SUBBANDING ESPRIT BY DAUBECHIES WAVELETS 1. Introduction Nowadays

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SUBBANDING ESPRIT BY DAUBECHIES WAVELETS 1. Introduction Nowadays Powered By Docstoc
					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,
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 [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-
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 [8] Rao, S. and Pearlman, W.A., Analysis of linear prediction, coding, and spectral estimation
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 [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-
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[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

				
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