Spatial Polarimetric Time-Frequency Distributions ... - Semantic Scholar

Spatial Polarimetric Time-Frequency Distributions and Applications to Direction-of-Arrival Estimation Yimin Zhang, Moeness G. Amin, and Baha A. Obeidat Center for Advanced Communications Villanova University, Villanova, PA 19085 ABSTRACT Time-frequency distributions (TFDs) have evolved to be a powerful technique for nonstationary signal analysis and synthesis. With the use of a multi-sensor array, spatial time-frequency distributions (STFDs) have been developed and successfully applied to high-resolution direction-of-arrival (DOA) estimation and blind recovery of the source waveforms. In this paper, the polarimetric dimension is introduced to the STFDs resulting in the spatial polarimetric time-frequency distributions (SPTFDs) as a platform for the processing of non-stationary polarized signals. In the SPTFD platform, polarized signals are decomposed (projected) into two orthogonal polarization components, such as horizontal and vertical, and later processed in a manner where their polarization characteristics are exploited. This empowers the STFDs with additional degrees of freedom and improves the robustness of the signal and noise subspaces, and therefore, serving to enhance DOA estimation, signal recovery, and source separation performance. To demonstrate the advantages of the SPTFDs, the polarimetric time-frequency MUSIC (PTF-MUSIC) method for DOA estimation is proposed based on the SPTFD platform and is shown to outperform the time-frequency, polarimetric, and conventional MUSIC methods. Keywords: Time-frequency distributions, array signal processing, polarization, smart antennas, MUSIC.

1. INTRODUCTION Over the past two decades, time-frequency distributions (TFDs) have been equipped for nonstationary signal analysis and synthesis in the areas of speech, biomedicine, automotive industry, and machine monitoring.1, 2 Most recently, the spatial dimension has been incorporated, along with the time and frequency variables, into quadratic and higher-order TFDs, and led to the introduction of spatial time-frequency distributions (STFDs) for sensor signal processing. STFDs improve direction-of-arrival (DOA) estimations and blind recovery of the source waveforms impinging on a multi-sensor receiver, specifically those of nonstationary temporal characteristics.3–8 Polarization and polarization diversities, on the other hand, have been proven to be very effective in wireless communications and various types of radar systems. Polarization has also been incorporated in array antennas for improved estimation of signal parameters, including DOA.9, 10 Despite the extensive research work performed in time-frequency (t-f) signal representations and polarimetric signal processing methods, these two important areas have not been coupled or considered within the same platform. In this paper, we develop the spatial polarimetric time-frequency distributions (SPTFDs) for direction finding applications. These techniques utilize not only the time-varying Doppler frequency signatures, but also the polarization signatures, whether they are stationary or time-varying. The signal polarization information empowers the STFDs, as it retains the integrity of eigenstructure methods and improves the robustness of the respective signal and noise subspaces under low signal-to-noise ratios (SNRs) and in a coherent signal environment. To demonstrate such advantages, the polarimetric time-frequency MUSIC (PTF-MUSIC) method is proposed based on the SPTFD platform and is shown to outperform the MUSIC techniques based on the time-frequency, polarimetric, and conventional methods. This work was supported in part by the ONR under Grant No. N00014-98-1-0176 and DARPA under Grant No. MDA972-02-1-0022. The content of the information does not necessarily reflect the position or policy of the Government, and no official endorsement should be inferred.

This paper is organized as follows. Section 2 discusses the signal model utilized, while Section 3 introduces the spatial polarimetric time-frequency distributions (SPTFDs). The polarimetric time-frequency MUSIC (PTFMUSIC) is proposed in Section 4. Spatial and polarization averaging methods for coherent signal decorrelation are investigated in Section 5. Simulations that clearly demonstrate the effectiveness of the proposed methods are provided in Section 6. Notations A a (.)T (.)H

(.)∗ (.)[i] (.)(i)

matrix vector transpose Hermitian

complex conjugate polarization index subarray index

2. SIGNAL MODEL Consider a narrowband direction finding problem, where n signals arrive at an m-element array. The following linear data model is assumed, x(t) = A(Θ)s(t) + n(t) (1) where the m × n spatial matrix A(Θ) = [a(θ1 ), a(θ2 ), . . . , a(θn )] is a structured mixing matrix. The elements of the m × 1 vector x(t), which represents the measured or sensor data, are multicomponent signals, while the element of the n × 1 vector s(t) are often monocomponent signals, where s(t) = [s1 (t) s2 (t) . . . sn (t)]T . n(t) is an m × 1 additive noise vector, which consists of independent zero-mean, white and Gaussian distributed processes. The STFD of a data vector x(t) is expressed as6 Z Z τ τ Dxx (t, f ) = φ(t − u, τ )x(t + )xH (t − )e−j2πf τ dudτ 2 2

(2)

where the (i, j)th element of Dxx (t, f ) is defined as [Dxx (t, f )]ij = Dxi xj (t, f ), i, j = 1, 2, . . . , m. The kernel φ(t, τ ) uniquely defines the TFD and is a function of the time and lag variables. In this paper, all the integrals are from −∞ to ∞. The auto- and cross-sensor TFDs are, respectively, defined as Z Z Dxx (t, f ) =

φ(t − u, τ )x(t +

τ ∗ τ )x (t − )e−j2πf τ dudτ 2 2

(3)

and Z Z Dx1 x2 (t, f ) =

φ(t − u, τ )x1 (t +

τ ∗ τ )x2 (t − )e−j2πf τ dudτ. 2 2

(4)

Due to the linear data model, the noise-free STFD is obtained by substituting (1) in (2), Dxx (t, f ) = A(Θ)Dss (t, f )AH (Θ)

(5)

where Dss (t, f ) is the TFD matrix of s(t) which consists of auto- and cross-source TFDs as its elements. With the presence of the noise, which is uncorrelated with the signals, the expected values of the above equations yields E[Dxx (t, f )] = A(Θ)E[Dss (t, f )]AH (Θ) + σI. (6) In the above equation, σ is the noise power, I is the identity matrix, and E[.] denotes the statistical expectation operator.

Equations (5) and (6) are similar to the mathematical formula which has been commonly used in narrowband array processing problems, relating the source correlation matrix to the sensor spatial correlation matrix. However, the correlation matrices are replaced by the source and sensor TFD matrices. The two subspaces spanned by the principle eigenvectors of Dxx (t, f ) and the columns of A(Θ) are, therefore, identical. In3, 5, 8 it is further shown that, by only selecting the t-f points with highly localized signal energy, the eigenvalues and eigenvectors estimated from Dxx (t, f ) are more robust to noise than their counterparts obtained from the corresponding data covariance matrix Rxx = E[x(t)xH (t)]. This implies that key problems in various array processing applications can be addressed and solved using a new formulation that is more tuned to nonstationary signal environments.

3. SPATIAL POLARIMETRIC TIME-FREQUENCY DISTRIBUTIONS 3.1. Polarimetric Modeling The receiver array is constructed from m dual polarization sensors. Assuming that the ith received signal, si (t), is a plane electromagnetic wave with two orthogonal polarization components, denoted as p and q. Then the signal polarization components are [p]

si (t)

= si (t) cos(γi )

[q] si (t)

= si (t) sin(γi )ejφi

(7)

where φi is defined as the phase difference between the two polarization components, and the parameter γi denotes the respective amplitude ratio which is defined as ¯! Ã¯ ¯ s[p] (t) ¯ ¯ i ¯ γi = arctan ¯ [q] ¯ (8) ¯ s (t) ¯ i

where stationary polarization is assumed in the observation period.

3.2. Polarimetric Time-Frequency Distributions In passive radar, sonar, and most communication problems, the received signal with dual polarizations can be expressed as x(t) = [x[p] (t) x[q] (t)]T (9) where the superscripts (.)[p] and (.)[q] , respectively, denote two orthogonal polarizations. They can be, for example, vertical and horizontal polarizations, or right-hand and left-hand circular polarizations. In active radar and sonar applications, the received signal with dual transmit and dual receive polarizations can be expressed as x(t) = [x[pp] (t) x[pq] (t) x[qp] (t) x[qq] (t)]T (10) where the first letter of the superscript denotes the transmit polarization, and the second letter denotes the receive polarization. For notation simplicity and uniformity, we focus only on the pp and qq components, and let x[p] (t) and x[q] (t) denote x[pp] (t) and x[qq] (t), respectively. In this way, equation (9) can be used to represent both passive and active signal processing. The self- and cross-polarized TFD is expressed as ZZ τ τ Dx[i] x[i] (t, f ) = φ(t − u, τ )x[i] (t + )(x[i] (t − ))∗ e−j2πf τ dudτ 2 2

(11)

and ZZ Dx[i] x[k] (t, f ) =

φ(t − u, τ )x[i] (t +

τ τ )(x[k] (t − ))∗ e−j2πf τ dudτ 2 2

(12)

respectively, where the superscripts i and k denote either p or q. The self-polarized TFD is useful to obtain the time-frequency signature of each polarization component. The self- and cross-polarized TFDs together define the polarization TFD matrix, ZZ Dxx (t, f ) =

φ(t − u, τ )x(t +

τ H τ )x (t − )e−j2πf τ dudτ. 2 2

(13)

The polarization TFD matrix is of dimension 2×2, although it can be 4×4 if the full four element representation in eq. (10) is used. The off-diagonal elements of Dxx (t, f ) are the cross-polarized terms Dxi xj (t, f ). Accordingly, the polarization TFD matrix can be used to estimate the self- and cross-polarization signatures of propagation channels.

3.3. Spatial Polarimetric Time-Frequency Distributions Equations (9)–(13) correspond to a single dual-polarization sensor case. With an m-sensor array, the data vector, for each polarization i, i = p, q, is expressed as, x[i] (t) = A[i] (Θ)s[i] (t) + n[i] (t).

(14)

It is noted that the mixing matrix is polarization-independent, i.e., A[p] (Θ) = A[q] (Θ) = A(Θ). In the following, A(Θ) is abbreviated as A, for notation simplicity. The generalization of eq. (12) to the multi-sensor receiver is obtained using eq. (14). The STFD matrix introduced in eq. (5) can be defined for each polarization Z Z τ τ Dx[i] x[i] (t, f ) = φ(t − u, τ )x[i] (t + )(x[i] (t − ))H e−j2πf τ dudτ (15) 2 2 In the noise-free environment

Dx[i] x[i] (t, f ) = ADs[i] s[i] (t, f )AH .

(16)

In a similar manner, the cross-polarization STFD matrix between the data vectors with two different polarizations can be defined as,

= =

D [i] [k] (t, f ) Z xZ x τ τ φ(t − u, τ )x[i] (t + )(x[k] (t − ))H e−j2πf τ dudτ 2 2 H ADs[i] s[k] (t, f )A .

(17)

We are now in a position to tie the polarization, the spatial, and the t-f properties of the signals incident on the antenna array. Based on eq. (14), the following vector can be constructed for both polarizations, · [p] ¸ · ¸· [p] ¸ · [p] ¸ x (t) A 0 s (t) n (t) x(t) = = + 0 A x[q] (t) s[q] (t) n[q] (t) = Bs(t) + n(t), ·

(18)

¸

A 0 is block-diagonal and s[i] (t), i = p, q, are the source signal vectors for polarization i. The 0 A STFD of the dual-polarization vector, x(t), can therefore be defined as where B =

D (t, f ) Z xx Z τ τ = φ(t − u, τ )x(t + )xH (t − )e−j2πf τ dudτ 2 2 · ¸· ¸· ¸H A 0 Ds[p] s[p] (t, f ) Ds[p] s[q] (t, f ) A 0 = . 0 A Ds[q] s[p] (t, f ) Ds[q] s[q] (t, f ) 0 A

(19)

Dxx (t, f ) is referred to as the spatial polarimetric time-frequency distribution (SPTFD) matrix. This distribution serves as a general framework within which typical problems in array processing can be addressed. Its effectiveness is shown in Section 4 by proposing the PTF-MUSIC method.

4. POLARIMETRIC TIME-FREQUENCY MUSIC Time-frequency MUSIC has been proposed for improved spatial resolution for signals with clear t-f signatures.7, 8 The PTF-MUSIC is an extension of the t-f MUSIC for diversely polarized signals and a polarized array. It is based on the eigen-decomposition of post-processed SPTFD matrices corresponding to multiple t-f points. Timefrequency averaging and joint block-diagonalization are two techniques proposed to integrate SPTFD matrices at multiple t-f points.7, 8 The SNR enhancement, achieved via the selection t-f points of high energy concentration pertaining to all or some of the sources, allows the t-f MUSIC algorithms to be much more robust than the conventional MUSIC algorithm.8 By performing joint block-diagonalization of the SPTFD matrices Dxx (t, f ) over several (t, f ) points, we obtain the signal and noise subspaces, represented as matrices Us and Un , respectively.7 Consider the following steering matrix · B(θ) =

a(θ) 0 0 a(θ)

¸ (20)

where a(θ) is the n × 1 steering vector representing each polarization. If we define a(θ) to be of unit norm, that is, aH (θ)a(θ) = 1, then BH (θ)B(θ) becomes the 2 × 2 identity matrix. In order to exploit the contributions of the two polarization components, we design the following extended normalized steering vector, B(θ)c b(θ, c) = = B(θ)c (21) kB(θ)ck T

where the vector c = [c1 c2 ] is a unit norm vector with unknown coefficients. In (21), we have used the fact 1 1 that kB(θ)ck = [cH BH (θ)B(θ)c] 2 = (cH c) 2 = 1. The PTF-MUSIC spectrum is given by the following function, P (θ)

=

−1 [min bH (θ, c)Un UH n b(θ, c)] c

−1 = [min cH BH (θ)Un UH n B(θ)c] c

(22)

where in evaluating the MUSIC spectrum one has to minimize the term in brackets by selection of c. Finding the minimum of this quadratic surface is equivalent to finding the the minimum eigenvalue of the 2 × 2 matrix BH (θ)Un UH n B(θ). Thus, a computationally expensive search in polarization is avoided by performing an eigen decomposition on a 2 × 2 matrix. The MUSIC spectrum can be equivalently expressed as, P (θ) =

−1 [λmin (BH (θ)Un UH n B(θ))]

(23)

At the DOAs corresponding to peaks in the PTF-MUSIC spectrum, the polarization parameters of the source signals can be estimated from c(θk ) =

vmin (BH (θk )Un UH n B(θk ))

(24)

k = 1, 2, . . . , n, where vmin is the eigenvector corresponding to the minimum eigenvalue of the 2 × 2 matrix.

5. SPATIAL AND POLARIMETRIC AVERAGING FOR DECORRELATION 5.1. Spatial Averaging In applying the MUSIC algorithms, when the source signals are coherent or highly correlated, the correlation matrix becomes rank deficient, and the sources cannot be resolved. This remains the same when the STFD or SPTFD matrix is used in the t-f MUSIC or PTF-MUSIC algorithms. To remedy this problem, spatial smoothing11 and polarization averaging12 have been employed to decorrelate the source signals prior to signal and noise subspace estimations. Both methods act on recovering the full-rank of the rank deficient source correlation matrix. In the underlying problem, these methods are applied to recover the rank of the SPTFD matrix of the source signals. Spatial smoothing involves dividing the cross-polarized array of m sensors into Pn overlapping sub-arrays of m − Pn + 1 sensors, and averaging the now Pn SPTFD matrices. The received vector of signals at the ith sub-array is expressed as · (i)[p] ¸ · [p] ¸ · (i)[p] ¸ x (t) n (t) (i) s (t) x(i) (t) = (i)[q] = B1 + (i)[q] (25) x (t) s[q] (t) n (t) i = 1, 2, . . . , m − Pn + 1, where

·

(i) B1

A1 Di−1 = 0

0 A1 Di−1

¸ (26)

with A1 being the new (m − Pn + 1) × n steering matrix, and D being the following n × n diagonal matrix h i D = diag e−j2π sin(θ1 ) , · · · , e−j2π sin(θn ) . (27) (i)

Denoting Dxx as the SPTFD matrix corresponding to x(i) (t), the spatially smoothed SPTFD matrix is defined (i) by averaging, Dxx , i = 1, 2, . . . , Pn , i.e. DxxSS (t, f ) =

Pn 1 X D(i) (t, f ). Pn i=1 xx

(28)

5.2. Polarimetric Averaging Polarimetric averaging is another preprocessing method aimed to combat the rank deficiency in the source SPTFD matrix, Dss (t, f ). It is worth noting that polarization averaging can be used in both the presence of sub-array averaging and its absence. In the former, the polarimetric averaged SPTFD matrix is defined as DxxP A (t, f ) =

1 [D [p] [p] (t, f ) + Dx[q] x[q] (t, f )] . 2 x x

(29)

In the later, the subarray and polarization averaged SPTFD matrix becomes Pn h i 1 X (i) (i) Dx[p] x[p] (t, f ) + Dx[q] x[q] (t, f ) , 2Pn i=1

DxxSSP A (t, f ) = (i)

(30)

(i)

where Dx[p] x[p] (t, f ) and Dx[q] x[q] (t, f ) are the STFDs corresponding to x(i)[p] (t) and x(i)[q] (t), respectively.

6. SIMULATIONS 6.1. Uncorrelated Sources Scenario In the first set of simulations, we consider a uniform linear array of four (m = 4) dual-polarization cross-dipoles with half-wavelength inter-element spacing. To fully demonstrate the advantages of the proposed SPTFD framework and the PTF-MUSIC algorithm, we consider two far-field sources (sources 1 and 2) with chirp waveforms in the presence of an undesired sinusoidal signal (source 3). Table 1 shows three respective normalized

starting and end frequencies, DOAs (measured from the the broadside), and the two polarization parameters, γ and φ. All signals have the same signal power (SNR=13dB). The data length is 256 samples and the length of the rectangular window kernel of the Pseudo Wigner-Ville distribution (PWVD) is 65 samples. The PWVD averaged over the four sensors is shown in Fig. 1 and 2, respectively, for the horizontal and vertical polarizations. Because the sources are closely spaced, the crossterm suppression effect is limited. To further suppress the crossterms, we can utilize both the spatial and polarization information. Fig. 3 shows the WVD averaged over the four sensors and both polarizations. In this example, because source 1 and source 2 have orthogonal polarizations, their crossterms are completely suppressed. In this part, we compute PTF-MUSIC spectrum and the results are compared with the conventional MUSIC, polarimetric MUSIC, and t-f MUSIC. For both t-f MUSIC and PTF-MUSIC, the array averaged PWVD is used to mitigate crossterms. Based on the averaged PWVD, only the t-f points on the chirp signatures of sources 1 and 2 are considered for STFD and SPTFD matrix construction. In Fig. 4, the MUSIC spectra are plotted and compared for different methods. Results from three independent trials are shown. For conventional and t-f MUSIC, only the vertical polarization components are used. While the t-f MUSIC utilizes source discrimination and increased SNR, and the polarimetric MUSIC takes advantage of polarization diversity, only the PTF-MUSIC enjoys both properties. It is evident that only the proposed PTF-MUSIC accurately estimates the DOAs of the two sources, whereas all other methods fail. Table 1. Signal parameters

source 1 source 2 source 3

start freq. 0.20 0.22 0.10

end freq. 0.40 0.42 0.10

DOA (deg.) −3 3 5

γ (deg.) 45 45 20

φ (deg.) 0 180 0

0 0.05

Normalized frequency

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

50

100 150 Time (samples)

200

250

Figure 1. PWVD averaged over the horizontal sensors.

6.2. Coherent Sources Scenario In the second set simulations, we consider a uniform linear array of five (m = 5) dual-polarization cross-dipoles with half-wavelength inter-element spacing. Three far-field sources are considered. The first two sources (sources 1 and 2) are of identical chirp signatures, whereas the third one is an undesired sinusoidal signal (source 3). Table 2 shows the signal parameters. All signals have the same signal power (SNR=10dB). The data length is 256 samples.

0 0.05


0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

50


200

250

Figure 2. PWVD averaged over the vertical sensors.

0 0.05


0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

50


200

250

Figure 3. PWVD averaged over sensors and polarizations.

The PWVD averaged over the five cross-polarization sensors is shown in Fig. 5, where the length of the kernel window is set equal to 65 samples. The suppression of crossterms is highly dependent of the spatial and polarization correlations among the signals. A. Polarimetric Averaging Polarimetric averaging of the STFD matrices of the data samples across the horizontal and the vertical polarizations can successfully decorrelate coherent sources. Fig. 6 shows the spectra of the polarimetric MUSIC and PTF-MUSIC, respectively, each for three independent trials, where polarimetric averaging was employed on the five vertical and five horizontal antennas. For the PTF-MUSIC method, only the two coherent sources (i.e., sources 1 and 2) are selected. It can be seen that both methods successfully decorrelate the two coherent sources. However, it is only the PTF-MUSIC that clearly shows an exemplary performance due to the source selection capability. From Fig. 6 we learn that polarization averaging is an effective tool for decorrelating coherent sources. B. Subarray and Polarization Averaging In this simulation in addition to the polarimetric averaging, spatial smoothing preprocessing is performed with two subarrays of four sensors. The estimated spectra for polarimetric MUSIC and PTF-MUSIC are shown

Conventional MUSIC

Polarimetric MUSIC

60

35 30

50 25

Magnitude (dB)

Magnitude (dB)

40

30

20 15 10

20 5 10 0 0 −20

−3 3 5 Direction−of−arrival (degrees)

−5 −20

20

−3 3 5 Direction−of−arrival (degrees)

TF−MUSIC

20

PTF−MUSIC

40

35

35

30

30

25

Magnitude (dB)

Magnitude (dB)

25 20 15

20 15 10

10 5

5

0

0 −5 −20

−3 3 Direction−of−arrival (degrees)

20

−5 −20


20

Figure 4. MUSIC spectra. Table 2. Signal parameters

source 1 source 2 source 3

start freq. 0.20 0.20 0.10

end freq. 0.50 0.50 0.10

DOA (deg.) −6 6 12

γ (deg.) 35 45 25

φ (deg.) 5 170 −90

in Fig. 7. For comparison, in Fig. 8 we plotted the spectra computed from conventional MUSIC method using a 10 vertical-polarization antenna array, with spatial smoothing preprocessing. Due to the close orientations of the second and third sources, the performance of all non-TF-based methods was severely hindered. Only the PTF-MUSIC spectrum, which discriminated the third signal from consideration, shows sharp and less biased peaks at the DOAs of the two coherent sources.

7. CONCLUSION Spatial polarimetric time-frequency distributions (SPTFDs) have been introduced. The SPTFDs are a powerful platform to utilize the polarization, spatial, and temporal signatures of signal arrivals for sophisticated array processing. To demonstrate the advantage of the SPTFD platform, we have considered the direction finding problem and proposed the polarimetric time-frequency MUSIC (PTF-MUSIC) method as an example of its applications, in such scenarios where the signals are highly localized in the time-frequency domain and diversely

0 0.05


0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

50


200

250

Figure 5. PWVD averaged over sensors and polarizations.

PTF−MUSIC 35

35

30

30

25

25

20

Magnitude (dB)

Magnitude (dB)

Polarimetric MUSIC 40

20 15

15 10

10

5

5

0

0 −20


12

20

−5 −20

(a) Polarimetric MUSIC


20

(b) PTF-MUSIC (2 signals chosen)

Figure 6. Polarimetric MUSIC and PTF-MUSIC spectra with polarization averaging.

polarized. The proposed PTF-MUSIC significantly outperforms the other existing MUSIC methods, including conventional MUSIC, time-frequency MUSIC and polarimetric MUSIC.

REFERENCES 1. L. Cohen, “Time-frequency distributions — a review,” Proc. IEEE, vol. 77, no. 7, pp. 941–981, July 1989. 2. S. Qian and D. Chen, Joint Time-Frequency Analysis — Methods and Applications, Englewood Cliffs, NJ: Prentice Hall, 1996. 3. Y. Zhang, W. Mu, and M. G. Amin, “Time-frequency maximum likelihood methods for direction finding,” J. Franklin Inst., vol. 337, no. 4, pp. 483–497, July 2000. 4. A. R. Leyman, Z. M. Kamran, and K. Abed-Meraim, “Higher-order time frequency-based blind source separation technique,” IEEE Signal Processing Letters, vol. 7, no. 7, pp. 193–196, July 2000. 5. M. G. Amin and Y. Zhang, “Direction finding based on spatial time-frequency distribution matrices,” Digital Signal Processing, vol. 10, no. 4, pp. 325–339, Oct. 2000.

Polarimetric MUSIC

PTF−MUSIC

60

50 45

50

40 35

Magnitude (dB)

Magnitude (dB)

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30 25 20 15 10

10

5 0 −20


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(a) Polarimetric MUSIC


20

(b) PTF-MUSIC (2 signals chosen)

Figure 7. Polarimetric and PTF-MUSIC spectra with spatial smoothing and polarimetric averaging.

Conventional MUSIC 30 25

Magnitude (dB)

20 15 10 5 0 −5 −10 −20


12

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Figure 8. 10-sensor conventional MUSIC with spatial smoothing.

6. A. Belouchrani and M. G. Amin, “Blind source separation based on time-frequency signal representations,” IEEE Trans. Signal Processing, vol. 46, no. 11, pp. 2888–2897, Nov. 1998. 7. A. Belouchrani and M. G. Amin, “Time-frequency MUSIC,” IEEE Signal Processing Letters, vol. 6, pp. 109–110, May 1999. 8. Y. Zhang, W. Mu, and M. G. Amin, “Subspace analysis of spatial time-frequency distribution matrices,” IEEE Trans. Signal Processing, vol. 49, no. 4, pp. 747–759, April 2001. 9. E. R. Ferrara and T. M. Parks, “Direction finding with an array of antennas having diverse polarizations,” IEEE Trans. Antennas Propagat., vol. 31, pp. 231–236, March 1983. 10. J. Li and R. J. Compton, “Angle and polarization estimation using ESPRIT with a polarization sensitive array,” IEEE Trans. Antennas Propagat., vol. 39, pp. 1376–1383, Sept. 1991. 11. T. Shan, M. Wax and T. Kailath, “On Spatial Smoothing for Direction-of-arrival Estimation of Coherent Signals,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 806–811, Aug. 1985. 12. H. Yamada, K. Onishi and Y. Yamaguchi, “Polarimetric superresolution technique for 2-D radar target imaging,” SPIE Proc., vol. 3120, pp. 317–326.