Document Sample

INTERNATIONAL JOURNAL OF c 2007 Institute for Scientiﬁc INFORMATION AND SYSTEMS SCIENCES Computing and Information Volume 3, Number 4, Pages 623–631 SUBBANDING ESPRIT BY DAUBECHIES WAVELETS YANBO XUE, JINKUAN WANG, AND ZHIGANG LIU Abstract. A novel subbanding ESPRIT method by Daubechies wavelets is suggested in this paper. The proposed approach ﬁlters the signals into dif- ferent subbands with reduced wavelet packet ﬁlters and then applies standard ESPRIT to the subband signals. The rotational invariance in the subarrays is proven and the mapping method from the subband to the fullband is also formulated. Experimental results show that in sense of root mean square error (RMSE) reduction and output gain enhancement, the suggested method out- performs ESPRIT in scenarios of highly correlated signals and/or low signal- to-noise ratio (SNR). The RMSE and output gain versus Daubechies wavelets with diﬀerent lengths also show the impacts of choice of Daubechies wavelets on the performance of the new method. Key Words. ESPRIT, Daubechies wavelets, subband, DOA, and reduced wavelet packet ﬁlters. 1. Introduction Nowadays the ESPRIT [1] (estimation of signal parameters via rotational invari- ance technique) algorithm has won great success in the direction-of-arrival (DOA) estimation problem, and it has been used for DOA estimation, harmonic analysis, frequency estimation, delay estimation, and combinations thereof. ESPRIT works well at high signal-to-noise ratio (SNR), long data sequences, and in the context of uncorrelated signals. However, the performance of ESPRIT degrades greatly when these conditions are not met. Recently wavelet theory has been introduced to array signal processing in two ways, the wavelet denoising methods [2, 3] and subbanding methods [4, 5, 6], in which the latter will be more eﬃcient for correlated signals because they can decorrelate the correlated signals into diﬀerent subbands. The subbanding method, in essence, is a beamspace approach [7]. In this paper, a novel subbanding ESPRIT method by Daubechies wavelets is proposed to decorrelated the signals in beamspace. The new method enjoys the ad- vantages of subband decomposition [8]. In order to hold the rotational invariance, we need to modify the Daubechies wavelets to compose reduced wavelet packet ﬁlters. The decorrelation ability of the suggested algorithm relies on the decompo- sition level of subband, and is better than ESPRIT. Simulation results, showing the performance improvement of the suggested approach in sense of root mean square error (RMSE) and output gain for correlated signals at low SNR, is also given. The RMSE and output gain versus Daubechies wavelets with diﬀerent length also Received by the editors June 1, 2006 and, in revised form, July 22, 2006. This work is supported by the doctor foundation from the Ministry of Education of China, under Grant no. 20050145019, and the Directive Plan from the Department of Education of Hebei Province, China, under Grant no. Z2004103. 623 624 Y. XUE, J. WANG, AND Z. LIU show the impacts of choice of Daubechies wavelets on the performance of the new method. 2. Problem Formulation Consider a uniform linear array (ULA) with M isotropic sensors spaced by half wavelength d = λ/2, and there are D (D < M ) narrowband plane waves cen- tered at frequency ω0 with propagation speed c, impinging from the directions θ = {θ1 , θ2 , · · · , θD }. The signal received by the array can be expressed as (1) x(n) = A(θ)s(n) + w(n), where x(n), s(n) and w(n) denote respectively the M × 1 received signal vec- tor, D × 1 wavefront vector, and M × 1 additive noise vector. If we deﬁne ωi = ω0 sin θi d/c as the equivalent spatial frequency of the i-th wavefront, the mixing ma- trix A(ω) ∈ C M ×D can be expressed as A(ω) = [a(ω1 ), a(ω2 ), · · · , a(ωD )], where a(ωi ) = [1, e−jωi , · · · , e−j(M −1)ωi ]T denotes the steering vector corresponding to the spatial frequency ωi , and superscript T denotes transpose. Assume that the signals are zero mean wide sense stationary (WSS) processes, and wm (n) is the zero mean white Gaussian noise (WGN) which is uncorrelated to the signals and has identical variance σ 2 in each sensor. Then the output covariance matrix is given by Rxx = E[xxH ] (2) = ARss AH + σ 2 I, where Rss = E[ssH ] is the D × D signal covariance matrix and I is an M × M identity matrix. Superscript H denotes Hermitian transpose. It is shown that Rss is nondiagonal and nonsingular when the signals are spatially correlated, and nondiagonal and singular when some signals are coherent [9], in both cases the standard ESPRIT degrades greatly. 3. Subbanding ESPRIT by Reduced Wavelet Packet Filters 3.1. Reduced Wavelet Packet Filters. Wavelet packet is the magniﬁcent ex- tension of wavelet. In the process of wavelet packet transform, both the details and approximations are decomposed into two parts, which oﬀers the richest analysis compared with wavelet transform. The decomposing procedure can be done simply via Mallat’s pyramid algorithm. In the standard wavelet ﬁlters, the length of the ﬁltered vector is half of the original vector length. We have found that, however, the standard wavelet is not suitable for the array signal because the circular ﬁltering does not make any sense and even contaminates the array structure. So we modify the lowpass and highpass matrixes H and G from dimension M/2 × M to Mf × M , where Mf = (M − 2L + 2)/2 with 2L denotes the ﬁlter length, given by T (3) H = hT 0 hT 1 ··· hT f −1 M and T T T T (4) G = g0 g1 ··· gMf −1 , where hi = [h0 , ..., h2L−1 , 0, ..., 0]T 2i and gi = [g0 , ..., g2L−1 , 0, ..., 0]T 2i , in which 2L M −2L 2L M −2L the nonzero parts denote the wavelet packet ﬁlter coeﬃcients. T denotes an M ×M circular shift matrix [10], given by SUBBANDING ESPRIT BY DAUBECHIES WAVELETS 625 0 1 0 0 ··· 0 0 0 0 0 1 0 ··· 0 0 0 0 0 0 1 ··· 0 0 0 (5) T = . . . . .. . . . . . . . . . . . . . . . . . . . 0 0 0 0 ··· 0 0 1 1 0 0 0 ··· 0 0 0 With reduced wavelet packet, we won’t lose the orthonormality of standard wavelet packet, while enjoy most of the advantages of wavelet packet transform, like multiresolution analysis (MRA) and energy decomposition. The computational load also decreases with reduced wavelet packet, which can be used in practical im- plementation. Then we can decompose the measured data of (1) into the low frequency subband and high frequency subband using the ﬁlters H and G, formulated by (6) ˆ xh (n) = As(n) + wh (n) and (7) ˘ xg (n) = As(n) + wg (n), ˆ ˘ where A = HA and A = GA denote respectively the Mf × D low subband and high subband mixing matrix, and the ﬁltered matrixes are named low frequency matrix xh (n) := Hx(n) and high frequency matrix xg (n) := Gx(n). In an eﬀort to have an l levels decomposition, the sensors number M must meet the requirement l−1 of mod (M − 2L + 2 i=1 2i , 2l ) = 0, where mod (a, b) means the remainder of a dividing b. Thus we have two Mf × 1 matrixes xh (n) and xg (n) and each of them can be used to compose two subarrays. 3.2. Rotational Invariance. To apply ESPRIT algorithm to each subband, it is important to exploit the rotational invariance between two subarrays. Taking xh (n) for an example, the lowpass matrix can be rewritten as ˆ H | O2 h0 (8) H= = , hMf −1 O2 | ˆ H ˆ where H ∈ R(Mf −1)×(M −2) is also a reduced wavelet packet ﬁlter, which has the similar structure as H in (3), and O2 is an (Mf − 1) × 2 zero matrix. The mixing matrix A can also be expressed as T A = bT (ω) 0 bT (ω) · · · 1 bT −1) (ω) (M T = AT 1 | bT −2) (ω) bT −1) (ω) (M (M T (9) = bT (ω) bT (ω) 0 1 | AT 2 , in which we deﬁne bk (ω) = [e−jkω1 , e−jkω2 , · · · , e−jkωD ] as the spatial frequency response vector corresponding to the k-th sensor, A1 and A2 are composed from the ﬁrst (M − 2) rows and last (M − 2) rows of the Vondermonde matrixes A. If the ﬁrst subarray consists of the ﬁrst to the (Mf −1)-th sensors and the second ˆ subarray consists of the second to the Mf -th sensors, the mixing matrixes AX and Aˆ Y of two subarrays are given by (10) ˆ ˆ AX = H O2 A = HA1 ˆ 626 Y. XUE, J. WANG, AND Z. LIU and (11) ˆ AY = O2 ˆ ˆ H A = HA2 , ˆ ˆ It is easy to say that AX and AY can be related by a diagonal matrix Φ = diag{e −j2ω1 −j2ω2 ,e ,··· ,e −j2ωD ˆ ˆ }, i.e., AY = AX Φ. Here Φ is the rotational invari- ance in the subband signals. By exploiting the diagonal elements of Φ using standard ESPRIT, we can obtain the spatial frequency in the subbands without having to know the mixing matrix A. ˆ Same can be said for the subarrays corresponding to the high frequency subband xg . So the validity of the subband ESPRIT is shown with a 1-level reduced wavelet packet decomposition. It can also been easily proven with an any-level decompo- sition. As we desire the fullband frequencies, we need to map the frequencies from subbands back to fullband. To an any-level reduced wavelet packet decomposition, we map the frequencies as follows ˆ ω ωl,k + (k − 1)πsgn(ˆ l,k ) l , k = 1, 3, 5, · · · (12) ωf b = 2 ωl,k − kπsgn(ˆ l,k ) ˆ ω , k = 2, 4, 6, · · · 2l ω ˆ ˆ where sgn(ˆ l,k ) denotes the sign of ωl,k , ωl,k is the subband frequency of the l-th level k-th node. 3.3. Algorithm Summary. Step 1: Form the matrix X = [x(1), x(2), · · · , x(N )] by taking N snapshots of model (1). Step 2: Filter X with reduced wavelet packet ﬁlters H and G to yield two matrixes Xh = HX and Xg = GX. Step 3: Determine the number of signals [11] of the mother node X and its two children nodes Xh and Xg . Accept the children nodes and goto Step 2 if there are no modes lost. Otherwise stop the decomposition at the mother node. Step 4: Prune the binary tree using the best bases selection method [12] to ﬁnd the optimal leaf nodes. Step 5: Divide each leaf nodes into two subarrays and apply ESPRIT to estimate the subband spatial frequency. Step 6: Map the subband frequency back to the fullband frequency using (12) and then the DOA from θi = arcsin {ωf b · λ/d} . 3.4. Computational Complexity. For very large arrays, the computational loads for real-implementation of signal-based algorithm are expensive, which require O(M 3 ) eigendecomposition. When we introduce the subbanding method, we can enjoy the computational savings of beamspace separation. Take two subbands as 3 an example, we require only 2 × O(Mf ) eigendecomposition, in which Mf is nearly 3 the half of M when M >> L. The computations can also be reduced to O(Mf ) because all the subbands can be processed in parallel. 4. Experimental Results Computer simulations are conducted in this section to assess the validity of the subband ESPRIT by Daubechies wavelets. Common to all experiments, a ULA with M = 32 isotropic sensors is selected and N = 100 snapshots are taken. Four sources from −45◦ , −20◦ , 10◦ and 70◦ emit narrow-band signals with the same power. We use Monte Carlo method to obtain 50 independent runs for each SUBBANDING ESPRIT BY DAUBECHIES WAVELETS 627 0 −2 −4 Output Gain (dB) −6 −8 −10 −12 −14 −90 −45 −20 10 70 90 DOA (deg) Figure 1. ESPRIT estimates for signals from −45◦ , −20◦ , 10◦ and 70◦ with SNR = −15 dB, c13 = 0 and 50 trial runs. 0 −2 −4 Output Gain (dB) −6 −8 −10 −12 −14 −90 −45 −20 10 70 90 DOA (deg) Figure 2. Subband ESPRIT estimates for signals from −45◦ , −20◦ , 10◦ and 70◦ with SNR = −15 dB, c13 = 0 and 50 trial runs. example and Daubechies wavelet db5 is chosen for all the cases except for the last simulation. The signals are assumed to be correlated. The correlation coeﬃcient, let we say between the ﬁrst signal s1 and the third signal s3 , is deﬁned as c13 = cov(s1 , s3 )/ cov(s1 , s1 )cov(s3 , s3 ) with cov(a, b) denotes the covariance of a and b. In the ﬁrst example, we consider the case of uncorrelated signals (c13 = 0) and low SNR (SNR = −15 dB). Fig. 1 and Fig. 2 display the simulation results with conventional ESPRIT and our proposed method respectively, in which we notice the estimates with subband ESPRIT are closely distributed along the DOA’s, while those with the conventional one are sparsely distributed. In the second example, we consider the case of highly correlated signals (c13 = 0.9) and low SNR (SNR = −12 dB). The results are shown in Fig. 3 and Fig. 4, in which we can see that our method can estimate both the correlated and the uncorrelated signals, while the standard ESPRIT fails to resolve the DOA’s 628 Y. XUE, J. WANG, AND Z. LIU 0 −2 −4 Output Gain (dB) −6 −8 −10 −12 −14 −90 −45 −20 10 70 90 DOA (deg) Figure 3. ESPRIT estimates for signals from −45◦ , −20◦ , 10◦ and 70◦ with SNR = −12 dB, c13 = 0.9 and 50 trial runs. 0 −2 −4 Output Gain (dB) −6 −8 −10 −12 −14 −90 −45 −20 10 70 90 DOA (deg) Figure 4. Subband ESPRIT for signals from −45◦ , −20◦ , 10◦ and 70◦ with SNR = −12 dB, c13 = 0.9 and 50 trial runs. of correlated signals. The decorrelation ability of the subbanding method is easily proven by this example. To show the performance improvement of our suggested method in diﬀerent correlation coeﬃcient, further simulations are given next. Fig. 5 is plotted to show the RMSE versus correlation coeﬃcient at low SNR (SNR = −12dB). in which RMSE curve with conventional ESPRIT is depicted in dash line with diamond markers ( ) and the curve with our approach is in solid line with asterisk markers (∗). It is easily to conclude that the new method outperforms the standard ESPRIT for all correlation coeﬃcient settings at low SNR, and we can decrease the RMSE with the subbanding version. The output gain versus correlation coeﬃcient at low SNR is also given in Fig. 6. It is clear that our method has higher output gain than ESPRIT at low SNR’s and in highly correlated signals. It is shown also in both Fig. 5 and Fig. 6 that our method is robust to coeﬃcients, which means it works better in high correlated signals where the standard ESPRIT fails. SUBBANDING ESPRIT BY DAUBECHIES WAVELETS 629 2.5 ESPRIT ESPRIT + Daub 2 1.5 RMSE (deg) 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Correlation Coefficient Figure 5. RMSE v.s. c13 for ESPRIT and subband ESPRIT for signals from −45◦ , −20◦ , 10◦ and 70◦ with SNR = −12 dB. −0.5 −1 −1.5 Output Gain (dB) −2 −2.5 −3 −3.5 −4 ESPRIT ESPRIT + Daub −4.5 0 0.2 0.4 0.6 0.8 1 Correlation Coefficient Figure 6. Output gain v.s. c13 for ESPRIT and subband ESPRIT for signals from −45◦ , −20◦ , 10◦ and 70◦ with SNR = −12 dB. Another simulation is conducted to show the eﬀect of selection of Daubechies wavelets on the performance, given in Fig. 7 and Fig. 8. One can obvious see that db2 − 9 wavelets perform better than the other wavelets in sense of RMSE, and db4 − 7 and db12 − 13 wavelets have higher output gain than other wavelets. These properties conﬁrm the applicability of db5 to our experimental settings. 5. Conclusions Daubechies wavelets have been introduced in this paper to subbanding ESPRIT in the context of correlated signals and/or low SNR. We prove the rotational in- variance in the subarrays when we use the reduced wavelet packet ﬁlters. Mapping method from the subband to the fullband is also formulated in (12). With the proposed method, we can also enjoy the computational savings. Simulation results show that our approach has better performance than the standard ESPRIT in sense 630 Y. XUE, J. WANG, AND Z. LIU 1.5 1 RMSE (deg) 0.5 0 1 3 5 7 9 11 13 Daubechies Wavelets db(n) Figure 7. RMSE v.s. Daubechies wavelets for ESPRIT and sub- band ESPRIT for signals from −45◦ , −20◦ , 10◦ and 70◦ with SNR = −12 dB. −0.7 −0.75 −0.8 Output Gain (dB) −0.85 −0.9 −0.95 −1 −1.05 −1.1 1 3 5 7 9 11 13 Daubechies Wavelets db(n) Figure 8. Output gain v.s. Daubechies wavelets for ESPRIT and subband ESPRIT for signals from −45◦ , −20◦ , 10◦ and 70◦ with SNR = −12 dB. of RMSE and output gain, and can be used as a potential alternative to the prepro- cessing methods in the context of correlated signals. The RMSE and output gain versus Daubechies wavelets with diﬀerent length also show the impacts of choice of Daubechies wavelets on the performance of the new method. References [1] Roy, R.H., ESPRIT-Estimation of Signal Parameters via Rotational Invariance Technique, Ph.D Dissertation, Stanford University, 1987. [2] Sathish, R. and Anand, G.V., Wavelet denoising for plane wave DOA estimation by MUSIC, Proc. IEEE TENCON’03, vol. 1, pp. 104-108, 2003. [3] Rao, A.M. and Jones, D.L., A denoising approach to multisensor signal estimation, IEEE Trans. Signl. Proce., vol. 48, no. 5, pp. 1225-1234, 2000. SUBBANDING ESPRIT BY DAUBECHIES WAVELETS 631 [4] Wang, B., Wang, Y., and Chen, H., Spatial wavelet transform preprocessing for direction- of-arrival estimation, IEEE Antennas Propag. Societ. Internat. Sympos., no. 4, pp. 672-675, June 2002. [5] Xue, Y.B., Wang, J.K., and Liu, Z.G., Wavelet packets-based direction-of-arrival estimation, Proc. IEEE ICASSP’04, vol. 2, pp. 505-508, May 2004. [6] Xue, Y.B., Wang, J.K., and Zhang, Y.M., Non-mapping back SB-ESPRIT for coherent sig- nals, Proc. IEEE ICASSP’05, vol. 4, pp. 633-636, March 2005. [7] Xu, G., Silverstein, S.D., Roy, R.H., and Kailath, T., Beamspace ESPRIT, IEEE Trans. Signal Proces., vol. 42, no. 2, pp. 349-356, Feb. 1994. [8] Rao, S. and Pearlman, W.A., Analysis of linear prediction, coding, and spectral estimation from subbands, IEEE Trans. Infor. Theor., vol. 42, no.4, pp. 1160-1178, April 1996. [9] Shan, T.J., Wax, M., and Kailath, T., On spatial smoothing for direction-of-arrival estimation of coherent signals, IEEE Trans. ASSP, vol. 33, no. 4, pp. 806-811, Aug. 1983. [10] Percival, D.B. and Walden, A.T., Wavelet methods for time series analysis, Cambridge Uni- versity Press, England, 2000. [11] Wax, M. and Kailath, T., Detection of signals by information theoretic criteria, IEEE Trans. Acous. Speec. Signl. Proce., vol. 33, no. 2, pp. 387-392, Feb. 1985. [12] Coifman, R.R. and Wickerhauser, M.V., Entropy-based algorithms for best basis selection, IEEE Trans. Info. Theo., vol. 38, pp. 713-718, 1992. School of Information Science and Engineering, Northeastern University, Shenyang, 110004, China E-mail: {yxue,wjk,zliu}@mail.neuq.edu.cn URL: http://www.neuq.edu.cn/sasp

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 12 |

posted: | 5/27/2011 |

language: | English |

pages: | 9 |

OTHER DOCS BY fdh56iuoui

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.