QRC-ESPRIT Method for Wideband Signals

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In this paper, a new algorithm for a high resolution
Direction Of Arrival (DOA) estimation method for multiple
wideband signals is proposed. The proposed method proceeds
in two steps. In the first step, the received signals data is
decomposed in a Toeplitz form using the first-order statistics.
In the second step, The QR decomposition is applied on the
constructed Toeplitz matrix. Compared with existing schemes,
the proposed scheme provides several advantages. First, it
requires computing the triangular matrix R or the orthogonal
matrix Q to find the DOA; these matrices can be computed
with O(n2) operation. However, most of the existing schemes
required eignvalue decomposition (EVD) for the covariance
matrix or singular value decomposition (SVD) for the data
matrix; using EVD or SVD requires much more complex
computational O(n3) operation. Second, the proposed scheme
is more suitable for high-speed communication since it
requires first-order statistics and a single snapshot. Third,
the proposed scheme can estimate the correlated wideband
signals without using spatial smoothing techniques; whereas,
already-existing schemes do not. Accuracy of the proposed
wideband DOA estimation method is evaluated through
computer simulation in comparison with a conventional
method.

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QRC-ESPRIT Method for Wideband Signals

  1. 1. ACEEE Int. J. on Communication, Vol. 02, No. 03, Nov 2011 QRC-ESPRIT Method for Wideband Signals Nizar Tayem Electrical Engineering Department Prince Mohammad Bin Fahd University Al Khobar 31952, Saudi Arabia ntayem@pmu.edu.saAbstract – In this paper, a new algorithm for a high resolution proposed for non-coherent signals requires lessDirection Of Arrival (DOA) estimation method for multiple computational load. A class of representative high-resolutionwideband signals is proposed. The proposed method proceeds DOA estimation approaches for wideband sources when thein two steps. In the first step, the received signals data is signals are highly correlated due to multipath or reflection isdecomposed in a Toeplitz form using the first-order statistics. the coherent signal subspace method (CSSM) in [9] and [10].In the second step, The QR decomposition is applied on the The CSSM converts the wideband signal subspaces into aconstructed Toeplitz matrix. Compared with existing schemes,the proposed scheme provides several advantages. First, it predefined narrowband subspace by focusing matrices;requires computing the triangular matrix R or the orthogonal subsequently, the narrowband subspace-based DOAmatrix Q to find the DOA; these matrices can be computed estimation methods, e.g. MUSIC, can be applied. However,with O(n2) operation. However, most of the existing schemes the CSSM method requires initial DOAs, and its performancerequired eignvalue decomposition (EVD) for the covariance is sensitive to these initial values. Two other classes ofmatrix or singular value decomposition (SVD) for the data wideband source localization approaches are based onmatrix; using EVD or SVD requires much more complex orthogonality of subspace. In one class the test ofcomputational O(n3) operation. Second, the proposed scheme orthogonality of projected subspace (TOPS) [11] is referredis more suitable for high-speed communication since it to as a non-coherent method. The method requires singularrequires first-order statistics and a single snapshot. Third,the proposed scheme can estimate the correlated wideband value decomposition (SVD) to get the signal subspace ofsignals without using spatial smoothing techniques; whereas, each frequency point. However, using the SVD required thealready-existing schemes do not. Accuracy of the proposed computational cost of O(n3) operations. In the other class,wideband DOA estimation method is evaluated through the test of orthogonality of frequency subspace (TOFS) [12]computer simulation in comparison with a conventional has been proposed for non-coherent wideband signals . TOFSmethod. constructs the searching steering vector of every possible DOA and frequency. However, TOFS cannot resolve theIndex terms – array signal processing, uniform linear array, desired DOAs when the signal to noise ratio (SNR) is low.Wideband signal, Toeplitz matrix. The objective of this paper is to provide an efficient algorithm to estimate DOAs for multiple wideband sources and to I. INTRODUCTION reduce the computational complexity when compared with the existing schemes. To achieve these objectives, the Source localization using a sensor array is one of the key proposed method proceeds in two steps. In the first step, thetechniques in mobile communication. It plays an importantrole in many application fields such as radar, sonar, mobile received signals data is decomposed in a Toeplitz form usingcommunication, Multiple-Input Multiple-Output (MIMO) the first-order statistics. In the second step, the QR (nosystem, biomedical, and wireless sensor networks [1], [2]. previous reference for an acronym) decomposition is appliedNumerous classical subspace methods e.g., Multiple Signal on the constructed Toeplitz matrix. Compared with existingClassification (MUSIC) [3], Estimation of Signal Parameters schemes, the proposed scheme provides several advantages.via Rotational Invariance Techniques (ESPRIT) [4], and First, it requires computing the triangular matrix R or theMaximum Likelihoods method (ML) [5] have been developed orthogonal matrix Q to find the DOA. The computation ofto estimate the DOA of the narrowband signals. Many of QR for n  n with the Toeplitz matrix [13-14] requires O(n2);these narrowband methods are not applicable to wideband whereas, the existing schemes require eigenvaluessignal directly. The reason is that the energies of narrowband decomposition (EVD) or singular value decomposition (SVD)signals are concentrated in the frequency band that is for an n  n cross-spectral matrix. The major drawback ofrelatively small compared with center frequency. The SVD or EVD is that they require much more computationalwideband signals encountered in many applications such as complexity O(n3). In addition, the proposed method is morehigh-speed data communication, passive sonars, and speech suitable for high-speed communication first order statistics,signal processing . A wide range of techniques have been a single snapshot, and requires computing the triangularproposed to deal with the wideband source localization matrix R or Q only, which further reduces the computationalproblem. The maximum likelihood extended for the wideband complexity. However, the existing method involves multi-signal scenario in [6] and [7] provides optimal statistic dimensional research or high order statistics which increasesproperties, but it requires a multi-dimensional search and is the complexity and computational load even more. In addition,highly non-linear. The one dimensional search method in [8] the proposed method is able to resolve the DOAs of highly© 2011 ACEEE 1DOI: 01.IJCOM.02.03. 32
  2. 2. ACEEE Int. J. on Communication, Vol. 02, No. 03, Nov 2011correlated sources. This paper is organized as follows: in model of wideband and nonstationary source localization issection II, the proposed method is introduced based on first similar to the one of narrowband and stationary case at each and second order statistics. In section III, we present the time and frequency bin. For the first step in the proposedsimulation results. A conclusion is given in Section IV method, we convert the output data vector X  k  into a Toeplitz Hermitian data matrix with II. SYSTEM MODEL AND PROPOSED METHOD dimensions N  1  N  1 . The advantage of introducing Consider a symmetric uniform linear array (ULA) the Toeplitz Hermitian data matrix is that it has a rank that isconsisting of M=2N+1 identical omni-directional antennas, related to the DOA of the sources irrespective of whether thewhere the distance between adjacent antennas is d. Consider sources are coherent or not. This matrix, denoted Z, is givenL wideband signal sources from the directions 1 , 2 , , L . byThen the received signal by the m-th sensor can be expressedas  x0 k  x1 k  x 2 k   x N k       L  x1  k x0 k  x1  k   x( N 1) k  xm (t )   sl (t   m  l )  nm (t ) (1),   x2 k  x1 k  x0 k   x ( N  2 ) k  l 1   (8)where sl(t) is the l-th source signal; nm(t) is the additive white       Gaussian Noise (AWGN) at the m-th sensor; and  x   x   x    x0  k    N k N 1 k N 2 k  m l   m d cos  l / is the delay, where v and d denote the The signal space of the Toeplitz data matrix in (8) has full rankpropagation speed of source signals and the distance which implies that all the incident sources can be detectedbetween sensors, respectively. A narrowband DOA whether they are coherent or non-coherent. The second stepestimation and source localization problem requires that the of the proposed method is to calculate the R data matrix or Qinter-sensor spacing should satisfy the so-called half- data matrix from the received Toeplitz data matrix (8) to estimatewavelength constraint to avoid the phase ambiguity caused the DOA for form coherent multiple wideband sources. Byby the multi-valued property of the cosine function. However, using the Toeplitz data matrix in (8) the necessary informationhalf-wavelength constraint is not valid to a multiple wideband about the noise subspace or the signal subspace can besource data model. Applying the Discrete Fourier Transform extracted using QR factorization. One of the reasons that QR(DFT) to (1), we obtain factorization [15-17] is widely used in adaptive applications is that in RRQR the signal information can be effectively updated making the algorithm suitable for tracking moving sources. In addition, the computation of QR for the n  nIf we use the element at the center of the array as a reference Toeplitz matrix [13-14] requires O(n2) operations. In thepoint, then the 2 N  1  1 output vector can be written as proposed method, we calculate the triangular factorization R to find the DOA for the wideband signal. Using the shiftX  k   x N ωk , , x0 ωk , , x N ωk T (3), invariance property of Toeplitz matrix we can partition the data matrix  in two ways:where the superscript .T represents transpose. We canexpress X  k  as  x   y T  (9) L  0 k X k    sl k a k ,l   nk  (4),  z  F1   l 1where a ,l  is the array response vector in spatial-  F1 yK  (10),frequency domain and is defined as or    z KT  x0 (k )     a  k ,   e j k  N    1  e j k N    (5). where F1 is an (n-1)x(n-1) submatrix of  ,The matrix formulation of (4) can be written asX  k   A k s  k   n k  (6),where sk   s1k ,, sLk T and nk   nN k ,, nN k Tare the DTFT data vectors of the source and noise,respectively. The 2 N 1  L matrix Let R be the upper triangular factor of the Cholesky A( k )  a  k ,1 , , a  k , L  (7) factorization of T  ,is the array manifold matrix in the spatial-frequency domain. RT R  T  (11),Equation (6) means that in the time-frequency domain, the© 2011 ACEEE 2DOI: 01.IJCOM.02.03. 32
  3. 3. ACEEE Int. J. on Communication, Vol. 02, No. 03, Nov 2011where R is the upper triangular matrix from the QR Since R22 is very small, the basis of the noise space can bedecomposition of  . The matrix R can be partitioned in twodifferent ways obtained from the above R factor as follows. Let V be a permutation matrix which represents the row and column (12) interchange. Then, given the small norm of the matrix R22 , r r fr  R   1,1 the upper triangular matrix R can be written as O Rb   R   R11 R12 V T (21), where the matrix R in (21) defines the null space of Y. Let G  Rt r1c  represent any vector in the null space of R, i.e. RG  0 . To oor R  (13),  O rN , N  find the structure of G, we partition G into g1 with L wh er e r fr  r1, 2  r1, N , r1c  r1, N     rN 1, N ; Rb i s components and g 2 with (N-L+1) components. Then GR=0the bottom submatrix and Rt is the top submatrix have a implies thatdimension ( N  1)  ( N  1) . From equations (9), (11) and (12)  g1 we get  R11 R12     0 (22)  g2  r12 r1,1rfr  x0 k 2  zT z x0 k  yT  zTT1T ,1  so that R11g1  R12 g 2  0 . Since R11 is a non-singular T T    (14).rfrr1,1 rfrrfr  Rb Rb   x0k y T1z T   yyT T1T1  T  matrix, g1 can be written in terms of g 2 as followsSimilarly, from equations (10), (11) and (13) we get  g1   R111R12 g2 (23).RtT Rt RtT r1c   T1T1  zK zKT T z x0 k  T y K T R T    RT  Then G can be written as   yRTyR x0k 2  (15)r1c Rt r1T r1c  rn,n  y T 1 x0 k c 2     g    R 1 R Comparing the left upper submatrices in (15) we get G   1    11 12  g 2 (24).  g 2   I N  L 1   RtT Rt  T T1T1  K KT z z (16); To find the basis of null space R, we choose any set of N-L+1while comparison of the lower right matrices on both sides of linearly independent vectors; for example, the columns of(14) gives us the N-L+1 dimensional identity matrix. The basis for the nullr T r fr  Rb Rb = yy T  T1T1 T T (17). space of the upper triangular matrix R, which is also the null fr space of Y, is therefore given byFinally, (16) and (17) give the main relation T T T K KT T   R 1 R Rb Rb  Rt Rt  yy  z z  r fr r fr (18) H qr   11 12  (25).  I N  L 1   where from (14), rf r is given by It is important to observe that the columns of the null space x0 k  y T  z T T1 of H qr are not orthonormal. This is in contrast to the nullrfr  (19) space which can be derived from the SVD or EVD techniques. x    z z  0 k 2 T 1/ 2 We now note that the subspace spanned by the columns ofEquations (18) and (19) represent the base in calculating the H qr is orthogonal to the subspace spanned by the columnsmatrix R. For more detailed information on obtaining theelements of R matrix return to [13] and [14]. After the matrix R of B, where the columns of B contain the information aboutis calculated, the DOA of the wideband signal can be found, the AOAs of incident sources. This is similar to the well-as discussed in the following paragraph. Now the calculated known MUSIC algorithm in which the eigenvector of the noise subspace is orthogonal to the steering vector of the signaldata matrix R with dimensions N  1 N  1 can be subspace. To find the AOAs, we search for the minimumpartitioned as H peaks of H qr B( ) . Since the basis of is not orthonormal, R R12 R   11  (20), we use the orthogonal projection onto this subspace in order  0 R22  to improve the performance by making the basis of the nullwhere R 11 is L  L  ; R 12 is L   N  1  L  ; and R 22 is space of orthonormal. So the columns of the null space of become orthonormal as followsN  L  1 N  L  1 . The QR factorization in (34) is called (26).a rank-revealing QR factorization if R22 has a small norm.© 2011 ACEEE 3DOI: 01.IJCOM.02.03. 32
  4. 4. ACEEE Int. J. on Communication, Vol. 02, No. 03, Nov 2011We now employ the root MUSIC algorithm to find the IV. CONCLUSIONwideband DOAs of the incident sources. The proposed QRC- A new approach is introduced for the estimation of DOAESPRIT using root MUSIC [18]can be written as for wideband signals. The observation output is arranged into a Toeplitz matrix which enables us to perform the estimation using one or more snapshots and estimates the (27). coherent source without any additional processing compared The roots of the polynomial are obtained from (27) for all the to TLS-ESPRIT and Root MUSIC that need spatial smoothingfrequency bins k, and then they are averaged to derive the techniques. The proposed methods employ QR factorizationfinal DOA of the incident signals to estimate an R matrix or Q matrix which has a much lower 1 kH complexity and computational cost O(n2) compared to existing ˆ  k     k  K k kL (28), methods, which require EVD or SVD and O(n3) operation. Through simulations we have shown that the proposedwhere k L , k H  denotes the index of frequency ranges which method exhibit significant performance improvement over many subarray methods such as TLS-ESPRIT and Rootsatisfy K  k H  k L  1 . MUSIC. These advantages make our proposed method appropriate for real-time implementation. III. SIMULATION RESULTS Performance of the proposed method is compared REFERENCESwith that of the TLS-ESPRIT and Root MMUSIC with spatial [1] J. Krim and M. Viberg. “Two decades of array signal processingsmoothing, which are considered one of the best DOA research: the parametric approach.” IEEE signal Processing. Mag.estimation algorithms. The specifications of the simulation Vol.13:3, pp. 67-94. Jul.1996.are shown in Table I. [2] J. C. Chen, K. Yao, and R. E. Hudson. “Source Localization and beamforming.” IEEE signal Processing. Mag. Vol.19:2, pp. 30-39. TABLE I. Mar. 2002. SPECIFICATIONS OF SIMULATION [3] R. O. Schmidt. “Multiple Emitter Location and Signal Parameter Estimation.” IEEE Trans. Antennas and Propagation. Vol. 34, pp. 276-280. Mar.1986. [4] R. Roy and T. Kailath. “ESPRIT—Estimation of Signal Parameters via Rotational Invariance Techniques.” IEEE Trans. Acoust., Speech, and Signal Processing. Vol.37:7, pp.984-995. July 1989. [5] I.Ziskind and M. Wax. “Maximum likelihood localization of multiple sources by alternating projection.” IEEE Trans. Acoustic., Speech, Signal Process. Vol.36, pp.1553-1560. Oct. 1988. [6] M. A. Doron, A. J.Weiss, and H. Messer. “Maximum-likelihoodFigures 1 and 2 show the root-mean-square-error (RMSE) in direction finding of wide-band sources.” IEEE Trans. Signal Process.degree for Wideband DOA estimation error versus SNR form Vol. 41:1, pp. 411–414. Jan. 1993.0 to 20 dB for two coherent sources at 30º and 45º. It is observed [7] J. C. Chen, R. E. Hudson, and K. Yao. “Maximum-likelihoodthat the proposed methods result in lower RMSE compared source localization and unknown sensor locationestimation for wideband signals in the near-field.” IEEE Trans. Signal Process.to the TLS-ESPRIT and Root MUSIC with spatial smoothing. Vol. 50:8, pp. 1843–1854. Aug. 2002.In addition, the proposed method provides low complexity [8] W.J. Zeng and Xi. Lin Li. “High-Resolution Multiple Widebandand computational cost O(n2) compared to the method in [11] and Nonstationary Source Localization with Unknown Number ofwhich requires EVD or SVD and O(n3) operation. Furthermore, Sources. IEEE Trans. Signal Process. Vol. 58:6, pp. 3125–3136.the proposed method does not require spatial smoothing Jun. 2010.techniques to estimate the coherent sources; whereas, TLS- [9] H. Wang and M. Kaveh. “Coherent signal-subspace processingESPRIT and Root MUSIC do. In Table II, the computational for the detection and estimation of angles of arrival of multiplecost of the proposed method is evaluated in comparison with wideband sources.” IEEE Trans. Acoust., Speech, Signal Process.TLS-ESPRIT and Root-MUSIC methods. Note that the unit Vol. ASSP-33, pp. 823–831. Aug. 1985.of computation time is in seconds. We observe that the [10] S. Valaee and P. Kabal. “Wideband array processing using a two-sided correlation transformation.” IEEE Trans. Signal Process.proposed method requires smaller computational cost Vol. 43, pp. 160–172. Jan. 1995.compared with TLS-ESPRIT and Root-MUSIC. [11] Y. Yoon, L. M. Kaplan, and J. H. McClellan. “TOPS: New TABLE II. DOA Estimation for Wideband Signals.” IEEE Trans. Signal COMPUTIONAL COST Processing, Vol. 54:6, pp. 1977-1988. June 2006. [12] H. Yu and J. L. Huang. “A new Method for wideband DOA Estimation.” Wireless communication, Networking and Mobile Computing 2007. WiCom 2007, pp. 598-601. Sep. 2007. [13] D. R. Sweet. “Fast Toeplitz orthogonalization.” Numer. Math. Vol. 53, pp. 1-21. 1984.© 2011 ACEEE 4DOI: 01.IJCOM.02.03.32
  5. 5. ACEEE Int. J. on Communication, Vol. 02, No. 03, Nov 2011[14] A.W. Bojanczyk, R. P. Brent, and F. R. de Hoog. “QRFactorization of Toeplitz matrices.” Numer. Math. Vol. 49, pp. 81-94. 1986.[15] C. H. Bischof and G. M. Shroff. “On updating signalsubspaces.” IEEE Transactions on Acoustics, Speech, and SignalProcessing. Vol. 40, pp. 96-105. Jan 1992.[16] M. P. Fargues and M. P. Ferreira. “Investigations in thenumerical behavior of the adaptive rank-revealing QR factorization.” IEEE Transactions on Acoustics, Speech, and Signal Processing.Vol. 43, pp. 2787-2791. Nov. 1995.[17] M. Bouri and S. Bourennane. “High resolution methods basedon rank revealing triangular factorization.” Transactions onEngineering, Computing and Technology. Vol. 2, pp. 35-38. Dec2004.[18] A. J. Barabel. “Improved the resolution performance ofeigenstructure-based direction-finding algorithm.” In Proc. IEEEICASSP 83, pp. 336-339. 1983. Figure 2. RMSE for wideband DOA angle estimation errors versus SNR for Source 2 at 45º by using the proposed method, Root MUSIC, and TLS-ESPRIT algorithms. Figure 1. RMSE for wideband DOA angle estimation errors versus SNR for Source 1 at 30º by using the proposed method, Root MUSIC, and TLS-ESPRIT algorithms© 2011 ACEEE 5DOI: 01.IJCOM.02.03. 32

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