2-D unitary matrix pencil method for efficient direction of ...€¦ · Utilization of a unitary transform for efficient computation in the matrix pencil method to find the direction

Digital Signal Processing 16 (2006) 767–781

www.elsevier.com/locate/dsp

2-D unitary matrix pencil method for efficient direction ofarrival estimation

Nuri Yilmazer ∗, Tapan K. Sarkar

Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244-1240, USA

Available online 18 July 2006

Abstract

In this study, we extended the one-dimensional (1-D) unitary matrix pencil method (UMP) [N. Yilmazer, J. Koh, T.K. Sarkar,Utilization of a unitary transform for efficient computation in the matrix pencil method to find the direction of arrival, IEEE Trans.Antennas Propagat. 54 (1) (2006) 175–181] to two-dimensional case, where 2-D matrix pencil (MP) method are used to find the 2-Dpoles corresponding to the direction of arrival (DOA), azimuth and elevation angles, of the far field sources impinging on antennaarrays. This technique uses MP method to compute the DOA of the signals using a very efficient computational procedure in whichthe complexity of the computation can be reduced significantly by using a unitary matrix transformation. This method applies thetechnique directly to the data without forming a covariance matrix. Using real computations through the unitary transformation forthe 2-D matrix pencil method leads to a very efficient computational methodology for real time implementation on a DSP chip.The numerical simulation results are provided to see the performance of the method.© 2006 Elsevier Inc. All rights reserved.

Keywords: Unitary transform; Centro-hermitian matrix; Real valued computations; Matrix pencil method; Simultaneous azimuth elevationangle estimation; 2-D matrix pencil method

1. Introduction

The problem of estimating the direction of arrival of the various sources impinging on an antenna array has receivedconsiderable attention in many fields including radar, sonar, radio astronomy, and mobile communications.

In this paper, unitary transform for the 1-D MP method has been extended for the 2-D case to find the DOA ofthe signals, azimuth and elevation angles, using an efficient computational procedure in which the complexity ofthe computation can be reduced significantly by using a unitary matrix transformation, hence only doing real val-ued computations. Unitary transform can convert the complex matrix to a real matrix along with their eigenvectorsand thereby reducing the computational cost at least by a factor of four, since multiplication of two complex num-bers require four real multiplications and two additions. This reduction in the number of computations is achievedby using a transformation, which maps centro-hermitian matrices to real matrices [1,5,18]. It is very important toincrease the resolution of the DOA estimation as well as to reduce their computational complexity. In MP method,based on the spatial samples of the data, the analysis is done on a snapshot-by-snapshot basis, and therefore nonsta-

* Corresponding author.E-mail address: [email protected] (N. Yilmazer).

1051-2004/$ – see front matter © 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.dsp.2006.06.005

768 N. Yilmazer, T.K. Sarkar / Digital Signal Processing 16 (2006) 767–781

tionary environments can be handled easily. Unlike the conventional covariance matrix techniques, the MP methodcan find DOA easily in the presence of multi-path coherent signal without performing additional processing of spatialsmoothing.

Increasing the accuracy of DOA estimation as well as reducing the computational complexity is vital in real timesystems. Capon’s minimum variance technique attempts to overcome the poor resolution problems associated withthe delay-and-sum method [12]. More advanced approaches are so-called super resolution techniques that are basedon the eigen-structure of the input covariance matrix including multiple signal classification (MUSIC), root-MUSIC,and estimation of signal parameters via rotational invariance techniques (ESPRIT) provides the high resolution DOAestimation. Music algorithm proposed by Schmidt [7] returns the pseudo-spectrum at all frequency samples. Root-MUSIC [13] returns the estimated discrete frequency spectrum, along with the corresponding signal power estimates.Root-MUSIC is one of the most useful approaches for frequency estimation of signals made up of a sum of exponen-tials embedded in white Gaussian noise.

The conventional signal processing algorithms using the covariance matrix works on the assumption that the signalsimpinging on the array are not coherent. Under uncorrelated conditions, the source covariance matrix satisfies the fullrank condition, which is the basis of the eigen-decomposition. Many techniques involve modification of the covariancematrix through a preprocessing scheme called spatial smoothing [14,17]. Sarkar and Hua [8,15] utilized the matrixpencil to get the DOA of the signals in a coherent multipath environment.

On the other hand, some efforts have been done to reduce the computational complexity of the calculations. Huangand Yeh [16] have developed a unitary transform, which can convert a complex matrix to a real matrix along with theireigenvectors. Their simple transformation reduces the processing time by dealing with only real valued computations.The processing time could be reduced almost four times, since the complex multiplication cost four times more thanthat of real multiplications. More work has been done by Haardt, and Nossek [6,11], and they applied the method toESPRIT to successfully reduce the computational burden.

In many applications, such as radar imaging, nuclear magnetic resonance imaging, and wave number estimationrequires the estimation of two-dimensional (2-D) poles in 2-D data [9,10]. 1-D unitary transformation has been suc-cessfully applied to 1-D MP method to reduce the computational complexity. In this paper, we extended unitarytransform for the 2-D case. In 2-D MP method, 2-D estimation problem is reduced to two 1-D problems, each one ofthe poles for each direction is estimated separately and the estimated poles needs to be paired to get the correct pairwhich corresponds to correct elevation, and azimuth angles for direction of arrival problem.

The rest of the paper is organized as follows. In Section 2, the unitary transform and the related theorems are given.In Section 3, the signal model for the 2-D case is presented and 2-D MP estimation technique and the pole pairingis revisited. Unitary 2-D MP method is introduced in Section 6. The computer simulation is provided in Section 8,followed by the conclusions.

2. Unitary transform

A square matrix, BN×N , is called unitary matrix, if it satisfies BBH = I . The superscript H denotes the complexconjugate transpose of a matrix, where I is the identity matrix N × N . Any matrix A, where A ∈ CP×S , is calledcentro-hermitian [1,2], if it satisfies

A =∏P

A∗ ∏S

, (1)

where∏

P is called the exchange matrix and defined as

∏P

=

⎡⎢⎢⎢⎢⎣

0 · · · 0 0 10 · · · 0 1 00 · · · 1 0 0...

. . ....

......

1 · · · 0 0 0

⎤⎥⎥⎥⎥⎦

P×P

. (2)

Here∏

is a P × P square matrix, and A∗ is conjugate of A.
P

N. Yilmazer, T.K. Sarkar / Digital Signal Processing 16 (2006) 767–781 769

Theorem. If the matrix A is centro-hermitian, then QHP AQS is a real matrix. Here the matrix Q is unitary, whose

columns are conjugate symmetric and has a sparse structure [16], for P even, we have

QP = 1√2

[I iI∏ −i

∏]. (3)

Here I and∏

are matrices that have the dimension of P/2 and i = √−1.When P is odd, we have

QP = 1√2

[I 0 iI

0√

2 0∏0 −i

∏]

. (4)

Here I and∏

are matrices that have the dimension of (P − 1)/2, and 0 is a (P − 1)/2 × 1 vector whose elementsare 0.

Proof. Using∏

P

∏P = I , the conjugate of QH

P AQS is(QH

P AQS

)∗ = QTP A∗Q∗

S = QTP

∏P

∏P

A∗ ∏S

∏S

Q∗S.

Since,∏

Q∗ = Q, and∏

P A∗ ∏S = A(

QHP AQS

)∗ = QHP AQS. (5)

Therefore, QHP AQS is a real matrix. Other related theorems can be found in [1,2].

3. 1-D unitary matrix pencil method revisited

Let us consider the uniform linear array (ULA) case, where there are N omni directional antenna elements, and M

impinging signals coming to the array. The voltage values at the feeding point of antenna arrays are {x(0), x(1), . . . ,

x(N − 1)}. One can write the signal as

x(k) =M∑i=1

Riejωik, k = 0,1, . . . ,N − 1. (6)

This column data vector can be written into Hankel matrix form to obtain the matrix Y [2].

Y =

⎡⎢⎢⎣

x(0) x(1) · · · x(L)

x(1) x(2) · · · x(L + 1)...

.... . .

...

x(N − L − 1) x(N − L) · · · x(N − 1)

⎤⎥⎥⎦

(N−L)×(L+1)

, (7)

where L is the pencil parameter. The matrix pencil from the Hankel matrix Y can be written as

J2Y − λJ1Y. (8)

The matrices J1, and J2 are called selection matrices and defined as follows:

J1 =

⎡⎢⎢⎣

1 0 · · · 0 00 1 · · · 0 0...

.... . .

......

0 0 · · · 1 0

⎤⎥⎥⎦

(N−L−1)×(N−L)

and J2 =

⎡⎢⎢⎣

0 1 0 · · · 00 0 1 · · · 0...

......

. . ....

0 0 · · · 0 1

⎤⎥⎥⎦

(N−L−1)×(N−L)

. (9)

The matrices J1, and J2 are used to select the first and last (N − 1) components of the matrix Y as discussed in [2].Equation (8) can be written as

QHJ2QQHYQ = λQHJ1QQHYQ. (10)

Note that QQH = I , and QHYQ = Xr is real [1], when matrix Y is centro-hermitian.

QHJ2QXr = λQHJ1QXr. (11)


It can be shown that∏

P

∏P = I ,

∏P QP = Q∗, and QH

P

∏P = QT,

∏P J2

∏P+1 = J1, and therefore, (11) can

be represented by

QH∏∏

J2

∏∏QXr = QH

∏J1

∏QXr = QTJ1Q

∗Xr = (QHJ1Q

)∗Xr. (12)

Hence, equation converts to(QHJ1Q

)∗Xr = λQHJ1QXr. (13)

Therefore,

tan

(ωi

2

)Re

(QHJ1Q

)Xr = Im

(QHJ1Q

)Xr. (14)

The singular value decomposition (SVD) of real matrix Xr can be written as

Xr = UΣV T. (15)

Here U and V are orthogonal matrices whose elements are the eigenvectors, Σ is a diagonal matrix with itselement σi , which are called singular values of the matrix Xr.

U = [u1 u2 . . . uM . . .],V = [v1 v2 . . . vM . . .],Σ = diag(σ1 � σ2 � · · ·σM � σM+1 · · ·).

Let us define the Es = [u1 u2 . . . uM ], such that the elements of Es are the first M singular vectors corre-sponding to the largest singular values σ1, σ2, . . . , σM . M is number of the signals and can be estimated from thesingular values based on the criteria defined in [4]. In order to reduce the effect of the noise, one can write (14) as

tan

(ωi

2

)Re

(QHJ1Q

)Es = Im

(QHJ1Q

)Es, i = 1, . . . ,M. (16)

So, tan(ωi/2) can be solved as the generalized eigenvalue of the matrix pair {Im(QHJ1Q)Es , Re(QHJ1Q)Es}.The solution to the problem can be reduced to an ordinary eigenvalue problem, where tan(ωi/2) is the eigenvalue of[Re(QHJ1Q)Es]−1 Im(QHJ1Q)Es . In this algorithm all computations are made using real numbers and therefore, novariable is complex including the eigenvalues and the eigenvectors in this procedure.

One should state that premultiplying and postmultiplying Y by QH and Q require only additions and scaling. Forexample, in the case of N sensor elements, and for a given pencil parameter L, the computations done for the unitarytransformation, Xr = QHYQ, requires around (N − L)∗2(L) real additions. This is negligible as compared to thecomputations done for the computation involved in the eigen-decomposition. Eigen-structure based methods for esti-mating DOA of the sources impinging on a ULA requires complex calculations in computing the eigenvectors and theeigenvalues. The matrix pencil method, in addition, requires the computation of a singular value decomposition (SVD)of the complex-valued data. It should be stated that eigen-decomposition with complex-valued data matrix is quitecomputation intensive. The eigen-decomposition process consists of a large portion of the whole computational load.To reduce the computational complexity during eigen-decomposition, application of a unitary transformation is pro-posed for DOA estimation by using a real-valued SVD. Computing the eigen-components of the unitary transformeddata matrix requires only real computations.

The unitary matrix pencil method is thus a completely real-valued algorithm, as it requires only real-valued com-putations. Apart from finding the singular values and vectors, the rest of the calculations are also real computations asopposed to the ones done in the conventional MP method. A big portion of the computational load is occupied by themultiplication operations, so transforming the data can save a noticeable amount of computations and the processingtime is reduced greatly.

4. Summary of the 1-D unitary matrix pencil method

The algorithm can be summarized as follows:


(1) Convert the complex data matrix into real matrix Xr, by using Xr = QHYQ.(2) Compute the SVD of Xr and calculate Es , which is the M principal singular vectors of Xr.(3) Evaluate Re(QHJ1Q), and Im(QHJ1Q).(4) Calculate the generalized eigenvalues, λ1, λ2, . . . , λM of Im(QHJ1Q)Es , and Re(QHJ1Q)Es .(5) Calculate ωi = 2 tan−1(λi), i = 1, . . . ,M .

5. 2-D signal model and matrix pencil method

In many applications, such as radar imaging, and wave number estimation requires the estimation of two-dimensional (2-D) poles in 2-D data. Let us consider the 2-D uniform rectangular array (URA). The noiseless dataz(m,n) measured at the feeding points of the omni directional antennas defined as

z(m,n) =P∑

p=1

apejϕpej 2πλ

�x sin θp cosφpmej 2πλ

�y sin θp sinφpn, 0 � m � M − 1, 0 � n � N − 1 (17)

it can be simplified to

z(m,n) =P∑

p=1

αpxmp yn

p,

where

αp = apejϕ, xp = ej 2πλ

�x sin θp cosφp , yp = ej 2πλ

�y sin θp sinφp , xp = ejωx , yp = ejωy ,

where

ωx = 2π

λ�x sin θp cosφp, ωy = 2π

λ�y sin θp sinφp. (18)

Basically, in 2-D MP method, 2-D problem is divided into two 1-D problems, and then solved for each pole ineach dimension and pair them together to get the correct DOA angles. The data matrix z(m,n) can be enhanced andwritten in Hankel block matrix structure as defined in [3]. The distance between the antenna elements are in x- andy-direction, �x = �y = λ/2. The signal model has P , 2-D exponential signals, where ap and ϕp are the magnitudesand the phases respectively. The objective is to find the (xp, yp) pairs, which correspond to the azimuth and elevationangles. The formulation of the 2-D matrix pencil method is discussed in detail in [3]. The noiseless data (18) can bewritten in matrix form as

D =

⎡⎢⎢⎣

z(0;0) z(0;1) · · · z(0;N − 1)

z(1;0) z(1;1) · · · z(1;N − 1)...

.... . .

...

z(M − 1;0) z(M − 1;1) · · · z(M − 1;N − 1)

⎤⎥⎥⎦ . (19)

Basically, in 2-D matrix pencil method, the problem is divided into 1-D MP case, and it is solved for each of thedimension separately and these estimated poles are paired to find the corresponding DOA angles. The following isintroduced and studied deeply in [3] by Hua. The data matrix z(m,n) can be enhanced and written in Hankel blockmatrix structure as follows:

De =

⎡⎢⎢⎣

D0 D1 · · · DM−B

D1 D2 · · · DM−B+1...

.... . .

...

DB−1 DB · · · DM−1

⎤⎥⎥⎦ . (20)

The original data matrix z(m,n) is an M × N complex matrix. Extending the original data matrix by stacking inHankel structure De in (21), where each element of De is also an Hankel matrix, which is obtained by windowing therows of the original data matrix z(m,n).

Dm =

⎡⎢⎢⎣

z(m;0) z(m;1) · · · z(m;N − C)

z(m;0) z(m;1) · · · z(m;N − C + 1)...

.... . .

...

⎤⎥⎥⎦ . (21)

z(m;C − 1) z(m;C) · · · z(m;N − 1)


The B and C are the window pencil parameters used to obtain Hankel matrix in (20) and (21).The matrix Dm can be written as Dm = YCGXm

d YR Here

YC =

⎡⎢⎢⎣

1 1 · · · 1y1 y2 · · · yP

......

. . ....

y(C−1)1 y

(C−1)2 · · · y

(C−1)P

⎤⎥⎥⎦

(CXP)

, (22)

YR =

⎡⎢⎢⎢⎣

1 y1 · · · y(N−C)1

1 y2 · · · y(N−C)2

......

. . ....

1 yP · · · y(N−C)P

⎤⎥⎥⎥⎦

P×(N−C+1)

, (23)

Xd = diag(x1, x2, . . . , xP ), and G is the diagonal matrix, where G = diag(a1, a2, . . . , aP ),Dm is an C × (N − C

+ 1) Hankel matrix. The extended matrix De could be written as

De = ECGER. (24)

De is an B × (M − B + 1) Hankel block matrix

EC =⎡⎢⎣

YC

YCXd

· · ·YCXB−1

d

⎤⎥⎦ , (25)

ER = [YR,XdYR, . . . ,XdYR

]. (26)

By multiplying the EC by the Shuffling matrix P gives ECP , where

ECP =⎡⎢⎣

XC

XCYd

· · ·XCYB−1

d

⎤⎥⎦ , (27)

XC =

⎡⎢⎢⎣

1 1 · · · 1x1 x2 · · · xP...

.... . .

...

x(C−1)1 x

(C−1)2 · · · x

(C−1)P

⎤⎥⎥⎦

(CXP)

, (28)

Yd = diag(y1, y2, . . . , yP ).

As can be seen, we need to repeat this procedure for each of the dimensions, such as x- and y-direction in rectan-gular coordinates. The eigen-structure of the matrix De is found by taking the SVD

De = UsΣsVHs + UnΣnV

Hn . (29)

The Us , Σs , and V Hs are in the signal subspace corresponding the P principal components whereas Un, Σn, and V H

n

are in the noise subspace. One can write Us = ECT [3], where T is a nonsingular matrix. The matrix pencil can bewritten along the x-direction

Ux2 − λUx1, (30)

Ux1 = Us with last C rows deleted, (31)

Ux2 = Us with first C rows deleted. (32)

The rank reducing numbers of the matrix pencils, Ux2 − λUx1, are the poles of Xd = diag(x1, x2, . . . , xP ). So theparameter ωx can be estimated from the eigenvalues of the matrix (Ux1)

+Ux2. (Ux1)+ is called pseudo-inverse and

defined as

(Ux1)+ = (

UH Ux1)−1

UH = pinv(Ux1). (33)
x1 x1


For the pole in y-direction, we can define,

USY = SECT . (34)

Here matrix S is the shuffling matrix, see [3]. So, the matrix pencil can be written as

Uy2 − λUy1, (35)

Uy1 = USY with last B rows deleted, (36)

Uy2 = USY with first B rows deleted. (37)

The rank reducing numbers of the matrix pencil, Uy2 − λUy1, are the poles of Yd = diag(y1, y2, . . . , yP ). So theparameter ωy can be estimated from the matrix (Uy1)

+Uy2. (Uy1)+ is called pseudo-inverse and defined as (Uy1)

+ =(UH

y1Uy1)−1UH

y1 = pinv(Uy1). The necessary conditions for choosing pencil parameters B and C are pointed outin [3].

(B − 1)C � P, B(C − 1) � P, (M − B + 1)(N − C + 1) � P. (38)

After finding the poles xP and yP , which are not in pair and it needs to be paired to get the correct pairs, whichcorresponds to the true azimuth and elevation angles. By exploiting the property that signal space is orthogonal-to-noise subspace, which implies that eL ⊥Un. The pairs can be matched together by maximizing the criterion below [3].

Jsp(i, j) =P∑

p=1

∥∥uHpeL(xi, yj )

∥∥2, p = 1,2, . . . ,P , (39)

eL = xL ⊗ yL, (40)

where ⊗ is the Kronecker product, xL = [1, x, . . . , xB−1]T, and yL = [1, y, . . . , yC−1]T. Here {up; i = 1,2, . . . ,P }are the principles eigenvectors.

6. 2-D unitary matrix pencil method

We now can extend the 1-D UMP method for the 2-D MP method. So by using the unitary transform, the complexvalued data matrix can be converted to real matrix [2]. By using the centro-symmetry of the antenna arrays or anycomplex matrix could be written centro-hermitian matrix form. If the data matrix De is centro-hermitian or any matrixcan also be written in centro-hermitian matrix form as

Dch = [De

...�D∗e �

]. (41)

By using Theorem 1, we can write

QHDchQ = Dr, (42)

where Dr is real valued matrix. Let us introduce the selection matrices that will be used to write matrix pencils. Thematrices J3, J4, J5, and J6 are called selection matrices and used to select the rows of the Dr real matrix in order towrite the matrix pencil along the x- and y-direction in rectangular coordinates.

J3 = [I(BC−C)

...0(BC−C)×(C)

](BC−C)×(BC)

, J4 = [0(BC−C)×(C)

...I(BC−C)

](BC−C)×(BC)

, (43)

J5 = [I(BC−B)

...0(BC−B)×(B)

](BC−B)×(BC)

, J6 = [0(BC−B)×(B)

...I(BC−B)

](BC−B)×(BC)

. (44)

One can easily write the matrix pencil as similar to in (9) for the 2-D case along the x-direction as

(J4)(De) − λ(J3)(De).

It is reduced to the form below by using (10)–(13)

tan

(ωxp

)Re

(QHJ3Q

)Esx = Im

(QHJ3Q

)Esx, i = 1, . . . ,P , (45)

2


Fig. 1. The scatter plot of direction of arrival angles of 2 impinging signals, SNR = 5 dB, 200 Monte Carlo simulation.

where P is the number of impinging signals arriving to the URA, Esx is the left singular eigenvector correspond-ing to P principal eigenvalues. So tan(ωxp/2) can be solved as the generalized eigenvalue of the matrix pair{Im(QHJ3Q)Esx , Re(QHJ3Q)Esx}. In a similar way, one can write the matrix pencil along the y-direction as

(J6)(S)(De) − λ(J5)(S)(De).

It is reduced to the form below by using (10)–(13)

tan

(ωyp

2

)Re

(QHJ5Q

)Esy = Im

(QHJ5Q

)Esy (46)

to find the poles, yp , along the y-direction. Esy is the left singular eigenvector corresponding to P principal eigen-values which is found by applying the shuffling matrix to the Dr. So, tan(ωyp/2) can be solved as a generalizedeigenvalue of the matrix pair{Im(QHJ5Q)Esy , Re(QHJ5Q)Esy}.

SVD of Dr can be written as

Dr = UΣV T.

U = [u1 u2 . . . uP . . .] and V = [v1 v2 . . . vP . . .] are orthogonal matrices.Σ is a diagonal matrix with its element σi , σ1 � σ2 � · · ·σp � σp+1 · · ·, which are called singular values. Let us

define the Esx = [u1 u2 . . . uP ], such that the elements of Esx are the first P singular vectors corresponding tothe largest singular values σ1, σ2, . . . , σP . P is number of the signals and can be estimated from the singular valuesbased on the criteria defined in [4].

6.1. Pole paring for 2-D unitary matrix pencil method

After finding the poles xp and yp , it needs to be paired to get the correct pairs, which corresponds to the trueazimuth and elevation angles. The difference between the 2-D MP and 2-D UMP is that the pairing in UMP willbe real valued computations in addition to computing the SVD with real valued computations. By exploiting theproperty that signal space is orthogonal to noise subspace, which implies eL ⊥Un. The pairs can be matched togetherby maximizing the criterion below. But the difference here again is that the pairing algorithm is real valued whichreduce the computation time as well.

Jsp(i, j) =P∑∥∥uH

peL(xi, yj )∥∥2

, p = 1,2, . . . ,P . (47)
p=1




where eL = xL ⊗ yL, ⊗ is the Kronecker product. xL = [1, x, . . . , xB−1]T, and yL = [1, y, . . . , yC−1]T. Here {up; i =1,2, . . . ,P } are the principles eigenvectors.

Once we find the poles and pair them, we can find the elevation and azimuth angles θ and φ. By using (18), we canfind the azimuth angles as

φp = a tan

(wy

wx

). (48)

In the same way, we can find the elevation angle as

θp = a sin(√

w2x + w2

y

). (49)



Fig. 5. The variance −10 log10(var(φ1)) of 2-D UMP, and 2-D MP are plotted against the SNR, 800 Monte Carlo.

6.2. Computational complexity

In 2-D MP method, most of the computation load is due to the SVD computation, matrix pencil computation, andpairing computation. In 2-D MP method all the computations are complex valued computations, but in 2-D UMP, allthese computations are done with real valued computations, which reduce the complexity. The difference between the2-D UMP and 2-D MP method comes into play at this point, because all the computations of SVD are real valuedin UMP unlike MP method, which reduce the complexity greatly. As a simple example, the multiplication of twocomplex numbers requires 4 real multiplications and two additions. Computing the SVD, and left singular vectors ofa matrix [De]BC×(M−B+1)(N−C+1) requires 17

3 B3C3 + B2C2(M − B + 1)(N − C + 1) multiplications [3,19]. Forthe pairing method, the number of computations required are 1P 3BC [3]. The computation required for the matrix
2


Fig. 6. The variance −10 log10(var(φ2)) of 2-D UMP, and 2-D MP are plotted against the SNR, 800 Monte Carlo.

Fig. 7. The variance −10 log10(var(θ1)) of 2-D UMP, and 2-D MP are plotted against the SNR, 800 Monte Carlo.

pencil for the two of them is 3P 2BC. For the 2-D UMP method, all these computations are done with real valued asopposed to 2-D MP where these computations are complex valued operations.

7. Summary of the 2-D unitary matrix pencil method

The algorithm can be summarized as follows:

(1) Convert the complex data matrix into real matrix Dr, by using Dr = QHDchQ.(2) Compute the SVD of Dr, to calculate Esx , and Esy which are the P principal singular vectors of Dr.(3) Evaluate tan(ωxp/2)Re(QHJ3Q)Esx = Im(QHJ3Q)Esx .


Fig. 8. The variance −10 log10(var(θ2)) of 2-D UMP, and 2-D MP are plotted against the SNR, 800 Monte Carlo.

Table 1Summary of signal features incident on the 2-Dantenna array

Signal 1 Signal 2

φ 30◦ 45◦θ 40◦ 55◦

(4) Evaluate tan(ωyp/2)Re(QHJ5Q)Esy = Im(QHJ5Q)Esy .(5) Pair them by using the criterion defined.

(6) Find φp = a tan(wy/wx) and θp = a sin(√

w2x + w2

y

).

8. Simulation results

In this section, computer simulation results are provided to illustrate the performance of this new technique. Thenoise contaminated signal model is formulated as

z̃(m,n) = z(m,n) + w(m,n), 0 � m � M − 1, 0 � n � N − 1, (50)

where w(m,n) is a zero mean Gaussian white noise with variance σ 2. A two-dimensional rectangular array of omnidirectional isotropic point sensors are considered in this study. The distance between the antenna elements are �x =�y = λ/2. It is assumed that there are two signals impinging on URA. The signals have a phase of γi = 0 degrees.The number of the antenna elements are M = N = 20. The scatter plot of the estimated elevation and azimuth anglesare shown in Figs. 1–4 for different signal-to-noise (SNR) ratios of SNR = 5, 10, 15, and 20 dB. The results arebased on 200 Monte Carlo simulations. As it is expected, when the SNR increases, the estimated values approach toits true values in the scatter plot. The pencil parameters are chosen to be K = L = 9.

The inverse of the sample variance of the estimates of φ (azimuth angle), θ (elevation angle), for the 2-D UMP iscompared against the 2-D MP versus signal-to-noise ratio (SNR) of the incoming signals and are plotted in Figs. 5–8.Different values of SNR are plotted along the x-axis and the variance of the estimated azimuth, elevation angles arein logarithmic domain, is shown along the y-axis. For the low SNR values the 2-D MP performs better than the 2-DUMP method, after some certain threshold their performance are the same. The results are based on 800 Monte Carlosimulations.


Fig. 9. Bias of the estimator for φ1, azimuth angle versus SNR (diamond shape: 2-D UMP, star shape: 2-D MP).

Fig. 10. Bias of the estimator for φ2, azimuth angle versus SNR (diamond shape: 2-D UMP, star shape: 2-D MP).

The bias of the estimator has also been studied to see the efficiency of the new method. For the elevation andazimuth angles, the bias of the estimator is computed. The bias is calculated as

bias(φ) = ε(φ̂) − φ, (51)

bias(θ) = ε(θ̂) − θ, (52)

where ε(·) denotes the expected value, φ̂, and θ̂ are the estimates of φ, and θ , respectively. The bias (in dB) of theestimator for azimuth and elevation angles, versus SNR is shown in Figs. 9–12.


Fig. 11. Bias of the estimator for θ1, elevation angle versus SNR (diamond shape: 2-D UMP, star shape: 2-D MP).

Fig. 12. Bias of the estimator for θ2, elevation angle versus SNR (diamond shape: 2-D UMP, star shape: 2-D MP).

9. Conclusion

A unitary transform for the two-dimensional matrix pencil method has been successfully formulated and utilizedto convert the complex data matrix to a real matrix, hence reducing the computational complexity significantly inDOA estimation problem. It is seen that for lower SNR of the data, 2-D MP performs better than 2-D UMP. After acertain threshold, both the MP and the new UMP method can be used to model a given data set by a sum of complexexponentials and the UMP can be implemented on a DSP chip using only real arithmetic.

References

[1] A. Lee, Centrohermitian and skew-centrohermitian matrices, Linear Algebra Appl. 29 (1980) 205–210.


[2] N. Yilmazer, J. Koh, T.K. Sarkar, Utilization of a unitary transform for efficient computation in the matrix pencil method to find the directionof arrival, IEEE Trans. Antennas Propagat. 54 (1) (2006) 175–181.

[3] Y. Hua, Estimating two-dimensional frequencies by matrix enhancement and matrix pencil, IEEE Trans. Signal Process. 40 (9) (1992) 2267–2280.

[4] T.A. Sarkar, M.C. Wicks, M. Salazar-Palma, Smart Antennas, Wiley, 2003.[5] L. Datta, D.M. Salvatore, Some results on matrix symmetries and a pattern recognition application, IEEE Trans. Acoust. Speech Signal

Process. ASSP-34 (4) (1986) 992–993.[6] M. Haardt, J.A. Nossek, Unitary ESPRIT: How to obtain increased estimation accuracy with a reduced computational burden, IEEE Trans.

Signal Process. 43 (5) (1995) 1232–1242.[7] R.O. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propagat. 34 (3) (1986) 276–280.[8] T.K. Sarkar, O. Pereira, Using the matrix pencil method to estimate the parameters of a sum of complex exponentials, IEEE Antennas Propagat.

Mag. 37 (1) (1994) 48–54.[9] Y. Li, J. Razavilar, K.J. Liu, A high resolution technique for multidimensional NMR spectroscopy, IEEE Trans. Biomed. Eng. 45 (1) (1998)

78–86.[10] A.J. van der Veen, P. Ober, E.F. Deprettere, Azimuth and elevation computation in high resolution DOA estimation, IEEE Trans. Sig-

nal Process. 40 (7) (1992) 1828–1832.[11] M. Haardt, K. Hueper, J.B. Moore, J.A. Nossek, Simultaneous Schur Decomposition of several matrices to achieve automatic pairing in

multidimensional harmonic retrieval problem, in: Proceedings on EUSIPCO, 1996, pp. 531–534.[12] J. Capon, Maximum likelihood spectral estimation, Nonlinear Methods Spectral Anal. (1979) 155–179.[13] A.J. Barabell, Improving the resolution performance of eigen-structure-based direction finding algorithm, in: Proceedings of the IEEE Inter-

national Conference on Acoustics, Speech, and Signal Processing, 1983, pp. 336–339.[14] K. Takao, N. Jijuma, An adaptive array utilizing an adaptive spatial averaging technique for multi-path environments, IEEE Trans. Antennas

Propagat. 35 (12) (1987) 1389–1396.[15] Y. Hua, T.K. Sarkar, Generalized pencil-of-function method for extracting poles of an EM system from its transient response, IEEE Trans.

Antennas Propagat. 37 (2) (1989) 229–234.[16] K.C. Huang, C.C. Yeh, Unitary transformation method for angle of arrival estimation, IEEE Trans. Signal Process. 39 (4) (1991) 975–977.[17] R. Bachl, The forward and backward averaging technique applied to TLS-ESPRIT processing, IEEE Trans. Signal Process. 43 (11) (1995)

2691–2699.[18] G. Xu, R.H. Roy, T. Kailath, Detection of number of sources via exploitation of centro-symmetric property, IEEE Trans. Signal Process. 42 (1)

(1994) 102–111.[19] G.H. Golub, C. Van Loan, Matrix Computations, Johns Hopkins Univ. Press, Baltimore, 1983.

Nuri Yilmazer was born in Silifke, Turkey. He received the B.S. degree from Cukurova University, Adana, Turkey, in 1996,and the M.S. degree from University of Florida, Gainesville, Florida, in 2000. He is currently working towards the Ph.D. degreein the Department of Electrical Engineering at Syracuse University, Syracuse, New York. His current research interests includeadaptive signal processing and smart antennas. He received the Outstanding Teaching Assistant award from Syracuse University in2006.

Tapan K. Sarkar received the B.Tech. degree from the Indian Institute of Technology, Kharagpur, in 1969, the M.Sc.E. de-gree from the University of New Brunswick, Fredericton, NB, Canada, in 1971, and the M.S. and Ph.D. degrees from SyracuseUniversity, Syracuse, NY, in 1975. He was a Research Fellow at the Gordon McKay Laboratory, Harvard University, Cambridge,MA, from 1977 to 1978. He is now a Professor in the Department of Electrical and Computer Engineering, Syracuse University.His current research interests deal with numerical solutions of operator equations arising in electromagnetics and signal processingwith application to system design.

Documents

2-D unitary matrix pencil method for efficient direction of ...€¦ · Utilization of a unitary transform for efficient computation in the matrix pencil method to find the direction