Matrix pencil method for simultaneously estimating azimuth

Digital Signal Processing 16 (2006) 796–816 www.elsevier.com/locate/dsp Matrix pencil method for simultaneously estimating azimuth and elevation angles of arrival along with the frequency of the incoming signals Nuri Yilmazer ∗ , Raul Fernandez-Recio, Tapan K. Sarkar Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244-1240, USA Available online 17 July 2006 Abstract In this paper we describe a method for simultaneously estimating the direction of arrival (DOA) of the signal along with its unknown frequency. In a typical DOA estimation problem it is often assumed that all the signals are arriving at the antenna array at the same frequency which is assumed to be known. The antenna elements in the array are then placed half wavelength apart at the frequency of operation. However, in practice seldom all the signals arrive at the antenna array at a single pre-specified frequency, but at different frequencies. The question then is what to do when there are signals at multiple frequencies, which are unknown. This paper presents an extension of the matrix pencil method to simultaneously estimate the DOA along with the operating frequency of each of the signals. This novel approach involves approximating the voltages that are induced in a three-dimensional antenna array, by a sum of complex exponentials by jointly estimating the direction of arrival (both azimuth and elevation angles) along with the carrier frequencies of multiple far-field sources impinging on the array by using the three-dimensional matrix pencil method. The matrix pencil method is a direct data domain method for approximating a function by a sum of complex exponentials in the presence of noise. The variances of the estimates computed by the matrix pencil method are quite close to the Cramer–Rao bound. Finally, we illustrate how to carry out the broadband DOA estimation procedure using realistic antenna elements located in a conformal array. Some numerical examples are presented to illustrate the applicability of this methodology in the presence of noise. It is shown that the variance decreases as the SNR increases. The Cramer–Rao bound for the estimators are also provided to illustrate the accuracy and the computational efficiency of this new methodology. © 2006 Elsevier Inc. All rights reserved. Keywords: Matrix pencil method; 3-D matrix pencil method; Simultaneous azimuth; Elevation angles and frequency estimation; Wavelength estimation; Direction of arrival estimation; Angle of arrival estimation; DOA estimation; Estimation of frequency; Broadband DOA estimation; Conformal array 1. Introduction In contemporary literature, most of the efforts have primarily been directed for estimating the two dimensional spa- tial frequencies (namely, the azimuth angle φ and the elevation angle θ) of plane waves that are arriving at an antenna * Corresponding author. E-mail address: nyilmaze@syr.edu (N. Yilmazer). 1051-2004/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.dsp.2006.05.009 N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 797 array. It is generally assumed that all the signals have the same frequency of operation even though they are arriving from different directions and that the antenna elements in a linear array that are uniformly spaced at 0.5λ, where λ is the wavelength at the frequency of operation. We now describe a methodology for simultaneously estimating the frequency of operation and the DOA of the signals using a three-dimensional antenna array. The voltages induced in the antenna elements of the three-dimensional antenna arrays are used to estimate the frequency of operation and the DOA of the signal simultaneously using the matrix pencil method. The problem of estimating the DOA of multiple far field sources can be found in many application areas such as in radar, sonar, and radio communication and is being studied extensively by many researchers [1–4]. The matrix pencil method, a high-resolution DOA estimation algorithm, has been applied to find the azimuth, and elevation angles of the narrow band sources, which are in the far field of the receiver antennas. In some cases the carrier frequency of the multiple signals may not be known, in that case, the carrier and the DOA angles, azimuth, and elevation must be estimated simultaneously. In the one-dimensional (1-D) uniform linear array (ULA) [1,7], it has been assumed that the frequency of the signals is known. As an extension to 1-D and 2-D matrix pencil method, 3-D matrix pencil method is presented for estimating three-dimensional (3-D) frequencies is explained in Section 2. Many efforts have been devoted for estimating the two-dimensional (2-D) frequencies for a 2-D data set. The applications cover a wide range of fields from synthetic aperture radar imaging or frequency, wavenumber estimation in array processing, nuclear magnetic resonance imaging, radar, array signal processing, radar surveillance systems, and space-division-multiple-access (SDMA) for mobile communication. At the same time, high resolution direction of arrival (DOA) estimation techniques are very important in systems such as radar or sonar applications too. In the current situation, we have a 3-D data set or, as we will see later, a data set placed in a 3-D space such as two 2-D data sets (e.g., planes) placed orthogonal to each other. In Section 2, the formulation of the problem is introduced and the signal model is given. In Section 3 the ma- trix pencil method is described for the 3-D case. In Section 4, the pairing method for the 3-D poles is introduced. Sample numerical results through computer simulations are provided in Section 5. Finally, Section 6 is the conclu- sions. 2. Problem formulation Let us consider a 3-D data cube denoted as ν(a;b;c) that consists of I , 3-D exponential sinusoids, which can be written as ν(a;b;c) = I  i=1 M i e (jγ i +j2πf 1i a+j 2πf 2i b+j 2πf 3i c) , (1) where a = 1, ,A, b = 1, ,B, and c = 1, ,C. Each of the 3-D sinusoids has amplitude of M i along with the respective phases γ i and three frequencies (f 1i ,f 2i ,f 3i ). The ultimate goal is to estimate correctly each of the three frequencies for each 3-D sinusoid and group them appropriately. The estimation of the complex amplitudes of the signals can be reduced to the solution of a least square problem. The paper is focused on how to obtain the frequencies for all the sinusoids (f 1i ,f 2i ,f 3i ), i = 1, ,I, and how to group them to identify each sinusoid correctly. For the DOA estimation problem they will be elevation, and azimuth angles and the carrier frequencies of each of the far-field sources. The original data, from which we can extract the frequencies from, can be a data cube or two planes of data as will be explained later on, situated along three dimensions. These three dimensions can be the three spatial coordinates or data from two spatial coordinates at different instants of time. Graphically, we can represent the two-dimensional data as a plane and three-dimensional data as a cube. A cube, in this context, is a set of planes situated along one direction, as it is shown in Fig. 1a. As we have mentioned, the original data can also be along two planes situated along two orthogonal planes and this situation is illustrated graphically in Fig. 1b. We can rewrite Eq. (1) as follows: ν(a;b;c) = I  i=1 α i x a i y b i z c i , (2) 798 N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 Fig. 1. Illustration of the three-dimensional data possibilities for the 3-D matrix pencil method. where {α i ; i =1, ,I} are the complex amplitudes of each of the 3-D sinusoids and {(x i ,y i ,z i ); i = 1, ,I} are the 3-D poles that we want to extract from the data. The frequencies can be obtained uniquely from the 3-D poles, i.e., f 1i = 1 2π Im  log(x i )  ,f 2i = 1 2π Im  log(y i )  ,f 3i = 1 2π Im  log(z i )  . (3) In the first stage of the computations, we need to build three pencils of matrices to extract the three sets of poles. In the second stage we need to group correctly the three frequencies that have been extracted individually and which correspond to each of the 3-D sinusoids. Let us consider a cube of dimensions A ×B × C. A three-dimensional grid is generated with a separation between elements of the grid of  x ,  y , and  z in the x-, y-, and z-directions, respec- tively. In this case,  x ,  y , and  z are the distance between the antenna elements in the corresponding directions. Let us assume that there are I far field sources for the 3-D antenna elements. This problem is equivalent to estimating I undamped signals with different frequencies, elevation and azimuth angles for the DOA problem. The induced voltage in each of the antenna elements in the three-dimensional grid due to I 3-D sinusoids is represented by (1). The goal is to estimate correctly each of the three frequencies for each 3-D sinusoid and group them adequately. We can rewrite Eq. (1) as follows: ν(a;b;c) = I  i=1 α i x a i y b i z c i , (4) α i =M i exp(j γ i ), (5) x i =exp(j 2πf 1i ), (6) y i =exp(j 2πf 2i ), (7) z i =exp(j 2πf 3i ). (8) Consider a three-dimensional array in space with the axes oriented along the Cartesian coordinates, e.g., ˆx, ˆy, and ˆz. The distances between the elements are  x ,  y , and  z for each of the three coordinates and the number of sensors are A, B, and C along the ˆx, ˆy, and ˆz. Let us also suppose that I signals arrive with different azimuth and elevation angles {(φ i ,θ i ); i = 1, ,I} and with different wavelengths {λ i ; i = 1, ,I}. We assume that the signals are narrowband centered in λ i . Therefore, we have ν(a;b;c) = I  i=1 M i e j  γ i + 2π λ i  x cos φ i sin θ i a+ 2π λ i  y sin φ i sin θ i b+ 2π λ i  z cos θ i c  , (9) α i =M i exp(j γ i ), (10) N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 799 x i =exp  j 2π λ i  x cos φ i sin θ i  , (11) y i =exp  j 2π λ i  y sin φ i sin θ i  , (12) z i =exp  j 2π λ i  z cos θ i  . (13) Once the poles are found along each plane, the corresponding elevation, azimuth angles and the wavelength of the signals θ , φ, and λ are obtained for each source as G = Im  ln x i 2π x  , (14) E = Im  ln y i 2π y  , (15) F = Im  ln z i 2π z  , (16) φ = arc cot  G E  , (17) θ =arctan  √ G 2 +E 2 F  , (18) λ =  1 G 2 +E 2 +F 2 . (19) 3. 3-D matrix pencil theory The 3-D data matrix can be enhanced by using the partition-and-stacking process [5–7]. The enhanced-matrix matrix pencil technique for estimating the three-dimensional frequencies is constructed for handling the estimation of the three-dimensional frequency. The column vectors along the x-direction is enhanced by a pencil parameter L and it is stacked to get the matrix D y,z , which is has a Hankel structure. D y,z = ⎡ ⎢ ⎢ ⎣ ν(0;y;z) ν(1;y;z) ··· ν(A −L;y;z) ν(1;y;z) ν(2;y;z) ··· ν(A −L +1;y;z) . . . . . . . . . . . . ν(L −1;y;z) ν(L;y;z) ··· ν(A −1;y;z) ⎤ ⎥ ⎥ ⎦ L(A−L+1) . (20) After obtaining the matrix D y,z in (19), we enhance the matrix along the y direction with the pencil parameter K using a similar procedure as D z = ⎡ ⎢ ⎢ ⎣ D 0,z D 1,z ··· D B−K,z D 1,z D 2,z ··· D B−K+1,z . . . . . . . . . . . . D K−1,z D K,z ··· D B−1,z ⎤ ⎥ ⎥ ⎦ LK(A−L+1)(B−K+1) . (21) So for the next step, the D z matrix is generated in a similar fashion with pencil parameter R using the data along the z-direction as D e = ⎡ ⎢ ⎢ ⎣ D 0 D 1 ··· D C−R D 1 D 2 ··· D C−R+1 . . . . . . . . . . . . D R−1 D R ··· D C−1 ⎤ ⎥ ⎥ ⎦ LKR(A−L+1)(B−K+1)(C−R+1) . (22) 800 N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 The enhanced data matrix D e is now used to obtain the three-dimensional poles. The matrix D y,z can be decomposed as D y,z =X 1 AY y d Z z d X 2 , (23) where A is a diagonal matrix of dimension (I ×I) and the diagonal elements contain the complex amplitudes of the I signals, X 1 = ⎡ ⎢ ⎢ ⎣ 11··· 1 x 1 x 2 ··· x I . . . . . . . . . . . . x L−1 1 x L−1 2 ··· x L−1 I ⎤ ⎥ ⎥ ⎦ , (24) Y d =diag(y 1 ,y 2 , ,y I ), (25) Z d =diag(z 1 ,z 2 , ,z I ), (26) X 2 = ⎡ ⎢ ⎢ ⎢ ⎣ 1 x 1 ··· x A−L 1 1 x 2 ··· x A−L 2 . . . . . . . . . . . . 1 x I ··· x A−L I ⎤ ⎥ ⎥ ⎥ ⎦ , (27) D z =E 1 AZ z d E 2 . (28) The matrix D z can be decomposed as E 1 = ⎡ ⎢ ⎢ ⎣ X 1 X 1 Y d . . . X 1 Y K−1 d ⎤ ⎥ ⎥ ⎦ . (29) Matrix E 1 has the dimension of KL ×I.HereZ d is a diagonal matrix defined as Z d =diag(z 1 ,z 2 , ,z I ), (30) E 2 =  X 2 Y d X 2 Y B−K d X 2  , (31) and E 2 has the dimension I ×(A −L + 1)(B −K + 1). Finally, the enhanced matrix D e can be written as D e =F 1 AF 2 , (32) where F 1 is defined as F 1 = ⎡ ⎢ ⎢ ⎣ E 1 E 1 Z d . . . E 1 Z R−1 d ⎤ ⎥ ⎥ ⎦ . (33) The dimensions of the matrix F 1 are LKR ×I , and the matrix F 2 is defined by F 2 =  E 2 Z d E 2 Z C−R d E 2  . (34) The dimensions of the matrix F 2 are I ×(A −L +1)(B −K +1)(C − R + 1). 4. Eigenstructure of the data matrix D e It is better to study the eigenstructure of the matrix D e , in order to observe how to estimate the 3-D poles. A singular value decomposition (SVD) [1,5] must be first applied to remove the effects of noise before applying the matrix pencil N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 801 technique. The matrix D e can be written as D e =UΛV H =U s Λ s V H s +U n Λ n V H n , (35) where the superscript H denotes the conjugate transpose, and the subindexes s and n stands for the principal com- ponents of the signal and noise component, respectively. Here U and V are unitary matrices whose columns are the eigenvectors of (D e )(D e ) H , and (D e ) H (D e ), respectively. Λ is the singular values of D e , which are located on the main diagonals in the descending order σ 1  σ 2  ··· σ I . If the data is noiseless, the first I singular values are nonzero, the rest is zero, where σ i > 0fori =0, 1, ,I, σ i =0fori =I +1, ,min  LKR(A −L +1)(B −K +1)(C −R +1)  . If the data is noisy, I needs to be estimated, and how to do it is illustrated in [1]. The ratio of each of the singular value to the largest one determines the value of I . After computing the SVD, the data is split along the signal subspace and the noise subspace. The signal subspace has dimension I (the number of signals) that corresponds to the main eigenvalues of Λ and the noise subspace that is related to the rest of eigenvalues [8]. They are represented by U s =[u 1 ,u 2 , ,u I ], (36) Λ s =diag(σ 1 ,σ 2 , ,σ I ), (37) V s =[ν 1 ,ν 2 , ,ν I ], (38) U n =[u I +1 ,u I +2 , ,u min(LKR(A−L+1)(B−K+1)(C−R+1)) ], (39) Λ n =diag(σ I +1 ,σ I +2 , ,σ min(LKR(A−L+1)(B−K+1)(C−R+1)) ), (40) V n =[ν I +1 ,ν I +2 , ,ν min(LKR(A−L+1)(B−K+1)(C−R+1)) ]. (41) The matrix D e has the dimension of (LKR) ×(A −L +1)(B −K +1)(C −R +1). We know that the rank(D e ) = I if and only if rank(F 1 ) = I is also equal to rank(F 2 ) = I . The rank of F 1 depends on the three parameters K, L, and R. The same thing applies for the rank(F 2 ). The necessary condition for F 1 to be of full rank I is that the number of rows of F 1 is not less than I . So the rank(F 1 ) = I if and only if LKR  I . In the same way, for the F 2 , the rank(F 2 ) = I if and only if (A −L +1)(B −K + 1)(C − R + 1)  I . So the rank(D e ) = I if and only if LKR  I , and (A −L +1)(B − K +1)(C −R +1)  I . Next we define a permutation matrix that is used for the extraction of the poles [5]. The permutation (shuffling) matrix in this case will be: P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ s(1) s(1 +L) . . . s(1 +(KR −1)L) s(2) s(2 +L) . . . s(2 +(KR −1)L) . . . . . . s(L) s(L +L) . . . s(L +(KR − 1)L) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ LKR×LKR (42) 802 N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 and the other shuffling matrix is defined as J = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ s(1) . . . s(L) s(1 +LK) . . . s(L +LK) s(1 +LK(R −1)) . . . s(L +LK(R − 1)) s(1 +L) . . . s(2L) s(1 +L +LK) . . . s(2L +LK(R −1)) . . . . . . s(LK +LK(R −1)) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ LKR×LKR (43) where s(i) is the 1 ×LKR vector with one at the ith position and zero everywhere else. 4.1. Estimation of poles z i A matrix pencil is formed to estimate the poles along the z-direction from U 2 −λU 1 =0, (44) where U 1 and U 2 are constructed from U s as U 1 :U s with the last LK rows deleted, U 2 :U s with the first LK rows deleted. The generalized eigenvalues of Eq. (43) will be the poles {z i ; i =1, ,I}. They are computed as the eigenvalues of U + 1 U 2 , where U + 1 is the Moore–Penrose pseudo-inverse of U 1 and is defined as U + 1 = (U H 1 U 1 ) −1 U H 1 . Therefore the poles are computed as the eigenvalues of z i =eig((U H 1 U 1 ) −1 U H 1 U 2 ). 4.2. Estimation of poles y i Next, we define the following: U y s =JU s , (45) U y 2 −λU y 1 =0, (46) where U y 1 :U y s with the last LR rows deleted, U y 2 :U y s with the first LR rows deleted. The generalized eigenvalues of Eq. (45) will be the poles {y i ; i = 1, ,I} and are computed from the eigen- values of (U y 1 ) + U y 2 , where (U y 1 ) + is the Moore–Penrose pseudo-inverse of (U y 1 ) and is defined as (U y 1 ) + = ((U y 1 ) H (U y 1 )) −1 (U y 1 ) H . Therefore the poles are computed as the eigenvalues of y i =eig(((U y 1 ) H (U y 1 )) −1 (U y 1 ) H U y 2 ). N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 803 4.3. Estimation of poles x i Next we form the following: U x s =PU s , (47) U x 2 −λU x 1 =0, (48) where U x 1 :U x s with the last KR rows deleted, U x 2 :U x s with the first KR rows deleted. The generalized eigenvalues of Eq. (47) will be the poles {x i ; i = 1, ,I} and are computed from the eigen- values of (U x 1 ) + U x 2 , where (U x 1 ) + is the Moore–Penrose pseudo-inverse of (U x 1 ) and is defined as (U x 1 ) + = ((U x 1 ) H (U x 1 )) −1 (U x 1 ) H . Therefore the poles are computed as the eigenvalues of x i =eig(((U x 1 ) H (U x 1 )) −1 (U x 1 ) H U x 2 ). It is known that the range of the principal vector range(U s ) = range(F 1 ), [U s ]=[F 1 ][T ], where [T ] is a nonsingular unique matrix of size I . The necessary conditions when choosing the parameters K, L, and R will be as follows; By analyzing the shuffling matrices, we LK, LR, and KR rows are deleted from the U s , U y s , and U x s matrices correspondingly. The rows of these matrices is no less than I , so we can write that (R −1)LK  I, (K − 1)LR  I, (L −1)KR  I, (A −L +1)(B −K +1)(C − R + 1)  I. 4.4. Pole paring for 3-D matrix pencil method After finding the individual poles x i , y i , and z i one needs to associate the right components with each other to get the correct pairs, so that it will correspond to the true azimuth and elevation angles for the specific wavelength of the impinging far field sources. So there is a need for establishing a pairing algorithm. By exploiting the property that the signal space is orthogonal to the noise subspace implies that e L ⊥U n . The pairs can be matched together by maximizing the criterion below J sp (i,j,k)= I  p=1   u H p e L (x i ,y j ,z k )   2 ,i=1, 2, I, j =1, 2, I, k =1, 2, I, (49) e L =x L ⊗y L ⊗z L , (50) where ⊗ is the Kronecker product, and x L , y L , and z L are defined as follows: x L =[1,x, ,x K−1 ] T , y L =[1,y, , y L−1 ] T , z L =[1,z, ,z R−1 ] T .Here{u p ; p = 1, 2, ,I} are the I principles eigenvectors of the matrix D e . Pole pairing algorithm for three-dimensional matrix pencil method i = 1 while i<=I (number of signals) DO for j =1 size of the set containing x i poles DO for k = 1 size of the set containing y i poles DO Compute J s (i,j,k) from (48) End k End j Find the maximum J s (i,j,k) Remove the paired poles from the set (corresponding to the maximum). Increment i End 804 N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 5. Numerical results In this section, illustrative computer simulation results are provided to illustrate the performance of this novel technique. The noise contaminated signal model is formulated as ˜ν(a;b;c) = I  i=1 M i e j  γ i + 2π λ i  x cos φ i sin θ i a+ 2π λ i  y sin φ i sin θ i b+ 2π λ i  z cos θ i c  +w(a,b, c), (51) Table 1 Summary of the signal features incident on the antenna array Signal 1 Signal 2 Signal 3 Frequency 300 MHz 290 MHz 280 MHz φ 30 ◦ 40 ◦ 50 ◦ θ 45 ◦ 35 ◦ 25 ◦ Fig. 2. The variance −10 log 10 (var(φ 1 )), 3-D MP and the CRB are plotted against the SNR. Fig. 3. The variance −10 log 10 (var(θ 1 )), 3-D MP and the CRB are plotted against the SNR. N. Yilmazer et al. / Digital Signal Processing 16 (2006) 796–816 805 Fig. 4. The variance −10 log 10 (var(λ 1 )), 3-D MP and the CRB are plotted against the SNR. (a) (b) (c) (d) Fig. 5. The scatter plots of elevation and azimuth angles for (a) SNR = 5, (b) 10, (c) 15, and (d) 25 dB. [...]... inverse of the matrix F References [1] Y Hua, T.K Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise, IEEE Trans Acoust Speech Signal Process 38 (5) (1990) [2] Y Li, J Razavilar, K.J Liu, A high resolution technique for multidimensional NMR spectroscopy, IEEE Trans Biomed Eng 45 (1) (1998) 78–86 [3] A.J van der Veen, P Ober, E.F Deprettere, Azimuth and... averaging technique for multi-path environments, IEEE Trans Antennas Propagat 35 (12) (1987) 1389–1396 [5] Y Hua, Estimating two-dimensional frequencies by matrix enhancement and matrix pencil, IEEE Trans Signal Process 40 (9) (1992) [6] T.K Sarkar, M.C Wicks, M Salazar-Palma, R.J Bonneau, Smart Antennas, Wiley, New York, 2003 [7] N Yilmazer, J Koh, T.K Sarkar, Utilization of a unitary transform for efficient... estimator for azimuth angle versus SNR Fig 10 Bias of the estimator for elevation angle versus SNR Fig 11 Bias of the estimator for wavelength versus SNR 809 810 N Yilmazer et al / Digital Signal Processing 16 (2006) 796–816 problem, where one can estimate not only the azimuth and elevation angles of the various signals impinging on array but also their wavelengths of operation It has been shown how to form... bound (CRB) measures the goodness of an estimator This bound is the smallest limit for the variance of the estimated values under noisy measurements with white Gaussian noise The bound is found from using the Fisher information matrix, whose diagonal elements are the corresponding CRB of that element The Fisher information matrix and how it relates to the CRB is explained in Appendix A The simulation results... enhanced data matrix and by using this data matrix one then pair the respective poles to obtain the corresponding angle of arrivals and the wavelengths Numerical examples are provided to illustrate the validity of the new method The statistical performance parameters such as variance of the estimator have also been compared with the Cramer–Rao bound to observe the accuracy of the new method Acknowledgment... ϕI and ⎡ ⎤ Mi ⎢ γi ⎥ ⎢ ⎥ (57) ϕi = ⎢ fi ⎥ ⎣ ⎦ φi θi The I × I Fisher information matrix [F ] is defined by F(ϕ) ij = −E ∂2 log p(˜ |ϕ) ν ∂ϕi ∂ϕj (58) , where Fij is the (i, j )th element of the matrix F In addition the matrix F can be partitioned as F = {Fij , i = 1, , I, j = 1, , I } where Fij is a 5 × 5, (i, j )th block matrix of F E(·) is the expectation operator, and ∂/∂ϕi is the partial derivative... using the equations as outlined, the Fisher information matrix is formed It is a 5 × 5 matrix The Cramer–Rao bound (CRB) is defined as var ϕi ˆ F−1 (ϕ) ii (91) and var ϕi ˆ Fii , (92) where Fii is the ith diagonal element of F−1 The CRB on the variance of the unbiased estimate of the ith parameter ˆ ˆ ˆ ˆ ϕi is the ith diagonal element of the inverse of the matrix F−1 (ϕ) So, var(Mi ), var(γi ), var(fˆi... can be seen, for the increased SNR values, the estimated values approach to its true values in the histogram plot 808 N Yilmazer et al / Digital Signal Processing 16 (2006) 796–816 (a) (b) (c) (d) Fig 8 The histogram of wavelength for (a) SNR = 5, (b) 10, (c) 15, and (d) 25 dB The bias of the estimator has also been studied to see the efficiency of the new method For the elevation and azimuth angles... bias (in dB) of the estimator for azimuth and elevation angles, and the wavelength versus SNR is shown in Figs 9–11 For higher values of the SNR, the bias of the estimator decreases, as expected 6 Conclusions In this study, a new methodology has been presented to extract the 3-D frequencies from the sampled values of the voltages induced in an array This 3-D estimation method can easily be applied to... elevation and azimuth angles are shown in Fig 5 for different signal-to-noise (SNR) ratios of SNR = 5, 10, 15, and 25 dB The results are based on 200 Monte Carlo simulations As it is expected, when the SNR increases, the estimated values approach to its true values in the scatter plot The histogram of the estimated elevation and azimuth angles and the wavelength of the sources are shown in Figs 6–8 for different . 1-D and 2-D matrix pencil method, 3-D matrix pencil method is presented for estimating three-dimensional (3-D) frequencies is explained in Section 2. Many efforts have been devoted for estimating. computational efficiency of this new methodology. © 2006 Elsevier Inc. All rights reserved. Keywords: Matrix pencil method; 3-D matrix pencil method; Simultaneous azimuth; Elevation angles and frequency. =  1 G 2 +E 2 +F 2 . (19) 3. 3-D matrix pencil theory The 3-D data matrix can be enhanced by using the partition-and-stacking process [5–7]. The enhanced -matrix matrix pencil technique for estimating the three-dimensional