ABSTRACT. In this paper, we introduce the spatial polarimetric time-frequency distribution (SPTFD) as a platform to process nonstationary array signals with two ...
DIRECTION FINDING USING SPATIAL POLARIMETRIC TIME-FREQUENCY DISTRIBUTIONS Yimin Zhang, Moeness G. Amin, and Baha A. Obeidat Center for Advanced Communications Villanova University, Villanova, PA 19085, USA E-mail: {yimin.zhang, moeness.amin, baha.obeidat}@villanova.edu ABSTRACT In this paper, we introduce the spatial polarimetric time-frequency distribution (SPTFD) as a platform to process nonstationary array signals with two orthogonal polarization components, such as horizontal and vertical. The use of dual polarization empowers the STFDs and improves the robustness of the respective signal and noise subspaces. With the additional polarimetric information, improved direction finding performance can be achieved. To demonstrate such advantages, the polarimetric time-frequency MUSIC (PTFMUSIC) 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. 1. INTRODUCTION Over the past two decades, time-frequency distribution (TFDs) have evolved to be a powerful technique 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. The STFD has been successfully applied to highresolution direction-of-arrival (DOA) estimations and blind recovery of the source waveforms impinging on a multi-sensor receiver, specifically those of nonstationary temporal characteristics [3-7]. Polarization and polarization diversity, on the other hand, have been proven to be very effective in wireless communications and various types of radar systems. This work was supported in part by the ONR under Grant No. N00014-98-1-0176 and DARPA under Grant No. MDA97202-1-0022. The content of the information does not necessarily reflect the position or policy of the Government, and no official endorsement should be inferred.
Polarization has also been incorporated in array antennas for improved estimation of signal parameters, including the DOA [8, 9]. 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 ratio (SNR) and in a coherent signal environment. 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 (PTF-MUSIC) is proposed in Section 4. Simulations that clearly demonstrate the efficiency of this method are provided in Section 5. 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) 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. n(t) is an m × 1 additive noise vector, which consists of in-
dependent zero-mean, white and Gaussian distributed processes. The STFD of a data vector x(t) is expressed as [10] Dxx Z (t, Z f) =
τ H τ )x (t − )e−j2πf t 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 ), where i, j = 1, 2, . . . , m, and the superscript (.)H denotes the complex conjugate transpose of a matrix or vector. In this paper, all the integrals are from −∞ to ∞. Due to the linear data model, the noise-free STFD is obtained by substituting (1) in (2), φ(t − u, τ )x(t +
Dxx (t, f ) = A(Θ)Dss (t, f )AH (Θ)
(3)
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.
(4)
In the above equation, σ is the noise power, I is the identity matrix, and E[.] denotes the statistical expectation operator. In this equation, the sensor and source TFD matrices in (3) are now replaced by the sensor and source t-f spectrum matrices. Equations (3) and (4) 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. Here, these correlation matrices are replaced by source and sensor TFD matrices. The two subspaces spanned by the principle eigenvectors of Dxx (t, f ) and the columns of A are, therefore, identical. In [4, 6, 7], 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 means 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. POLARIMETRIC SPATIAL TIME-FREQUENCY DISTRIBUTIONS Consider a passive radar or sonar problems, the received signal with dual polarizations can be expressed as (5) x(t) = [xp (t) xq (t)]T
where the superscripts (.)p and (.)q , respectively, denote two orthogonal polarizations. They can be, for example, vertical and horizontal polarizations, or righthand and left-hand circular polarizations. With an m-sensor array, the data vector, for each polarization i, is expressed as, xi (t) = Ai (Θ)si (t) + ni (t).
(6)
It is noted that the mixing matrix is polarization-independent, i.e., Ai (Θ) = A(Θ) for both i. In the following, A(Θ) is abbreviated as A, for notation simplicity. Based on Eq. (6), the following vector can be constructed for all polarizations, · p ¸ · ¸· p ¸ · p ¸ x (t) A 0 s (t) n (t) x(t) = = + xq (t) 0 A sq (t) nq (t) =
Bs(t) + n(t).
(7)
When the noise is ignored, the STFD matrix introduced earlier can be defined for each polarization
= =
D i i (t, f ) Z xZx τ τ φ(t − u, τ )xi (t + )(xi (t − ))H e−j2πf t dudτ 2 2 ADsi si (t, f )AH . (8)
In a similar manner, the cross-polarization STFD matrix between the data vectors with two different polarizations can be defined as,
= =
Dxi xk (t, f ) Z Z τ τ φ(t − u, τ )xi (t + )(xk (t − ))H e−j2πf t dudτ 2 2 ADsi sk (t, f )AH . (9)
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. The STFD of the dual-polarization vector, x(t), can therefore be defined as
= =
D (t, f ) Z xx Z τ τ φ(t − u, τ )x(t + )xH (t − )e−j2πf t dudτ 2 2 · ¸· ¸· ¸H A 0 Dsp sp (t, f ) Dsp sq (t, f ) A 0 0 A Dsq sp (t, f ) Dsq sq (t, f ) 0 A (10)
Dxx (t, f ) is referred to as the spatial polarimetric timefrequency distribution (SPTFD) matrix. This distribution serves as a general framework within which typical problems in array processing can be addressed. As an example, the PTF-MUSIC method is proposed for DOA estimation.
4. POLARIMETRIC TIME-FREQUENCY MUSIC Time-frequency MUSIC has been proposed for improved spatial resolution for signals with t-f characteristics [3, 7]. It is based on the eigen-decomposition of postprocessed STFD matrices, defined in Eq. (2), corresponding to multiple t-f points. Averaging and joint block-diagonalization are two techniques proposed to integrate STFD matrices at multiple t-f points [3, 7]. The SNR enhancement achieved via the selection of energy concentrated t-f locations, and possibly through source number reduction and signal discriminations in the t-f domain, allows the t-f MUSIC algorithm to be much more robust than the conventional MUSIC algorithm [7]. By performing joint block-diagonalization of the STFD matrices Dxx (t, f ) of equations (10) over several (t, f ) points where the energy of the signal arrivals is concentrated, we obtain the signal and noise subspaces, represented as matrices Us and Un , respectively [3]. Also, denote the following steering matrix · ¸ a(θ) 0 B(θ) = (11) 0 a(θ) 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 kB(θ)ck
(12)
T
where the vector c = [c1 c2 ] is a unit norm vector with unknown coefficients. In (12), we have used the 1 1 fact 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,
source 1 source 2 source 3
c
H
= [min c B c
−1 (θ)Un UH . n B(θ)c]
Polar. (deg.) 45 −45 20
0
−1 = [min bH (θ, c)Un UH n b(θ, c)] H
Table 1: Signal parameters start end DOA freq. freq. (deg.) 0.20 0.40 −3 0.22 0.42 3 0.10 0.10 5
0.05 0.1
(13)
It can be shown that the minimum value within the brackets is given by the minimum eigenvalue of the matrix BH (θ)Un UH n B(θ), and the minimizing vector c is the corresponding eigenvector [8].
Normalized frequency
P (θ)
the conventional MUSIC, polarimetric MUSIC, and tf MUSIC. In the Simulations, we consider a uniform linear array of four dual-polarization antennas with half-wavelength inter-element spacing. To fully demonstrate the advantages of the proposed PTF-MUSIC, 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 their respective normalized starting and end frequencies, DOAs (measured from the the broadside), and the linear polarization angle (measured from the vertical axis to horizontal axis). All signals have the same signal power (SNR=13dB). The data length is 256 samples. The Wigner-Ville distribution (WVD) averaged over the four sensors and both polarizations is shown in Fig. 1. For both t-f MUSIC and PTF-MUSIC, the array averaged WVD is used to mitigate crossterms. Based on the averaged WVD, only the t-f points on the chirp signatures of sources 1 and 2 are considered for STFD and SPTFD matrix construction. In Figs. 2–5, the MUSIC spectra are plotted and compared for different methods. Results from five random 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.
0.15 0.2 0.25 0.3 0.35 0.4 0.45
5. SIMULATIONS The above expressions are used to calculate the PTFMUSIC spectrum and the results are compared with
0.5
Figure 1
50
100 150 Time (samples)
200
WVD averaged over sensors.
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Conventional MUSIC
Time−Frequency MUSIC
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Figure 2
6. CONCLUSION In this paper, the spatial polarimetric time-frequency distributions (SPTFDs) were introduced. Based on SPTFD, we proposed the polarimetric time-frequency MUSIC method which, in such scenarios where the signals are highly localized in the time-frequency domain and diversely polarized, significantly outperforms the other existing MUSIC methods, including timefrequency 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] A. Belouchrani and M. G. Amin, “Time-frequency MUSIC,” IEEE Signal Processing Letters, vol. 6, pp. 109–110, May 1999. [4] 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.
−5 −20
−3 3 Direction−of−arrival (degrees)
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MUSIC spectra. [5] 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. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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.
