Progress In Electromagnetics Research, PIER 102, 31 48, 2010

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1 Progress In Electromagnetics Research, PIER 102, 31 48, 2010 ACCURATE PARAMETER ESTIMATION FOR WAVE EQUATION F. K. W. Chan, H. C. So, S.-C. Chan, W. H. Lau and C. F. Chan Department of Electronic Engineering City University of Hong Kong 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China Abstract Waves arise in many physical phenomena which have applications such as describing the voltage along a transmission line and medical imaging modality of elastography. In this paper, estimating the parameters for two forms of lossy wave equations, which correspond to multi-mode and multi-dimensional waves, are tackled. By exploiting the linear prediction property of the noise-free signals, an iterative quadratic maximum likelihood (IQML) approach is devised for accurate parameter estimation. Simulation results show that the estimation performance of the proposed IQML algorithms can attain the optimal benchmark, namely, Cramér-Rao lower bound, at sufficiently high signal-to-noise ratio and/or large data size conditions. 1. INTRODUCTION Consider the electromagnetic wave equation for a homogeneous and isotropic medium at a fixed frequency: 2 U(x, y, z) + k 2 U(x, y, z) = 0 (1) U(x, y, z) is the complex field amplitude, k is the propagation constant, 2 is the Laplacian operator of the form 2 = 2 x y z 2 (2) The solution can be expressed as a linear combination of plane waves, that is [1] U(x, y, z) = [A m exp(j(k x,m x + k y,m y + k z,m z)) m=1 +B m exp( j(k x,m x + k y,m y + k z,m z))] (3) Corresponding author: F. K. W. Chan (ckwf@hkexperts.com).

2 32 Chan et al. [k x,m k y,m k z,m ] T is the wave number vector of the mth wave and A m and B m are the associated forward and backward complex amplitudes, respectively. When the wave equation is applied to a channel waveguide, the propagation direction is confined to one-dimension (1-D), say, z, the guided field measured along the propagation direction z can be described by reducing (3) to M U(z) = [A m exp(jk z,m z) + B m exp( jk z,m z)] (4) m=1 M is the number of nondegenerate modes supported. Examples of channel waveguide include optical fibers, ridge waveguides, and coaxial cables. As (4) is a realistic model of the physical phenomenon in many scenarios [2] such as describing the voltage along a transmission line and medical imaging modality of elastography, it is of interest to find the complex-valued parameters, namely, {A m }, {B m } and {k z,m }. To the best of our knowledge, Oliphant [2] was the first to tackle this parameter estimation problem with the use of a nonlinear least squares () algorithm. Although the estimator can attain the maximum likelihood (ML) performance under white Gaussian noise environment, it is hard to implement in practice as its objective function is multi-modal. Recently, So et al. [3] have devised an iterative quadratic maximum likelihood (IQML) [4 6] algorithm for the 1-D wave equation in white Gaussian noise by making use of the linear prediction (LP) property [7, 8] in U(z). Apart from more computationally attractive, it is demonstrated in [3] that the IQML scheme can provide optimum accuracy even with a smaller threshold signal-to-noise ratio (SNR) than that of the estimator. However, both [2, 3] only address the simplest scenario of M = 1 in (4) which corresponds to the 1-D single-mode waveguide. For M > 1, the waveguide is multi-mode and a generalized treatment is required, which will be discussed in detail in this work. Moreover, when the wave equation is applied to planar waveguides or free-space, the propagation direction is no longer confined to 1-D. That is, the field is expressed by (3) without further simplification and its parameter estimation will also be investigated. The rest of the paper is organized as follows. In Section 2, we develop an IQML-based parameter estimator for the wave equation of (4) in the presence of white Gaussian noise, which can be considered as an extension of [3]. The basic idea is to solve for {k z,m } in an iterative manner first and then estimate {A m } and {B m } according to linear least squares (LLS). In Section 3, we generalize (3) to the P -dimensional (P -D) model with P 2. Prior to employing the

3 Progress In Electromagnetics Research, PIER 102, IQML approach, we exploit the subspace technique [9] to use the principal singular vectors as the inputs to the algorithm. Cramér- Rao lower bound () [10] computation for the investigated signal models, which gives a lower bound on the variance attainable by any unbiased estimator using the same input data, is provided in Section 4. Simulation results are included in Section 5 to evaluate the estimation performance of the proposed approach by comparing with the. Finally, conclusions are drawn in Section 6. A list of mathematical symbols that are used in the paper is given in Table 1. Table 1. List of symbols. Symbol C M N T H 1 vec E I i Meaning set of M N complex matrices transpose conjugate transpose complex conjugate inverse vectorization Kronecker product expectation i i identity matrix 0 i j i j zero matrix R(a) I(a) â [a] i [A] i,j A diag ( ) Toeplitz (a, b T ) real part of a imaginary part of a estimate of a ith element of a (i, j) entry of A derivative of A diagonal matrix Toeplitz matrix with first column a and first row b T 2. ESTIMATION FOR MULTI-MODE SCENARIOS In this Section, we develop the parameter estimation algorithm for lossy wave equation corresponding to 1-D multi-mode waveguide. Based on uniformly sampling a noisy version of (4), the observed signal model is R(n) = U(n) + Q(n) (5)

4 34 Chan et al. U(n) = M [A m exp(jk m n) + B m exp( jk m n)], n = 1, 2,..., N (6) m=1 The k m represents the wave number we drop the subscript z for notation simplicity, while A m and B m are the associated amplitudes determined by the auxiliary or boundary conditions, of the mth wave. All k m, A m and B m, m = 1, 2,..., M, are unknown complex-valued constants to be estimated, while the number of modes, M, is assumed known. The additive noises {Q(n)} are complex uncorrelated white Gaussian processes with variances σ 2. The task is to find all unknown parameters given the N measurements of {R(n)}. Our methodology is first to find the nonlinear parameters of wave numbers. The linear {A m } and {B m } are then estimated using an LLS [10] fit. We notice that each mode of U(n) can be expressed as A m exp(jk m n) + B m exp( jk m n) = 0.5(A m + B m + A m B m ) exp(jk m n) +0.5(B m + A m A m + B m ) exp( jk m n) =(A m +B m ) cos(k m n)+j(a m B m ) sin(k m n)=α m cos(k m n+φ m ) (7) α m = 2 A m B m (8) ( ) φ m = tan 1 j(bm A m ) (9) A m + B m From (7), it is clear that U(n) is a sum of M sinusoids with complexvalued frequencies. Making use of the sinusoidal LP relationship that U(n) can be expressed as a linear combination of its past 2M samples, we have [8]: 2M n=0 λ n U(n 2M) = 0, λ n = λ 2M n, λ 0 = 1, n = 2M + 1,..., N (10) {λ n } are called the LP coefficients. The wave numbers {k m } are related to the following polynomial constructed from {λ n }: 2M n=0 λ n z 2M n = 0 (11) whose roots are z = exp(±jk m ), m = 1, 2,..., M [8]. With the use of λ n = λ 2M n, (10) is expressed in matrix form as u = Uλ (12)

5 Progress In Electromagnetics Research, PIER 102, u = µ 1 µ 2M+1 (13) U = [µ 2 + µ 2M µ 3 + µ 2M 1... µ M+2 + µ M µ M+1 ] (14) λ = [λ 1 λ 2... λ M ] T (15) µ n = [U(n) U(n + 1)... U(N 2M 1 + n)] T (16) As {U(n)} are not available, we have to perform estimation from {R(n)}. Let R and r be the noisy versions of U and u constructed from {R(n)}. According to (12), we now have: r Rλ (17) Employing the weighted least squares (WLS) technique, the estimate of λ, denoted by ˆλ, is [10] ( ) H ( ) ˆλ = arg min R λ r W R λ r = ( R H WR ) 1 R H Wr (18) λ W is the weighting matrix. To achieve the best accuracy, we employ the Markov estimate [10] of W and its inverse is derived as: { W 1 = E (Rλ r) (Rλ r) H} = E{Λrr H Λ H } = ΛE { (u + q)(u + q) H} Λ H = ΛE { qq H} Λ H = σ 2 ΛΛ H (19) Λ = Toeplitz( [1 01 (N 2M 1) ] T, [ 1 λ1... λ M λ M 1... λ (N 2M 1) ] ) (20) r = [R(1) R(2)... R(N)] T (21) u = [U(1) U(2)... U(N)] (22) q = [Q(1) Q(2)... Q(N)] T (23) Note that Λu = 0 (N 2M 1) 1 and it is unnecessary to know σ 2 as it will be canceled out in (18), that is, we only need (19) up to a multiplying scalar. As W is parameterized by the unknown parameter λ, we estimate λ in an iterative manner and the estimation procedure is summarized as follows. (i) Set W = I N 2M (ii) Calculate ˆλ using (18) (iii) Compute an updated version of W using (19) with λ = ˆλ

6 36 Chan et al. (iv) Repeat Steps (ii) (iii) until a stopping criterion is reached (v) Solve the roots of (11) with λ = ˆλ to obtain the estimate of k m, namely, ˆk m, m = 1, 2,..., M Upon convergence of ˆλ, the bias of (18) can be investigated by premultiplying both sides by R H WR to construct: f(ˆλ) = R H W(Rˆλ r) = 0 (N 2M) 1 (24) W is a function of ˆλ. For sufficiently large SNR and/or data size, ˆλ will be located at a reasonable proximity of λ. Denoting λ = ˆλ λ and using Taylor s series to expand f(ˆλ) around λ yields: 0 (N 2M) 1 f(λ) + ( R H W (I M (Rλ r)) + R H WR ) λ (U + Q) H W ((U + Q)λ (u + q)) ( (U+Q) H W (I M ((U+Q)λ u q))+(u+q) H W(U+Q) ) λ (25) Q and q are the noise components of R and r, respectively. With the use of Uλ u = 0 (N 2M) 1 and ignoring the second-order terms whose values are much smaller than those of the first-order terms, we obtain: U H W (Qλ q) ( U H W (I M (Uλ u)) + U H WU ) λ λ = {( U H W (I M (Uλ u))+u H WU )} 1 U H W (Qλ q) (26) W = [W λ 1 W λ 2... W λ m ] and W λ i = W λ i. As all elements in Q and q are of zero-mean, we have E{ λ} 0 M 1. That is, ˆλ is an asymptotically unbiased estimate of λ. Employing {ˆk m } and (5) (6), the LLS estimates of {A m } and {B m }, denoted by {Âm} and { ˆB m }, are [10]: ] [Â1 ˆB1... Â M ˆBM = r T G ( G T G ) 1 (27) exp(jˆk 1 ) exp(j2ˆk 1 )... exp(jn ˆk T 1 ) exp( jˆk 1 ) exp( 2jˆk 1 )... exp( jn ˆk 1 ) G = (28) exp(jˆk M ) exp(j2ˆk M )... exp(jn ˆk M ) exp( jˆk M ) exp( j2ˆk M )... exp( jn ˆk M ) which are optimum for independent and identically distributed (IID) {Q(n)}. Note that when there is numerical instability in computing

7 Progress In Electromagnetics Research, PIER 102, (18) or (27), we can use singular value decomposition (SVD) to calculate the matrix inverse or add a small number to the diagonal elements in case of rank deficiency. 3. ESTIMATION FOR MULTI-DIMENSIONAL SCENARIOS In this Section, the multi-dimensional wave equation is addressed. Based on uniformly sampling a generalized version of (3), the signal model is: R(n 1, n 2,..., n P ) = U(n 1, n 2,..., n P ) + Q(n 1, n 2,..., n P ), n p = 1, 2,..., N p (29) U(n 1, n 2,..., n P ) = A exp j k p n p + B exp j k p n p (30) Here, the single-mode is considered for simplicity. The noise-free signal U(n 1, n 2,..., n P ) corresponds to a P -D lossy wave with P 2, k p represents the pth dimension wave number while A and B are unknown complex constants, and we drop the subscript m for notation simplicity. The size in the pth dimension is denoted by N p, p = 1, 2..., P. The {Q(n 1, n 2,..., n P )} are P -D additive white Gaussian noises with identical variance of σ 2 and the powers of all P dimensions are the same. The task is to find k 1, k 2,..., k P, A and B, given the N 1 N 2... N P measurements in {R(n 1, n 2,..., n P )}. Following (7) (9), we rewrite U(n 1, n 2,..., n P ) as U(n 1, n 2,..., n P ) = α cos k p n p + φ (31) and Let ũu n1,...,n m 1,n m+1,...,n P = α = 2 AB (32) ( ) j(b A) φ = tan 1 B + A [ U(n 1,..., n m 1, 1, n m+1,..., n P )... (33) U(n 1,..., n m 1, N m, n m+1,..., n P )] T (34)

8 38 Chan et al. Furthermore, we construct U m C N m N m by stacking ũu n1,...,n m 1,n m+1,...,n P, n p = 1, 2,..., N p, p = 1, 2,..., m 1, m+1,..., P N m = P,p m n p. From (30), any column of U m can be expressed as ũu n1,...,n m 1,n m+1,...,n P = Ψ m η m (35) [ ] T exp(jkm ) exp(j2k Ψ m = m )... exp(jn m k m ) (36) exp( jk m ) exp( j2k m )... exp( jn m k m ) [ ( η m = A exp j ) ( P,p m k pn p B exp j )] T P,p m k pn p (37) As a result, U m is of rank 2. On the other hand, factorizing U m with SVD yields U m = U m,s S m,s V H m,s + U m,n S m,n V H m,n (38) S m,s = diag(s m,1, s m,2 ) and S m,n = diag(s m,3, s m,4,..., s m,nm ) with singular values s m,1 > s m,2 > s m,3 =... = s m,nm = 0 while U m,s = [u m,1 u m,2 ] C Nm 2 and V m,s = [v m,1 v m,2 ] C N m 2 are orthonormal matrices whose columns are the corresponding left and right singular vectors of the signal subspace, respectively. In addition, U m,n = [u m,3 u m,4... u m,nm ] C N m (N m 2) and V m,s = [v m,3 v m,4... v m,n m ] C N m (N m 2) are orthonormal matrices whose columns are the corresponding left and right singular vectors of the noise subspace, respectively. As U m,s and Ψ m span the same subspace, we have [9] U m,s = Ψ m Ω m (39) Ω m C 2 2 is unknown. It can be deduced from (39) that [u m,i ] l is U(l) in (6) with M = 1, A = [Ω m ] 1,i and B = [Ω m ] 2,i, i = 1, 2. Nevertheless, in the presence of noise, U(n 1, n 2,..., n P ) is replaced by R(n 1, n 2,..., n P ) and the equal sign in (39) is replaced by approximately equal sign, leading to U m λ m um (40) U m = [U T m,1 U T m,2] T (41) U m,i = [ [u m,i ] 2 [u m,i ] 3... [u m,i ] Nm 1] T (42) um = [ u T m,1 u T m,2] T (43) um,i = [ [u m,i ] 1 +[u m,i ] 3 [u m,i ] 2 +[u m,i ] 4... [u m,i ] Nm 2 +[u m,i ] Nm ] T (44) λ m = 2 cos(k m ) (45)

9 Progress In Electromagnetics Research, PIER 102, As a result, the WLS estimate of λ m, denoted by ˆλ m, is ( ) H ( ) ˆλ m = arg min U m λm um Wm U m λm um λ m = (U H mw m U m ) 1 U H mw m um (46) In the Appendix A, we have derived the optimal weighting matrix W m as [ ( ) 1 W m = diag s 2 m,1, s 2 m,2 Λ m Λm] H (47) Λ m = Toeplitz ( [1 01 (Nm 3)] T, [ 1 λm (Nm 3)] ) (48) which is parameterized by the unknown parameter λ m. As (47) is block diagonal, (46) can be simplified to 2 i=1 ˆλ m = s2 m,i U H m,i W m um,i 2 i=1 s2 m,i U H m,i W (49) m U m,i W m = ( Λ m Λ H m) 1 (50) Similar to Section 2, we estimate λ m, m = 1, 2,..., P, in a separable and iterative manner as follows: (i) Set W m = I Nm 2 (ii) Calculate ˆλ m using (49) (iii) Compute an updated version of W m using (50) with λ m = ˆλ m (iv) Repeat Steps (ii) (iii) until a stopping criterion is reached (v) Use the finalized ˆλ m to compute the estimate of k m as ˆk m = cos 1 ( ˆλ m /2) After all {ˆk m } are available, the optimal estimates of A and B, denoted by  and ˆB, are then computed using LLS as [ ˆB] = r T G ( G T G ) 1 (51) G = exp(jnt ˆk) 1 exp(jn T ˆk) 2... exp(jn T Npˆk) P exp( jn T ˆk) 1 exp( jn T ˆk) 2... exp( jn T Npˆk) P T (52) Here, ˆk = [ˆk 1 ˆk2... ˆk P ] T, r C P N p denotes the vector containing all elements in {R(n 1, n 2,..., n P )} and n l Z P contains the indices with [n l ] p equals n p in [r] l.

10 40 Chan et al. For the case of a multi-mode multi-dimensional wave, we can combine the previous developments for (6) and (30) to achieve parameter estimation. When the number of modes is M, the rank of U m in (38) becomes 2M and we need to utilize the first 2M left singular vectors and singular values for estimating the wave numbers each column of U m,s now corresponds to a M-mode 1-D wave. It is worthy to point out that the major difficulty lies in pairing the wave numbers in the P -dimension [11]. Using P = 3 as an illustration, we have to determine the triplets of (k x,m, k y,m, k z,m ) after independently obtaining k x,m, k y,m and k z,m, m = 1, 2,..., M. 4. DERIVATION OF CRAMÉR-RAO LOWER BOUND In this Section, the s for parameters in (5) (6) and (29) (30) are derived. We first address the 1-D multiple waves. As Q(n), n = 1, 2,..., N are IID, the, denoted by H, is computed from the inverse of the corresponding Fisher information matrix (FIM): H 1 = 2 σ 2 R ( FF H) (53) the columns of F contain U(n) θ and θ = [ θk T θa T T B] θt with θ k = [R(k 1 ) I(k 1 )... R(k M ) I(k M )] T, θ A = [R(A 1 ) I(A 1 )... R(A M ) I(A M )] T and θ k = [R(B 1 ) I(B 1 )... R(B M ) I(B M )] T. The values of U(n) θ are calculated as: U(n) R(k m ) = jna m exp(jk m n) jnb m exp( jk m n) (54) U(n) I(k m ) = na i exp(jk m n) + nb i exp( jk m n) (55) U(n) R(A m ) = exp(jk mn) (56) U(n) I(A m ) = j exp(jk mn) (57) U(n) R(B m ) = exp( jk mn) (58) U(n) I(B m ) = j exp( jk mn) (59) The of k m, A m, B m are [H] 2m 1,2m 1 + [H] 2m,2m, [H] 2M+2m 1,2M+2m 1 + [H] 2M+2m,2M+2m, [H] 3M+2m 1,3M+2m 1 + [H] 3M+2m,3M+2m, m = 1, 2,..., M, respectively.

11 Progress In Electromagnetics Research, PIER 102, For the signal model of (29) (30), the expression of FIM is equal to (53) except that the elements in F are now modified as U(n 1,n 2,...,n P ) θ with θ = [R(k 1 ) I(k 1 )... R(k P ) I(k P ) R(A) I(A) R(B) I(B)] T, n p = 1, 2,..., N p, p = 1, 2,..., P. The values for U(n 1,n 2,...,n P ) θ are computed as: U(n 1, n 2,..., n P ) = jn p A exp j k p n p jn p B exp j k p n p R(k p ) U(n 1, n 2,..., n P ) I(k p ) U(n 1, n 2,..., n P ) R(A m ) U(n 1, n 2,..., n P ) I(A m ) U(n 1, n 2,..., n P ) R(B m ) U(n 1, n 2,..., n P ) I(B m ) = n p A exp j = exp j = j exp j k p n p + n p B exp j (60) k p n p (61) k p n p (62) = exp j k p n p (63) = j exp j k p n p (64) k p n p (65) The of k p is [H] p,p + [H] p+p,p+p, p = 1, 2,..., P, and those of A and B are [H] 2P +1,2P +1 + [H] 2P +2,2P +2 and [H] 2P +3,2P +3 + [H] 2P +4,2P +4, respectively. 5. SIMULATION RESULTS Computer simulations have been conducted to evaluate the lossy wave parameter estimation performance of the proposed approach for 1-D dual waves and 3-D single wave in zero-mean white complex Gaussian noise, that is M = 2 and P = 3. The stopping criterion of the proposed methodology is a fixed number of iterations. In both scenarios, we iterate the algorithms for 3 times as no obvious improvement is observed for more iterations. To illustrate the

12 42 Chan et al. comparative performance, we include the results of the estimators which are realized by the Newton s method with the true values as initial estimates. We scale Q(n) and Q(n 1, n 2, n 3 ) to produce different SNR conditions SNR = N n=1 U(n) 2 /(Nσ 2 ) and SNR = N1 N2 N3 n 1 =1 n 2 =1 n 3 =1 U(n 1, n 2, n 3 ) 2 /(N 1 N 2 N 3 σ 2 ) in the former and latter cases, respectively. All results provided are averages of 1000 independent runs. In the first test, the mean square error (MSE) performance of wave numbers versus SNR for 1-D multiple lossy waves is examined. The parameters of U(n) are k 1 = 1 + j0.02, k 2 = 2 j0.01, A 1 = 1 + j0.6, A 2 = j0.65, B 1 = j0.8 and B 2 = j0.6 while a data length of N = 10 is assigned. Figures 1 to 6 plot E{ k 1 ˆk 1 2 }, E{ k 2 ˆk 2 2 }, E{ A 1 Â1 2 }, E{ B 1 ˆB 1 2 }, Mean Square Error of k (db) Mean Square Error of k 2 (db) Figure 1. Mean square error for k 1 versus SNR with 1-D multiple waves. -50 Figure 2. Mean square error for k 2 versus SNR with 1-D multiple waves. Mean Square Error of A (db) Mean Square Error of B 1 (db) Figure 3. Mean square error for A 1 versus SNR with 1-D multiple waves. Figure 4. Mean square error for B 1 versus SNR with 1-D multiple waves.

13 Progress In Electromagnetics Research, PIER 102, Mean Square Error of A (db) Figure 5. Mean square error for A 2 versus SNR with 1-D multiple waves. Attenuation coefficient of k True value 0 SNR Figure 7. Mean attenuation coefficient estimate for k 1 versus SNR with 1-D multiple waves. Mean Square Error of B (db) Figure 6. Mean square error for B 2 versus SNR with 1-D multiple waves. Attenuation coefficient of k True value SNR Figure 8. Mean attenuation coefficient estimate for k 2 versus SNR with 1-D multiple waves. E{ A 2 Â2 2 } and E{ B 2 ˆB 2 2 }, respectively. It is seen that the accuracy of both approaches attains the for SNR 8 db. The proposed and methods have almost the same performance except in the estimation of k 2 and A 1 at SNR 6 db. It is worthy to note that in practice, the method will give poorer performance when its initial estimates are not sufficiently close to the global solution. Regarding computational complexity, the average computation times of the proposed and algorithms for a single trial are measured as s and s, respectively, indicating the former is more computationally attractive. The mean attenuation coefficients obtained from ˆk 1 and ˆk 2 are plotted in Figures 7 and 8. We see that both algorithms give nearly unbiased estimates when the SNR is sufficiently high and the proposed method exhibits a larger bias

14 44 Chan et al. in Figure 7 when SNR 6 db. Comparing Figures 1, 2, 7 and 8, we deduce that the variance in the IQML estimator is smaller than that in the method. This test is repeated with white uniform noise to investigate the algorithm robustness and the MSE results for k 1 and k 2 are plotted in Figures 9 and 10. It is observed that the performance of the both algorithms in uniform noise is comparable with that of the Gaussian noise case. We have also repeated the first experiment with N = 30 to study the effect of larger data length and the MSEs for k 1 and k 2 are shown in Figures 11 and 12. We see that the performance of both schemes improves and their threshold SNR is reduced to 4 db. Combining the findings, the proposed estimator is able to achieve optimum performance for sufficiently high SNRs and/or large data lengths. Mean Square Error of k 1 (db) Mean Square Error of k 2 (db) Figure 9. Mean square error for k 1 versus SNR in uniform noise. -50 Figure 10. Mean square error for k 2 versus SNR in uniform noise. Mean Square Error of k (db) Figure 11. Mean square error for k 1 versus SNR at N = 30. Mean Square Error of k (db) Figure 12. Mean square error for k 2 versus SNR at N = 30.

15 Progress In Electromagnetics Research, PIER 102, Mean Square Error of k (db) Proppsed -70 Figure 13. Mean square error for k 1 versus SNR with 3-D single wave. Mean Square Error of k (db) Mean Square Error of k (db) Proppsed -70 Figure 14. Mean square error for k 2 versus SNR with 3-D single wave. Proppsed -70 Figure 15. Mean square error for k 3 versus SNR with 3-D single wave. In the second test, the wave number estimation performance of the proposed approach for 3-D single-mode is evaluated. Now the parameter settings are k 1 = 1 + j0.02, k 2 = 2 j0.01, k 3 = j0.01, A = 1 + j0.6, B = j0.8 and N 1 = N 2 = N 3 = 10. Figures 13 to 15 show the MSEs of the wave numbers in the three dimensions. We observe that the wave number estimation performance achieves for SNR 4 db. Note that the results for A and B are not included as similar findings are obtained. 6. CONCLUSION An iterative parameter estimation approach for two forms of lossy wave equations, which correspond to multi-mode and multi-dimensional waves, has been devised. In the algorithm development, linear

16 46 Chan et al. prediction and weighted least squares techniques are utilized in both scenarios. Furthermore, we make use of the subspace methodology to achieve parameter estimation from the principal singular vectors of the multi-dimensional data. It is demonstrated that the estimation performance of the proposed algorithms attains Cramér-Rao lower bound for sufficiently high signal-to-noise ratio (SNR) and/or large data size conditions. As an illustration, for a one-dimensional dualmode wave with an observation length of 10, the threshold SNR is 10 db. ACKNOWLEDGMENT The work described in this paper was supported by a grant from CityU (Project No ). APPENDIX A. In this Appendix, the derivation of W m in (47) is provided. Let ũu m,1,2 and u m,1,2 be the noise-free version of u m,1,2 = [ u T m,1 u T T m,2] and the corresponding perturbation, respectively. Using (I 2 Λ m )ũu m,1,2 = 0 2Nm 4, we first show that the inverse of W m is Wm 1 =E {(U m λ m um) (U m λ m um) H} =E { (I 2 Λ m ) (ũu m,1,2 + u m,1,2 )(ũu m,1,2 + u m,1,2 ) H ( I 2 Λ H )} m =E { (I 2 Λ m ) u m,1,2 u H ( m,1,2 I2 Λ H )} m (A1) As E { u m,1,2 u H m,1,2} in (A1) involve perturbations of singular vectors, we need to relate it with the noise Q(n 1, n 2,..., n P ). Let Q m be the noise counterpart of U m. The first-order perturbation of u m,i is [12] u m,i = U m,s D m,i U H m,sq m v m,i s m,i + U m,s D m,i S m,s V H m,sq H mu m,i 3 + U m,n U H m,nq m v m,i s 1 m,i = t m,i,j (A2) j=1 ( t m,i,1 = s m,i v T m,i U m,s D m,i U H m,s) vec(qm ) (A3) t m,i,2 = ( u T m,i U m,s D m,i S m,s V H ) ( ) m,s vec Q H m (A4)

17 Progress In Electromagnetics Research, PIER 102, t m,i,3 = s 1 ( m,i v T m,i U m,n U H m,n) vec (Qm ) = s 1 m,i ( v T m,i I Nm ) vec(qm ) s 1 m,i ( v T m,i U m,s U H m,s) vec(qm ) (A5) Here, D m,i is a diagonal matrix with the (i, i) entry being zero and the other diagonal entry equals 1/(s 2 m,3 i s2 m,i ). Note that t m,i,j, j = 1, 2, 3, are obtained by the formula vec(abc) = (C T A)vec(B). Using (A B)(C D) = AC BD and Λ m U m,s = 0 (Nm 2) 2, (A1) becomes [ s 1 m,1 (I 2 Λ m ) (vt m,1 I ] N m ) s 1 m,2 (vt m,2 I E ( vec(q N m ) m )vec(q m ) H) [ s 1 m,1 (v m,1 I N m ) s 1 m,2 (v m,2 I N m ) ] ( I 2 Λ H ) m [ s = σ 2 1 m,1 (I 2 Λ m ) (vt m,1 I ] N m ) s 1 m,2 (vt m,2 I N m ) [ s 1 m,1 (v m,1 I N m ) s 1 m,2 (v m,2 I N m ) ] ( I 2 Λ H ) m ( ( ) ) (I2 = σ 2 (I 2 Λ m ) diag s 2 m,1, s 2 m,2 I Nm Λ H ) m ( ) = σ 2 diag s 2 m,1, s 2 m,2 Λ m Λ H m (A6) which corresponds to (47). REFERENCES 1. Jackson, J. D., Classical Electrodynamics, John Wiley & Sons, Oliphant, T. E., On parameter estimates for the lossy wave equation, IEEE Transactions on Signal Processing, Vol. 56, No. 1, 49 60, Jan So, H. C., M. T. Amin, and F. K. W. Chan, Accurate and simple estimator for lossy wave equation, IEEE Signal Processing Letters, Vol. 16, No. 9, , Sep Goodwin, G. C. and R. L. Payne, Dynamic System Identification: Experiment Design and Data Analysis, Academic Press, New York, Kumaresan, R., L. L. Scharf, and A. K. Shaw, An algorithm for pole-zero modeling and spectral analysis, IEEE Transactions on Acoustic, Speech and Signal Processing, Vol. 34, , Jun

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