Correlation-based Algorithm for Multi-Dimensional Single-Tone Frequency Estimation

Transcription

1 Correlation-based Algorithm for Multi-Dimensional Single-Tone Frequency Estimation Weize Sun, H.C. So and Lanxin Lin Department of Electronic Engineering, City University of Hong Kong Abstract In this paper, parameter estimation for a R-dimensional (R-D) single cisoid with R 2 in additive white Gaussian noise is addressed. By exploiting the correlation of the data samples, we construct R single-tone sequences which contain the R-D frequency parameters. Based on linear prediction and weighted linear squares techniques, two proposals are developed for fast and accurate frequency estimation from each constructed sequence. The two devised estimators are proved to be asymptotically unbiased while their variances achieve Cramér-Rao lower bound when the signal-to-noise ratio and/or data length tend to infinity. Computer simulations are also included to compare the proposed approach with conventional R-D harmonic retrieval schemes in terms of mean square error performance and computational complexity. Keywords: frequency estimation, multi-dimensional spectral analysis, fast algorithm, correlation 1. Introduction Estimating the parameters of sinusoidal components from a finite number of noisy discrete-time measurements has been an important research topic be- Preprint submitted to Signal Processing September 25, 2012

2 cause it corresponds to many science and engineering problems [1] [3]. Apart from the standard one-dimensional signal model [4] [6], multi-dimensional spectral estimation [7] in fact has many applications such as array processing [8] [9], nuclear magnetic resonance (NMR) spectroscopy [10], wireless communication channel estimation [11] [12] as well as detection and localization of multiple targets using multiple-input multiple-output (MIMO) radar [13]. The crucial step in R-dimensional (R-D) sinusoidal parameter estimation, withr 2, istofindthefrequency parameterswhicharenonlinearintheobserved data. For 2-D frequency estimation, maximum-likelihood (ML) [14] [15] and subspace [8] [13], [16] [17] approaches are commonly used choices. For higher dimensional signals, the latter seems to be the only practical solution although its computational requirement is very high. State-of-the-art subspace methods include MUSIC [9], decoupled root-music [17], unitary ESPRIT (UE) [8], rank reduction estimator (RARE) [12], multi-dimensional folding (MDF) [11] and improved MDF (IMDF) [16], and they usually need eigenvalue decomposition (EVD) or singular value decomposition (SVD) in the algorithm implementation. By effectively utilizing the correlation of the observed data, we contribute to devising an accurate frequency estimator for a R-D single-tone with less computational load in this work. The rest of the paper is organized as follows. To facilitate the understanding of our underlying idea, we first develop the correlation-based approach for a 2-D cisoid in the presence of white Gaussian noise in Section 2. By exploiting correlation on the 2-D raw data, a novel single-tone sequence whose frequency corresponds to one of the two dimensions is constructed. The generalized weighted linear predictor (GWLP) [18], which can provide optimum 2

3 performance for 1-D single-tone in white noise, is then applied for estimation of the two frequency parameters in a separable manner. Note that linear prediction (LP) and weighted least squares (WLS) are two key ingredients in the GWLP. Two estimators are devised, and their mean and variance expressions are produced. It is worthy to point out that the derivation of the weighting matrix and performance analysis are different from those of [18]. Section 3 generalizes the proposed solutions to higher dimensional signals with the use of tensor algebra [19]. Simulation results are included in Section 4 to corroborate the theoretical development and to compare the correlation-based approach with the approximate iterative quadratic ML (AIQML) [15], IMDF [16] and UE [8] algorithms as well as Cramér-Rao lower bound (CRLB) [20]. It is demonstrated that the proposed methods are more computationally efficient and their performance is close to CRLB particularly when the signalto-noise ratio (SNR) is sufficiently high. Finally, conclusions are drawn in Section Two-dimensional Frequency Estimation We first define the notation as follows. Scalars, vectors, matrices and tensors are denoted by italic, bold lower-case, bold upper-case and bold calligraphic symbols, respectively. The magnitude, angle, real and imaginary parts of a complex scalar a are a, (a), R{a} and I{a}, while the variable, noise-free value and estimate of a vector a are represented by ã, ā and â, respectively. The mean and variance of â are denoted by E{â} and var(â). The I i is the i i identity matrix and 0 i j is the i j zero matrix. The operators for transpose, complex conjugate, conjugate transpose, inverse and 3

4 Khatri-Rao product are T,, H, 1, and, respectively. The mth element of a M 1 and (m,n) entry of A M N are denoted by [a] m and [A] m,n, respectively, whileweusea m1,m 2,...,m R torepresent the(m 1,m 2,...,m R )entry of a R-D tensor A M 1 M 2 M R where R 3. where In matrix form, the 2-D signal model with size M 1 M 2 is expressed as: Y = X+Q (1) [X] m1,m 2 = γe j(m 1ω 1 +m 2 ω 2 ), m r = 1,2,,M r, r = 1,2 (2) is the noise-free tone sample, and [Q] m1,m 2 is a zero-mean complex white Gaussian process with unknown variance σq 2. Moreover, γ is the complex amplitude while ω 1 ( π,π) and ω 2 ( π,π) are the frequencies, and they are all unknown constants. Here, the task is to determine ω 1 and ω 2 from the observed matrix Y with M = M 1 M 2 samples. The main idea of the algorithm development is to exploit correlation to convert Y into a vector which contains the information of ω 1 or ω 2. That is, ω 1 and ω 2 are estimated in a separable manner. We notice that X is of rank 1 and can be factorized as X = γg 1 g2 T (3) [ ] T where g r = e jωr e j2ωr e jmrωr, r = 1,2 (4) ] Defining X = [x 1 x 2 x M2 and writing Y and Q in the same way, we easily obtain the LP property: x 1 = e jω 2 γg 1 (5) x m = e jω 2 x m 1 (6) 4

5 As a result, pre-multiplying X by x H 1 gives γ 2 M 1 e jω 2 g2 T, which is a singletone vector with frequency ω 2. Since X is not available, we use its noisy version which is given by the zero-lag correlation between all columns of Y and its first column vector to achieve frequency estimation: z T (2) = yh 1 Y (7) with elements: [z (2) ] m2 = y H 1 y m2 = y H 1 (x m2 +q m2 ) (8) Note that the subscript of (2) in z refers to ω 2. According to the LP property, we obtain y H 1 x m = e jω 2 y H 1 x m 1. Hence (8) satisfies: [z (2) ] m2 e jω 2 [z (2) ] m2 1 = y H 1 ( qm2 e jω 2 ) q m2 1, m2 = 2,3,,M 2 (9) Denotingz (2)1 = [ T ] T [z (2) ] 1 [z (2) ] 2 [z (2) ] M2 1] andz(2)2 = [[z (2) ] 2 [z (2) ] 3 [z (2) ] M2 and assigning ρ 2 = e jω 2, we construct the LP error vector based on (9): e (2) = z (2)2 ρ 2 z (2)1 (10) The ω 2 is then estimated by minimizing the following WLS cost function: J( ρ 2 ) = ( ) H ( ) z (2)2 ρ 2 z (2)1 W M2 1 z(2)2 ρ 2 z (2)1 (11) where W M2 1 C (M 2 1) (M 2 1) is a weighting matrix. Differentiating J( ρ 2 ) with respect to ρ 2 and then setting the resultant expression to zero, we get the estimate of ρ 2 : ˆρ 2 = zh (2)1 W M 2 1z (2)2 z H (2)1 W M 2 1z (2)1 (12) 5

6 In Appendix A, we have derived the optimal weighting matrix, denoted by W M2 1(ω 2 ), which is a function of the unknown frequency ω 2 : W M2 1(ω 2 ) = [ E{e (2) e H (2)} ] 1 = P 1 (ω 2 ) (13) where 2+(M/M 2 1)/(SNR+1) e jω e jω 2 2 e jω P(ω 2 ) = e jω 2 2 e jω e jω 2 2 (14) with SNR = γ 2 /σq 2. In the following, two iterative algorithms are proposed to approximate the conceptual solution of (12). As SNR is unknown, our first proposal is to set [P(ω 2 )] 1,1 = 2 which is valid particularly when SNR is sufficiently large, namely, SNR >> M/M 2. Underthisapproximation, P(ω 2 )hasaclosed-form inverse and the phase of z H (2)1 W M 2 1(ω 2 )z (2)1 in (12) is zero [18]. Employing the substitution of ˆρ 2 = e jˆω 2, (12) is simplified to ˆω 2 = (z H (2)1 W M 2 1(ω 2 )z (2)2 ) (15) where the elements of W M2 1(ω 2 ) are: [W M2 1(ω 2 )] m,n = M 2min(m,n) mn M 2 e j(m n)ω 2 (16) m = 1,2,,M 2 1, n = 1,2,,M 2 1 6

7 In this study, we apply the GWLP [18] algorithm which updates ˆω 2 and W M2 1(ω 2 ) in an iterative manner. We refer this estimator to as correlation- 1 (C-1) method. The steps of finding ˆω 2 are summarized in Table 1. The estimation of ˆω 1 is performed using the same procedure on Y T. (i) Compute the z (2) using (7) (ii) Obtain an initial value of ˆω 2 using (15) with [W M2 1(ω 2 )] m,n = 0 for m n in (16), which is in fact the weighted linear predictor estimate [21] (iii) Construct W M2 1(ω 2 ) according to (16) with ω 2 = ˆω 2 (iv) Compute an updated ˆω 2 using (15) (v) Repeat Steps (iii) (iv) until a stopping criterion is reached. In this study, we stop for τ iterations. Table 1: Estimation algorithm for ω 2 In Appendix B, we have shown that when the frequency estimation error is sufficiently small, the mean and variance of ˆω r, r = 1,2, are: E{ˆω r } ω r (17) var(ˆω r ) 6 ( γ 2 +σ 2 Q) σ 2 Q ( M/Mr γ 2 +σ 2 Q) Mr (M 2 r 1) γ 2 (18) Equation (17) means that the algorithm is an asymptotically unbiased estimator. The CRLB for ω r, denoted by CRLB(ω r ), is [20]: CRLB(ω r ) = 6σ 2 Q M(M 2 r 1) γ 2 (19) 7

8 To study the limit of the C-1 estimator, we define the efficiency, which is the ratio of CRLB to the variance, denoted by: eff 1 (ω r ) = CRLB(ω ( r) γ 2 +M r /Mσ 2 ( ) Q) = (SNR+M r/m) var(ˆω r ) γ 2 +σq 2 (SNR+1) (20) whosevalueisbetween 0and1. Itiseasy toseethatthec-1methodachieves optimum performance when SNR, with eff 1 (ω r ) = 1. On the other hand, the above development may not be appropriate when M/M r, r = 1,2, is large particularly when SNR >> M/M r does not hold. To tackle this scenario, our second approximation on (12) is to discard [z (2) ] 1 of (7) in the computation. Let z (r)s be z (r) without the first element and follow (9) (16), we obtain ˆω r = (z H (r)s1 W M r 2(ω r )z (r)s2 ) (21) [ T where z (r)s1 = [z (r)s ] 1 [z (r)s ] 2 [z (r)s ] Mr 2] (22) [ T z (r)s2 = [z (r)s ] 2 [z (r)s ] 3 [z (r)s ] Mr 1] (23) We refer this estimator to as correlation-2 (C-2) method. As a result, the estimate of (21) is obtained in a similar manner as in Table 1. Following Appendix A and replacing M r by M r 1, we see that the frequency estimate is also asymptotically unbiased and its variance is: var(ˆω r ) 6 ( γ 2 +σ 2 Q) σ 2 Q ( M/Mr γ 2 +σ 2 Q) (Mr 1)[(M r 1) 2 1] γ 2 (24) Analogous to (20), the efficiency of this estimator is eff 2 (ω r ) = (M r 1)[(M r 1) 2 1](SNR+M r /M) M r (M 2 r 1)(SNR+1) (0,1) (25) and we see that eff 2 (ω r ) = 1 when both SNR and M r tend to infinity. 8

9 3. Extension to Higher Dimension In this section, we generalize our development to higher dimensional signals with R 3, where tensor representation is employed. The observed R-D sinusoidal signal is modeled as: Y = X +Q (26) where x m1,m 2,...,m R = γ R e jωrmr, m r = 1,2,...,M r, r = 1,2,...,R, (27) r=1 The Y M 1 M 2 M R is the tensorial structured data set with length M r along the rth dimension. The tensor X is the signal component where γ and ω r ( π,π) represent the unknown complex amplitude and frequency in the rth dimension. On the other hand, the entries in Q are zero-mean complex white Gaussian noises with unknown variances σq 2. Now the task is to find {ω r } from the M = R r=1 M r samples of Y. [ ] T, By writing g r = e jωr e jωr2 e jωrmr to align with the presentation in Section 2 we define the rth unfolding of X as the transpose version of [19]: [X] (r) = γ(g r+1 g R g 1 g 2 g r 1 )gr T ] = [x (r)1 x (r)2 x (r)mr (28) and multiplying [X] (r) by x H (r)1 : c (r) = x H (r)1 [X] (r) = M M r γ 2 e jωr g T r (29) 9

10 ] Analogousto(7) (9),wedefine[Y] (r) = [y (r)1 y (r)2 y (r)mr and[q] (r) = ] [q (r)1 q (r)2 q (r)mr and construct where z T (r) = y H (r)1[y] (r) (30) [z (r) ] mr = e jω 2 [z (r) ] mr 1 +y H (r)1( q(r)mr e jω 2 q (r)mr 1), mr = 2,3,,M r Generalizing (11), the WLS cost function is: J( ρ r ) = ( ) H ( ) z (r)2 ρ r z (r)1 W Mr 1 z(r)2 ρ r z (r)1 (31) (32) [ T where ρ r = e j ωr while z (r)1 = [z (r) ] 1 [z (r) ] 2 [z (r) ] Mr 1] and z(r)2 = [ ] T. [z (r) ] 2 [z (r) ] 3 [z (r) ] Mr Following (10) (15) and Appendix A with the assumption of SNR >> M/M r, the C-1 estimate of ω r, r = 1,2,,R, is: ˆω r = (z H (r)1w Mr 1(ω r )z (r)2 ) (33) Similarly, by discarding the first element in z (r), r = 1,2,,R, the C-2 frequency estimate has the same form as in (14). In Appendix B, we have proved that the higher dimensional C-1 and C-2 estimates are asymptotically unbiased and their variances equal (18) and (24), respectively. In summary, according to the development, the C-1 estimator is more preferable when SNR >> M/M r while the C-2 method is a better choice if M r >> 1, r = 1,2,,R. Finally, the complexity orders of the proposed methods are investigated as follows. In our study, the iterative procedure stops when the number of 10

11 iterations reaches the value of τ. According to Table 1, the major computations in the rth dimension has a complexity of O(τM r ), implying that the orders of complexity for the C-1 and C-2 algorithms are O( R r=1 τm r) and O( R r=1 τ(m r 1)), respectively. As a comparison, the complexity order of the AIQML [15] method is O( R r=1 τm r) while the IMDF [16] and UE [8] schemes have complexities of O(4k t (L E L E /L 1 )K E ) and O(2k t L E K E ), respectively, wherek t isaconstantdependsonthedesignofthesvdalgorithm, L E r +KE r 1 = M r, L E r > 1, KE r > 1, L E = R r=1 LE r and KE = R r=1 KE r. It can be roughly seen that when τ is small, the correlation-based methods are more computationally efficient. 4. Numerical Examples Computer simulations have been conducted to evaluate the R-D singetone frequency estimation performance of the correlation-based approach in terms of mean square frequency error (MSFE) and complexity performance. We also include comparisons with the AIQML [15], IMDF [16] and UE [8] algorithms as well as CRLB [20]. Three iterations, that is, τ = 3, are employed as the stopping criterion in the proposed and AIQML methods because no significant improvement is observed for more iterations. The tone magnitude is γ = 1 and its phase varies from π to π in each independent trial. We properly scale the zero-mean white Gaussian noise data to produce different SNR conditions, where SNR = γ 2 /σq 2 = 1/σ2, and all results provided are averages of 1000 independent. In the first test, we solely compare the performance of the C-1 and C-2 methods. Figures 1 and 2 show the MSFEs versus SNR for the 2-D and 3-D 11

12 ] [ ] ] [ ] cases with [ω 1 ω 2 = π and [ω 1 ω 2 ω 3 = π, respectively. The derived variances and CRLB are also included to verify the theoretical findings and investigate the algorithm optimality. Three choices of N, namely, N = 5, N = 10 and N = 20, are studied where M r = N for all dimensions. We see from both figures that the performance of the C-1 and C-2 estimators agrees with the analytic expressions of (18) and (24), respectively, when SNR 4dB. Although the C-1 method is generally superior to the C-2 algorithm, it is observed that the latter can give smaller MSFE at SNR = 10dB when N = 20 at R = 3. Furthermore, comparing to the situation when the data length is smaller, namely, N=5, the performance of the C-2 method is close to the CRLB for a larger N, say N=20, which also align with the development in Section 2. Inthesecond test, comparison ismadewiththeaiqml andimdfmethods in 2-D estimation with M 1 = M 2 = 10. The frequency parameters are ] [ ] [ω 1 ω 2 = π and the results are shown in Figure 3. We observe that the IMDF method has the best threshold performance at SNR = 6dB, while the AIQML and proposed methods have threshold SNRs at 2dB and 0dB, respectively. Moreover, the MSFEs of the AIQML and C-1 schemes attain the CRLB at SNR 10dB. On the other hand, the average computational times of the AIQML, IMDF, C-1 and C-2 algorithms for a single run are s, s, s and s, respectively, indicating the attractiveness of the proposed approach in terms of complexity. It is seen that the computational times also agree with the complexity order analysis in Section 3. In the third test, a comparative study in 3-D estimation is made with 12

13 the UE and IMDF algorithms. The results for M 1 = M 2 = M 3 = 10 and ] [ ] [ω 1 ω 2 ω 3 = π are plotted in Figure 4. At SNR 2dB, the UE and IMDF methods outperform the proposed estimators. However, when SNR > 5dB, the MSFE of the C-1 scheme achieves the CRLB. The average computational times for the UE, IMDF, C-1 and C-2 algorithms for a single run are s, s, s and s, which also agree with the complexity study in Section 3. Finally, we perform comparison for different R [3,5] with M r = 4, r = 1,2,,R, and the results are shown in Figure 5. The SNR is set to 20dB and all the frequencies varies from 0.99π to 0.99π in each independent run. It is seen that the C-1 algorithm performs the best and the theoretical calculations of (18) and (24) are again validated. Their average single run computational times are tabulated in Table 2, which agree with Section 3 and indicate the computational advantage of the proposed methodology particularly when the data dimension is large. R UE IMDF C-1 C Table 2: Average computational times versus R 5. Conclusion A fast and accurate multi-dimensional single frequency estimation approach based on correlation has been devised. Two estimators, C-1 and C-2, 13

14 which are ideal for high SNR and/or large sample conditions, are proposed. The main idea in the algorithm development is to construct single-tone sequences which contain the frequency parameters by exploiting correlation of the observed data. Frequency estimation for each dimension is performed separately by using LP and WLS techniques. It is proved that the variances of the C-1 and C-2 estimators attain CRLB in the presence of white Gaussian noise, when the SNR and/or data length tend to infinity. Furthermore, the proposed correlation approach is more computationally attractive than the AIQML, UE and IMDF schemes, and provides superior estimation accuracy at higher SNRs. As a future work, we will extend the correlation-based methodology to estimation of multiple tones. Appendix A We now derive the weighting matrix for the general signal model of (26). According to (31), the LP residue error vector for the rth dimension is e (r) = z (r)1 ρ r z (r)2 (A.1) where[e (r) ] nr = y(r)1 H ξ (r)n r,n r = 1,2,,M r 1,ξ (r)nr = ( ) q (r)(nr+1) ρ r q (r)nr andρ r = e jωr. Applying thegauss-markov theorem [22], the optimal weighting matrix which is a function of ρ r, denoted by W Mr 1(ρ r ), is: W Mr 1(ρ r ) = [ E { e (r) e(r)}] H 1 = B 1 (ρ r ) (A.2) where [B(ρ r )] m,n =E{[e (r) ] m [e (r) ] n} = E{y H (r)1 ξ (r)m ξ H (r)n y (r)1 } =[K 1 ] m,n +[K 2 ] m,n +[K 3 ] m,n +[K 4 ] m,n, m,n = 1,2,,M r 1 (A.3) 14

15 with [K 1 ] m,n =E{x H (r)1 ξ (r)mξ H (r)n x (r)1} (A.4) [K 2 ] m,n =E{q H (r)1 ξ (r)mξ H (r)n q (r)1} (A.5) [K 3 ] m,n =E{x H (r)1 ξ (r)mξ H (r)n q (r)1} (A.6) [K 4 ] m,n =E{q H (r)1 ξ (r)mξ H (r)n x (r)1} (A.7) On the other hand, since Q contains zero-mean independent and identically distributed Gaussian samples, we get: E{x H (r)1 q (r)p} = E{q H (r)p q (r)q} = 0 (A.8) and E{q H (r)pq (r)p } = M M r σ 2 Q (A.9) for any p q, p,q = 1,2,,M r, which result in K 3 = K 4 = 0 (Mr 1) (M r 1) and E{ξ (r)m ξ H (r)n} = σq 2(1+ ρ r 2 )I M, m = n Mr σq 2ρ r I, M Mr m = n+1 σq 2ρ ri M, Mr m = n 1 0, otherwise (A.10) Therefore (1+ ρ r 2 ) ρ r ρ K 1 = M r (1+ ρ r 2 ) ρ r 0 0 γ 2 σq M r 0 0 ρ r (1+ ρ r 2 ) ρ ρ r (1+ ρ r 2 ) (A.11) 15

16 TocalculateK 2, wenoticethatwhenp 1, q (r)1 andξ (r)p areindependent to each other, meaning that E{q H (r)1 ξ (r)p} = 0. Hence [K 2 ] m,n = [K 1 ] m,n σq 2/ γ 2 for any m 1 or n 1, and [K 2 ] 1,1 = E{q H (r)1 ξ (r)1ξ H (r)1 q (r)1} = M/M r (1 + M/M r ρ r 2 )σq 4. Substituting K 1, K 2 and ρ r = e jωr into (A.3) yields B(ω r ) = M M r ( γ 2 +σ 2 Q)σ 2 QP(ω r ) (A.12) where 2+(M/M r 1)σQ 2/( γ 2 +σq 2) ejωr e jωr 2 e jωr 0 0 P(ω r ) = e 2 e jωr e jωr 2 (A.13) As M/M r ( γ 2 + σq 2)σ2 Q is just a scalar, we can simply write W M r 1(ρ r ) = W Mr 1(ω r ) = P 1 (ω r ). Note that (7) (8) correspond to the special case of 2-D estimation. Appendix B The bias and variance of ˆω r, r = 1,2,,R, for the general model of (26) are now analyzed. Let the bias of ˆω r be ω r = ˆω r ω r. Following [18], for sufficiently high SNR conditions, we have ω r var(ˆω r ) κ r z H (r)1 W M r 1(ω r ) z (r)1 E{κ 2 r } (E{ z H (r)1 W M r 1(ω r ) z (r)1 }) 2 (B.1) (B.2) 16

17 where κ r I { e jωr ( z H (r)1w Mr 1(ω r ) z (r)2 + z H (r)1w Mr 1(ω r ) z (r)2 )} (B.3) with z (r)i = z (r)i + z (r)i, i = 1,2. Notice that z T (r) = (x (r)1 +q (r)1 ) H [x (r)1 x (r)2 x (r)mr ]and z T(r) = (x (r)1+q (r)1 ) H [q (r)1 q (r)2 q (r)mr ]. According to (A.8) (A.9), we obtain E{x H (r)(m) (x (r)1 +q (r)1 )(x (r)1 +q (r)1 ) H x (r)(n) } =x H (r)(m)x (r)1 x H (r)1x (r)(n) +x H (r)(m)e{q (r)1 )(q (r)1 ) H }x (r)(n) =e j(m n) M M r ( M M r γ 2 +σ 2 Q ) γ 2 (B.4) and [ ] T E{ z (r) } = M/M r σq (B.5) Therefore the expected value of the denominator of (B.1) is: E{ z H (r)1 W M r 1(ω r ) z (r)1 } M r 1 m=1 M r 1 n=1 E{x H (r)m (x (r)1 +q (r)1 )(M r min(m,n) mn)e j(m n)ω (x (r)1 +q (r)1 ) H x (r)n } ( = M ( M Mr 1 γ 2 +σ 2 Mr 2 Q M ) γ 2 (M r m m 2 )+ r m=1 ) M r 1 n 1 (M r m mn) n=1 m=1 M r 1 m=1 m 1 (M r n mn)+ = M(M2 r 1) ( M γ 2 +σ 12 M Q) γ 2 2 (B.6) r On the other hand, κ r can be written as: n=1 κ r =t r1 +t r2 (B.7) 17

18 where tr1 = e jωr z H (r)1 W M r 1(ω r ) z (r)2 e jωr z H (r)2 W M r 1(ω r ) z (r)1 2j = zh (r)2 W M r 1(ω r )( z (r)2 ρ r z (r)1 ) 2j (B.8) t r2 = e jωr z T (r)2 W M r 1 (ω r) z (r)1 ejωr z T (r)1 W M r 1 (ω r) z (r)2 2j = ( z (r)2 ρ r z (r)1 ) H W Mr 1(ω r ) z (r)2 2j (B.9) because z (r)2 = ρ r z (r)1, ρ r = e jωr and W Mr 1(ω r ) = W H M r 1 (ω r). As E{ z (r)2 } = 0 (Mr 1) 1, we have: E{t r1 } = 1 2j E{ρ r z H (r)2 W M r 1(ω r ) z (r)1 }+0 = 1 2j E{ zh (r)1 W M r 1(ω r ) z (r)1 } = 1 2j M r 1 m=1 M r 1 n=1 (x (r)1 +q (r)1 ) H q (r)n } = 1 2j = 1 2j = 1 2j M r 1 m=1 M r 1 m=1 M r 1 n=1 E{x H (r)m (x (r)1 +q (r)1 )(M r min(m,n) mn)e j(m n)ω E{x H (r)m(x (r)1 +q (r)1 )(M r m)e j(1 m)ω (x (r)1 +q (r)1 ) H q (r)1 }+0 (M r m) E{x H (r)1 (x (r)1 +q (r)1 )(x (r)1 +q (r)1 ) H q (r)1 } (M r n) E{x H (r)1 x (r)1q H (r)1 q (r)1}+0 =j M2 (M r 1) 4M r γ 2 σ 2 Q (B.10) and E{t r2 } = E{t r1 }. This means that E{ ω r } E{κ r } = 0 when SNR is sufficiently high, indicating the asymptotically unbiasedness of ˆω r. To calculate (B.2), we notice that E{κ 2 r } =E{t r1t r2 }+E{t r2t r1 }+E{t r1t r1 }+E{t r2t r2 } =2E{t r1 t r2 }+E{t 2 r1 }+E{t2 r2 } (B.11) 18

19 where E{t 2 r1} = E{t 2 r2} = M4 (M r 1) 2 γ 4 σ 4 16Mr 2 Q (B.12) and E{t r1 t r2 } = 1 4 E{ zh (r)2 W M r 1(ω r )( z (r)2 ρ r z (r)1 )( z (r)2 ρ r z (r)1 ) H W Mr 1(ω r ) z (r)2 } 1 4 E{ zh (r)2 W M r 1(ω r )B(ω r )W Mr 1(ω r ) z (r)2 } = 1 4 E{ zh (r)2 W M r 1(ω r ) M M r ( γ 2 +σ 2 Q )σ2 Q W 1 M r 1 (ω r)w Mr 1(ω r ) z (r)2 } = M 4M r ( γ 2 +σ 2 Q )σ2 Q E{ zh (r)1 W M r 1(ω r ) z (r)1 } (B.13) For sufficiently high SNRs, E{t 2 r1} E{t r1 t r2 }, and substituting (B.6) and (B.11) (B.13) into (B.2), we obtain (18). Acknowledgement The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU ]. References [1] S.L. Marple, Digital Spectral Analysis with Applications, Englewood Cliffs, NJ: Prentice-Hall, 1987 [2] S.M. Kay, Modern Spectral Estimation: Theory and Application, Englewood Cliffs, NJ: Prentice-Hall, 1988 [3] P. Stoica and R. Moses, Spectral Analysis of Signals, Upper Saddle River, NJ: Prentice-Hall,

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23 0 10 C 1 C 2 (13) (17) CRLB Mean Square Error (db) N=10 N=5 60 N= SNR (db) Figure 1: Average mean square frequency error versus SNR at M 1 = M 2 = N 23

24 Mean Square Error (db) N=5 N=10 N=20 C 1 C 2 (13) (17) CRLB SNR (db) Figure 2: Average mean square frequency error versus SNR at M 1 = M 2 = M 3 = N 24

25 Mean Square Error (db) C 1 C 2 AIQML [15] IMDF [16] (13) (17) CRLB SNR (db) Figure 3: Average mean square frequency error versus SNR for 2-D case 25

26 Mean Square Error (db) C 1 C 2 UE [8] IMDF [16] (13) (17) CRLB SNR (db) Figure 4: Average mean square frequency error versus SNR for 3-D case 26

27 Mean Square Error (db) C 1 C 2 UE [8] IMDF [16] (13) (17) CRLB Number of Dimensions Figure 5: Average mean square frequency error versus R at M r = 4 and SNR= 20dB 27