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A Root-MUSIC-Like Direction Finding Method for Cyclostationary Signals

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A Root-MUSIC-Like Direction Finding Method for Cyclostationary Signals
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  EURASIP Journal on Applied Signal Processing 2005:1, 69–73c  2005 Hindawi Publishing Corporation ARoot-MUSIC-LikeDirectionFindingMethodforCyclostationarySignals PascalCharg´e LESIA, DGEI, INSA Toulouse, 135 Avenue de Rangueil, 31077 Toulouse Cedex 4, FranceEmail: pascal.charge@insa-toulouse.fr  YideWang IREENA/SETRA, ´Ecole Polytechnique de l’Universit ´e de Nantes, La Chantrerie, 44306 Nantes, FranceEmail: yide.wang@polytech.univ-nantes.fr Received 4 December 2003; Revised 2 June 2004 We propose a new root-MUSIC-like direction finding algorithm that exploits cyclostationarity in order to improve the direction-of-arrival estimation. The proposed cyclic method is signal selective, it allows to increase the resolution power and the noiserobustness significantly, and it is also able to handle more sources than the number of sensors. Computer simulations are used toshow the performance of the algorithm. Keywords and phrases:  array processing, direction finding, MUSIC, cyclostationary signals. 1. INTRODUCTION The aim of this paper is the estimation of the direction-of-arrival (DOA) of impinging signals in the telecommunica-tions systems area, where almost all signals exhibit the cyclo-stationarity property [1]. The cyclostationarity has been firstintroduced into array processing by Gardner [2]. We can find in the literature several algorithms (see [1]) that exploit cy-clostationarity to improve the performances of the conven-tional methods. Instead of using the correlation matrix as inthe conventional methods, these algorithms require the esti-mation of the cyclic correlation matrix that reflects the cy-clostationarity of incoming signals, assuming they have baudrates and/or are carrier modulated signals as they would bein radar and radio communication applications. Recently,an extended cyclic MUSIC algorithm has been proposedin [3] that provides a rather good estimation performance. In [4] an extended root-MUSIC (extension of the root- MUSIC [5]) algorithm has been proposed in the noncircular source case.In this paper, we propose a new extended cyclic directionfinding method that allows to select desired signals and to ig-nore interferences, by exploiting the cyclostationarity prop-erty of the signals of interest (SOI). The proposed methodis inspired from the extended root-MUSIC method [4], and then is restricted to linear uniformly spaced arrays. But it hasthe distinct advantage over [3] in that it does not require a search over parameter space. Instead, our algorithm hererequires calculation of the roots of a polynomial, which is asimple process and has low computation cost. 2. DATAMODEL In this paper, we consider a uniform linear array of   L  anten-nas. Suppose there are  K   electromagnetic waves impingingon the array from angular directions  θ  k ,  k  =  1, ... , K  . Theincident waves are assumed to be plane waves, as generatedfrom far-field point sources. Furthermore, the signals are as-sumed to be narrowband. In our study, we assume that  K  α sources emit cyclostationary signals with cycle frequency   α (with  K  α  ≤  K  ). In the following, we consider that  s ( t  ) con-tainsonlythe K  α  signalsthatexhibitcyclefrequency  α ,andallof the remaining K  − K  α  signals (that have no cycle frequency  α ) and any noise are lumped into a vector  i ( t  ). Using this as-sumption, the signal received by the array from the emittingnarrowband sources can be written as z ( t  ) = As ( t  ) +  i ( t  ), (1)where the vector  s ( t  ) = [ s 1 ( t  ), ... , s K  α ( t  )] T  contains the tem-poral signals that have cycle frequency   α , while the vec-tor  i ( t  ) represents interfering sources and noise. The matrix  A  =  [ a ( θ  1 ), ... , a ( θ  K  α )] contains the steering vectors of theimpinging SOI. We assume that the received signals are sam-pled at  N   distinct times  t  n ,  n = 1,2, ... , N  .  70 EURASIP Journal on Applied Signal ProcessingThe cyclic autocorrelation matrix and the cyclic conju-gate autocorrelation matrix at cycle frequency   α  for some lagparameter  τ   are then nonzero and can be estimated by  R  α zz ( τ  ) = 1 N  N   n = 1 z  t  n  +  τ  2  z H   t  n − τ  2  e −  j 2 παt  n , (2) R  α zz ∗ ( τ  ) = 1 N  N   n = 1 z  t  n  +  τ  2  z T   t  n − τ  2  e −  j 2 παt  n , (3)where the superscript  ∗  denotes the complex conjugate,  H  denotes the conjugate transpose, and  T   denotes the trans-pose.We form the following extended data vector: z CE ( t  ) =   z ( t  ) z ∗ ( t  )  .  (4)The cyclic correlation matrix for this extended data mod-el can be estimated as R  α CE ( τ  ) = 1 N  N   n = 1 I α 2 L  t  n  z CE  t  n  +  τ  2  z H  CE  t  n − τ  2  , (5)where the time-dependent matrix   I α 2 L ( t  ) is defined by  I α 2 L ( t  ) =  I L e −  j 2 παt  00 I L e +  j 2 παt    (6)and  I L  is the  L -dimensional identity matrix. This extendedcyclic correlation matrix can be developed as R  α CE ( τ  ) =  R  α zz ( τ  )  R  α zz ∗ ( τ  ) R  α ∗ zz ∗ ( τ  )  R  α ∗ zz  ( τ  )  , (7)where the matrices  R  α zz ( τ  ) and  R  α zz ∗ ( τ  ) are estimated by  (2) and (3), respectively. By choosing the cycle frequency parameter  α  in the es-timation of the extended correlation matrix to be the cyclefrequency of the  K  α  SOI, the contribution to the cyclic corre-lation matrix from the other  K   − K  α  signals (assumed not tohave the same cycle frequency) and from any noise convergesto zero as the integration time used in the estimate tends toinfinity. 3. DOAESTIMATION By computing the SVD of   R  α CE ( τ  ) similarly to the cyclic MU-SIC algorithm, a left signal subspace can be defined by the K   α  left singular vectors associated with the  K   α  nonzero sin-gular values. These singular vectors form the column vec-tors of the matrix   U s . In the same way a left null space isspanned by the remaining 2 L  −  K   α  singular vectors asso-ciated with the zero singular values of   R  α CE ( τ  ), and thesesingular vectors are the column vectors of the matrix   U n .Notethatinthepractice,therearenozerosingularvaluesbutonly small singular values, and the dimension on the signalsubspace can be estimated by the MDL criterion [6]. Some cyclostationary signals have a rank two in the signal subspacespanned by column vectors of the matrix   U s  and others haveonly a rank one, so that the dimension of the signal subspaceis equal to  K   α  with  K  α  ≤ K   α  ≤ 2 K  α  (see [3] for more details). From this statement, we can conclude that in the worstcase if all signals have a rank two, we can only detect up to L − 1 possible DOA. But when all signals have a rank one, thenumber of incoming signals that can be detected is twice thatin the worst case.As in [3], an extended steering vector corresponding to this data model can be defined as follows: b ( θ  , k  ) =  a ( θ  )  00 a ∗ ( θ  )  k  , (8)where  k   is a (2 × 1) vector. Vector  k   allows to form a singlesteering vector from  a ( θ  ) and  a ∗ ( θ  ), and then it allows to ex-ploit simultaneously the information provided by both thecyclic autocorrelation matrix and the cyclic conjugate auto-correlation matrix.According to the subspace processing principle and by using this extended steering vector, the DOA of the SOI isgiven by the minima of the following function:¯ P  ( θ  , k  ) =  U H n  b ( θ  , k  )  2 = k  H  Mk  , (9)where  M  is a (2 × 2) matrix: M =  a H  ( θ  ) U n 1 U H n 1 a ( θ  )  a H  ( θ  ) U n 1 U H n 2 a ∗ ( θ  ) a T  ( θ  ) U n 2 U H n 1 a ( θ  )  a H  ( θ  ) U ∗ n 2 U T n 2 a ( θ  )  , (10)and  U n 1  and  U n 2  are two submatrices of the same dimension U n  =  U n 1 U n 2  .  (11)It can be shown [3] that  U ∗ n 2 U T n 2  =  U n 1 U H n 1 . So, the twodiagonal elements of the matrix   M  are equal. Note also thatthe two nondiagonal elements form a complex conjugatepair.The minimum of the quadratic form in (9) for a particu-lar  θ   is given by the smallest eigenvalue of the matrix   M . Thiseigenvalue is always nonnegative since the quadratic form isnonnegative. When  θ   takes exactly the value of one of thetrue DOA, function (9) equals zero. Then, in this case, thesmallest eigenvalue of  M is equal to zero and the determinantof the matrix   M  equals zero too.We now define the complex variable  z  : z  = e  j (2 πδ/λ )sin( θ  ) , (12)  A Root-MUSIC-Like Direction Finding Method 71where  δ   denotes the distance between two adjacent antennasand  λ  denotes the wavelength of impinging SOI. Then  a ( θ  )can be written as a ( z  ) =  1, z  , z  2 , ... , z  L − 1  T  , (13)and the matrix   M  is a function of   z  . We estimate the DOAsby finding the values of   z   such thatdet { M }= 0 .  (14)The left-hand side of (14) is a polynomial of   z  . The DOAestimation problem is then transformed into a polynomialrooting problem that can be solved using computationally e ffi cient root-solving algorithms.The polynomial of   z   is given by the following form:det { M }= m 211 − m 12 m 21 , (15)where m 11  = a T   1 z   U n 1 U H n 1 a ( z  ), m 12  = a T   1 z   U n 1 U H n 2 a  1 z   , m 21  = a T  ( z  ) U n 2 U H n 1 a ( z  ) . (16)Therefore,  m 11  is a polynomial in  z   whose  l  th coe ffi -cient is given by the sum of the elements of the  l  th diago-nal of   U n 1 U H n 1 , where  l   = − L  + 1 indicates the lowest di-agonal and  l   =  L  −  1 indicates the highest diagonal. Let c  =  [ c 1 , ... , c 2 L − 1 ] T  be the column vector of the coe ffi cientsof the polynomial  m 11 . Then we have m 11  =  z  − L +1 , ... , z  − 1 ,1, z  , ... , z  L − 1  c = 2 L − 1   p = 1 c  p z   p − L (17)with c  p  = min[ L ,2 L −  p ]  i = max[1, L −  p +1]  U n 1 U H n 1  i ,  p + i − L .  (18)Then, we can write that m 211  =  z  − L +1 , ... , z  L − 1  cc T   z  − L +1 , ... , z  L − 1  T  .  (19)Hence, the coe ffi cients of the polynomial  m 211  are equalto the sum of the antidiagonal elements of the matrix   cc T  .Let  s = [ s 1 , ... , s 4 L − 3 ] T  be the vector containing these 4 L − 3coe ffi cients. For  p = 1, ... ,4 L − 3, we have s  p  = min[2 L − 1,  p ]  i = max[1,  p − 2 L +2]  cc T   i ,  p − i +1 .  (20)We obtain the following: m 211  = 4 L − 3   p = 1 s  p z   p − (2 L − 1) .  (21)Since the matrix   U n 1 U H n 1  is a Hermitian matrix, the ele-ments of the vector  c  have the symmetry property   c i  = c ∗ 2 L − i .Since  cc T  is a symmetrical matrix, the coe ffi cients of thepolynomial  m 211  keep the same property of symmetry   s  p  = s ∗ 4 L − 2 −  p .In the same way, let  u  be the column vector containingthe sum of the 2 L − 1 antidiagonal elements of the matrix  U n 1 U H n 2  such that for  p = 1, ... ,2 L − 1, u  p  = min[ L ,  p ]  i = max[1,  p − L +1]  U n 1 U H n 2  i ,  p − i +1 .  (22)Then, we can show that m 12  =  1, z  − 1 , ... , z  − (2 L − 2)  u , m 21  =  1, z  , ... , z  2 L − 2  u ∗ , m 12 m 21  =  1, ... , z  − 2 L +2  uu H   1, ... , z  2 L − 2  T  . (23)Let  r  be the column vector whose elements are the sum of the diagonal elements of the matrix   uu H  . For  p = 1, ... ,4 L − 3, the coe ffi cients are r   p  = min[2 L − 1,4 L −  p − 2]  i = max[1,2 L −  p ]  uu H   i ,  p + i − (2 L − 1) .  (24)Hence, m 12 m 21  = 4 L − 3   p = 1 r   p z   p − (2 L − 1) .  (25)Since the matrix   uu H  is a Hermitian matrix, the coe ffi -cients of the polynomial  m 12 m 21  also have the property of symmetry   r   p  = r  ∗ 4 L − 2 −  p .Equation (14) can now be written as det { M }= 4 L − 3   p = 1  s  p − r   p  z   p − (2 L − 1) = 0 .  (26)The roots of the polynomial det { M }  can be computedusing any polynomial rooting algorithm. The DOA estimatesare obtained using (12): θ  k  = arcsin   λ 2 πδ   arg  z  n  , (27)  72 EURASIP Journal on Applied Signal Processing 10 . 80 . 60 . 40 . 20       C    y    c      l      i    c    a    u     t    o    c    o    r    r    e      l    a     t      i    o    n      f    u    n    c     t      i    o    n 420 − 2 − 4 − 6 α    (  M  H  z  )   − 0 . 2  − 0 . 1 0 0 . 1 0 . 2 0 . 3  τ  (  µ s  ) Figure  1: Magnitude of the BPSK cyclic autocorrelation functionversus  α  and  τ  . where  z  n  represents one of the  K  α  roots selected for DOA es-timation. Due to the symmetry property of the polynomialcoe ffi cients, roots appear in reciprocal conjugate pairs  z  i  and1  /z  ∗ i  . In each pair, one root is inside the unit circle while theother is outside the unit circle (the two roots coincide if they are on the unit circle). Either one of the two can be usedfor DOA estimation, since they have the same angle in thecomplex plane. We have decided to use the roots inside theunit circle. According to (12), the modulus of the roots cor- responding to incoming SOI should be equal to one. In thepractice in presence of noise, modulus are not necessary onebut are expected to be close to one. We then select the  K  α roots that are the nearest to the unit circle as being the rootscorresponding to the DOA estimates.Note that the degree of the polynomial det { M }  is 4 L  −  4(4 L  −  3 coe ffi cients). Hence, the number of roots is 4 L  −  4,and since the roots appear in reciprocal pairs, the proposedprocedure allows in some conditions (e.g., BPSK, AM sig-nals), to determine until 2( L  −  1) signals. This point shouldbe emphasized since the number of signals estimated by theproposed method may be larger than the number of sensors.This characteristic is due to our proposed data model. 4. SIMULATIONRESULTS In this section, we present some simulation results that il-lustrate the performance of the proposed algorithm. We alsocompare the simulation results of the proposed procedurewith those of the classical root-MUSIC algorithm.We consider here a linear uniformly spaced array with6 sensors spaced by a half wavelength of the incoming sig-nals. Incoming BPSK cyclostationary signals are generatedwith noise, and the signal-to-noise ratio (SNR) is 0dB foreach signal. The bit rate of the BPSK SOI is 4Mbps. Othersignals are considered as interferers, which are BPSK mod-ulated signals with a 3 . 2Mbps bit rate. In order to choosecorrectly the parameters  α  and  τ  , we have estimated themagnitude of the cyclic correlation function (Figure 1) fora 4Mbps BPSK modulated signals sampled with the fre-quency 32MHz during 25microseconds. It can be notedthat the magnitude of the cyclic autocorrelation functionand that of the conjugate cyclic autocorrelation function are 03060901201501802102402703003300 . 20 . 40 . 60 . 81 Figure  2: Selected polynomial roots by the proposed method. 03060901201501802102402703003300 . 20 . 40 . 60 . 81 Figure  3: Selected polynomial roots by the classical root-MUSICmethod. equal for a BPSK signal. According to this result, the pro-posed cyclic method is simulated with  α  =  4MHz and  τ   = 0 . 125microseconds. In the next simulations, the averagingtime is equal to 25microseconds and the sample frequency is32MHz. The contribution of both the interferer signals andthat of the noise are theoretically zero in the two cyclic corre-lation matrices.The behavior of the proposed method and that of theclassical root-MUSIC method can be compared by drawingthe selected roots by these two methods in the polar plane.Figure 2 shows the distribution of the selected roots by theproposed method obtained from 500 Monte Carlo simula-tions when two BPSK SOI arrive from  − 3 ◦ and 3 ◦ , and oneinterferer BPSK source from 15 ◦ DOA. Figure 3 shows thedistributionoftheselectedrootsbytheclassicalroot-MUSICmethod obtained from 500 Monte Carlo simulations in thesame situation.  A Root-MUSIC-Like Direction Finding Method 73 Table  1: Mean and standard deviations for both methods.Method True DOAs ( ◦ ) SOI #1 SOI #2 Interferer − 3 ◦ 3 ◦ 15 ◦ Proposed method Mean  − 2 . 973 3 . 009 —Standard deviation  − 0 . 355 0 . 260 —Root method Mean  − 2 . 583 2 . 635 14 . 907Standard deviation 1 . 151 1 . 828 0 . 429 Table  2: Mean and standard deviations for the proposed methodwith seven SOI and seven interferer signals.SOI True DOAs ( ◦ ) Mean Std. deviationSOI #1  − 70  − 69 . 649 1 . 012SOI #2  − 50  − 49 . 986 0 . 512SOI #3  − 30  − 30 . 005 0 . 442SOI #4  − 10  − 10 . 005 0 . 289SOI #5 20 19 . 989 0 . 623SOI #6 40 40 . 017 0 . 542SOI #7 60 59 . 945 0 . 734 These figures show that the proposed cyclic method al-lows to perfectly select the two SOI, and ignores the inter-ferer signal. The polynomial roots selected by the proposedcyclic method are all located in two small areas. Those ob-tained from the classical root-MUSIC are more scattered. Itcan be deduced that the estimation provided by the proposedmethod is more accurate.The performance of the estimators can also be obtainedfrom 500 Monte Carlo simulations by calculating the meanand the standard deviation of DOA estimates. Table 1 showsthe simulation results when two BPSK SOI arrive from  − 3 ◦ and 3 ◦ , and one interferer BPSK source from 15 ◦ DOA.The proposed cyclic algorithm performs better than theclassical root-MUSIC thanks to the exploitation of bothcyclic correlation matrices; more information about sourcesis used and the observation dimension space (i.e., the size of the extended covariance matrix   R α CE ( τ  )) is doubled.Table 2 provides simulation results for only the proposedprocedurewhensevenSOIandseveninterfererBPSKsourcesimpingeonthe6-sensorsarray.DOAsofSOIare − 70 ◦ , − 50 ◦ , − 30 ◦ ,  − 10 ◦ , 20 ◦ , 40 ◦ , and 60 ◦ . DOAs of interferer signals are − 35 ◦ , − 20 ◦ ,0 ◦ ,15 ◦ ,25 ◦ ,30 ◦ ,and45 ◦ .Themethodalwaysig-nores interferer signals and gives rather accurate estimationsof the SOI DOA. These last results show that the proposedmethod is signal selective and is able to handle more sourcesof interest than the number of sensors. 5. CONCLUSION We have described a signal-selective procedure for DOA es-timation. By assuming that the incoming signals are BPSKmodulated signals, the algorithm uses the cyclostationary property of the signals to improve the estimations perfor-mance. The proposed method is able to handle more sourcesof interest than the number of sensors. Moreover, by using apolynomial rooting technique, the proposed algorithm doesnot require an explicit search procedure, and hence consid-erably reduces the computational requirements. REFERENCES [1] W. A. Gardner,  Cyclostationarity in Communications and Signal Processing  , IEEE Press, New York, NY, USA, 1993.[2] W. A. Gardner, “Simplification of MUSIC and ESPRIT by ex-ploitation of cyclostationarity,”  Proceedings of the IEEE  , vol. 76,no. 7, pp. 845–847, 1988.[3] P. Charg´e, Y. Wang, and J. Saillard, “An extended cyclic MUSICalgorithm,”  IEEE Trans. Signal Processing  , vol. 51, no. 7, pp.1695–1701, 2003.[4] P. Charg´e, Y. Wang, and J. Saillard, “A non-circular sourcesdirection finding method using polynomial rooting,”  Signal Processing  , vol. 81, no. 8, pp. 1765–1770, 2001.[5] A. J. Barabell, “Improving the resolution performance of eigenstructure-based direction-finding algorithms,” in  Proc.IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP ’83) , vol. 8, pp. 336–339, Cambridge, Mass, USA, April 1983.[6] M. Wax and T. Kailath, “Detection of signals by informationtheoretic criteria,”  IEEE Trans. Acoustics, Speech, and Signal Processing  , vol. 33, no. 2, pp. 387–392, 1985. PascalCharg´e  was born on January 2, 1973.He received the Ph.D. degree in electricalengineering from the University of Nantes,France, in 2001. Until 2003, he was withthe IRCCyN Laboratory, where he was aResearch Fellow with the Electronic Sys-tems, Telecommunications, and Signal Pro-cessing Division. Since September 2003, hehas been an Assistant Professor at the Na-tional Institute of Applied Sciences (INSA),Toulouse, and he is Research Fellow with the LESIA Laboratory.His research interests include sensors array processing, statisticalsignal processing, and signal processing for wireless communica-tions. Yide Wang   received the B.S. degree in elec-trical engineering from Beijing University of Post and Telecommunication (BUPT),Beijing, China, in 1984, the M.S. and thePh.D. degrees in signal processing from theUniversity of Rennes, France, in 1986 and1989, respectively. He is now a Professorat the Polytechnic School, the University of Nantes, France. His research interestsinclude array processing, spectral analysis,and mobile wireless communications systems.
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