FACTA UNIVERSITATIS (NIS)

SER.: ELEC. ENERG. vol. 23, no. 3, December 2010, 367-378

Wideband Cyclic MUSIC Algorithms: AFrequency-Domain Approach

Ivan P. Pokrajac and Desimir Vucic

Abstract: In this paper, we propose two spectral cyclic MUSIC DOA estimation al-gorithms for wideband cyclostationary signals: the wideband spectral cyclic conju-gate MUSIC (WSCCM) algorithm and the extended wideband spectral cyclic MUSIC(EWSCM) algorithm. The proposed algorithms can be applied to estimate the spectralcyclic conjugate correlation matrix and the extended spectral cyclic correlation matrixof wideband cyclostationary signals. The presented algorithms do not require anyknowledge of the optimal time lag parameter, which appears in the similar algorithmsin time domain. In the proposed algorithms, the DOAs are estimated at each spectralcomponent of the selected frequency band for a cyclic frequency of interest. This canbe used to form the azimuth cyclic frequency diagram, similar to the azimuth fre-quency diagram in wideband direction-finding systems. Some simulation results areprovided to confirm the theoretical findings.

Keywords: Cyclostationarity, direction of arrival (DOA), MUSIC, spectral cyclic(conjugate) correlation matrix.

1 Introduction

MANY signal selective algorithms for DOA estimation have been developed toovercome some limitations of the standard approaches, such as the possi-bility to automatically classify signals as desired or undesired. The first signal se-lective algorithm based on cyclostationarity properties is proposed by Gardner [1],and named Cyclic MUSIC. Since then, many algorithms for DOA estimation havebeen developed to improve performance of Cyclic MUSIC [2, 3, 4]. An Extended

Manuscript received on September 10, 2010.I. Pokrajac is with the Military Academy, Department of Telecommunication, Pavla Jurisica

Sturma 33, 11000 Belgrade, Serbia (e-mail: ivan.pokrajac@vs.rs). D. Vucic is with theSchool of Computing, Union University, Knez Mihajlova 6, 11000 Belgrade, Serbia (e-mail:dvucic@raf.edu.rs).

367

368 I. Pokrajac and D. Vucic:

Cyclic MUSIC algorithm [2] has been proposed to improve the performance of theCyclic MUSIC algorithm by exploiting both cyclic correlation and cyclic conju-gate correlation. However, Cyclic MUSIC and Extended Cyclic MUSIC algorithmswere developed only for DOA estimation of narrowband signals. The most seriouslimitation of the algorithms presented in [1, 2] is reflected in the fact that both al-gorithms used estimation of the cyclic correlation matrix and the extended cycliccorrelation matrix in time domain using one time lag parameter . To overcomethis limitation, [3] proposes the averaged (conjugate) cyclic MUSIC (ACM) algo-rithm by averaging the cyclic or cyclic conjugate correlation matrix over differenttime lags (time delay). The proposed algorithms are designed for wideband cyclo-stationary signals. The Extended Wideband Cyclic MUSIC (EWCM) algorithm,which exploits both the averaged cyclic and averaged cyclic conjugate correlationmatrix, is also presented in [3].

Another possible approach to solve the problem of optimal time lag knowledgeis proposed in [4] based on the Fourier transformation of the cyclic correlation ma-trix. Using this idea, in [5], an algorithm for DOA estimation based on cyclostation-arity properties of narrowband signals in frequency domain is proposed. Instead ofthe cyclic correlation matrix, the spectral cyclic correlation matrix (SCCM) is eval-uated in the proposed algorithm. However, the algorithms proposed in [4] and [5]exploit only the cyclic property for DOA estimation, and they do not have the po-tential performance merit as that of the Averaged Conjugate Cyclic MUSIC andExtended Cyclic MUSIC.

In this paper, we propose the wideband spectral cyclic conjugate MU-SIC (WSCCM) algorithm and the extended wideband spectral cyclic MUSIC(EWSCM) algorithm. The algorithm presented in [5] exploits only cyclic propertiesof signals for DOA estimation, but not cyclic conjugate properties. It is important tonote that complete analysis of wideband cyclostationary signals requires estimationof both the spectral cyclic and spectral cyclic conjugate correlation matrix, becausesome signals have both nonzero spectral cyclic correlation and nonzero spectralcyclic conjugate correlation, e.g., binary phase shift keying (BPSK) or minimumshift keying (MSK). Because of that, we propose WSCCM and EWSCM algo-rithms modifying the algorithm in [5] with the idea of [2, 4]. These algorithms per-form signal selective DOA estimation of wideband and narrowband signals exploit-ing the spectral cyclic conjugate correlation matrix and extended spectral cycliccorrelation matrix. The spectral cyclic properties of cyclostationary signals corre-spond to the spectral correlation at some frequency associated with the multiple ofthe baud rate. The spectral cyclic conjugate properties of cyclostationary signalscorrespond to the spectral correlation at some frequency associated with the car-rier frequency. The proposed algorithms can be used in wideband direction findingsystems in order to form the azimuth cyclic frequency diagram in the process of

Wideband Cyclic MUSIC Algorithms: A Frequency-Domain Approach 369

spectrum segmentation. Both of our methods work well when signals exploit cyclicproperties at the same cyclic frequency.

2 Background

2.1 Wideband data model in frequency domain

Consider an antenna array composed of L omnidirectional elements, which receivesK uncorrelated wideband signals sk(t), k = 1, . . . ,K in a selected frequency sub-band fBW at central frequency fC. The wavefronts of K signals impinge on theantenna array from directions 1,2, . . . ,k where k represents azimuth k andelevation of the kth signal. Let Ka from K signals exhibit cyclic or cyclic conju-gate property at a selected cycle frequency a and let the other KKa signals exhibitcyclic or cyclic conjugate features at different cycle frequencies. If we consider thatKa signals are contained in the vector sss(t), then KKa signals, which exhibit cyclicor cyclic conjugate features at cycle frequencies different from and any noise arelumped into the vector nnn(t). If the wideband assumption holds, in the temporalfrequency domain, the sensor outputs can be represented in a matrix-vector form asfrequency dependent narrowband signals [4, 5].

XXX( f ) = AAA( , f ) SSS( f )+NNN( f ) (1)

where the frequency-dependent steering matrix is AAA(, f ) = [aaa(1, f )aaa(2, f ) . . .aaa(K , f )] whose columns are steering vectors corresponding tothe signals of interest (SOIs). XXX( f ) = [X1( f ) X2( f ) . . .XL( f )]T where []T denotesthe transpose, is the finite time Fourier transform of the wideband signals receivedin the observed interval T . SSS( f ) = [S1( f ) S2( f ) . . .SK ( f )]

T is the finite timeFourier transform of Ka signals down converted to base band by frequency fCin the same observed interval, NNN( f ) = [N1( f ) N2( f ) . . .NL( f )]T is the finite timeFourier transform of the interfering signals and the white Gaussian noise.

If the frequency in (1) is fixed, it is possible to use narrowband signal subspacemethods such as MUSIC. By the discrete Fourier transformation (DFT) we canperform a narrowband decomposition to obtain sensor outputs with a fixed temporalfrequency. The decomposed signal can be represented as:

XXX( fh) = AAA(, fh) SSS( fh)+NNN( fh) (2)

where the fh frequency component is denoted by fh, h [1,H] , where H is thetotal number of spectral components for the selected frequency sub-band fBW .

370 I. Pokrajac and D. Vucic:

2.2 Wideband spectral cyclic MUSIC algorithm

Under the condition that the contribution to the SCCM from K Ka signals andfrom any noise converges to zero as the integration time tends to infinity, the SCCMof the received signal is given by [4, 5]:

RRRxx( fh) = E[

XXX(

fh +2

)

XH(

fh 2

)

]

= AAA(

, fh +2

)

RSS( fh)A(

, fh 2

)H(3)

where ()H denotes the conjugate transpose, and where RRRSS( fh) is the spectral cycliccorrelation matrix of the signal in the base-band at the hth spectral component. Theselected cyclic frequency denotes the distance between the two spectral compo-nents f1 = fh + /2 and f2 = fh /2.

One of the advantages of the Wideband Spectral Cyclic MUISC algorithm infrequency domain is the ability to increase the number of detectable signals byestimating SCCM on each spectral component from the selected frequency sub-band at the selected cyclic frequency of interest. The number of detectable signals,which exhibit cyclic features at the selected cycle frequency, is L1 at each spectralcomponent. Using this algorithm, it is possible to estimate DOAs for more signalsthat use the same frequency sub-band simultaneously and exhibit cyclic features atthe same or different cycle frequencies.

3 Wideband Spectral Cyclic Conjugate Music (WSCCM) Algorithm

Instead of the calculation of SCCM like in [4] and [5], we propose to calculateSCCCM and estimate DOAs based on conjugate cyclic properties of signals. Ourgoal is to obtain a matrix which has a form similar to (3).

Under the condition that the contribution to the SCCCM from K-K signals andfrom any noise converges to zero as the integration time tends to infinity, the esti-mation of the spectral cyclic conjugate correlation matrix of the real-valued signalscan be expressed as:

RRRxx( fh) = E[

XXX(

fh +2

)

XXXT(

fh 2

)

]

= AAA(

, fh +2

)

RRRSS( fh)AAA(

, fh 2

)T(4)

whereRRRSS( fh) = E

[

SSS(

fh +2

)

SSST(

fh 2

)

]

is the spectral cyclic conjugate correlation matrix of real-valued signals at the hthspectral component for the cyclic frequency of interest .

Wideband Cyclic MUSIC Algorithms: A Frequency-Domain Approach 371

If we observe a bi-frequency plane ( f ,), all signals in a selected frequencyband exhibit conjugate cyclostationarity properties around the intermediate fre-quency of the selected frequency sub-band, or around the zero-intermediate fre-quency after down-conversion. However, in the case of complex-valued signals,the spectral cyclic conjugate correlation matrix can not be estimated using (4), be-cause there is no symmetry in the Fourier transform of the received signal aroundthe zero-intermediate frequency. In the case of the complex-valued signals, the es-timation of the spectral cyclic conjugate correlation matrix for cyclic frequency can be expressed as:

RRRxx( fi) = E[

XXX(

2

)

XXXT(

2

)

]

= AAA(

,2

)

RRRSS( fi)AAA(

,2

)T(5)

where fi denotes the ith frequency component that corresponds to the zero-intermediate frequency, = 2 fh, and

RRRxx( fi) = E[

SSS(

2

)

SSST(

2

)

]

is the spectral cyclic conjugate correlation matrix of signals in the base-band at theith spectral component for the cyclic frequency of interest .

Digital Frequency Smoothed Method DFSM [6] can be used in order to fre-quency smooth the spectral cyclic conjugate correlation matrix in a frequency rangearound the zero-intermediate frequency. In this case, the frequency-smoothed spec-tral cyclic conjugate correlation matrix in the range |m| M/2 can be expressedas:

RRRxx( fi) = E[

XXX(

2+ m

)

XXXT(

2m

)

]

(6)

where m corresponds to the m fBW/H frequency component.The proposed algorithm for DOA estimation of wideband cyclostationary sig-

nals based on estimation of SCCCM can be described in few steps.

Step 1. The received continuous-time signals xl(t), l = 1, . . . ,L are down convertedto the base-band by IQ mixing at each antenna and then converted to discrete-time by sampling with the sampling frequency fs = fBW .

Step 2. The received discrete-time signals xl(t) are divided into Q vectors, xql (t),

l = 1, . . . ,L, q = 1, . . . ,Q (the total number of vectors is LQ).

Step 3. The H-point DFT for each of the LQ vectors is calculated Xql ( f ) =DFT[xql (n)], l = 1, . . . ,L, q = 1, . . . ,Q. The number of points for the DFTdepends on the cyclic resolution rez = fBW/H .

372 I. Pokrajac and D. Vucic:

Step 4. The spectral cyclic conjugate correlation matrix is evaluated for a selectedcyclic frequency = 2 fh.

RRRxx( fi) =1Q

Q

q=1

m

XXXq(

2+ m

)

XXXq(

2m

)T(7)

A simplified spectral cyclic conjugate correlation matrix computational flowgraph of the proposed algorithm at a selected cycle frequency is shown inFig. 1.

Fig. 1. Simplified SCCCM computational flow graph for the proposed algorithm.

Step 5. The number of signals K , which exploit the cyclic conjugate properties atthe selected cyclic frequency, is determined by using Canonical CorrelationSignificance Test (CCST).

Step 6. DOA is estimated using MUSIC algorithm, at the selected cyclic fre-quency.

PMUSIC(

,2

)

=aaa(

,2

)

aaa(

,2

)

aaa(

,2

)

EEEn,

2 (8)

where aaa(,/2) is the steering vector evaluated at the frequency componentthat corresponds to /2, and EEEn,a is the noise subspace matrix of RRRxx( fi).

Wideband Cyclic MUSIC Algorithms: A Frequency-Domain Approach 373

Step 7. In order to estimate DOA at all cyclic frequencies repeat steps from step 4.to step 6 for each of the H frequency components, fh = /2.

Step 8. The final estimates of DOAs can be performed averaging those spectralcomponents which are joined in the same signals

4 Extended Wideband Spectral Cyclic MUSIC (EWSCM) Algorithm

To develop our EWSCM algorithm, an extended spectral wideband cyclic correla-tion matrix will be constructed. In order to exploit both the spectral cyclic correla-tion matrix and spectral cyclic conjugate correlation matrix for DOA estimation infrequency domain, the following extended data vector is formed:

XXXE ( fh) =

XXX( fh +2)

XXX( fh 2)

(9)

where XXXE ( fh) is an extended data vector of the signal received at the selected fre-quency component fh and the cycle frequency of interest . XXXE( fh) is a 2L 1matrix. The extended spectral cyclic correlation matrix for the extended data vec-tor can be calculated as:

RE( fh) = E[XXXE ( fh)XXX

He ( fh)]

=

E[XXX( fh +2)XXXH( fh

2)] E[XXX( fh +

2)XXXT ( fh +

2)]

E[XXX( fh 2)XXXH( fh

2)] E[XXX( fh

2)XXXT ( fh +

2)]

(10)

where RRRE ( fh) is the extended spectral cyclic correlation matrix with dimensions2L2L.

Based on the definition of the SCCM (2) and SCCCM (5), we can replace thelower two sub-matrices with:

(

RRRxx ( fh))

= E[

XXX( fh 2

)XXXT ( fh +2

)]

(11)

and(

RRR1xx( fi))

= E[

XXX( fh 2

)XXXH( fh 2

)]

(12)

where 1 = 2 fh .In order to exploit cyclic and cyclic conjugate properties of the wideband sig-

nal for DOA estimation, the SCCM and SCCCM have to be estimated at the samecyclic frequency of interest . Based on (10) and (11), it can be concluded that

374 I. Pokrajac and D. Vucic:

SCCM and SCCCM matrices are not estimated at the same cyclic frequency. Toestimate SCCM and SCCCM matrices at the same cyclic frequency it is neces-sary that the carrier frequency of the selected signal corresponds to the central fre-quency of the selected frequency sub-band. In that case, after the down conversionin the base-band, the signal exhibits cycle conjugate properties at the same cyclicfrequency as the cycle properties for some cyclostationary signal like the BPSKsignal. Signals in the base-band exhibit cyclic and cyclic conjugate properties atspectral components fi which correspond to the zero-intermediate frequency.

Based on (10) and (11), and by replacing fh with fi it is possible to constructthe ESCCM matrix similarly to [3]:

RRRE ( fi) =

[

RRRxx( fi) RRRxx( fi)

(

RRRxx ( fi)) (

RRRxx ( fi))

]

(13)

This ESCCM matrix can be written in the following form:

RRRE ( fi) =

AAA(,2) 000

000 AAA(,2)

[

RRRxx( fi) RRRxx( fi)

(

RRRxx ( fi)) (

RRRxx ( fi))

]

AAA(,2) 000

000 AAA(,2)

H (14)

The matrix AAA(,/2) is evaluated at the frequency fC +/2 because it is sup-posed that the carrier frequency of the selected signal is the same as the centralfrequency of the selected frequency sub-band.

The DOAs are estimated by using the spatial spectrum of the Extended Wide-band Cyclic MUSIC method proposed in [3]. The spatial spectrum has the form:

PPP =[

min[BBBH(,2)UUUNUUU

HNBBB(,

2)]]1

(15)

where min[] denotes the minimum eigenvalue of the matrixBBBH(,/2)UUU NUUUHNBBB(,/2), UUUN is the noise subspace of RRRE ( fi), estimated byapplying the SVD, and BBB(,/2) is a matrix defined as:

BBB(,2) =

aaa(,2) 000

000 aaa(,2)

(16)

Wideband Cyclic MUSIC Algorithms: A Frequency-Domain Approach 375

5 Simulation Results

Three simulations have been carried out in order to illustrate the effectiveness of theproposed algorithms. In all simulated examples it is supposed that the widebandsignals are received by a linear, uniformly spaced array with five antennas, spacedby a half wavelength of frequency fA = 20MHz (A = c/ fA = 2d) where d denotesthe distance between two adjacent antennas in the observed time interval aroundT = 5ms (96256 samples). Frequency sub-band fBW = 19.2MHz is selected atthe central frequency fC = 20MHz. In all simulations the length of the FFT wasH=1024.

In the first example, the possibility to estimate DOAs exploiting both cyclo-stationarity and conjugate cyclostationarity is tested. The results obtained by theproposed algorithm are compared with the Wideband Spectral Cyclic MUSIC algo-rithm and Wideband Spectral Cyclic Conjugate MUSIC algorithm. It is supposedthat SOIs arrive at the antenna array from the azimuth of 0, 30 and 15 and ele-vation 0. The carrier frequencies of SOIs are fC1 = 20MHz, fC2 = 21.6MHz, andfC3 = 20MHz The signal to noise ratio (SNR) is 10 dB for each signal.

20 15 10 5 0 5 10 15 20 25 30 35 40

10

0

10

20

30

40

50

60

70

Azimuth [degrees]

Pm

usic

[dB

]

WSCMWSCCMEWSCM

Fig. 2. Spatial spectra for environment containing three wideband SOIs with 0,15 and 30 DOA estimated by using WSCM, WSCCM and EWSCM.

The first SOI is a wideband BPSK signal with the symbol interval of T0 =0.10417s which exhibits the cyclic property at the cyclic frequency 1 = 1/T0 =9.6MHz and the conjugate cyclic property at cycle frequencies 2,3 = 2( fC1 fC)1/T0 = 9.6MHz. The second SOI is also a wideband BPSK signal with the sym-bol interval of T0 = 0.15625s which exhibits the cyclic property at the cyclic

376 I. Pokrajac and D. Vucic:

frequency 1 = 1/T0 = 6.4MHz and the conjugate cyclic properties at cycle fre-quencies 2,3 = 2( fC1 fC) 1/T0 2 = 9.6MHz, 3 = 3.2MHz. The third SOIis an MSK signal with the symbol interval of T0 = 0.10417s which exhibits thecyclic property at cyclic frequency 1 = 1/T0 = 9.6MHz and the conjugate cyclicproperties at cycle frequencies 2,3 = 2( fC1 fC)1/T0 = 4.8MHz.

Fig. 2 shows the DOAs estimation using the algorithms based on the estimationof ESCCM, SCCM and SCCCM matrices at the cycle frequency = 9.6MHz.Based on the results shown in Fig. 2, it can be concluded that using the ExtendedWideband Spectral Cyclic MUSIC algorithm in frequency domain, it is possible toestimate DOAs that exploit both cyclostationarity and conjugate cyclostationarityof the received signals. Using the Wideband Spectral Cyclic MUSIC and WidebandSpectral Cyclic Conjugate MUSIC algorithms it is not possible to detect all signalswhich exhibit cyclostationarity and conjugate cyclostationarity at a selected cyclicfrequency.

Fig. 3. 3-D spatial spectra for environment containing three wideband SOIs with 0, 15 and30 DOA estimated by using WSCCM algorithm.

The DOAs estimation using the proposed WSCCM algorithm, based on theestimation of SCCCM, is shown in Fig. 3. Based on the results shown in Fig. 3,it can be concluded that the proposed algorithms enable detection and estimationDOAs of wideband cyclostationary signals that exhibit cyclic conjugate properties.Using the proposed algorithms three SOIs are detected. Two pairs of three cyclic

Wideband Cyclic MUSIC Algorithms: A Frequency-Domain Approach 377

frequencies with the same estimated DOAs correspond to BPSK signals and twocyclic frequencies with the same DOAs correspond to MSK signals.

In the following example, the dependence of the mean error (RMSE) of theestimated DOA on the signal bandwidth to carrier frequency ratio is studied. TheRMSE of the estimated DOA for the algorithm proposed in [5] varies with thechosen frequency for evaluating the steering vector. The wideband spectral cyclicconjugate MUSIC algorithm and extended wideband spectral cyclic MUSIC algo-rithm should not have this problem. In order to see this effect, in the example, thesignal bandwidth to carrier frequency ratio is varied from 3.33% to 66.66%. It issupposed that the SOI is a BPSK signal which arrives at the antenna array from theazimuth of 10. The DOA is estimated using the proposed algorithms for the caseof wideband signals and their forms for narrowband signals. In case of narrowbandsignals, the steering vector is evaluated at the carrier frequency of the signal. Forwideband signals, the steering vector is evaluated at the frequency which dependedon the chosen cyclic frequency and carrier frequency.

Fig. 4. RMSE of the estimated DOA versus signal bandwidth to carrier frequency ratio

The algorithms are run 100 times and the RMSE of the estimated DOA versusthe signal bandwidth to carrier frequency ratio is shown in Fig. 4. Based on theresults shown in Fig. 4, we can notice that the RMSE of the estimated DOA for theproposed wideband algorithms is almost the same for different signal bandwidth tocarrier frequency ratios. In the case of narrowband algorithms, RMSE of the esti-mated SOI changes proportionally with the signal bandwidth to carrier frequency

378 I. Pokrajac and D. Vucic:

ratio. This effect is not present when using the extended narrowband spectral cyclicMUSIC algorithm. Based on the obtained results, we can notice that the widebandand narrowband extended spectral cyclic MUSIC algorithms gave almost the sameRMSE of the estimated DOA up to 33.33% of the signal bandwidth to carrier fre-quency ratio.

6 Conclusion

This paper has presented a frequency domain approach for wideband Cyclic MU-SIC algorithms. The proposed algorithms can be applied to estimate the spectralcyclic conjugate correlation matrix and the extended spectral cyclic correlation ma-trix of wideband cyclostationary signals. In these algorithms DOAs are estimatedat each spectral component of the selected frequency band for the cyclic frequencyof interest. This can be used to form the azimuth versus cyclic frequency diagram,similar to the azimuth versus frequency diagram in wideband direction-finding sys-tems. The effectiveness of the proposed algorithms has been demonstrated by sim-ulation results.

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