivedsignals can be expressed by

x (119905) =119870+119875

sum

119896=1

a (120579119896) 119904119896 (119905) + n (119905) (1)

where 119904119896(119905) denotes the 119896th signal component 119896 = 1 2 119870

denotes the target components and 119896 = 119870+1119870+2 119870+

119875 are interference components The a(120579119896) in (1) denotes thesteering vector of the array in direction 120579119896 which is given by

a (120579119896) = [1 119890minus1198952120587119889 sin(120579119896) 119890minus1198952120587119889(119872minus1) sin(120579119896)]

119879 (2)

and n(119905) denotes the noise vector with zero mean and crosscovariance

119864 [n (1199051)n119867(1199052)] = 120590

2120575 (1199051 minus 1199052) I (3)

where I is the identity matrixSuppose that the received vector x(119905) is sampled at 119899 119899 =

1 2 119871 and the received signal can be expressed by (4) inthe matrix notation Consider

X = A (120579) S + N (4)

where X and N are119872times 119871matrices

X = [x (1) x (2) x (119871)]

N = [n (1) n (2) n (119871)] (5)

A(120579) is a119872times119870matrix as follows

A (120579) = [a (1205791) a (1205792) a (120579119870)] (6)

And S is a 119870 times 119871matrix

S = [s (1) s (2) s (119871)] (7)

International Journal of Distributed Sensor Networks 3

3 MSNWF DOA Estimation

Compared with the SBDOA estimation technique stated in[8] the proposed MSNWF DOA estimation technique inthis paper uses the same uniform linear antenna array atthe receiving end and the geometry of the array is similarto that used in ESPRIT techniques The antenna array isdecomposed into two equal-sized subarrays where the twosubarrays are used in conjunctionwith two subarrayMSNWFadaptive beamformers to obtain an optimal estimation of aphase-shift reference signal whose phase relative to that ofthe reference signal is a function of the target DOA Thetarget DOA is then computed from the estimated phase-shift between the reference signal r119896 and the phase-shiftedreference signal 119890119895120601119896r119896 In order to avoid the inversioncomputation of covariance matrix when getting the optimalweight vector of the beamformer the two beamformers as inFigure 1 in [18] are replaced with Multistage Nested WienerFilters The block diagram of the MSNWF DOA estimationsystem is illustrated in Figure 1

31 Subarray Signal Formation Consider that the array iscomposed of a ULA of119872 element as a receiver and decom-posed into two sets of119872minus1 element virtual subarraysA andB The downconverted baseband signal received by the 119898th119898 = 1 2 119872 element of the antenna array is expressed by

119909119898 (119899) =

119870+119875

sum

119896=1

119890119895(119898minus1)120601119896119904119896 (119899) (8)

The vectors of the A and B are given by

y119860 = [1199091 (119899) 1199092 (119899) 119909119872minus1 (119899)]119879

y119861 = [1199092 (119899) 1199093 (119899) 119909119872 (119899)]119879

(9)

respectively Let

b (120579119896) = [1 119890119895120601119896 119890

119895120601119896(119872minus2)]119879 (10)

and then the subarray signals y119860 and y119861 can be written as

y119860 (119899) =119870+119875

sum

119896=1

b (120579119896) 119904119896 (119899) + n119860 (119899)

y119861 (119899) =119870+119875

sum

119896=1

119890119895120601119896b (120579119896) 119904119896 (119899) + n119861 (119899)

(11)

where vectors n119860(119899) and n119861(119899) are the background noise atthe subarray respectively The phase-shift factor between the119896th components of signals y119860(119899) and y119861(119899) which forms the119896th signal is given by

119890119895120601119896 = 119890

minus1198952120587119889 sin(120579119896)120582 (12)

Sampling y119860(119899) and y119861(119899) obtains

Y119860 = [y119860 (1) y119860 (2) y119860 (119871)]

Y119861 = [y119861 (1) y119861 (2) y119861 (119871)] (13)

Adaptive beamformer AReferencesignals rk

Phase-shiftcomputationWeight vectors wk

MSNWF

MSNWF

Adaptive beamformer B

1

2

M

yA

yB

k

rk

Figure 1 Block diagram of the MSNWF DOA estimation system

32 Recursion Algorithm of MSNWF

321 Data Level Recursion of Match Filters In the Wienerfilter the estimation of the desired signal 1198890(119899) from anobservation vector x0(119899) is optimal in the minimum meansquare error (MMSE) sense The weight vector wX0 of theWiener filter can be obtained via solving the followingWiener-Hopf equations

Rx0wx0 = rx0d0 (14)

where Rx0 is the covariance matrix of observation vectorx0(119899) and rx0 is the cross correlation vector between theobservation vector x0(119899) and the desired signal 1198890(119899) Thecovariance matrix Rx0 cannot be readily estimated if x0(119899)is of high dimension Based on this Goldstein and Reedproposed that if the observation signal x0(119899) is prefiltered bya full-rank matrix T isin C119872times119872 to get a new observation signalz1(119899) = Tx0(119899) then the weight factor wz1 of Wiener filter isused to estimate the desired signal 1198890(119899) from z1(119899) results inthe same MSE [19ndash21]

The assumed full-rank prefilter matrix can be chosen as

T1 = [h1198671

B1] (15)

where119867 is the complex conjugate transpose operator Thus

z1 (119899) = [h1198671x0 (119899)

B1x0 (119899)] = [

1198891 (119899)

x1 (119899)] (16)

where B1 is referring to the blocking matrix B1h1 = 0 andh1 = rx0d0rx0d02

The solution of theWiener-Hopf equations relative to thetransformed system is

wz1 = Rminus1z1 rz1d0 = 1205721 [

1

minusRminus1x1 rx1d1] (17)

where Rz1 is the covariance matrix of the new observationsignal z1(119899) 1205721 = rx0d02(12059021198891 minus r119867x1d1R

minus1

x1 rx1d1) rz1d1 isthe cross correlation between d1(119899) and z1(119899) and Rx1 =

B1Rx0B1198671 1205902

1198891= h1198671Rx0h1 rx1d1 = B1Rx0h1

This process produces a new vector Wiener filter whichestimates the signal 1198891(119899) from the observation vector x1(119899)and a scalar Wiener filter is followed Repeating this process

4 International Journal of Distributed Sensor Networks

1205760(n)

d0(n)

x0(n)

++

++

minus

minus

d0(n)

d1(n)

d1(n)

1205761(n)

1205721

x1(n)

h1

B1 w1

Figure 2 First stage of the original MSNWF

x0(n)

tM

t2

t1d1(n)

d2(n)

dM(n)

d2(n)

d1(n)

d0(n)1205721

1205722

120572M

++

minus

++

minus

dMminus1(n)

Figure 3 Match filter bank structure of MSNWF

a nested structure can be obtained which is defined as theoriginal MSNWF [19ndash21]

In the original MSNWF the new desired signal 119889119894(119899) atthe output of the 119894th stage can be expressed as

119889119894 (119899) = x0 (119899)(119894minus1

prod

119896=1

B119867119896) h119894 = x119894minus1 (119899) t119894 (18)

According to (18) a filter t119894 is used to replace the 119894th stageWiener filter as Figure 2 shows which could be simply thecross correlation between the new observation x119894(119899) and thenew desired signal 119889119894(119899)

The new observation vector in Figure 3 is expressed as

d (119899) = [1198891 (119899) 1198892 (119899) 119889119873 (119899)]119879 (19)

which proved that this observation vector has a tridiagonalcovariance matrix [21]

Therefore the new desired signal 119889119894(119899) can be seen that itis the output of an119873 length filter t119894

t119894 = (

119894minus1

prod

119896=1

B119867119896) h119894 (20)

The filter t119894 is used to recover all the information of x119894minus1(119899)via 119889119894minus1(119899) The output 119889119894(119899) is gotten by the filter t119894+1 thus119889119894(119899) is correlated with 119889119894minus1(119899) and 119889119894+1(119899) However 119889119894+1(119899)is from the blocking matrix B119894+1 which is not correlatedwith 119889119894minus1(119899) Therefore 119889119894(119899) is only correlated with its twoneighbors And it is also required to be maximally correlatedwith 119889119894minus1(119899) Considering the orthogonality conditions themaximal correlation results in an optimization problem [25]as follows

t119894 = argmaxt

119864 [119889119894 (119899) 119889lowast

119894minus1(119899)]

st t119867t = 1 t119867t119896 = 0 119896 = 1 2 119894 minus 1

(21)

Using Lagrange multipliers the solution of (21) is

t119894 =(prod119894minus1

119896=1P119896)Rx0t119894minus1

10038171003817100381710038171003817(prod119894minus1

119896=1P119896)Rx0t119894minus1

100381710038171003817100381710038172

(22)

where P119896 = I119873 minus t119894t119867119894 Herein if B119894 is assumed to be equal to P119894 the filters t119894 are

an orthonormal basis for the Krylov subspace generated byrx0d0 and Rx0 [22] Therefore the result of recursion of theMSNWF can be obtained without B119894

At 119894th stage let

u119894 = Rx0t119894minus1 (23)

The filters t119894 of the recursion are computed as

t119894 = u119894 minus (t119867

119894minus1u119894) t119894minus1 minus (t

119867

119894minus2u119894) t119894minus2 (24)

In the recursion calculation process the filters t119894 arecalculatedwhich does not needB119894 and the inversion of covari-ance matrix and this reduces the complexity of computationThe calculation of t119894 only needs the last two members whichalso reduces the complexity of computation

322 Order Recursion At the stage (119872 minus 1) of the MSNWFthe orthogonal basis composed by the matcher filters t119894 isexpressed as

T(119872minus1) = [t1 t2 t119872minus1] (25)

The new observation vector obtained from the recursioncalculation can be written as

d(119872minus1) (119899) = [1198891 (119899) 1198892 (119899) 119889119872minus1 (119899)]

= [t1198671x0 (119899) t

119867

2x0 (119899) t

119867

119872minus1x0 (119899)]

= (T(119872minus1))119867

x0 (119899)

(26)

The covariance matrix can be written as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

(27)

The recursion coefficients are the components of Wienerfilter coefficients as (28) which is used to estimate 1198890(119899) fromd(119872minus1)(119899)

w(119872minus1)d = (R(119872minus1)d )minus1

r(119872minus1)dd0 = (R(119872minus1)d )minus1

(T(119872minus1))119867

rx0d0(28)

Then the coefficients of MSNWF can be expressed as

w(119872minus1)0

= T(119872minus1)w(119872minus1)d (29)

The MSE of the coefficients is

MSE(119872minus1) = 1205902

1198890minus rx0d0w

(119872minus1)

0(30)

which is updated withw(119872minus1)d and theMSE(119872minus1) at stage (119872minus

1) and with w(119872minus2)d and MSE(119872minus2) from the (119872 minus 2) stage

International Journal of Distributed Sensor Networks 5

According to (19) and its property the tridiagonal covari-ance matrix can be rewritten as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

= [

R11 R12R21 119903119872minus1119872minus1

] (31)

where

R11 = (T(119872minus2))119867

Rx0T(119872minus2)

R12 = [0119879 119903119872minus2119872minus1]119879

R21 = [0119879 119903lowast

119872minus2119872minus1]

(32)

The cross correlation vector between the new observationvector and desired signal 1198890(119899) is

r(119872minus1)dd0 = (T(119872minus1))119867

rx0d0 = [

1003817100381710038171003817rx0d010038171003817100381710038172

0] (33)

Given R(119872minus2)d from stage (119872 minus 2) the new elements ofR(119872minus1)d are calculated as

119903119872minus1119872minus1 = t119867119872minus1

Rx0t119872minus1

119903119872minus2119872minus1 = t119867119872minus2

Rx0t119872minus1(34)

According to (26) the (34) can be rewritten as

119903119872minus1119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus1(119899) 119889119872minus1 (119899)

119903119872minus2119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus2(119899) 119889119872minus1 (119899)

(35)

Consider the property that only the first element of thecross correlation vector 119903(119872minus1)

1198891198890is not equal to 0 Therefore

only the first column of the inverse of R(119872minus1)d is needed tocalculate the recursion coefficients via (28)

Let the inverse of R(119872minus1)d be noted as

C(119872minus1) = (R(119872minus1)d )minus1

= [c(119872minus1)1

c(119872minus1)2

c(119872minus1)119872minus1

]

= [

C(119872minus2) 0

0119879 0

] + 120573minus1

119872minus1b(119872minus1) (b(119872minus1))

119867

(36)

The various quantities in (36) are defined as in thefollowing equation

b(119872minus1) = [119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

]

120573119872minus1 = 119903119872minus1119872minus1 minus1003816100381610038161003816119903119872minus1119872minus1

1003816100381610038161003816

2119888(119872minus2)

119872minus2119872minus2

(37)

x0(n)

d0(n)

w1 w2

d1(n) d1(n)

x1(n)+

+

minus

++

minus

++minus +

+minus +

+minusMSE0

MSE1MSE2

tH1 tH2t1 t2

Figure 4 The structure of data level order recursive MSNWF

where 119888(119872minus2)119872minus2119872minus2

is the last element of the last column c(119872minus2)119872minus2

from the previous stageTherefore the column vector c(119872minus1)

1of C(119872minus1) can be

calculated as

c(119872minus1)1

= [c(119872minus2)1

0

] + 120573minus1

119872minus1(119888(119872minus2)

1119872minus2)lowast

[

1003816100381610038161003816119903119872minus2119872minus11003816100381610038161003816

2 c(119872minus2)119872minus2

minus119903lowast

119872minus2119872minus1

]

(38)

where 119888(119872minus2)1119872minus2

is the first element of c(119872minus2)119872minus2

It can be seen from (38) that c(119872minus1)

1depends on c(119872minus2)

1

from stage (119872 minus 2) and the new elements 119903119872minus2119872minus1 and119903119872minus1119872minus1 generated from the covariance matrix at stage (119872minus

1) And the coefficients of the Wiener filter w(119872minus1)d are alsodepending on that Moreover the last column vector c(119872minus1)

119872minus1

of C(119872minus1) depends on the last column c(119872minus2)119872minus2

from stage (119872minus

2) According to (36) the last column vector c(119872minus1)119872minus1

can beupdated as

c(119872minus1)119872minus1

= 120573minus1

119872minus1[minus119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

] (39)

It can be seen from (39) that the last column vector c(119872minus1)119872minus1

is only depending on the last column vector c(119872minus2)119872minus2

from stage(119872 minus 2) and the new element 119903119872minus2119872minus1 generated from thecovariance matrix at stage (119872 minus 1)

According to (38) and (39) it can be seen that in recursivecalculation process only c(119872minus1)

1and c(119872minus1)

119872minus1are needed to

be updated at each stage This avoids the calculation ofthe inversion of covariance matrix which also reduces thecomplexity of computation

As for the MSE expressed as in (30) it can be simple andcan be updated with 119888

(119872minus1)

11generated from the covariance

matrix at stage (119872 minus 1) as follows

MSE(119872minus1) = 1205902

1198890minus1003817100381710038171003817rx0d0

1003817100381710038171003817

2

2119888(119872minus1)

11 (40)

According to recursive algorithm about the calculation ofthe coefficients of thematch filters and the nest order the datalevel order recursive MSNWF DOA estimation structure canbe drawn as in Figure 4

4 MSNWF DOA Estimation System

41 Calculation of Weight Vector In the MSNWF DOAsystem the optimal estimation of the phase-shifted referencesignal 119890119895120601119896r119896 in the minimummean square error sense can be

6 International Journal of Distributed Sensor Networks

obtained at the output of the adaptive beamformer B whichuses the adaptive beamforming weights obtained from theadaptive beamformer A with the MSNWF structure

In the adaptive beamformer B consider the case wherethe phase-shifted reference signal 119890119895120601119896r119896 is the desired signaland the output of the adaptive beamformer B can be usedto estimate the desired signal Since the phase-shifted 119890

119895120601119896

is unknown both the phase-shifted reference signal and theweight vector of the adaptive beamformerB are not availableHowever the weight vector of the adaptive beamformer Bcan be obtained from the optimal weights of the adaptivebeamformer A

In the adaptive beamformer A the desired signal andobservation vector can be given by

1198891198600 (119899) = 119903119896 (119899) x1198600 (119899) = y119860 (119899) (41)

The optimal weight vector of adaptive beamformerA canbe readily obtained according to (42)

The flow diagram of calculation of weight vectors inadaptive beamformer A is as follows

rx1198600d1198600 = 119864 [x1198600 (119899) 119889lowast

1198600(119899)]

11990301119861 = 0 119888(1)

1119860= 119903minus1

11119860

MSE(1)119860

= 1205902

1198890119860minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(1)

1119860

FOR 119894 = 2 3 119872 minus 1

t119898119860 =119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) x119898minus1119860 (119899)

119889119894119860 (119899) = t119867119898119860

x119898minus1119860 (119899)

x119898119860 (119899) = x119898minus1119860 (119899) minus 119889119894119860 (119899) t119898119860

119903119898minus1119898119860 =

119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) 119889119898119860 (119899)

119903119898119898119860 =

119871minus1

sum

119899=0

119889lowast

119898119860(119899) 119889119898119860 (119899)

120573119894119860 = 119903119894119894119860 minus1003816100381610038161003816119903119894119894119860

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119860

c(119894)1119860

= [c(119894minus1)1119860

0

] + 120573minus1

119894119860(119888(119894minus1)

1198941119860)lowast

[

1003816100381610038161003816119903119894minus11198941198601003816100381610038161003816

2 c(119894minus1)119894119860

minus119903lowast

119894minus1119894119860

]

c(119894)119894119860

= 120573minus1

119894119860[minus119903119894minus1119894119860c

(119894minus1)

119894119860

1

]

MSE(119894)119860= 1205902

1198891198600minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(119894)

11119860

END

T(119872minus1)119860

= [t1119860 t2119860 t119872minus1119860]

w(119872minus1)1198600

= T(119872minus1)119860

c(119872minus1)1119860

(42)

In the adaptive beamformer B the phase-shifted desiredsignal and observation vector can be given by

1198891198610 (119899) = 119890119895120601119896119903119896 (43)

And the optimal weight vector of adaptive beamformer Bcan be obtained according to (42) as shown in (44)

The flow diagram of the calculation of weight vector inadaptive beamformer B is as follows

rx1198610d1198610 = 119864 [x1198610 (119899) 119889lowast

1198610(119899)]

11990301119861 = 0 119888(1)

1119861= 119903minus1

11119861

MSE(1)119861

= 1205902

1198890119861minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(1)

1119861

FOR 119894 = 2 3 119872 minus 1

t119898119861 =119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) x119898minus1119861 (119899)

119889119894119861 (119899) = t119867119898119861

x119898minus1119861 (119899)

x119898119861 (119899) = x119898minus1119861 (119899) minus 119889119894119861 (119899) t119898119861

119903119898minus1119898119861 =

119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) 119889119898119861 (119899)

119903119898119898119861 =

119871minus1

sum

119899=0

119889lowast

119898119861(119899) 119889119898119861 (119899)

120573119894119861 = 119903119894119894119861 minus1003816100381610038161003816119903119894119894119861

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119861

c(119894)1119861

= [c(119894minus1)1119861

0

] + 120573minus1

119894119861(119888(119894minus1)

1198941119861)lowast

[

1003816100381610038161003816119903119894minus11198941198611003816100381610038161003816

2 c(119894minus1)119894119861

minus119903lowast

119894minus1119894119861

]

c(119894)119894119861= 120573minus1

119894119861[minus119903119894minus1119894119861c

(119894minus1)

119894119861

1

]

MSE(119894)119861= 1205902

1198891198610minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(119894)

11119861

END

T(119872minus1)119861

= [t1119861 t2119861 t119872minus1119861]

w(119872minus1)1198610

= T(119872minus1)119861

c(119872minus1)1119861

(44)

Substituting (11) and (13) into (44) we have

w(119872minus1)1198600

= w(119872minus1)1198610

(45)

Therefore the weight vector w(119872minus1)1198610

can be obtained bycalculating the optimal weight of the adaptive beamformerA

42 Calculation of DOA The adaptive beamformer B basedon the structure of data level order recursive MSNWF can besimplified to a single stageWiener filter in virtue of obtaining

International Journal of Distributed Sensor Networks 7

its weight from the adaptive beamformer A Let r119896(119899) =

(w(119872minus1)1198610

)119867

y119861 denote the output signal of beamformer B Let

r119896 = [r119896 (1) r119896 (2) r119896 (119871)]119879 (46)

Thus r119896 is an optimal estimation of the phase-shiftedreference signal 119890119895120601119896r119896 in the MMSE sense which can bewritten as

r119896 = 119890119895120601119896r119896 + N119896 (47)

Let120601119896 denote an estimation of120601119896 which can be calculatedby using the least square method such that the square errorbetween the two signal vectors r119896 and r119896 is minimized

120601119896min

100381710038171003817100381710038171003817r119896 minus 119890

119895120601119896r1198961003817100381710038171003817100381710038172 (48)

In [18] Wang et al give the optimum solution of 120601119896

120601119896 = arg (r119896r119867

119896) (49)

According to (12) an estimation of the target DOA can beobtained then as

120579119896 = arcsin(minus120582120601119896

2120587119889) (50)

5 Simulation Results

In this section the performance of the proposed methodincluding the resolution capacity and accuracy of the datalevel order recursive MSNWF DOA techniques will beevaluated through numerical simulations In Sections 51 and52 the resolution and the capacity of the DOA estimationusing the data level order recursiveMSNWFDOA techniqueswill be illustrated and compared with other techniques suchas MUSIC ESPRIT SBDOA and original MSNWF DOAestimation techniques In Sections 53 and 54 the effectsof snapshot length and stage of data level order recursiveMSNWF on the estimation accuracy will be investigatedrespectively

51 Resolution of DOA Estimation Assume that a ULA of 10elements with a spacing of 119889 = 1205822 deployed at the receiverwas employed in the simulations to deal with a case wherethe DOAs of three signals and two interference signals areclosely distributed Further assume that the DOAs of thetarget related signal components are at minus2∘ 0∘ and 2∘ TheDOAs of the interference components are at minus4∘ and 4∘ Thebackground noise power spectral density ratio of the receivedsignal is set to 10 dB Snapshot length is fixed at 100 and thestage of MSNWF is set to 5 One thousand simulation runswere performed These simulation results are illustrated inFigure 5

The histograms of the resolution of DOA estimationobtained for these five techniques are shown in Figures5(a)ndash5(e) The histogram depicts the number of occurrencesestimated DOA as a function of DOA degrees In Figure 5(a)

the histogram of MUSIC technique shows two peak valueswhich deviate from the DOAs of the target signals InFigure 5(b) although the histogram of ESPIRT techniqueshows three peak values the peak values deviate from theDOAs of the target signals It is seen that the MUSICtechnique or ESPRIT technique cannot offer the desiredresults when the DOAs of target signals are very closeCorrespondingly in Figures 5(c) 5(d) and 5(e) the his-togram shows three peak values indicating that using theSBDOA original MSNWF DOA and the data level orderrecursive MSNWF DOA techniques all three DOAs aresuccessfully estimated Therefore it proved that the datalevel order recursive MSNWF DOA technique could obtaina better resolution than MUSIC and ESPRIT techniquesHowever the SBDOA requires 119874(1198723) operations the orig-inal MSNWF DOA technique needs 119874(21198722 + 9119872) oper-ations and the data level order recursive MSNWF DOAtechnique demands 119874(1198722 + 11119872) operations which signif-icantly reduce the complexity of computation In additionif the recursive order is enough the resolution of data levelorder recursive MSNWF DOA technique will be well asthat of SBDOA estimation and it is proved in Section 54The resolution and accuracy of data level order recursiveMSNWF are better than the original MSNWF which is dueto the update of the MSE at each stage

52 Capacity of DOA Estimation This simulation deals witha case where the number of target signals and interferenceis larger than that of antenna elements The simulationconditions are kept the same as those in Section 51 exceptfor the number of signal components considered The DOAsof 9 target signal components are set from minus40∘ to 40∘ withinterval 10∘ and the DOAs of 6 interference components areset from minus25∘ to 25∘ with interval 10∘ The simulation resultsare shown in Figure 6

Histograms of the obtained estimated DOAs are shownin Figures 6(a)ndash6(e) In Figures 6(a) and 6(b) the histogramsshow the deviated peak values anddemonstrate that these twotechniques cannot provide acceptable DOA estimation whenthe number of antenna elements is less than the total numberof target signals and interference In contrast in Figures6(c) 6(d) and 6(e) the histograms show that all 9 targetDOAs are successfully estimated when using the SBDOAoriginal MSNWF and data level order recursive MASNWFDOA techniques As can be seen the successful probability ofDOA estimation in data level order recursive MSNWF DOAtechnique is the same as that in the SBDOA and originalMSNWF DOA estimation techniques

53 Effects of Snapshot Length of MSNWF on DOA Estima-tion Accuracy In the simulation of snapshot length effectsthe snapshot length for adaptive beamformer A and DOAcalculation are set to different values such as 50 100 200 500and 1000 and the stages of both original MSNWF and datalevel order recursive MSNWF are set to 5 The DOA of thetarget signal is fixed at 0∘ and the DOAs of the interferenceare set from minus90∘ to 90∘ with interval 10∘ except 0∘ Theroot mean square error (RMSE) of the estimated target DOA

8 International Journal of Distributed Sensor Networks

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(a) Resolution of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(b) Resolution of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(c) Resolution of SBDOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus 3 minus 2 minus 1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(d) Resolution of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(e) Resolution of data level order recursive MSNWF DOA estimation

Figure 5 Comparison of the resolution of DOA estimation for signal sources that are closely distributed

averaged over one thousand simulation runs at different SNRconditions the RMSE of the estimated target DOA and thesnapshot length are illustrated in Figure 7

As can be seen in Figure 7 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 5∘when using a small snapshot

length such as 50 The simulation results show that when thesnapshot length is 500 the data level order recursiveMSNWFDOA estimation method will have estimation accuracy sim-ilar to that of the SBDOA technique However the RMSEof the data level order recursive MSNWF DOA technique isbetter than that of original MSNWF DOA technique under

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

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DistributedSensor Networks

International Journal of

2 International Journal of Distributed Sensor Networks

peaks in the MUSIC spectrum formed by the eigenvectorsof the noise subspace Thus the capacity of this DOAestimation is equal to the rank of the reciprocal subspaceof the selected noise subspace and is also less than thenumber of the antenna elements In ESPRIT techniques twovirtual subarrays structures are proposed to obtain two signalsubspaces The eigenvectors of the relevant signal subspacesare rotated for the DOAs of the signals As a result thecapacity of DOA estimation using ESPRIT is bounded by thenumber of subarraysThe application of the above techniquesis limited to cases where the number of signal sourcesis less than that of antenna elements These techniquesrequire subspace estimation eigendecomposition and thecomputation of covariance matrix inversion which leadsto high computational complexity and they are therebylimited to the applications where fast DOA estimation isrequired Furthermore in the presence of interference thesetechniques need to estimate the DOAs of all the target signalsand interference which decreases the accuracy of DOAestimation

The application of adaptive beamforming in DOA esti-mation has become the research focus on interference exis-tence [18] In [18] Wang et al developed a new structureof DOA estimation based on subarray beamforming Thistechnique has clear advantage on the DOA estimation wheninterference exists but it still needs the computation ofmatrixinversionwhich is not easy to be applied to a practical systemBased on this structure a DOA estimation technique basedon Multistage Nested Wiener Filter (MSNWF) [19ndash25] isproposed in [26] In [26] Yu stated an original MSNWFalgorithm [27] to estimate the DOAs which used a filter andblocking matrix to avoid the calculation of covariance matrixinversion In this technique however it cannot calculatethe coefficients of Wiener filter in backward recursion ifthe forward recursion does not finish the calculation of thematch filter and blockingmatrix And themean squared error(MSE) can not be determined when adding a new stageexcept for the last stage

In this paper a data level order recursive MSNWFDOA estimation technique that uses a reference signal isproposed in detail This DOA estimation technique uses twosubarray adaptive beamformers based on the data level orderrecursive MSNWF to construct the same array geometry forforming the phase-shift and rejecting interference at sametime The DOAs of the target signals are estimated from thephase-shifts by using reference signal after the rejection ofinterference Therefore the performance of DOA estimationis significantly improved This technique can be widely usedfor the implementation of hardware systems such as wirelesscommunication system active radar sonar and space-timeadaptive process (STAP) systems [28 29]

The advantages of the data level order recursive MSNWFDOA estimation are as follows (1) Since the use of data levelorder recursive MSNWF in this DOA estimation techniquerealizes the subspace eigendecomposition computation ofinversion of covariance matrix becomes unnecessary andthus reduces the complexity of computation the data levelMSNWF DOA estimation technique can be easily applied inhardware platform (2) An orthogonal basis for the Krylov

subspace spanned cross correlation vector and covariancematrix improves the computational efficiency when calcu-lating the weight vector of the match filter And the orderrecursion could update the weight vector of the match filterand the MSE at new stage

The paper is organized as follows In Section 2 the signalmodel is described In Section 3 the structure of the data levelorder recursive MSNWF DOA estimation system MSNWFbased adaptive beamforming including data level recursionof match filters and order recursion and DOA calculationof the proposed method are presented Design examples andsimulation results are given in Section 5 and conclusions aredrawn in Section 6

2 Signal Model

Consider a uniform linear array (ULA) system that uses 119872elements with adjacent element spacing 119889 deployed at a basestation Assume that 119870 narrowband signals and 119875 unknowninterference sources are received at the ULA with differentDOAs 120579119896 119896 = 1 2 119870 + 119875

Using complex envelope representation the receivedsignals can be expressed by

x (119905) =119870+119875

sum

119896=1

a (120579119896) 119904119896 (119905) + n (119905) (1)

where 119904119896(119905) denotes the 119896th signal component 119896 = 1 2 119870

denotes the target components and 119896 = 119870+1119870+2 119870+

119875 are interference components The a(120579119896) in (1) denotes thesteering vector of the array in direction 120579119896 which is given by

a (120579119896) = [1 119890minus1198952120587119889 sin(120579119896) 119890minus1198952120587119889(119872minus1) sin(120579119896)]

119879 (2)

and n(119905) denotes the noise vector with zero mean and crosscovariance

119864 [n (1199051)n119867(1199052)] = 120590

2120575 (1199051 minus 1199052) I (3)

where I is the identity matrixSuppose that the received vector x(119905) is sampled at 119899 119899 =

1 2 119871 and the received signal can be expressed by (4) inthe matrix notation Consider

X = A (120579) S + N (4)

where X and N are119872times 119871matrices

X = [x (1) x (2) x (119871)]

N = [n (1) n (2) n (119871)] (5)

A(120579) is a119872times119870matrix as follows

A (120579) = [a (1205791) a (1205792) a (120579119870)] (6)

And S is a 119870 times 119871matrix

S = [s (1) s (2) s (119871)] (7)

International Journal of Distributed Sensor Networks 3

3 MSNWF DOA Estimation

Compared with the SBDOA estimation technique stated in[8] the proposed MSNWF DOA estimation technique inthis paper uses the same uniform linear antenna array atthe receiving end and the geometry of the array is similarto that used in ESPRIT techniques The antenna array isdecomposed into two equal-sized subarrays where the twosubarrays are used in conjunctionwith two subarrayMSNWFadaptive beamformers to obtain an optimal estimation of aphase-shift reference signal whose phase relative to that ofthe reference signal is a function of the target DOA Thetarget DOA is then computed from the estimated phase-shift between the reference signal r119896 and the phase-shiftedreference signal 119890119895120601119896r119896 In order to avoid the inversioncomputation of covariance matrix when getting the optimalweight vector of the beamformer the two beamformers as inFigure 1 in [18] are replaced with Multistage Nested WienerFilters The block diagram of the MSNWF DOA estimationsystem is illustrated in Figure 1

31 Subarray Signal Formation Consider that the array iscomposed of a ULA of119872 element as a receiver and decom-posed into two sets of119872minus1 element virtual subarraysA andB The downconverted baseband signal received by the 119898th119898 = 1 2 119872 element of the antenna array is expressed by

119909119898 (119899) =

119870+119875

sum

119896=1

119890119895(119898minus1)120601119896119904119896 (119899) (8)

The vectors of the A and B are given by

y119860 = [1199091 (119899) 1199092 (119899) 119909119872minus1 (119899)]119879

y119861 = [1199092 (119899) 1199093 (119899) 119909119872 (119899)]119879

(9)

respectively Let

b (120579119896) = [1 119890119895120601119896 119890

119895120601119896(119872minus2)]119879 (10)

and then the subarray signals y119860 and y119861 can be written as

y119860 (119899) =119870+119875

sum

119896=1

b (120579119896) 119904119896 (119899) + n119860 (119899)

y119861 (119899) =119870+119875

sum

119896=1

119890119895120601119896b (120579119896) 119904119896 (119899) + n119861 (119899)

(11)

where vectors n119860(119899) and n119861(119899) are the background noise atthe subarray respectively The phase-shift factor between the119896th components of signals y119860(119899) and y119861(119899) which forms the119896th signal is given by

119890119895120601119896 = 119890

minus1198952120587119889 sin(120579119896)120582 (12)

Sampling y119860(119899) and y119861(119899) obtains

Y119860 = [y119860 (1) y119860 (2) y119860 (119871)]

Y119861 = [y119861 (1) y119861 (2) y119861 (119871)] (13)

Adaptive beamformer AReferencesignals rk

Phase-shiftcomputationWeight vectors wk

MSNWF

MSNWF

Adaptive beamformer B

1

2

M

yA

yB

k

rk

Figure 1 Block diagram of the MSNWF DOA estimation system

32 Recursion Algorithm of MSNWF

321 Data Level Recursion of Match Filters In the Wienerfilter the estimation of the desired signal 1198890(119899) from anobservation vector x0(119899) is optimal in the minimum meansquare error (MMSE) sense The weight vector wX0 of theWiener filter can be obtained via solving the followingWiener-Hopf equations

Rx0wx0 = rx0d0 (14)

where Rx0 is the covariance matrix of observation vectorx0(119899) and rx0 is the cross correlation vector between theobservation vector x0(119899) and the desired signal 1198890(119899) Thecovariance matrix Rx0 cannot be readily estimated if x0(119899)is of high dimension Based on this Goldstein and Reedproposed that if the observation signal x0(119899) is prefiltered bya full-rank matrix T isin C119872times119872 to get a new observation signalz1(119899) = Tx0(119899) then the weight factor wz1 of Wiener filter isused to estimate the desired signal 1198890(119899) from z1(119899) results inthe same MSE [19ndash21]

The assumed full-rank prefilter matrix can be chosen as

T1 = [h1198671

B1] (15)

where119867 is the complex conjugate transpose operator Thus

z1 (119899) = [h1198671x0 (119899)

B1x0 (119899)] = [

1198891 (119899)

x1 (119899)] (16)

where B1 is referring to the blocking matrix B1h1 = 0 andh1 = rx0d0rx0d02

The solution of theWiener-Hopf equations relative to thetransformed system is

wz1 = Rminus1z1 rz1d0 = 1205721 [

1

minusRminus1x1 rx1d1] (17)

where Rz1 is the covariance matrix of the new observationsignal z1(119899) 1205721 = rx0d02(12059021198891 minus r119867x1d1R

minus1

x1 rx1d1) rz1d1 isthe cross correlation between d1(119899) and z1(119899) and Rx1 =

B1Rx0B1198671 1205902

1198891= h1198671Rx0h1 rx1d1 = B1Rx0h1

This process produces a new vector Wiener filter whichestimates the signal 1198891(119899) from the observation vector x1(119899)and a scalar Wiener filter is followed Repeating this process

4 International Journal of Distributed Sensor Networks

1205760(n)

d0(n)

x0(n)

++

++

minus

minus

d0(n)

d1(n)

d1(n)

1205761(n)

1205721

x1(n)

h1

B1 w1

Figure 2 First stage of the original MSNWF

x0(n)

tM

t2

t1d1(n)

d2(n)

dM(n)

d2(n)

d1(n)

d0(n)1205721

1205722

120572M

++

minus

++

minus

dMminus1(n)

Figure 3 Match filter bank structure of MSNWF

a nested structure can be obtained which is defined as theoriginal MSNWF [19ndash21]

In the original MSNWF the new desired signal 119889119894(119899) atthe output of the 119894th stage can be expressed as

119889119894 (119899) = x0 (119899)(119894minus1

prod

119896=1

B119867119896) h119894 = x119894minus1 (119899) t119894 (18)

According to (18) a filter t119894 is used to replace the 119894th stageWiener filter as Figure 2 shows which could be simply thecross correlation between the new observation x119894(119899) and thenew desired signal 119889119894(119899)

The new observation vector in Figure 3 is expressed as

d (119899) = [1198891 (119899) 1198892 (119899) 119889119873 (119899)]119879 (19)

which proved that this observation vector has a tridiagonalcovariance matrix [21]

Therefore the new desired signal 119889119894(119899) can be seen that itis the output of an119873 length filter t119894

t119894 = (

119894minus1

prod

119896=1

B119867119896) h119894 (20)

The filter t119894 is used to recover all the information of x119894minus1(119899)via 119889119894minus1(119899) The output 119889119894(119899) is gotten by the filter t119894+1 thus119889119894(119899) is correlated with 119889119894minus1(119899) and 119889119894+1(119899) However 119889119894+1(119899)is from the blocking matrix B119894+1 which is not correlatedwith 119889119894minus1(119899) Therefore 119889119894(119899) is only correlated with its twoneighbors And it is also required to be maximally correlatedwith 119889119894minus1(119899) Considering the orthogonality conditions themaximal correlation results in an optimization problem [25]as follows

t119894 = argmaxt

119864 [119889119894 (119899) 119889lowast

119894minus1(119899)]

st t119867t = 1 t119867t119896 = 0 119896 = 1 2 119894 minus 1

(21)

Using Lagrange multipliers the solution of (21) is

t119894 =(prod119894minus1

119896=1P119896)Rx0t119894minus1

10038171003817100381710038171003817(prod119894minus1

119896=1P119896)Rx0t119894minus1

100381710038171003817100381710038172

(22)

where P119896 = I119873 minus t119894t119867119894 Herein if B119894 is assumed to be equal to P119894 the filters t119894 are

an orthonormal basis for the Krylov subspace generated byrx0d0 and Rx0 [22] Therefore the result of recursion of theMSNWF can be obtained without B119894

At 119894th stage let

u119894 = Rx0t119894minus1 (23)

The filters t119894 of the recursion are computed as

t119894 = u119894 minus (t119867

119894minus1u119894) t119894minus1 minus (t

119867

119894minus2u119894) t119894minus2 (24)

In the recursion calculation process the filters t119894 arecalculatedwhich does not needB119894 and the inversion of covari-ance matrix and this reduces the complexity of computationThe calculation of t119894 only needs the last two members whichalso reduces the complexity of computation

322 Order Recursion At the stage (119872 minus 1) of the MSNWFthe orthogonal basis composed by the matcher filters t119894 isexpressed as

T(119872minus1) = [t1 t2 t119872minus1] (25)

The new observation vector obtained from the recursioncalculation can be written as

d(119872minus1) (119899) = [1198891 (119899) 1198892 (119899) 119889119872minus1 (119899)]

= [t1198671x0 (119899) t

119867

2x0 (119899) t

119867

119872minus1x0 (119899)]

= (T(119872minus1))119867

x0 (119899)

(26)

The covariance matrix can be written as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

(27)

The recursion coefficients are the components of Wienerfilter coefficients as (28) which is used to estimate 1198890(119899) fromd(119872minus1)(119899)

w(119872minus1)d = (R(119872minus1)d )minus1

r(119872minus1)dd0 = (R(119872minus1)d )minus1

(T(119872minus1))119867

rx0d0(28)

Then the coefficients of MSNWF can be expressed as

w(119872minus1)0

= T(119872minus1)w(119872minus1)d (29)

The MSE of the coefficients is

MSE(119872minus1) = 1205902

1198890minus rx0d0w

(119872minus1)

0(30)

which is updated withw(119872minus1)d and theMSE(119872minus1) at stage (119872minus

1) and with w(119872minus2)d and MSE(119872minus2) from the (119872 minus 2) stage

International Journal of Distributed Sensor Networks 5

According to (19) and its property the tridiagonal covari-ance matrix can be rewritten as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

= [

R11 R12R21 119903119872minus1119872minus1

] (31)

where

R11 = (T(119872minus2))119867

Rx0T(119872minus2)

R12 = [0119879 119903119872minus2119872minus1]119879

R21 = [0119879 119903lowast

119872minus2119872minus1]

(32)

The cross correlation vector between the new observationvector and desired signal 1198890(119899) is

r(119872minus1)dd0 = (T(119872minus1))119867

rx0d0 = [

1003817100381710038171003817rx0d010038171003817100381710038172

0] (33)

Given R(119872minus2)d from stage (119872 minus 2) the new elements ofR(119872minus1)d are calculated as

119903119872minus1119872minus1 = t119867119872minus1

Rx0t119872minus1

119903119872minus2119872minus1 = t119867119872minus2

Rx0t119872minus1(34)

According to (26) the (34) can be rewritten as

119903119872minus1119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus1(119899) 119889119872minus1 (119899)

119903119872minus2119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus2(119899) 119889119872minus1 (119899)

(35)

Consider the property that only the first element of thecross correlation vector 119903(119872minus1)

1198891198890is not equal to 0 Therefore

only the first column of the inverse of R(119872minus1)d is needed tocalculate the recursion coefficients via (28)

Let the inverse of R(119872minus1)d be noted as

C(119872minus1) = (R(119872minus1)d )minus1

= [c(119872minus1)1

c(119872minus1)2

c(119872minus1)119872minus1

]

= [

C(119872minus2) 0

0119879 0

] + 120573minus1

119872minus1b(119872minus1) (b(119872minus1))

119867

(36)

The various quantities in (36) are defined as in thefollowing equation

b(119872minus1) = [119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

]

120573119872minus1 = 119903119872minus1119872minus1 minus1003816100381610038161003816119903119872minus1119872minus1

1003816100381610038161003816

2119888(119872minus2)

119872minus2119872minus2

(37)

x0(n)

d0(n)

w1 w2

d1(n) d1(n)

x1(n)+

+

minus

++

minus

++minus +

+minus +

+minusMSE0

MSE1MSE2

tH1 tH2t1 t2

Figure 4 The structure of data level order recursive MSNWF

where 119888(119872minus2)119872minus2119872minus2

is the last element of the last column c(119872minus2)119872minus2

from the previous stageTherefore the column vector c(119872minus1)

1of C(119872minus1) can be

calculated as

c(119872minus1)1

= [c(119872minus2)1

0

] + 120573minus1

119872minus1(119888(119872minus2)

1119872minus2)lowast

[

1003816100381610038161003816119903119872minus2119872minus11003816100381610038161003816

2 c(119872minus2)119872minus2

minus119903lowast

119872minus2119872minus1

]

(38)

where 119888(119872minus2)1119872minus2

is the first element of c(119872minus2)119872minus2

It can be seen from (38) that c(119872minus1)

1depends on c(119872minus2)

1

from stage (119872 minus 2) and the new elements 119903119872minus2119872minus1 and119903119872minus1119872minus1 generated from the covariance matrix at stage (119872minus

1) And the coefficients of the Wiener filter w(119872minus1)d are alsodepending on that Moreover the last column vector c(119872minus1)

119872minus1

of C(119872minus1) depends on the last column c(119872minus2)119872minus2

from stage (119872minus

2) According to (36) the last column vector c(119872minus1)119872minus1

can beupdated as

c(119872minus1)119872minus1

= 120573minus1

119872minus1[minus119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

] (39)

It can be seen from (39) that the last column vector c(119872minus1)119872minus1

is only depending on the last column vector c(119872minus2)119872minus2

from stage(119872 minus 2) and the new element 119903119872minus2119872minus1 generated from thecovariance matrix at stage (119872 minus 1)

According to (38) and (39) it can be seen that in recursivecalculation process only c(119872minus1)

1and c(119872minus1)

119872minus1are needed to

be updated at each stage This avoids the calculation ofthe inversion of covariance matrix which also reduces thecomplexity of computation

As for the MSE expressed as in (30) it can be simple andcan be updated with 119888

(119872minus1)

11generated from the covariance

matrix at stage (119872 minus 1) as follows

MSE(119872minus1) = 1205902

1198890minus1003817100381710038171003817rx0d0

1003817100381710038171003817

2

2119888(119872minus1)

11 (40)

According to recursive algorithm about the calculation ofthe coefficients of thematch filters and the nest order the datalevel order recursive MSNWF DOA estimation structure canbe drawn as in Figure 4

4 MSNWF DOA Estimation System

41 Calculation of Weight Vector In the MSNWF DOAsystem the optimal estimation of the phase-shifted referencesignal 119890119895120601119896r119896 in the minimummean square error sense can be

6 International Journal of Distributed Sensor Networks

obtained at the output of the adaptive beamformer B whichuses the adaptive beamforming weights obtained from theadaptive beamformer A with the MSNWF structure

In the adaptive beamformer B consider the case wherethe phase-shifted reference signal 119890119895120601119896r119896 is the desired signaland the output of the adaptive beamformer B can be usedto estimate the desired signal Since the phase-shifted 119890

119895120601119896

is unknown both the phase-shifted reference signal and theweight vector of the adaptive beamformerB are not availableHowever the weight vector of the adaptive beamformer Bcan be obtained from the optimal weights of the adaptivebeamformer A

In the adaptive beamformer A the desired signal andobservation vector can be given by

1198891198600 (119899) = 119903119896 (119899) x1198600 (119899) = y119860 (119899) (41)

The optimal weight vector of adaptive beamformerA canbe readily obtained according to (42)

The flow diagram of calculation of weight vectors inadaptive beamformer A is as follows

rx1198600d1198600 = 119864 [x1198600 (119899) 119889lowast

1198600(119899)]

11990301119861 = 0 119888(1)

1119860= 119903minus1

11119860

MSE(1)119860

= 1205902

1198890119860minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(1)

1119860

FOR 119894 = 2 3 119872 minus 1

t119898119860 =119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) x119898minus1119860 (119899)

119889119894119860 (119899) = t119867119898119860

x119898minus1119860 (119899)

x119898119860 (119899) = x119898minus1119860 (119899) minus 119889119894119860 (119899) t119898119860

119903119898minus1119898119860 =

119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) 119889119898119860 (119899)

119903119898119898119860 =

119871minus1

sum

119899=0

119889lowast

119898119860(119899) 119889119898119860 (119899)

120573119894119860 = 119903119894119894119860 minus1003816100381610038161003816119903119894119894119860

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119860

c(119894)1119860

= [c(119894minus1)1119860

0

] + 120573minus1

119894119860(119888(119894minus1)

1198941119860)lowast

[

1003816100381610038161003816119903119894minus11198941198601003816100381610038161003816

2 c(119894minus1)119894119860

minus119903lowast

119894minus1119894119860

]

c(119894)119894119860

= 120573minus1

119894119860[minus119903119894minus1119894119860c

(119894minus1)

119894119860

1

]

MSE(119894)119860= 1205902

1198891198600minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(119894)

11119860

END

T(119872minus1)119860

= [t1119860 t2119860 t119872minus1119860]

w(119872minus1)1198600

= T(119872minus1)119860

c(119872minus1)1119860

(42)

In the adaptive beamformer B the phase-shifted desiredsignal and observation vector can be given by

1198891198610 (119899) = 119890119895120601119896119903119896 (43)

And the optimal weight vector of adaptive beamformer Bcan be obtained according to (42) as shown in (44)

The flow diagram of the calculation of weight vector inadaptive beamformer B is as follows

rx1198610d1198610 = 119864 [x1198610 (119899) 119889lowast

1198610(119899)]

11990301119861 = 0 119888(1)

1119861= 119903minus1

11119861

MSE(1)119861

= 1205902

1198890119861minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(1)

1119861

FOR 119894 = 2 3 119872 minus 1

t119898119861 =119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) x119898minus1119861 (119899)

119889119894119861 (119899) = t119867119898119861

x119898minus1119861 (119899)

x119898119861 (119899) = x119898minus1119861 (119899) minus 119889119894119861 (119899) t119898119861

119903119898minus1119898119861 =

119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) 119889119898119861 (119899)

119903119898119898119861 =

119871minus1

sum

119899=0

119889lowast

119898119861(119899) 119889119898119861 (119899)

120573119894119861 = 119903119894119894119861 minus1003816100381610038161003816119903119894119894119861

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119861

c(119894)1119861

= [c(119894minus1)1119861

0

] + 120573minus1

119894119861(119888(119894minus1)

1198941119861)lowast

[

1003816100381610038161003816119903119894minus11198941198611003816100381610038161003816

2 c(119894minus1)119894119861

minus119903lowast

119894minus1119894119861

]

c(119894)119894119861= 120573minus1

119894119861[minus119903119894minus1119894119861c

(119894minus1)

119894119861

1

]

MSE(119894)119861= 1205902

1198891198610minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(119894)

11119861

END

T(119872minus1)119861

= [t1119861 t2119861 t119872minus1119861]

w(119872minus1)1198610

= T(119872minus1)119861

c(119872minus1)1119861

(44)

Substituting (11) and (13) into (44) we have

w(119872minus1)1198600

= w(119872minus1)1198610

(45)

Therefore the weight vector w(119872minus1)1198610

can be obtained bycalculating the optimal weight of the adaptive beamformerA

42 Calculation of DOA The adaptive beamformer B basedon the structure of data level order recursive MSNWF can besimplified to a single stageWiener filter in virtue of obtaining

International Journal of Distributed Sensor Networks 7

its weight from the adaptive beamformer A Let r119896(119899) =

(w(119872minus1)1198610

)119867

y119861 denote the output signal of beamformer B Let

r119896 = [r119896 (1) r119896 (2) r119896 (119871)]119879 (46)

Thus r119896 is an optimal estimation of the phase-shiftedreference signal 119890119895120601119896r119896 in the MMSE sense which can bewritten as

r119896 = 119890119895120601119896r119896 + N119896 (47)

Let120601119896 denote an estimation of120601119896 which can be calculatedby using the least square method such that the square errorbetween the two signal vectors r119896 and r119896 is minimized

120601119896min

100381710038171003817100381710038171003817r119896 minus 119890

119895120601119896r1198961003817100381710038171003817100381710038172 (48)

In [18] Wang et al give the optimum solution of 120601119896

120601119896 = arg (r119896r119867

119896) (49)

According to (12) an estimation of the target DOA can beobtained then as

120579119896 = arcsin(minus120582120601119896

2120587119889) (50)

5 Simulation Results

In this section the performance of the proposed methodincluding the resolution capacity and accuracy of the datalevel order recursive MSNWF DOA techniques will beevaluated through numerical simulations In Sections 51 and52 the resolution and the capacity of the DOA estimationusing the data level order recursiveMSNWFDOA techniqueswill be illustrated and compared with other techniques suchas MUSIC ESPRIT SBDOA and original MSNWF DOAestimation techniques In Sections 53 and 54 the effectsof snapshot length and stage of data level order recursiveMSNWF on the estimation accuracy will be investigatedrespectively

51 Resolution of DOA Estimation Assume that a ULA of 10elements with a spacing of 119889 = 1205822 deployed at the receiverwas employed in the simulations to deal with a case wherethe DOAs of three signals and two interference signals areclosely distributed Further assume that the DOAs of thetarget related signal components are at minus2∘ 0∘ and 2∘ TheDOAs of the interference components are at minus4∘ and 4∘ Thebackground noise power spectral density ratio of the receivedsignal is set to 10 dB Snapshot length is fixed at 100 and thestage of MSNWF is set to 5 One thousand simulation runswere performed These simulation results are illustrated inFigure 5

The histograms of the resolution of DOA estimationobtained for these five techniques are shown in Figures5(a)ndash5(e) The histogram depicts the number of occurrencesestimated DOA as a function of DOA degrees In Figure 5(a)

the histogram of MUSIC technique shows two peak valueswhich deviate from the DOAs of the target signals InFigure 5(b) although the histogram of ESPIRT techniqueshows three peak values the peak values deviate from theDOAs of the target signals It is seen that the MUSICtechnique or ESPRIT technique cannot offer the desiredresults when the DOAs of target signals are very closeCorrespondingly in Figures 5(c) 5(d) and 5(e) the his-togram shows three peak values indicating that using theSBDOA original MSNWF DOA and the data level orderrecursive MSNWF DOA techniques all three DOAs aresuccessfully estimated Therefore it proved that the datalevel order recursive MSNWF DOA technique could obtaina better resolution than MUSIC and ESPRIT techniquesHowever the SBDOA requires 119874(1198723) operations the orig-inal MSNWF DOA technique needs 119874(21198722 + 9119872) oper-ations and the data level order recursive MSNWF DOAtechnique demands 119874(1198722 + 11119872) operations which signif-icantly reduce the complexity of computation In additionif the recursive order is enough the resolution of data levelorder recursive MSNWF DOA technique will be well asthat of SBDOA estimation and it is proved in Section 54The resolution and accuracy of data level order recursiveMSNWF are better than the original MSNWF which is dueto the update of the MSE at each stage

52 Capacity of DOA Estimation This simulation deals witha case where the number of target signals and interferenceis larger than that of antenna elements The simulationconditions are kept the same as those in Section 51 exceptfor the number of signal components considered The DOAsof 9 target signal components are set from minus40∘ to 40∘ withinterval 10∘ and the DOAs of 6 interference components areset from minus25∘ to 25∘ with interval 10∘ The simulation resultsare shown in Figure 6

Histograms of the obtained estimated DOAs are shownin Figures 6(a)ndash6(e) In Figures 6(a) and 6(b) the histogramsshow the deviated peak values anddemonstrate that these twotechniques cannot provide acceptable DOA estimation whenthe number of antenna elements is less than the total numberof target signals and interference In contrast in Figures6(c) 6(d) and 6(e) the histograms show that all 9 targetDOAs are successfully estimated when using the SBDOAoriginal MSNWF and data level order recursive MASNWFDOA techniques As can be seen the successful probability ofDOA estimation in data level order recursive MSNWF DOAtechnique is the same as that in the SBDOA and originalMSNWF DOA estimation techniques

53 Effects of Snapshot Length of MSNWF on DOA Estima-tion Accuracy In the simulation of snapshot length effectsthe snapshot length for adaptive beamformer A and DOAcalculation are set to different values such as 50 100 200 500and 1000 and the stages of both original MSNWF and datalevel order recursive MSNWF are set to 5 The DOA of thetarget signal is fixed at 0∘ and the DOAs of the interferenceare set from minus90∘ to 90∘ with interval 10∘ except 0∘ Theroot mean square error (RMSE) of the estimated target DOA

8 International Journal of Distributed Sensor Networks

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(a) Resolution of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(b) Resolution of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(c) Resolution of SBDOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus 3 minus 2 minus 1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(d) Resolution of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(e) Resolution of data level order recursive MSNWF DOA estimation

Figure 5 Comparison of the resolution of DOA estimation for signal sources that are closely distributed

averaged over one thousand simulation runs at different SNRconditions the RMSE of the estimated target DOA and thesnapshot length are illustrated in Figure 7

As can be seen in Figure 7 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 5∘when using a small snapshot

length such as 50 The simulation results show that when thesnapshot length is 500 the data level order recursiveMSNWFDOA estimation method will have estimation accuracy sim-ilar to that of the SBDOA technique However the RMSEof the data level order recursive MSNWF DOA technique isbetter than that of original MSNWF DOA technique under

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

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Submit your manuscripts athttpwwwhindawicom

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 3

3 MSNWF DOA Estimation

Compared with the SBDOA estimation technique stated in[8] the proposed MSNWF DOA estimation technique inthis paper uses the same uniform linear antenna array atthe receiving end and the geometry of the array is similarto that used in ESPRIT techniques The antenna array isdecomposed into two equal-sized subarrays where the twosubarrays are used in conjunctionwith two subarrayMSNWFadaptive beamformers to obtain an optimal estimation of aphase-shift reference signal whose phase relative to that ofthe reference signal is a function of the target DOA Thetarget DOA is then computed from the estimated phase-shift between the reference signal r119896 and the phase-shiftedreference signal 119890119895120601119896r119896 In order to avoid the inversioncomputation of covariance matrix when getting the optimalweight vector of the beamformer the two beamformers as inFigure 1 in [18] are replaced with Multistage Nested WienerFilters The block diagram of the MSNWF DOA estimationsystem is illustrated in Figure 1

31 Subarray Signal Formation Consider that the array iscomposed of a ULA of119872 element as a receiver and decom-posed into two sets of119872minus1 element virtual subarraysA andB The downconverted baseband signal received by the 119898th119898 = 1 2 119872 element of the antenna array is expressed by

119909119898 (119899) =

119870+119875

sum

119896=1

119890119895(119898minus1)120601119896119904119896 (119899) (8)

The vectors of the A and B are given by

y119860 = [1199091 (119899) 1199092 (119899) 119909119872minus1 (119899)]119879

y119861 = [1199092 (119899) 1199093 (119899) 119909119872 (119899)]119879

(9)

respectively Let

b (120579119896) = [1 119890119895120601119896 119890

119895120601119896(119872minus2)]119879 (10)

and then the subarray signals y119860 and y119861 can be written as

y119860 (119899) =119870+119875

sum

119896=1

b (120579119896) 119904119896 (119899) + n119860 (119899)

y119861 (119899) =119870+119875

sum

119896=1

119890119895120601119896b (120579119896) 119904119896 (119899) + n119861 (119899)

(11)

where vectors n119860(119899) and n119861(119899) are the background noise atthe subarray respectively The phase-shift factor between the119896th components of signals y119860(119899) and y119861(119899) which forms the119896th signal is given by

119890119895120601119896 = 119890

minus1198952120587119889 sin(120579119896)120582 (12)

Sampling y119860(119899) and y119861(119899) obtains

Y119860 = [y119860 (1) y119860 (2) y119860 (119871)]

Y119861 = [y119861 (1) y119861 (2) y119861 (119871)] (13)

Adaptive beamformer AReferencesignals rk

Phase-shiftcomputationWeight vectors wk

MSNWF

MSNWF

Adaptive beamformer B

1

2

M

yA

yB

k

rk

Figure 1 Block diagram of the MSNWF DOA estimation system

32 Recursion Algorithm of MSNWF

321 Data Level Recursion of Match Filters In the Wienerfilter the estimation of the desired signal 1198890(119899) from anobservation vector x0(119899) is optimal in the minimum meansquare error (MMSE) sense The weight vector wX0 of theWiener filter can be obtained via solving the followingWiener-Hopf equations

Rx0wx0 = rx0d0 (14)

where Rx0 is the covariance matrix of observation vectorx0(119899) and rx0 is the cross correlation vector between theobservation vector x0(119899) and the desired signal 1198890(119899) Thecovariance matrix Rx0 cannot be readily estimated if x0(119899)is of high dimension Based on this Goldstein and Reedproposed that if the observation signal x0(119899) is prefiltered bya full-rank matrix T isin C119872times119872 to get a new observation signalz1(119899) = Tx0(119899) then the weight factor wz1 of Wiener filter isused to estimate the desired signal 1198890(119899) from z1(119899) results inthe same MSE [19ndash21]

The assumed full-rank prefilter matrix can be chosen as

T1 = [h1198671

B1] (15)

where119867 is the complex conjugate transpose operator Thus

z1 (119899) = [h1198671x0 (119899)

B1x0 (119899)] = [

1198891 (119899)

x1 (119899)] (16)

where B1 is referring to the blocking matrix B1h1 = 0 andh1 = rx0d0rx0d02

The solution of theWiener-Hopf equations relative to thetransformed system is

wz1 = Rminus1z1 rz1d0 = 1205721 [

1

minusRminus1x1 rx1d1] (17)

where Rz1 is the covariance matrix of the new observationsignal z1(119899) 1205721 = rx0d02(12059021198891 minus r119867x1d1R

minus1

x1 rx1d1) rz1d1 isthe cross correlation between d1(119899) and z1(119899) and Rx1 =

B1Rx0B1198671 1205902

1198891= h1198671Rx0h1 rx1d1 = B1Rx0h1

This process produces a new vector Wiener filter whichestimates the signal 1198891(119899) from the observation vector x1(119899)and a scalar Wiener filter is followed Repeating this process

4 International Journal of Distributed Sensor Networks

1205760(n)

d0(n)

x0(n)

++

++

minus

minus

d0(n)

d1(n)

d1(n)

1205761(n)

1205721

x1(n)

h1

B1 w1

Figure 2 First stage of the original MSNWF

x0(n)

tM

t2

t1d1(n)

d2(n)

dM(n)

d2(n)

d1(n)

d0(n)1205721

1205722

120572M

++

minus

++

minus

dMminus1(n)

Figure 3 Match filter bank structure of MSNWF

a nested structure can be obtained which is defined as theoriginal MSNWF [19ndash21]

In the original MSNWF the new desired signal 119889119894(119899) atthe output of the 119894th stage can be expressed as

119889119894 (119899) = x0 (119899)(119894minus1

prod

119896=1

B119867119896) h119894 = x119894minus1 (119899) t119894 (18)

According to (18) a filter t119894 is used to replace the 119894th stageWiener filter as Figure 2 shows which could be simply thecross correlation between the new observation x119894(119899) and thenew desired signal 119889119894(119899)

The new observation vector in Figure 3 is expressed as

d (119899) = [1198891 (119899) 1198892 (119899) 119889119873 (119899)]119879 (19)

which proved that this observation vector has a tridiagonalcovariance matrix [21]

Therefore the new desired signal 119889119894(119899) can be seen that itis the output of an119873 length filter t119894

t119894 = (

119894minus1

prod

119896=1

B119867119896) h119894 (20)

The filter t119894 is used to recover all the information of x119894minus1(119899)via 119889119894minus1(119899) The output 119889119894(119899) is gotten by the filter t119894+1 thus119889119894(119899) is correlated with 119889119894minus1(119899) and 119889119894+1(119899) However 119889119894+1(119899)is from the blocking matrix B119894+1 which is not correlatedwith 119889119894minus1(119899) Therefore 119889119894(119899) is only correlated with its twoneighbors And it is also required to be maximally correlatedwith 119889119894minus1(119899) Considering the orthogonality conditions themaximal correlation results in an optimization problem [25]as follows

t119894 = argmaxt

119864 [119889119894 (119899) 119889lowast

119894minus1(119899)]

st t119867t = 1 t119867t119896 = 0 119896 = 1 2 119894 minus 1

(21)

Using Lagrange multipliers the solution of (21) is

t119894 =(prod119894minus1

119896=1P119896)Rx0t119894minus1

10038171003817100381710038171003817(prod119894minus1

119896=1P119896)Rx0t119894minus1

100381710038171003817100381710038172

(22)

where P119896 = I119873 minus t119894t119867119894 Herein if B119894 is assumed to be equal to P119894 the filters t119894 are

an orthonormal basis for the Krylov subspace generated byrx0d0 and Rx0 [22] Therefore the result of recursion of theMSNWF can be obtained without B119894

At 119894th stage let

u119894 = Rx0t119894minus1 (23)

The filters t119894 of the recursion are computed as

t119894 = u119894 minus (t119867

119894minus1u119894) t119894minus1 minus (t

119867

119894minus2u119894) t119894minus2 (24)

In the recursion calculation process the filters t119894 arecalculatedwhich does not needB119894 and the inversion of covari-ance matrix and this reduces the complexity of computationThe calculation of t119894 only needs the last two members whichalso reduces the complexity of computation

322 Order Recursion At the stage (119872 minus 1) of the MSNWFthe orthogonal basis composed by the matcher filters t119894 isexpressed as

T(119872minus1) = [t1 t2 t119872minus1] (25)

The new observation vector obtained from the recursioncalculation can be written as

d(119872minus1) (119899) = [1198891 (119899) 1198892 (119899) 119889119872minus1 (119899)]

= [t1198671x0 (119899) t

119867

2x0 (119899) t

119867

119872minus1x0 (119899)]

= (T(119872minus1))119867

x0 (119899)

(26)

The covariance matrix can be written as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

(27)

The recursion coefficients are the components of Wienerfilter coefficients as (28) which is used to estimate 1198890(119899) fromd(119872minus1)(119899)

w(119872minus1)d = (R(119872minus1)d )minus1

r(119872minus1)dd0 = (R(119872minus1)d )minus1

(T(119872minus1))119867

rx0d0(28)

Then the coefficients of MSNWF can be expressed as

w(119872minus1)0

= T(119872minus1)w(119872minus1)d (29)

The MSE of the coefficients is

MSE(119872minus1) = 1205902

1198890minus rx0d0w

(119872minus1)

0(30)

which is updated withw(119872minus1)d and theMSE(119872minus1) at stage (119872minus

1) and with w(119872minus2)d and MSE(119872minus2) from the (119872 minus 2) stage

International Journal of Distributed Sensor Networks 5

According to (19) and its property the tridiagonal covari-ance matrix can be rewritten as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

= [

R11 R12R21 119903119872minus1119872minus1

] (31)

where

R11 = (T(119872minus2))119867

Rx0T(119872minus2)

R12 = [0119879 119903119872minus2119872minus1]119879

R21 = [0119879 119903lowast

119872minus2119872minus1]

(32)

The cross correlation vector between the new observationvector and desired signal 1198890(119899) is

r(119872minus1)dd0 = (T(119872minus1))119867

rx0d0 = [

1003817100381710038171003817rx0d010038171003817100381710038172

0] (33)

Given R(119872minus2)d from stage (119872 minus 2) the new elements ofR(119872minus1)d are calculated as

119903119872minus1119872minus1 = t119867119872minus1

Rx0t119872minus1

119903119872minus2119872minus1 = t119867119872minus2

Rx0t119872minus1(34)

According to (26) the (34) can be rewritten as

119903119872minus1119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus1(119899) 119889119872minus1 (119899)

119903119872minus2119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus2(119899) 119889119872minus1 (119899)

(35)

Consider the property that only the first element of thecross correlation vector 119903(119872minus1)

1198891198890is not equal to 0 Therefore

only the first column of the inverse of R(119872minus1)d is needed tocalculate the recursion coefficients via (28)

Let the inverse of R(119872minus1)d be noted as

C(119872minus1) = (R(119872minus1)d )minus1

= [c(119872minus1)1

c(119872minus1)2

c(119872minus1)119872minus1

]

= [

C(119872minus2) 0

0119879 0

] + 120573minus1

119872minus1b(119872minus1) (b(119872minus1))

119867

(36)

The various quantities in (36) are defined as in thefollowing equation

b(119872minus1) = [119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

]

120573119872minus1 = 119903119872minus1119872minus1 minus1003816100381610038161003816119903119872minus1119872minus1

1003816100381610038161003816

2119888(119872minus2)

119872minus2119872minus2

(37)

x0(n)

d0(n)

w1 w2

d1(n) d1(n)

x1(n)+

+

minus

++

minus

++minus +

+minus +

+minusMSE0

MSE1MSE2

tH1 tH2t1 t2

Figure 4 The structure of data level order recursive MSNWF

where 119888(119872minus2)119872minus2119872minus2

is the last element of the last column c(119872minus2)119872minus2

from the previous stageTherefore the column vector c(119872minus1)

1of C(119872minus1) can be

calculated as

c(119872minus1)1

= [c(119872minus2)1

0

] + 120573minus1

119872minus1(119888(119872minus2)

1119872minus2)lowast

[

1003816100381610038161003816119903119872minus2119872minus11003816100381610038161003816

2 c(119872minus2)119872minus2

minus119903lowast

119872minus2119872minus1

]

(38)

where 119888(119872minus2)1119872minus2

is the first element of c(119872minus2)119872minus2

It can be seen from (38) that c(119872minus1)

1depends on c(119872minus2)

1

from stage (119872 minus 2) and the new elements 119903119872minus2119872minus1 and119903119872minus1119872minus1 generated from the covariance matrix at stage (119872minus

1) And the coefficients of the Wiener filter w(119872minus1)d are alsodepending on that Moreover the last column vector c(119872minus1)

119872minus1

of C(119872minus1) depends on the last column c(119872minus2)119872minus2

from stage (119872minus

2) According to (36) the last column vector c(119872minus1)119872minus1

can beupdated as

c(119872minus1)119872minus1

= 120573minus1

119872minus1[minus119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

] (39)

It can be seen from (39) that the last column vector c(119872minus1)119872minus1

is only depending on the last column vector c(119872minus2)119872minus2

from stage(119872 minus 2) and the new element 119903119872minus2119872minus1 generated from thecovariance matrix at stage (119872 minus 1)

According to (38) and (39) it can be seen that in recursivecalculation process only c(119872minus1)

1and c(119872minus1)

119872minus1are needed to

be updated at each stage This avoids the calculation ofthe inversion of covariance matrix which also reduces thecomplexity of computation

As for the MSE expressed as in (30) it can be simple andcan be updated with 119888

(119872minus1)

11generated from the covariance

matrix at stage (119872 minus 1) as follows

MSE(119872minus1) = 1205902

1198890minus1003817100381710038171003817rx0d0

1003817100381710038171003817

2

2119888(119872minus1)

11 (40)

According to recursive algorithm about the calculation ofthe coefficients of thematch filters and the nest order the datalevel order recursive MSNWF DOA estimation structure canbe drawn as in Figure 4

4 MSNWF DOA Estimation System

41 Calculation of Weight Vector In the MSNWF DOAsystem the optimal estimation of the phase-shifted referencesignal 119890119895120601119896r119896 in the minimummean square error sense can be

6 International Journal of Distributed Sensor Networks

obtained at the output of the adaptive beamformer B whichuses the adaptive beamforming weights obtained from theadaptive beamformer A with the MSNWF structure

In the adaptive beamformer B consider the case wherethe phase-shifted reference signal 119890119895120601119896r119896 is the desired signaland the output of the adaptive beamformer B can be usedto estimate the desired signal Since the phase-shifted 119890

119895120601119896

is unknown both the phase-shifted reference signal and theweight vector of the adaptive beamformerB are not availableHowever the weight vector of the adaptive beamformer Bcan be obtained from the optimal weights of the adaptivebeamformer A

In the adaptive beamformer A the desired signal andobservation vector can be given by

1198891198600 (119899) = 119903119896 (119899) x1198600 (119899) = y119860 (119899) (41)

The optimal weight vector of adaptive beamformerA canbe readily obtained according to (42)

The flow diagram of calculation of weight vectors inadaptive beamformer A is as follows

rx1198600d1198600 = 119864 [x1198600 (119899) 119889lowast

1198600(119899)]

11990301119861 = 0 119888(1)

1119860= 119903minus1

11119860

MSE(1)119860

= 1205902

1198890119860minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(1)

1119860

FOR 119894 = 2 3 119872 minus 1

t119898119860 =119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) x119898minus1119860 (119899)

119889119894119860 (119899) = t119867119898119860

x119898minus1119860 (119899)

x119898119860 (119899) = x119898minus1119860 (119899) minus 119889119894119860 (119899) t119898119860

119903119898minus1119898119860 =

119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) 119889119898119860 (119899)

119903119898119898119860 =

119871minus1

sum

119899=0

119889lowast

119898119860(119899) 119889119898119860 (119899)

120573119894119860 = 119903119894119894119860 minus1003816100381610038161003816119903119894119894119860

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119860

c(119894)1119860

= [c(119894minus1)1119860

0

] + 120573minus1

119894119860(119888(119894minus1)

1198941119860)lowast

[

1003816100381610038161003816119903119894minus11198941198601003816100381610038161003816

2 c(119894minus1)119894119860

minus119903lowast

119894minus1119894119860

]

c(119894)119894119860

= 120573minus1

119894119860[minus119903119894minus1119894119860c

(119894minus1)

119894119860

1

]

MSE(119894)119860= 1205902

1198891198600minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(119894)

11119860

END

T(119872minus1)119860

= [t1119860 t2119860 t119872minus1119860]

w(119872minus1)1198600

= T(119872minus1)119860

c(119872minus1)1119860

(42)

In the adaptive beamformer B the phase-shifted desiredsignal and observation vector can be given by

1198891198610 (119899) = 119890119895120601119896119903119896 (43)

And the optimal weight vector of adaptive beamformer Bcan be obtained according to (42) as shown in (44)

The flow diagram of the calculation of weight vector inadaptive beamformer B is as follows

rx1198610d1198610 = 119864 [x1198610 (119899) 119889lowast

1198610(119899)]

11990301119861 = 0 119888(1)

1119861= 119903minus1

11119861

MSE(1)119861

= 1205902

1198890119861minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(1)

1119861

FOR 119894 = 2 3 119872 minus 1

t119898119861 =119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) x119898minus1119861 (119899)

119889119894119861 (119899) = t119867119898119861

x119898minus1119861 (119899)

x119898119861 (119899) = x119898minus1119861 (119899) minus 119889119894119861 (119899) t119898119861

119903119898minus1119898119861 =

119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) 119889119898119861 (119899)

119903119898119898119861 =

119871minus1

sum

119899=0

119889lowast

119898119861(119899) 119889119898119861 (119899)

120573119894119861 = 119903119894119894119861 minus1003816100381610038161003816119903119894119894119861

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119861

c(119894)1119861

= [c(119894minus1)1119861

0

] + 120573minus1

119894119861(119888(119894minus1)

1198941119861)lowast

[

1003816100381610038161003816119903119894minus11198941198611003816100381610038161003816

2 c(119894minus1)119894119861

minus119903lowast

119894minus1119894119861

]

c(119894)119894119861= 120573minus1

119894119861[minus119903119894minus1119894119861c

(119894minus1)

119894119861

1

]

MSE(119894)119861= 1205902

1198891198610minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(119894)

11119861

END

T(119872minus1)119861

= [t1119861 t2119861 t119872minus1119861]

w(119872minus1)1198610

= T(119872minus1)119861

c(119872minus1)1119861

(44)

Substituting (11) and (13) into (44) we have

w(119872minus1)1198600

= w(119872minus1)1198610

(45)

Therefore the weight vector w(119872minus1)1198610

can be obtained bycalculating the optimal weight of the adaptive beamformerA

42 Calculation of DOA The adaptive beamformer B basedon the structure of data level order recursive MSNWF can besimplified to a single stageWiener filter in virtue of obtaining

International Journal of Distributed Sensor Networks 7

its weight from the adaptive beamformer A Let r119896(119899) =

(w(119872minus1)1198610

)119867

y119861 denote the output signal of beamformer B Let

r119896 = [r119896 (1) r119896 (2) r119896 (119871)]119879 (46)

Thus r119896 is an optimal estimation of the phase-shiftedreference signal 119890119895120601119896r119896 in the MMSE sense which can bewritten as

r119896 = 119890119895120601119896r119896 + N119896 (47)

Let120601119896 denote an estimation of120601119896 which can be calculatedby using the least square method such that the square errorbetween the two signal vectors r119896 and r119896 is minimized

120601119896min

100381710038171003817100381710038171003817r119896 minus 119890

119895120601119896r1198961003817100381710038171003817100381710038172 (48)

In [18] Wang et al give the optimum solution of 120601119896

120601119896 = arg (r119896r119867

119896) (49)

According to (12) an estimation of the target DOA can beobtained then as

120579119896 = arcsin(minus120582120601119896

2120587119889) (50)

5 Simulation Results

In this section the performance of the proposed methodincluding the resolution capacity and accuracy of the datalevel order recursive MSNWF DOA techniques will beevaluated through numerical simulations In Sections 51 and52 the resolution and the capacity of the DOA estimationusing the data level order recursiveMSNWFDOA techniqueswill be illustrated and compared with other techniques suchas MUSIC ESPRIT SBDOA and original MSNWF DOAestimation techniques In Sections 53 and 54 the effectsof snapshot length and stage of data level order recursiveMSNWF on the estimation accuracy will be investigatedrespectively

51 Resolution of DOA Estimation Assume that a ULA of 10elements with a spacing of 119889 = 1205822 deployed at the receiverwas employed in the simulations to deal with a case wherethe DOAs of three signals and two interference signals areclosely distributed Further assume that the DOAs of thetarget related signal components are at minus2∘ 0∘ and 2∘ TheDOAs of the interference components are at minus4∘ and 4∘ Thebackground noise power spectral density ratio of the receivedsignal is set to 10 dB Snapshot length is fixed at 100 and thestage of MSNWF is set to 5 One thousand simulation runswere performed These simulation results are illustrated inFigure 5

The histograms of the resolution of DOA estimationobtained for these five techniques are shown in Figures5(a)ndash5(e) The histogram depicts the number of occurrencesestimated DOA as a function of DOA degrees In Figure 5(a)

the histogram of MUSIC technique shows two peak valueswhich deviate from the DOAs of the target signals InFigure 5(b) although the histogram of ESPIRT techniqueshows three peak values the peak values deviate from theDOAs of the target signals It is seen that the MUSICtechnique or ESPRIT technique cannot offer the desiredresults when the DOAs of target signals are very closeCorrespondingly in Figures 5(c) 5(d) and 5(e) the his-togram shows three peak values indicating that using theSBDOA original MSNWF DOA and the data level orderrecursive MSNWF DOA techniques all three DOAs aresuccessfully estimated Therefore it proved that the datalevel order recursive MSNWF DOA technique could obtaina better resolution than MUSIC and ESPRIT techniquesHowever the SBDOA requires 119874(1198723) operations the orig-inal MSNWF DOA technique needs 119874(21198722 + 9119872) oper-ations and the data level order recursive MSNWF DOAtechnique demands 119874(1198722 + 11119872) operations which signif-icantly reduce the complexity of computation In additionif the recursive order is enough the resolution of data levelorder recursive MSNWF DOA technique will be well asthat of SBDOA estimation and it is proved in Section 54The resolution and accuracy of data level order recursiveMSNWF are better than the original MSNWF which is dueto the update of the MSE at each stage

52 Capacity of DOA Estimation This simulation deals witha case where the number of target signals and interferenceis larger than that of antenna elements The simulationconditions are kept the same as those in Section 51 exceptfor the number of signal components considered The DOAsof 9 target signal components are set from minus40∘ to 40∘ withinterval 10∘ and the DOAs of 6 interference components areset from minus25∘ to 25∘ with interval 10∘ The simulation resultsare shown in Figure 6

Histograms of the obtained estimated DOAs are shownin Figures 6(a)ndash6(e) In Figures 6(a) and 6(b) the histogramsshow the deviated peak values anddemonstrate that these twotechniques cannot provide acceptable DOA estimation whenthe number of antenna elements is less than the total numberof target signals and interference In contrast in Figures6(c) 6(d) and 6(e) the histograms show that all 9 targetDOAs are successfully estimated when using the SBDOAoriginal MSNWF and data level order recursive MASNWFDOA techniques As can be seen the successful probability ofDOA estimation in data level order recursive MSNWF DOAtechnique is the same as that in the SBDOA and originalMSNWF DOA estimation techniques

53 Effects of Snapshot Length of MSNWF on DOA Estima-tion Accuracy In the simulation of snapshot length effectsthe snapshot length for adaptive beamformer A and DOAcalculation are set to different values such as 50 100 200 500and 1000 and the stages of both original MSNWF and datalevel order recursive MSNWF are set to 5 The DOA of thetarget signal is fixed at 0∘ and the DOAs of the interferenceare set from minus90∘ to 90∘ with interval 10∘ except 0∘ Theroot mean square error (RMSE) of the estimated target DOA

8 International Journal of Distributed Sensor Networks

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(a) Resolution of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(b) Resolution of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(c) Resolution of SBDOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus 3 minus 2 minus 1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(d) Resolution of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(e) Resolution of data level order recursive MSNWF DOA estimation

Figure 5 Comparison of the resolution of DOA estimation for signal sources that are closely distributed

averaged over one thousand simulation runs at different SNRconditions the RMSE of the estimated target DOA and thesnapshot length are illustrated in Figure 7

As can be seen in Figure 7 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 5∘when using a small snapshot

length such as 50 The simulation results show that when thesnapshot length is 500 the data level order recursiveMSNWFDOA estimation method will have estimation accuracy sim-ilar to that of the SBDOA technique However the RMSEof the data level order recursive MSNWF DOA technique isbetter than that of original MSNWF DOA technique under

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

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DistributedSensor Networks

International Journal of

4 International Journal of Distributed Sensor Networks

1205760(n)

d0(n)

x0(n)

++

++

minus

minus

d0(n)

d1(n)

d1(n)

1205761(n)

1205721

x1(n)

h1

B1 w1

Figure 2 First stage of the original MSNWF

x0(n)

tM

t2

t1d1(n)

d2(n)

dM(n)

d2(n)

d1(n)

d0(n)1205721

1205722

120572M

++

minus

++

minus

dMminus1(n)

Figure 3 Match filter bank structure of MSNWF

a nested structure can be obtained which is defined as theoriginal MSNWF [19ndash21]

In the original MSNWF the new desired signal 119889119894(119899) atthe output of the 119894th stage can be expressed as

119889119894 (119899) = x0 (119899)(119894minus1

prod

119896=1

B119867119896) h119894 = x119894minus1 (119899) t119894 (18)

According to (18) a filter t119894 is used to replace the 119894th stageWiener filter as Figure 2 shows which could be simply thecross correlation between the new observation x119894(119899) and thenew desired signal 119889119894(119899)

The new observation vector in Figure 3 is expressed as

d (119899) = [1198891 (119899) 1198892 (119899) 119889119873 (119899)]119879 (19)

which proved that this observation vector has a tridiagonalcovariance matrix [21]

Therefore the new desired signal 119889119894(119899) can be seen that itis the output of an119873 length filter t119894

t119894 = (

119894minus1

prod

119896=1

B119867119896) h119894 (20)

The filter t119894 is used to recover all the information of x119894minus1(119899)via 119889119894minus1(119899) The output 119889119894(119899) is gotten by the filter t119894+1 thus119889119894(119899) is correlated with 119889119894minus1(119899) and 119889119894+1(119899) However 119889119894+1(119899)is from the blocking matrix B119894+1 which is not correlatedwith 119889119894minus1(119899) Therefore 119889119894(119899) is only correlated with its twoneighbors And it is also required to be maximally correlatedwith 119889119894minus1(119899) Considering the orthogonality conditions themaximal correlation results in an optimization problem [25]as follows

t119894 = argmaxt

119864 [119889119894 (119899) 119889lowast

119894minus1(119899)]

st t119867t = 1 t119867t119896 = 0 119896 = 1 2 119894 minus 1

(21)

Using Lagrange multipliers the solution of (21) is

t119894 =(prod119894minus1

119896=1P119896)Rx0t119894minus1

10038171003817100381710038171003817(prod119894minus1

119896=1P119896)Rx0t119894minus1

100381710038171003817100381710038172

(22)

where P119896 = I119873 minus t119894t119867119894 Herein if B119894 is assumed to be equal to P119894 the filters t119894 are

an orthonormal basis for the Krylov subspace generated byrx0d0 and Rx0 [22] Therefore the result of recursion of theMSNWF can be obtained without B119894

At 119894th stage let

u119894 = Rx0t119894minus1 (23)

The filters t119894 of the recursion are computed as

t119894 = u119894 minus (t119867

119894minus1u119894) t119894minus1 minus (t

119867

119894minus2u119894) t119894minus2 (24)

In the recursion calculation process the filters t119894 arecalculatedwhich does not needB119894 and the inversion of covari-ance matrix and this reduces the complexity of computationThe calculation of t119894 only needs the last two members whichalso reduces the complexity of computation

322 Order Recursion At the stage (119872 minus 1) of the MSNWFthe orthogonal basis composed by the matcher filters t119894 isexpressed as

T(119872minus1) = [t1 t2 t119872minus1] (25)

The new observation vector obtained from the recursioncalculation can be written as

d(119872minus1) (119899) = [1198891 (119899) 1198892 (119899) 119889119872minus1 (119899)]

= [t1198671x0 (119899) t

119867

2x0 (119899) t

119867

119872minus1x0 (119899)]

= (T(119872minus1))119867

x0 (119899)

(26)

The covariance matrix can be written as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

(27)

The recursion coefficients are the components of Wienerfilter coefficients as (28) which is used to estimate 1198890(119899) fromd(119872minus1)(119899)

w(119872minus1)d = (R(119872minus1)d )minus1

r(119872minus1)dd0 = (R(119872minus1)d )minus1

(T(119872minus1))119867

rx0d0(28)

Then the coefficients of MSNWF can be expressed as

w(119872minus1)0

= T(119872minus1)w(119872minus1)d (29)

The MSE of the coefficients is

MSE(119872minus1) = 1205902

1198890minus rx0d0w

(119872minus1)

0(30)

which is updated withw(119872minus1)d and theMSE(119872minus1) at stage (119872minus

1) and with w(119872minus2)d and MSE(119872minus2) from the (119872 minus 2) stage

International Journal of Distributed Sensor Networks 5

According to (19) and its property the tridiagonal covari-ance matrix can be rewritten as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

= [

R11 R12R21 119903119872minus1119872minus1

] (31)

where

R11 = (T(119872minus2))119867

Rx0T(119872minus2)

R12 = [0119879 119903119872minus2119872minus1]119879

R21 = [0119879 119903lowast

119872minus2119872minus1]

(32)

The cross correlation vector between the new observationvector and desired signal 1198890(119899) is

r(119872minus1)dd0 = (T(119872minus1))119867

rx0d0 = [

1003817100381710038171003817rx0d010038171003817100381710038172

0] (33)

Given R(119872minus2)d from stage (119872 minus 2) the new elements ofR(119872minus1)d are calculated as

119903119872minus1119872minus1 = t119867119872minus1

Rx0t119872minus1

119903119872minus2119872minus1 = t119867119872minus2

Rx0t119872minus1(34)

According to (26) the (34) can be rewritten as

119903119872minus1119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus1(119899) 119889119872minus1 (119899)

119903119872minus2119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus2(119899) 119889119872minus1 (119899)

(35)

Consider the property that only the first element of thecross correlation vector 119903(119872minus1)

1198891198890is not equal to 0 Therefore

only the first column of the inverse of R(119872minus1)d is needed tocalculate the recursion coefficients via (28)

Let the inverse of R(119872minus1)d be noted as

C(119872minus1) = (R(119872minus1)d )minus1

= [c(119872minus1)1

c(119872minus1)2

c(119872minus1)119872minus1

]

= [

C(119872minus2) 0

0119879 0

] + 120573minus1

119872minus1b(119872minus1) (b(119872minus1))

119867

(36)

The various quantities in (36) are defined as in thefollowing equation

b(119872minus1) = [119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

]

120573119872minus1 = 119903119872minus1119872minus1 minus1003816100381610038161003816119903119872minus1119872minus1

1003816100381610038161003816

2119888(119872minus2)

119872minus2119872minus2

(37)

x0(n)

d0(n)

w1 w2

d1(n) d1(n)

x1(n)+

+

minus

++

minus

++minus +

+minus +

+minusMSE0

MSE1MSE2

tH1 tH2t1 t2

Figure 4 The structure of data level order recursive MSNWF

where 119888(119872minus2)119872minus2119872minus2

is the last element of the last column c(119872minus2)119872minus2

from the previous stageTherefore the column vector c(119872minus1)

1of C(119872minus1) can be

calculated as

c(119872minus1)1

= [c(119872minus2)1

0

] + 120573minus1

119872minus1(119888(119872minus2)

1119872minus2)lowast

[

1003816100381610038161003816119903119872minus2119872minus11003816100381610038161003816

2 c(119872minus2)119872minus2

minus119903lowast

119872minus2119872minus1

]

(38)

where 119888(119872minus2)1119872minus2

is the first element of c(119872minus2)119872minus2

It can be seen from (38) that c(119872minus1)

1depends on c(119872minus2)

1

from stage (119872 minus 2) and the new elements 119903119872minus2119872minus1 and119903119872minus1119872minus1 generated from the covariance matrix at stage (119872minus

1) And the coefficients of the Wiener filter w(119872minus1)d are alsodepending on that Moreover the last column vector c(119872minus1)

119872minus1

of C(119872minus1) depends on the last column c(119872minus2)119872minus2

from stage (119872minus

2) According to (36) the last column vector c(119872minus1)119872minus1

can beupdated as

c(119872minus1)119872minus1

= 120573minus1

119872minus1[minus119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

] (39)

It can be seen from (39) that the last column vector c(119872minus1)119872minus1

is only depending on the last column vector c(119872minus2)119872minus2

from stage(119872 minus 2) and the new element 119903119872minus2119872minus1 generated from thecovariance matrix at stage (119872 minus 1)

According to (38) and (39) it can be seen that in recursivecalculation process only c(119872minus1)

1and c(119872minus1)

119872minus1are needed to

be updated at each stage This avoids the calculation ofthe inversion of covariance matrix which also reduces thecomplexity of computation

As for the MSE expressed as in (30) it can be simple andcan be updated with 119888

(119872minus1)

11generated from the covariance

matrix at stage (119872 minus 1) as follows

MSE(119872minus1) = 1205902

1198890minus1003817100381710038171003817rx0d0

1003817100381710038171003817

2

2119888(119872minus1)

11 (40)

According to recursive algorithm about the calculation ofthe coefficients of thematch filters and the nest order the datalevel order recursive MSNWF DOA estimation structure canbe drawn as in Figure 4

4 MSNWF DOA Estimation System

41 Calculation of Weight Vector In the MSNWF DOAsystem the optimal estimation of the phase-shifted referencesignal 119890119895120601119896r119896 in the minimummean square error sense can be

6 International Journal of Distributed Sensor Networks

obtained at the output of the adaptive beamformer B whichuses the adaptive beamforming weights obtained from theadaptive beamformer A with the MSNWF structure

In the adaptive beamformer B consider the case wherethe phase-shifted reference signal 119890119895120601119896r119896 is the desired signaland the output of the adaptive beamformer B can be usedto estimate the desired signal Since the phase-shifted 119890

119895120601119896

is unknown both the phase-shifted reference signal and theweight vector of the adaptive beamformerB are not availableHowever the weight vector of the adaptive beamformer Bcan be obtained from the optimal weights of the adaptivebeamformer A

In the adaptive beamformer A the desired signal andobservation vector can be given by

1198891198600 (119899) = 119903119896 (119899) x1198600 (119899) = y119860 (119899) (41)

The optimal weight vector of adaptive beamformerA canbe readily obtained according to (42)

The flow diagram of calculation of weight vectors inadaptive beamformer A is as follows

rx1198600d1198600 = 119864 [x1198600 (119899) 119889lowast

1198600(119899)]

11990301119861 = 0 119888(1)

1119860= 119903minus1

11119860

MSE(1)119860

= 1205902

1198890119860minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(1)

1119860

FOR 119894 = 2 3 119872 minus 1

t119898119860 =119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) x119898minus1119860 (119899)

119889119894119860 (119899) = t119867119898119860

x119898minus1119860 (119899)

x119898119860 (119899) = x119898minus1119860 (119899) minus 119889119894119860 (119899) t119898119860

119903119898minus1119898119860 =

119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) 119889119898119860 (119899)

119903119898119898119860 =

119871minus1

sum

119899=0

119889lowast

119898119860(119899) 119889119898119860 (119899)

120573119894119860 = 119903119894119894119860 minus1003816100381610038161003816119903119894119894119860

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119860

c(119894)1119860

= [c(119894minus1)1119860

0

] + 120573minus1

119894119860(119888(119894minus1)

1198941119860)lowast

[

1003816100381610038161003816119903119894minus11198941198601003816100381610038161003816

2 c(119894minus1)119894119860

minus119903lowast

119894minus1119894119860

]

c(119894)119894119860

= 120573minus1

119894119860[minus119903119894minus1119894119860c

(119894minus1)

119894119860

1

]

MSE(119894)119860= 1205902

1198891198600minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(119894)

11119860

END

T(119872minus1)119860

= [t1119860 t2119860 t119872minus1119860]

w(119872minus1)1198600

= T(119872minus1)119860

c(119872minus1)1119860

(42)

In the adaptive beamformer B the phase-shifted desiredsignal and observation vector can be given by

1198891198610 (119899) = 119890119895120601119896119903119896 (43)

And the optimal weight vector of adaptive beamformer Bcan be obtained according to (42) as shown in (44)

The flow diagram of the calculation of weight vector inadaptive beamformer B is as follows

rx1198610d1198610 = 119864 [x1198610 (119899) 119889lowast

1198610(119899)]

11990301119861 = 0 119888(1)

1119861= 119903minus1

11119861

MSE(1)119861

= 1205902

1198890119861minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(1)

1119861

FOR 119894 = 2 3 119872 minus 1

t119898119861 =119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) x119898minus1119861 (119899)

119889119894119861 (119899) = t119867119898119861

x119898minus1119861 (119899)

x119898119861 (119899) = x119898minus1119861 (119899) minus 119889119894119861 (119899) t119898119861

119903119898minus1119898119861 =

119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) 119889119898119861 (119899)

119903119898119898119861 =

119871minus1

sum

119899=0

119889lowast

119898119861(119899) 119889119898119861 (119899)

120573119894119861 = 119903119894119894119861 minus1003816100381610038161003816119903119894119894119861

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119861

c(119894)1119861

= [c(119894minus1)1119861

0

] + 120573minus1

119894119861(119888(119894minus1)

1198941119861)lowast

[

1003816100381610038161003816119903119894minus11198941198611003816100381610038161003816

2 c(119894minus1)119894119861

minus119903lowast

119894minus1119894119861

]

c(119894)119894119861= 120573minus1

119894119861[minus119903119894minus1119894119861c

(119894minus1)

119894119861

1

]

MSE(119894)119861= 1205902

1198891198610minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(119894)

11119861

END

T(119872minus1)119861

= [t1119861 t2119861 t119872minus1119861]

w(119872minus1)1198610

= T(119872minus1)119861

c(119872minus1)1119861

(44)

Substituting (11) and (13) into (44) we have

w(119872minus1)1198600

= w(119872minus1)1198610

(45)

Therefore the weight vector w(119872minus1)1198610

can be obtained bycalculating the optimal weight of the adaptive beamformerA

42 Calculation of DOA The adaptive beamformer B basedon the structure of data level order recursive MSNWF can besimplified to a single stageWiener filter in virtue of obtaining

International Journal of Distributed Sensor Networks 7

its weight from the adaptive beamformer A Let r119896(119899) =

(w(119872minus1)1198610

)119867

y119861 denote the output signal of beamformer B Let

r119896 = [r119896 (1) r119896 (2) r119896 (119871)]119879 (46)

Thus r119896 is an optimal estimation of the phase-shiftedreference signal 119890119895120601119896r119896 in the MMSE sense which can bewritten as

r119896 = 119890119895120601119896r119896 + N119896 (47)

Let120601119896 denote an estimation of120601119896 which can be calculatedby using the least square method such that the square errorbetween the two signal vectors r119896 and r119896 is minimized

120601119896min

100381710038171003817100381710038171003817r119896 minus 119890

119895120601119896r1198961003817100381710038171003817100381710038172 (48)

In [18] Wang et al give the optimum solution of 120601119896

120601119896 = arg (r119896r119867

119896) (49)

According to (12) an estimation of the target DOA can beobtained then as

120579119896 = arcsin(minus120582120601119896

2120587119889) (50)

5 Simulation Results

In this section the performance of the proposed methodincluding the resolution capacity and accuracy of the datalevel order recursive MSNWF DOA techniques will beevaluated through numerical simulations In Sections 51 and52 the resolution and the capacity of the DOA estimationusing the data level order recursiveMSNWFDOA techniqueswill be illustrated and compared with other techniques suchas MUSIC ESPRIT SBDOA and original MSNWF DOAestimation techniques In Sections 53 and 54 the effectsof snapshot length and stage of data level order recursiveMSNWF on the estimation accuracy will be investigatedrespectively

51 Resolution of DOA Estimation Assume that a ULA of 10elements with a spacing of 119889 = 1205822 deployed at the receiverwas employed in the simulations to deal with a case wherethe DOAs of three signals and two interference signals areclosely distributed Further assume that the DOAs of thetarget related signal components are at minus2∘ 0∘ and 2∘ TheDOAs of the interference components are at minus4∘ and 4∘ Thebackground noise power spectral density ratio of the receivedsignal is set to 10 dB Snapshot length is fixed at 100 and thestage of MSNWF is set to 5 One thousand simulation runswere performed These simulation results are illustrated inFigure 5

The histograms of the resolution of DOA estimationobtained for these five techniques are shown in Figures5(a)ndash5(e) The histogram depicts the number of occurrencesestimated DOA as a function of DOA degrees In Figure 5(a)

the histogram of MUSIC technique shows two peak valueswhich deviate from the DOAs of the target signals InFigure 5(b) although the histogram of ESPIRT techniqueshows three peak values the peak values deviate from theDOAs of the target signals It is seen that the MUSICtechnique or ESPRIT technique cannot offer the desiredresults when the DOAs of target signals are very closeCorrespondingly in Figures 5(c) 5(d) and 5(e) the his-togram shows three peak values indicating that using theSBDOA original MSNWF DOA and the data level orderrecursive MSNWF DOA techniques all three DOAs aresuccessfully estimated Therefore it proved that the datalevel order recursive MSNWF DOA technique could obtaina better resolution than MUSIC and ESPRIT techniquesHowever the SBDOA requires 119874(1198723) operations the orig-inal MSNWF DOA technique needs 119874(21198722 + 9119872) oper-ations and the data level order recursive MSNWF DOAtechnique demands 119874(1198722 + 11119872) operations which signif-icantly reduce the complexity of computation In additionif the recursive order is enough the resolution of data levelorder recursive MSNWF DOA technique will be well asthat of SBDOA estimation and it is proved in Section 54The resolution and accuracy of data level order recursiveMSNWF are better than the original MSNWF which is dueto the update of the MSE at each stage

52 Capacity of DOA Estimation This simulation deals witha case where the number of target signals and interferenceis larger than that of antenna elements The simulationconditions are kept the same as those in Section 51 exceptfor the number of signal components considered The DOAsof 9 target signal components are set from minus40∘ to 40∘ withinterval 10∘ and the DOAs of 6 interference components areset from minus25∘ to 25∘ with interval 10∘ The simulation resultsare shown in Figure 6

Histograms of the obtained estimated DOAs are shownin Figures 6(a)ndash6(e) In Figures 6(a) and 6(b) the histogramsshow the deviated peak values anddemonstrate that these twotechniques cannot provide acceptable DOA estimation whenthe number of antenna elements is less than the total numberof target signals and interference In contrast in Figures6(c) 6(d) and 6(e) the histograms show that all 9 targetDOAs are successfully estimated when using the SBDOAoriginal MSNWF and data level order recursive MASNWFDOA techniques As can be seen the successful probability ofDOA estimation in data level order recursive MSNWF DOAtechnique is the same as that in the SBDOA and originalMSNWF DOA estimation techniques

53 Effects of Snapshot Length of MSNWF on DOA Estima-tion Accuracy In the simulation of snapshot length effectsthe snapshot length for adaptive beamformer A and DOAcalculation are set to different values such as 50 100 200 500and 1000 and the stages of both original MSNWF and datalevel order recursive MSNWF are set to 5 The DOA of thetarget signal is fixed at 0∘ and the DOAs of the interferenceare set from minus90∘ to 90∘ with interval 10∘ except 0∘ Theroot mean square error (RMSE) of the estimated target DOA

8 International Journal of Distributed Sensor Networks

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(a) Resolution of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(b) Resolution of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(c) Resolution of SBDOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus 3 minus 2 minus 1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(d) Resolution of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(e) Resolution of data level order recursive MSNWF DOA estimation

Figure 5 Comparison of the resolution of DOA estimation for signal sources that are closely distributed

averaged over one thousand simulation runs at different SNRconditions the RMSE of the estimated target DOA and thesnapshot length are illustrated in Figure 7

As can be seen in Figure 7 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 5∘when using a small snapshot

length such as 50 The simulation results show that when thesnapshot length is 500 the data level order recursiveMSNWFDOA estimation method will have estimation accuracy sim-ilar to that of the SBDOA technique However the RMSEof the data level order recursive MSNWF DOA technique isbetter than that of original MSNWF DOA technique under

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

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DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 5

According to (19) and its property the tridiagonal covari-ance matrix can be rewritten as

R(119872minus1)d = (T(119872minus1))119867

Rx0T(119872minus1)

= [

R11 R12R21 119903119872minus1119872minus1

] (31)

where

R11 = (T(119872minus2))119867

Rx0T(119872minus2)

R12 = [0119879 119903119872minus2119872minus1]119879

R21 = [0119879 119903lowast

119872minus2119872minus1]

(32)

The cross correlation vector between the new observationvector and desired signal 1198890(119899) is

r(119872minus1)dd0 = (T(119872minus1))119867

rx0d0 = [

1003817100381710038171003817rx0d010038171003817100381710038172

0] (33)

Given R(119872minus2)d from stage (119872 minus 2) the new elements ofR(119872minus1)d are calculated as

119903119872minus1119872minus1 = t119867119872minus1

Rx0t119872minus1

119903119872minus2119872minus1 = t119867119872minus2

Rx0t119872minus1(34)

According to (26) the (34) can be rewritten as

119903119872minus1119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus1(119899) 119889119872minus1 (119899)

119903119872minus2119872minus1 =

119871minus1

sum

119899=0

119889lowast

119872minus2(119899) 119889119872minus1 (119899)

(35)

Consider the property that only the first element of thecross correlation vector 119903(119872minus1)

1198891198890is not equal to 0 Therefore

only the first column of the inverse of R(119872minus1)d is needed tocalculate the recursion coefficients via (28)

Let the inverse of R(119872minus1)d be noted as

C(119872minus1) = (R(119872minus1)d )minus1

= [c(119872minus1)1

c(119872minus1)2

c(119872minus1)119872minus1

]

= [

C(119872minus2) 0

0119879 0

] + 120573minus1

119872minus1b(119872minus1) (b(119872minus1))

119867

(36)

The various quantities in (36) are defined as in thefollowing equation

b(119872minus1) = [119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

]

120573119872minus1 = 119903119872minus1119872minus1 minus1003816100381610038161003816119903119872minus1119872minus1

1003816100381610038161003816

2119888(119872minus2)

119872minus2119872minus2

(37)

x0(n)

d0(n)

w1 w2

d1(n) d1(n)

x1(n)+

+

minus

++

minus

++minus +

+minus +

+minusMSE0

MSE1MSE2

tH1 tH2t1 t2

Figure 4 The structure of data level order recursive MSNWF

where 119888(119872minus2)119872minus2119872minus2

is the last element of the last column c(119872minus2)119872minus2

from the previous stageTherefore the column vector c(119872minus1)

1of C(119872minus1) can be

calculated as

c(119872minus1)1

= [c(119872minus2)1

0

] + 120573minus1

119872minus1(119888(119872minus2)

1119872minus2)lowast

[

1003816100381610038161003816119903119872minus2119872minus11003816100381610038161003816

2 c(119872minus2)119872minus2

minus119903lowast

119872minus2119872minus1

]

(38)

where 119888(119872minus2)1119872minus2

is the first element of c(119872minus2)119872minus2

It can be seen from (38) that c(119872minus1)

1depends on c(119872minus2)

1

from stage (119872 minus 2) and the new elements 119903119872minus2119872minus1 and119903119872minus1119872minus1 generated from the covariance matrix at stage (119872minus

1) And the coefficients of the Wiener filter w(119872minus1)d are alsodepending on that Moreover the last column vector c(119872minus1)

119872minus1

of C(119872minus1) depends on the last column c(119872minus2)119872minus2

from stage (119872minus

2) According to (36) the last column vector c(119872minus1)119872minus1

can beupdated as

c(119872minus1)119872minus1

= 120573minus1

119872minus1[minus119903119872minus2119872minus1c

(119872minus2)

119872minus2

1

] (39)

It can be seen from (39) that the last column vector c(119872minus1)119872minus1

is only depending on the last column vector c(119872minus2)119872minus2

from stage(119872 minus 2) and the new element 119903119872minus2119872minus1 generated from thecovariance matrix at stage (119872 minus 1)

According to (38) and (39) it can be seen that in recursivecalculation process only c(119872minus1)

1and c(119872minus1)

119872minus1are needed to

be updated at each stage This avoids the calculation ofthe inversion of covariance matrix which also reduces thecomplexity of computation

As for the MSE expressed as in (30) it can be simple andcan be updated with 119888

(119872minus1)

11generated from the covariance

matrix at stage (119872 minus 1) as follows

MSE(119872minus1) = 1205902

1198890minus1003817100381710038171003817rx0d0

1003817100381710038171003817

2

2119888(119872minus1)

11 (40)

According to recursive algorithm about the calculation ofthe coefficients of thematch filters and the nest order the datalevel order recursive MSNWF DOA estimation structure canbe drawn as in Figure 4

4 MSNWF DOA Estimation System

41 Calculation of Weight Vector In the MSNWF DOAsystem the optimal estimation of the phase-shifted referencesignal 119890119895120601119896r119896 in the minimummean square error sense can be

6 International Journal of Distributed Sensor Networks

obtained at the output of the adaptive beamformer B whichuses the adaptive beamforming weights obtained from theadaptive beamformer A with the MSNWF structure

In the adaptive beamformer B consider the case wherethe phase-shifted reference signal 119890119895120601119896r119896 is the desired signaland the output of the adaptive beamformer B can be usedto estimate the desired signal Since the phase-shifted 119890

119895120601119896

is unknown both the phase-shifted reference signal and theweight vector of the adaptive beamformerB are not availableHowever the weight vector of the adaptive beamformer Bcan be obtained from the optimal weights of the adaptivebeamformer A

In the adaptive beamformer A the desired signal andobservation vector can be given by

1198891198600 (119899) = 119903119896 (119899) x1198600 (119899) = y119860 (119899) (41)

The optimal weight vector of adaptive beamformerA canbe readily obtained according to (42)

The flow diagram of calculation of weight vectors inadaptive beamformer A is as follows

rx1198600d1198600 = 119864 [x1198600 (119899) 119889lowast

1198600(119899)]

11990301119861 = 0 119888(1)

1119860= 119903minus1

11119860

MSE(1)119860

= 1205902

1198890119860minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(1)

1119860

FOR 119894 = 2 3 119872 minus 1

t119898119860 =119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) x119898minus1119860 (119899)

119889119894119860 (119899) = t119867119898119860

x119898minus1119860 (119899)

x119898119860 (119899) = x119898minus1119860 (119899) minus 119889119894119860 (119899) t119898119860

119903119898minus1119898119860 =

119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) 119889119898119860 (119899)

119903119898119898119860 =

119871minus1

sum

119899=0

119889lowast

119898119860(119899) 119889119898119860 (119899)

120573119894119860 = 119903119894119894119860 minus1003816100381610038161003816119903119894119894119860

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119860

c(119894)1119860

= [c(119894minus1)1119860

0

] + 120573minus1

119894119860(119888(119894minus1)

1198941119860)lowast

[

1003816100381610038161003816119903119894minus11198941198601003816100381610038161003816

2 c(119894minus1)119894119860

minus119903lowast

119894minus1119894119860

]

c(119894)119894119860

= 120573minus1

119894119860[minus119903119894minus1119894119860c

(119894minus1)

119894119860

1

]

MSE(119894)119860= 1205902

1198891198600minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(119894)

11119860

END

T(119872minus1)119860

= [t1119860 t2119860 t119872minus1119860]

w(119872minus1)1198600

= T(119872minus1)119860

c(119872minus1)1119860

(42)

In the adaptive beamformer B the phase-shifted desiredsignal and observation vector can be given by

1198891198610 (119899) = 119890119895120601119896119903119896 (43)

And the optimal weight vector of adaptive beamformer Bcan be obtained according to (42) as shown in (44)

The flow diagram of the calculation of weight vector inadaptive beamformer B is as follows

rx1198610d1198610 = 119864 [x1198610 (119899) 119889lowast

1198610(119899)]

11990301119861 = 0 119888(1)

1119861= 119903minus1

11119861

MSE(1)119861

= 1205902

1198890119861minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(1)

1119861

FOR 119894 = 2 3 119872 minus 1

t119898119861 =119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) x119898minus1119861 (119899)

119889119894119861 (119899) = t119867119898119861

x119898minus1119861 (119899)

x119898119861 (119899) = x119898minus1119861 (119899) minus 119889119894119861 (119899) t119898119861

119903119898minus1119898119861 =

119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) 119889119898119861 (119899)

119903119898119898119861 =

119871minus1

sum

119899=0

119889lowast

119898119861(119899) 119889119898119861 (119899)

120573119894119861 = 119903119894119894119861 minus1003816100381610038161003816119903119894119894119861

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119861

c(119894)1119861

= [c(119894minus1)1119861

0

] + 120573minus1

119894119861(119888(119894minus1)

1198941119861)lowast

[

1003816100381610038161003816119903119894minus11198941198611003816100381610038161003816

2 c(119894minus1)119894119861

minus119903lowast

119894minus1119894119861

]

c(119894)119894119861= 120573minus1

119894119861[minus119903119894minus1119894119861c

(119894minus1)

119894119861

1

]

MSE(119894)119861= 1205902

1198891198610minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(119894)

11119861

END

T(119872minus1)119861

= [t1119861 t2119861 t119872minus1119861]

w(119872minus1)1198610

= T(119872minus1)119861

c(119872minus1)1119861

(44)

Substituting (11) and (13) into (44) we have

w(119872minus1)1198600

= w(119872minus1)1198610

(45)

Therefore the weight vector w(119872minus1)1198610

can be obtained bycalculating the optimal weight of the adaptive beamformerA

42 Calculation of DOA The adaptive beamformer B basedon the structure of data level order recursive MSNWF can besimplified to a single stageWiener filter in virtue of obtaining

International Journal of Distributed Sensor Networks 7

its weight from the adaptive beamformer A Let r119896(119899) =

(w(119872minus1)1198610

)119867

y119861 denote the output signal of beamformer B Let

r119896 = [r119896 (1) r119896 (2) r119896 (119871)]119879 (46)

Thus r119896 is an optimal estimation of the phase-shiftedreference signal 119890119895120601119896r119896 in the MMSE sense which can bewritten as

r119896 = 119890119895120601119896r119896 + N119896 (47)

Let120601119896 denote an estimation of120601119896 which can be calculatedby using the least square method such that the square errorbetween the two signal vectors r119896 and r119896 is minimized

120601119896min

100381710038171003817100381710038171003817r119896 minus 119890

119895120601119896r1198961003817100381710038171003817100381710038172 (48)

In [18] Wang et al give the optimum solution of 120601119896

120601119896 = arg (r119896r119867

119896) (49)

According to (12) an estimation of the target DOA can beobtained then as

120579119896 = arcsin(minus120582120601119896

2120587119889) (50)

5 Simulation Results

In this section the performance of the proposed methodincluding the resolution capacity and accuracy of the datalevel order recursive MSNWF DOA techniques will beevaluated through numerical simulations In Sections 51 and52 the resolution and the capacity of the DOA estimationusing the data level order recursiveMSNWFDOA techniqueswill be illustrated and compared with other techniques suchas MUSIC ESPRIT SBDOA and original MSNWF DOAestimation techniques In Sections 53 and 54 the effectsof snapshot length and stage of data level order recursiveMSNWF on the estimation accuracy will be investigatedrespectively

51 Resolution of DOA Estimation Assume that a ULA of 10elements with a spacing of 119889 = 1205822 deployed at the receiverwas employed in the simulations to deal with a case wherethe DOAs of three signals and two interference signals areclosely distributed Further assume that the DOAs of thetarget related signal components are at minus2∘ 0∘ and 2∘ TheDOAs of the interference components are at minus4∘ and 4∘ Thebackground noise power spectral density ratio of the receivedsignal is set to 10 dB Snapshot length is fixed at 100 and thestage of MSNWF is set to 5 One thousand simulation runswere performed These simulation results are illustrated inFigure 5

The histograms of the resolution of DOA estimationobtained for these five techniques are shown in Figures5(a)ndash5(e) The histogram depicts the number of occurrencesestimated DOA as a function of DOA degrees In Figure 5(a)

the histogram of MUSIC technique shows two peak valueswhich deviate from the DOAs of the target signals InFigure 5(b) although the histogram of ESPIRT techniqueshows three peak values the peak values deviate from theDOAs of the target signals It is seen that the MUSICtechnique or ESPRIT technique cannot offer the desiredresults when the DOAs of target signals are very closeCorrespondingly in Figures 5(c) 5(d) and 5(e) the his-togram shows three peak values indicating that using theSBDOA original MSNWF DOA and the data level orderrecursive MSNWF DOA techniques all three DOAs aresuccessfully estimated Therefore it proved that the datalevel order recursive MSNWF DOA technique could obtaina better resolution than MUSIC and ESPRIT techniquesHowever the SBDOA requires 119874(1198723) operations the orig-inal MSNWF DOA technique needs 119874(21198722 + 9119872) oper-ations and the data level order recursive MSNWF DOAtechnique demands 119874(1198722 + 11119872) operations which signif-icantly reduce the complexity of computation In additionif the recursive order is enough the resolution of data levelorder recursive MSNWF DOA technique will be well asthat of SBDOA estimation and it is proved in Section 54The resolution and accuracy of data level order recursiveMSNWF are better than the original MSNWF which is dueto the update of the MSE at each stage

52 Capacity of DOA Estimation This simulation deals witha case where the number of target signals and interferenceis larger than that of antenna elements The simulationconditions are kept the same as those in Section 51 exceptfor the number of signal components considered The DOAsof 9 target signal components are set from minus40∘ to 40∘ withinterval 10∘ and the DOAs of 6 interference components areset from minus25∘ to 25∘ with interval 10∘ The simulation resultsare shown in Figure 6

Histograms of the obtained estimated DOAs are shownin Figures 6(a)ndash6(e) In Figures 6(a) and 6(b) the histogramsshow the deviated peak values anddemonstrate that these twotechniques cannot provide acceptable DOA estimation whenthe number of antenna elements is less than the total numberof target signals and interference In contrast in Figures6(c) 6(d) and 6(e) the histograms show that all 9 targetDOAs are successfully estimated when using the SBDOAoriginal MSNWF and data level order recursive MASNWFDOA techniques As can be seen the successful probability ofDOA estimation in data level order recursive MSNWF DOAtechnique is the same as that in the SBDOA and originalMSNWF DOA estimation techniques

53 Effects of Snapshot Length of MSNWF on DOA Estima-tion Accuracy In the simulation of snapshot length effectsthe snapshot length for adaptive beamformer A and DOAcalculation are set to different values such as 50 100 200 500and 1000 and the stages of both original MSNWF and datalevel order recursive MSNWF are set to 5 The DOA of thetarget signal is fixed at 0∘ and the DOAs of the interferenceare set from minus90∘ to 90∘ with interval 10∘ except 0∘ Theroot mean square error (RMSE) of the estimated target DOA

8 International Journal of Distributed Sensor Networks

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(a) Resolution of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(b) Resolution of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(c) Resolution of SBDOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus 3 minus 2 minus 1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(d) Resolution of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(e) Resolution of data level order recursive MSNWF DOA estimation

Figure 5 Comparison of the resolution of DOA estimation for signal sources that are closely distributed

averaged over one thousand simulation runs at different SNRconditions the RMSE of the estimated target DOA and thesnapshot length are illustrated in Figure 7

As can be seen in Figure 7 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 5∘when using a small snapshot

length such as 50 The simulation results show that when thesnapshot length is 500 the data level order recursiveMSNWFDOA estimation method will have estimation accuracy sim-ilar to that of the SBDOA technique However the RMSEof the data level order recursive MSNWF DOA technique isbetter than that of original MSNWF DOA technique under

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

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DistributedSensor Networks

International Journal of

6 International Journal of Distributed Sensor Networks

obtained at the output of the adaptive beamformer B whichuses the adaptive beamforming weights obtained from theadaptive beamformer A with the MSNWF structure

In the adaptive beamformer B consider the case wherethe phase-shifted reference signal 119890119895120601119896r119896 is the desired signaland the output of the adaptive beamformer B can be usedto estimate the desired signal Since the phase-shifted 119890

119895120601119896

is unknown both the phase-shifted reference signal and theweight vector of the adaptive beamformerB are not availableHowever the weight vector of the adaptive beamformer Bcan be obtained from the optimal weights of the adaptivebeamformer A

In the adaptive beamformer A the desired signal andobservation vector can be given by

1198891198600 (119899) = 119903119896 (119899) x1198600 (119899) = y119860 (119899) (41)

The optimal weight vector of adaptive beamformerA canbe readily obtained according to (42)

The flow diagram of calculation of weight vectors inadaptive beamformer A is as follows

rx1198600d1198600 = 119864 [x1198600 (119899) 119889lowast

1198600(119899)]

11990301119861 = 0 119888(1)

1119860= 119903minus1

11119860

MSE(1)119860

= 1205902

1198890119860minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(1)

1119860

FOR 119894 = 2 3 119872 minus 1

t119898119860 =119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) x119898minus1119860 (119899)

119889119894119860 (119899) = t119867119898119860

x119898minus1119860 (119899)

x119898119860 (119899) = x119898minus1119860 (119899) minus 119889119894119860 (119899) t119898119860

119903119898minus1119898119860 =

119871minus1

sum

119899=0

119889lowast

119898minus1119860(119899) 119889119898119860 (119899)

119903119898119898119860 =

119871minus1

sum

119899=0

119889lowast

119898119860(119899) 119889119898119860 (119899)

120573119894119860 = 119903119894119894119860 minus1003816100381610038161003816119903119894119894119860

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119860

c(119894)1119860

= [c(119894minus1)1119860

0

] + 120573minus1

119894119860(119888(119894minus1)

1198941119860)lowast

[

1003816100381610038161003816119903119894minus11198941198601003816100381610038161003816

2 c(119894minus1)119894119860

minus119903lowast

119894minus1119894119860

]

c(119894)119894119860

= 120573minus1

119894119860[minus119903119894minus1119894119860c

(119894minus1)

119894119860

1

]

MSE(119894)119860= 1205902

1198891198600minus10038171003817100381710038171003817rx1198600d1198600

10038171003817100381710038171003817

2

2119888(119894)

11119860

END

T(119872minus1)119860

= [t1119860 t2119860 t119872minus1119860]

w(119872minus1)1198600

= T(119872minus1)119860

c(119872minus1)1119860

(42)

In the adaptive beamformer B the phase-shifted desiredsignal and observation vector can be given by

1198891198610 (119899) = 119890119895120601119896119903119896 (43)

And the optimal weight vector of adaptive beamformer Bcan be obtained according to (42) as shown in (44)

The flow diagram of the calculation of weight vector inadaptive beamformer B is as follows

rx1198610d1198610 = 119864 [x1198610 (119899) 119889lowast

1198610(119899)]

11990301119861 = 0 119888(1)

1119861= 119903minus1

11119861

MSE(1)119861

= 1205902

1198890119861minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(1)

1119861

FOR 119894 = 2 3 119872 minus 1

t119898119861 =119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) x119898minus1119861 (119899)

119889119894119861 (119899) = t119867119898119861

x119898minus1119861 (119899)

x119898119861 (119899) = x119898minus1119861 (119899) minus 119889119894119861 (119899) t119898119861

119903119898minus1119898119861 =

119871minus1

sum

119899=0

119889lowast

119898minus1119861(119899) 119889119898119861 (119899)

119903119898119898119861 =

119871minus1

sum

119899=0

119889lowast

119898119861(119899) 119889119898119861 (119899)

120573119894119861 = 119903119894119894119861 minus1003816100381610038161003816119903119894119894119861

1003816100381610038161003816

2119888(119894minus1)

1119894minus1119861

c(119894)1119861

= [c(119894minus1)1119861

0

] + 120573minus1

119894119861(119888(119894minus1)

1198941119861)lowast

[

1003816100381610038161003816119903119894minus11198941198611003816100381610038161003816

2 c(119894minus1)119894119861

minus119903lowast

119894minus1119894119861

]

c(119894)119894119861= 120573minus1

119894119861[minus119903119894minus1119894119861c

(119894minus1)

119894119861

1

]

MSE(119894)119861= 1205902

1198891198610minus10038171003817100381710038171003817rx1198610d1198610

10038171003817100381710038171003817

2

2119888(119894)

11119861

END

T(119872minus1)119861

= [t1119861 t2119861 t119872minus1119861]

w(119872minus1)1198610

= T(119872minus1)119861

c(119872minus1)1119861

(44)

Substituting (11) and (13) into (44) we have

w(119872minus1)1198600

= w(119872minus1)1198610

(45)

Therefore the weight vector w(119872minus1)1198610

can be obtained bycalculating the optimal weight of the adaptive beamformerA

42 Calculation of DOA The adaptive beamformer B basedon the structure of data level order recursive MSNWF can besimplified to a single stageWiener filter in virtue of obtaining

International Journal of Distributed Sensor Networks 7

its weight from the adaptive beamformer A Let r119896(119899) =

(w(119872minus1)1198610

)119867

y119861 denote the output signal of beamformer B Let

r119896 = [r119896 (1) r119896 (2) r119896 (119871)]119879 (46)

Thus r119896 is an optimal estimation of the phase-shiftedreference signal 119890119895120601119896r119896 in the MMSE sense which can bewritten as

r119896 = 119890119895120601119896r119896 + N119896 (47)

Let120601119896 denote an estimation of120601119896 which can be calculatedby using the least square method such that the square errorbetween the two signal vectors r119896 and r119896 is minimized

120601119896min

100381710038171003817100381710038171003817r119896 minus 119890

119895120601119896r1198961003817100381710038171003817100381710038172 (48)

In [18] Wang et al give the optimum solution of 120601119896

120601119896 = arg (r119896r119867

119896) (49)

According to (12) an estimation of the target DOA can beobtained then as

120579119896 = arcsin(minus120582120601119896

2120587119889) (50)

5 Simulation Results

In this section the performance of the proposed methodincluding the resolution capacity and accuracy of the datalevel order recursive MSNWF DOA techniques will beevaluated through numerical simulations In Sections 51 and52 the resolution and the capacity of the DOA estimationusing the data level order recursiveMSNWFDOA techniqueswill be illustrated and compared with other techniques suchas MUSIC ESPRIT SBDOA and original MSNWF DOAestimation techniques In Sections 53 and 54 the effectsof snapshot length and stage of data level order recursiveMSNWF on the estimation accuracy will be investigatedrespectively

51 Resolution of DOA Estimation Assume that a ULA of 10elements with a spacing of 119889 = 1205822 deployed at the receiverwas employed in the simulations to deal with a case wherethe DOAs of three signals and two interference signals areclosely distributed Further assume that the DOAs of thetarget related signal components are at minus2∘ 0∘ and 2∘ TheDOAs of the interference components are at minus4∘ and 4∘ Thebackground noise power spectral density ratio of the receivedsignal is set to 10 dB Snapshot length is fixed at 100 and thestage of MSNWF is set to 5 One thousand simulation runswere performed These simulation results are illustrated inFigure 5

The histograms of the resolution of DOA estimationobtained for these five techniques are shown in Figures5(a)ndash5(e) The histogram depicts the number of occurrencesestimated DOA as a function of DOA degrees In Figure 5(a)

the histogram of MUSIC technique shows two peak valueswhich deviate from the DOAs of the target signals InFigure 5(b) although the histogram of ESPIRT techniqueshows three peak values the peak values deviate from theDOAs of the target signals It is seen that the MUSICtechnique or ESPRIT technique cannot offer the desiredresults when the DOAs of target signals are very closeCorrespondingly in Figures 5(c) 5(d) and 5(e) the his-togram shows three peak values indicating that using theSBDOA original MSNWF DOA and the data level orderrecursive MSNWF DOA techniques all three DOAs aresuccessfully estimated Therefore it proved that the datalevel order recursive MSNWF DOA technique could obtaina better resolution than MUSIC and ESPRIT techniquesHowever the SBDOA requires 119874(1198723) operations the orig-inal MSNWF DOA technique needs 119874(21198722 + 9119872) oper-ations and the data level order recursive MSNWF DOAtechnique demands 119874(1198722 + 11119872) operations which signif-icantly reduce the complexity of computation In additionif the recursive order is enough the resolution of data levelorder recursive MSNWF DOA technique will be well asthat of SBDOA estimation and it is proved in Section 54The resolution and accuracy of data level order recursiveMSNWF are better than the original MSNWF which is dueto the update of the MSE at each stage

52 Capacity of DOA Estimation This simulation deals witha case where the number of target signals and interferenceis larger than that of antenna elements The simulationconditions are kept the same as those in Section 51 exceptfor the number of signal components considered The DOAsof 9 target signal components are set from minus40∘ to 40∘ withinterval 10∘ and the DOAs of 6 interference components areset from minus25∘ to 25∘ with interval 10∘ The simulation resultsare shown in Figure 6

Histograms of the obtained estimated DOAs are shownin Figures 6(a)ndash6(e) In Figures 6(a) and 6(b) the histogramsshow the deviated peak values anddemonstrate that these twotechniques cannot provide acceptable DOA estimation whenthe number of antenna elements is less than the total numberof target signals and interference In contrast in Figures6(c) 6(d) and 6(e) the histograms show that all 9 targetDOAs are successfully estimated when using the SBDOAoriginal MSNWF and data level order recursive MASNWFDOA techniques As can be seen the successful probability ofDOA estimation in data level order recursive MSNWF DOAtechnique is the same as that in the SBDOA and originalMSNWF DOA estimation techniques

53 Effects of Snapshot Length of MSNWF on DOA Estima-tion Accuracy In the simulation of snapshot length effectsthe snapshot length for adaptive beamformer A and DOAcalculation are set to different values such as 50 100 200 500and 1000 and the stages of both original MSNWF and datalevel order recursive MSNWF are set to 5 The DOA of thetarget signal is fixed at 0∘ and the DOAs of the interferenceare set from minus90∘ to 90∘ with interval 10∘ except 0∘ Theroot mean square error (RMSE) of the estimated target DOA

8 International Journal of Distributed Sensor Networks

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(a) Resolution of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(b) Resolution of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(c) Resolution of SBDOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus 3 minus 2 minus 1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(d) Resolution of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(e) Resolution of data level order recursive MSNWF DOA estimation

Figure 5 Comparison of the resolution of DOA estimation for signal sources that are closely distributed

averaged over one thousand simulation runs at different SNRconditions the RMSE of the estimated target DOA and thesnapshot length are illustrated in Figure 7

As can be seen in Figure 7 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 5∘when using a small snapshot

length such as 50 The simulation results show that when thesnapshot length is 500 the data level order recursiveMSNWFDOA estimation method will have estimation accuracy sim-ilar to that of the SBDOA technique However the RMSEof the data level order recursive MSNWF DOA technique isbetter than that of original MSNWF DOA technique under

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 7

its weight from the adaptive beamformer A Let r119896(119899) =

(w(119872minus1)1198610

)119867

y119861 denote the output signal of beamformer B Let

r119896 = [r119896 (1) r119896 (2) r119896 (119871)]119879 (46)

Thus r119896 is an optimal estimation of the phase-shiftedreference signal 119890119895120601119896r119896 in the MMSE sense which can bewritten as

r119896 = 119890119895120601119896r119896 + N119896 (47)

Let120601119896 denote an estimation of120601119896 which can be calculatedby using the least square method such that the square errorbetween the two signal vectors r119896 and r119896 is minimized

120601119896min

100381710038171003817100381710038171003817r119896 minus 119890

119895120601119896r1198961003817100381710038171003817100381710038172 (48)

In [18] Wang et al give the optimum solution of 120601119896

120601119896 = arg (r119896r119867

119896) (49)

According to (12) an estimation of the target DOA can beobtained then as

120579119896 = arcsin(minus120582120601119896

2120587119889) (50)

5 Simulation Results

In this section the performance of the proposed methodincluding the resolution capacity and accuracy of the datalevel order recursive MSNWF DOA techniques will beevaluated through numerical simulations In Sections 51 and52 the resolution and the capacity of the DOA estimationusing the data level order recursiveMSNWFDOA techniqueswill be illustrated and compared with other techniques suchas MUSIC ESPRIT SBDOA and original MSNWF DOAestimation techniques In Sections 53 and 54 the effectsof snapshot length and stage of data level order recursiveMSNWF on the estimation accuracy will be investigatedrespectively

51 Resolution of DOA Estimation Assume that a ULA of 10elements with a spacing of 119889 = 1205822 deployed at the receiverwas employed in the simulations to deal with a case wherethe DOAs of three signals and two interference signals areclosely distributed Further assume that the DOAs of thetarget related signal components are at minus2∘ 0∘ and 2∘ TheDOAs of the interference components are at minus4∘ and 4∘ Thebackground noise power spectral density ratio of the receivedsignal is set to 10 dB Snapshot length is fixed at 100 and thestage of MSNWF is set to 5 One thousand simulation runswere performed These simulation results are illustrated inFigure 5

The histograms of the resolution of DOA estimationobtained for these five techniques are shown in Figures5(a)ndash5(e) The histogram depicts the number of occurrencesestimated DOA as a function of DOA degrees In Figure 5(a)

the histogram of MUSIC technique shows two peak valueswhich deviate from the DOAs of the target signals InFigure 5(b) although the histogram of ESPIRT techniqueshows three peak values the peak values deviate from theDOAs of the target signals It is seen that the MUSICtechnique or ESPRIT technique cannot offer the desiredresults when the DOAs of target signals are very closeCorrespondingly in Figures 5(c) 5(d) and 5(e) the his-togram shows three peak values indicating that using theSBDOA original MSNWF DOA and the data level orderrecursive MSNWF DOA techniques all three DOAs aresuccessfully estimated Therefore it proved that the datalevel order recursive MSNWF DOA technique could obtaina better resolution than MUSIC and ESPRIT techniquesHowever the SBDOA requires 119874(1198723) operations the orig-inal MSNWF DOA technique needs 119874(21198722 + 9119872) oper-ations and the data level order recursive MSNWF DOAtechnique demands 119874(1198722 + 11119872) operations which signif-icantly reduce the complexity of computation In additionif the recursive order is enough the resolution of data levelorder recursive MSNWF DOA technique will be well asthat of SBDOA estimation and it is proved in Section 54The resolution and accuracy of data level order recursiveMSNWF are better than the original MSNWF which is dueto the update of the MSE at each stage

52 Capacity of DOA Estimation This simulation deals witha case where the number of target signals and interferenceis larger than that of antenna elements The simulationconditions are kept the same as those in Section 51 exceptfor the number of signal components considered The DOAsof 9 target signal components are set from minus40∘ to 40∘ withinterval 10∘ and the DOAs of 6 interference components areset from minus25∘ to 25∘ with interval 10∘ The simulation resultsare shown in Figure 6

Histograms of the obtained estimated DOAs are shownin Figures 6(a)ndash6(e) In Figures 6(a) and 6(b) the histogramsshow the deviated peak values anddemonstrate that these twotechniques cannot provide acceptable DOA estimation whenthe number of antenna elements is less than the total numberof target signals and interference In contrast in Figures6(c) 6(d) and 6(e) the histograms show that all 9 targetDOAs are successfully estimated when using the SBDOAoriginal MSNWF and data level order recursive MASNWFDOA techniques As can be seen the successful probability ofDOA estimation in data level order recursive MSNWF DOAtechnique is the same as that in the SBDOA and originalMSNWF DOA estimation techniques

53 Effects of Snapshot Length of MSNWF on DOA Estima-tion Accuracy In the simulation of snapshot length effectsthe snapshot length for adaptive beamformer A and DOAcalculation are set to different values such as 50 100 200 500and 1000 and the stages of both original MSNWF and datalevel order recursive MSNWF are set to 5 The DOA of thetarget signal is fixed at 0∘ and the DOAs of the interferenceare set from minus90∘ to 90∘ with interval 10∘ except 0∘ Theroot mean square error (RMSE) of the estimated target DOA

8 International Journal of Distributed Sensor Networks

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(a) Resolution of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(b) Resolution of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(c) Resolution of SBDOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus 3 minus 2 minus 1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(d) Resolution of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(e) Resolution of data level order recursive MSNWF DOA estimation

Figure 5 Comparison of the resolution of DOA estimation for signal sources that are closely distributed

averaged over one thousand simulation runs at different SNRconditions the RMSE of the estimated target DOA and thesnapshot length are illustrated in Figure 7

As can be seen in Figure 7 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 5∘when using a small snapshot

length such as 50 The simulation results show that when thesnapshot length is 500 the data level order recursiveMSNWFDOA estimation method will have estimation accuracy sim-ilar to that of the SBDOA technique However the RMSEof the data level order recursive MSNWF DOA technique isbetter than that of original MSNWF DOA technique under

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Distributed Sensor Networks

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(a) Resolution of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(b) Resolution of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(c) Resolution of SBDOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus 3 minus 2 minus 1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(d) Resolution of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

minus4minus5 minus3 minus2 minus1 0 1 2 3 4 5

Estimated DOA (deg)

Occ

urre

nces

(e) Resolution of data level order recursive MSNWF DOA estimation

Figure 5 Comparison of the resolution of DOA estimation for signal sources that are closely distributed

averaged over one thousand simulation runs at different SNRconditions the RMSE of the estimated target DOA and thesnapshot length are illustrated in Figure 7

As can be seen in Figure 7 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 5∘when using a small snapshot

length such as 50 The simulation results show that when thesnapshot length is 500 the data level order recursiveMSNWFDOA estimation method will have estimation accuracy sim-ilar to that of the SBDOA technique However the RMSEof the data level order recursive MSNWF DOA technique isbetter than that of original MSNWF DOA technique under

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 9

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(a) Capacity of MUSIC DOA estimation

160

140

120

100

80

60

40

0

20

minus40 minus20minus60 60400 20

Occ

urre

nces

Estimated DOA (deg)

(b) Capacity of ESPRIT DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(c) Capacity of SBDOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(d) Capacity of original MSNWF DOA estimation

160

140

120

100

80

60

40

0

20

Occ

urre

nces

minus40 minus20minus60 60400 20

Estimated DOA (deg)

(e) Capacity of data level order recursive MSNWF DOA estimation

Figure 6 Comparison of the capacity of DOA estimation when the number of signal and interference sources exceeds the number of antennaelements

various snapshot lengths which is due to the update of MSEat each stage to obtain the optimal weight vector The RMSEobviously decreases as the snapshot length increases suchas the RMSE which will be less than 1∘ when using onethousand snapshot length of the signal This demonstrates

that the fast DOA tracking can be implemented by usingthe data level order recursive MSNWF DOA technique andthat the estimation accuracy will be improved when usingmore sample data And the simulation also proved thatthe capacity of data level order recursive MSNWF DOA

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Distributed Sensor Networks

L = 50

L = 100

L = 200

L = 500

L = 1000

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

RMSE

of e

stim

ated

DO

A (d

eg)

5

4

3

2

1

0

Figure 7 RMSE of the estimated DOA for different snapshot length119871 and the SNR

estimation technique can be larger than the number of thesensor elements

54 Effects of the Stage of MSNWF on DOA EstimationAccuracy In the simulation of the stage effect both stages oforiginal MSNWF and data level order recursive MSNWF foradaptive beamformer A are set to the same values such as 35 and 9 The snapshot length is set to 200 And other sim-ulation conditions are kept the same as those in Section 53The RMSE of the estimated target DOA averaged over onethousand simulations runs at different SNR conditions TheRMSE of the estimated target DOA with different stages ofMSNWF and SNR is demonstrated in Figure 8

As can be seen from Figure 8 both the original MSNWFand the data level order recursive MSNWF DOA techniqueslead to a RMSE of less than 3∘ when using different stagesof MSNWF and the RMSE decreases as the MSNWF stageincreases However the RMSE of the data level order recur-siveMSNWFDOAestimation technique is better than that oforiginal MSNWF DOA estimation technique under variousstages which ismainly due to the update ofMSE at each stage

Moreover in the same simulation conditions the RMSEof SBDOA estimation technique is less than 15∘ In contrastthe RMSE of data level order recursive MSNWF DOAestimation technique is almost equal to that of SBDOAwhen using 9 stages However the original MSNWF DOAestimation technique requires more stages to obtain similarestimation accuracy

6 Conclusion

A novel DOA estimation method based on data level orderrecursive MSNWF has been proposed in this paper In this

SBDOAOriginal MSNWFProposed method

minus5 0 5 10 15 20

SNR (dB)

3

25

2

15

1

05

0

Stage = 3

Stage = 5

Stage = 9

SBDOA

RMSE

of e

stim

ated

DO

A (d

eg)

Figure 8 RMSE of the estimated DOA for different MSNWF stagesand SNR

technique two subarray adaptive beamformers based onthe MSNWF are used to form the phase-shift and rejectinterference at the same time The DOAs of target signalsare estimated from the phase-shift by using reference signalafter interference rejection Therefore the performance ofDOA estimation such as resolution capacity and accuracy issignificantly improved And the complexity of computationis also significantly reduced by avoiding the calculation ofcovariance matrix inversion when getting the optimal weightvector of the beamformer This technique can be widelyused for the implementation of hardware systems such aswireless communication system active radar sonar andSTAP systems Numerical simulations demonstrating theeffectiveness and advantage of this technique are presented

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J C Liberti and T S Rappaport Smart Antennas for WirelessCommunication IS-95 and Third Generation CDMA Applica-tions Prentice Hall Englewood Cliffs NJ USA 1999

[2] H X Yu X F Zhang X Q Chen and H L Wu ldquoCom-putationally efficient DOA tracking algorithm in monostaticMIMO radar with automatic associationrdquo International Journalof Antennas and Propagation vol 2014 Article ID 501478 10pages 2014

[3] X Zhang and X Wang ldquoL-shaped-sensor-array-based local-ization and tracking method for 3D maneuvering targetrdquo

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 11

International Journal of Distributed Sensor Networks vol 2013Article ID 741284 8 pages 2013

[4] S Phoha J Koch E Grele C Griffin and B Madan ldquoSpace-time coordinated distributed sensing algorithms for resourceefficient narrowband target localization and trackingrdquo Interna-tional Journal of Distributed Sensor Networks vol 1 no 1 pp81ndash99 2005

[5] Y M Zhang M G Amin and S Kaushik ldquoLocalization andtracking of passive RFID tags based on direction estimationrdquoInternational Journal of Antennas and Propagation vol 2007Article ID 17426 9 pages 2007

[6] Y Wang X Duan D Tian J Zhou Y Lu and G Lu ldquoABayesian compressive sensing vehicular location method Basedon three-dimensional radio frequencyrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 483613 13pages 2014

[7] H Jiang C Liu Y Zhang and H J Cui ldquoFast 3D nodelocalization in multipath for UWB wireless sensor networksusing modified propagator methodrdquo International Journal ofDistributed Sensor Networks vol 2014 Article ID 312535 8pages 2014

[8] K Xiong Z Liu and W Jiang ldquoSAGE-based algorithm fordirection-of-arrival estimation and array calibrationrdquo Interna-tional Journal of Antennas and Propagation vol 2014 ArticleID 217482 8 pages 2014

[9] J S Yang X Z Wu and Q Wang ldquoChannel parameterestimation for scatter cluster model using modified MUSICalgorithmrdquo International Journal of Antennas and Propagationvol 2012 Article ID 619817 6 pages 2012

[10] C L Chang and G S Huang ldquoLow-complexity spatial-temporal filtering method via compressive sensing for inter-ference mitigation in a GNSS receiverrdquo International Journal ofAntennas and Propagation vol 2014 Article ID 501025 8 pages2014

[11] Y Doisy L Deruaz and R Been ldquoInterference suppression ofsubarray adaptive beamforming in presence of sensor disper-sionsrdquo IEEE Transactions on Signal Processing vol 58 no 8 pp4195ndash4212 2010

[12] L C Godara ldquoApplication of antenna arrays to mobile commu-nications II Beam-forming and direction-of-arrival considera-tionsrdquo Proceedings of the IEEE vol 85 no 8 pp 1195ndash1245 1997

[13] A Klouche-Djedid and M Fujita ldquoAdaptive array sensorprocessing applications for mobile telephone communicationsrdquoIEEE Transactions on Vehicular Technology vol 45 no 3 pp405ndash416 1996

[14] M S Bartlett ldquoPeriodogram analysis and continuous spectrardquoBiometrika vol 37 no 1-2 pp 1ndash16 1950

[15] J Capon ldquoHigh-resolution frequency-wave-number spectrumanalysisrdquo Proceedings of IEEE vol 57 no 8 pp 1408ndash1418 1969

[16] R O Schmidt ldquoMultiple emitter location and signal parameterestimationrdquo IEEE Transactions on Antennas and Propagationvol 34 no 3 pp 276ndash280 1986

[17] R Roy and T Kailath ldquoESPRIT-Estimation of signal parametersrotational invariance techniquesrdquo IEEE Transactions on Acous-tics Speech and Signal Processing vol 37 no 7 pp 984ndash9951989

[18] N Y Wang P Agathoklis and A Antoniou ldquoA new DOAestimation technique based on subarray beamformingrdquo IEEETransactions on Signal Processing vol 54 no 9 pp 3279ndash32892006

[19] J S Goldstein and I S Reed ldquoA newmethod of wiener filteringand its application to interference mitigation for communica-tionsrdquo in Proceedings of the MILCOM Conference vol 3 pp1087ndash1091 Monterey Calif USA November 1997

[20] J Scott Goldstein and I S Reed ldquoReduced-rank adaptivefilteringrdquo IEEE Transactions on Signal Processing vol 45 no 2pp 492ndash496 1997

[21] J S Goldstein I S Reed and L L Scharf ldquoA multistage repre-sentation of the wiener filter based on orthogonal projectionsrdquoIEEE Transactions on Information Theory vol 44 no 7 pp2943ndash2959 1998

[22] M L Honig and W M Xiao ldquoPerformance of reduced-rank linear interference suppressionrdquo IEEE Transactions onInformation Theory vol 47 no 5 pp 1928ndash1946 2001

[23] M L Honig and J S Goldstein ldquoAdaptive reduced-rankinterference suppression based on the multistage Wiener filterrdquoIEEE Transactions on Communications vol 50 no 6 pp 986ndash994 2002

[24] M D Zoltowski and E Santos ldquoAdvance in reduced-rankadaptive beamformingrdquo in Defense and Security Symposiumvol 5540 of Proceedings of SPIE Orlando Fla USA April 2004

[25] M D Zoltowski M Joham and S Chowdhury ldquoRecentadvances in reduced-rank adaptive filtering with applicationto high-speed wireless communicationsrdquo in Digital WirelessCommunication III vol 4395 of Proceedings of SPIE pp 482ndash485 April 2001

[26] J Yu DOA estimation technique research based on the wave ofthe known signal [MS dissertation] University of ElectronicScience and Technology of China Chengdu China 2010

[27] D Ricks and J S Goldstein ldquoEfficient implementation of multi-stage adaptive Weiner filtersrdquo in Proceedings of the AntennaApplications Symposium Allerton Park Ill USA September2000

[28] W L Myrick M D Zoltowski and J S Goldstein ldquoLow-sample performance of reduced-rank power minimizationbased jammer suppression for GPSrdquo in Proceedings of the IEEE6th International Symposium on Spread Spectrum Techniques ampApplications (ISSSTA rsquo00) vol 1 pp 93ndash97 IEEE ParsippanyNJ USA September 2000

[29] W LMyrick M D Zoltowski and J Scott Goldstein ldquoAdaptiveanti-jam reduced-rank space-time pre-processor algorithm forGPSrdquo in Institute of Navigation (ION) Conference pp 321ndash336Salt Lake City Utah USA September 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of