Singular Value Decomposition-Based Source Enumeration of Uniform-Linear Array with Hankel Matrix of Output Correlations

ZHANG Zhengchao, YAO Guijin, LI Yue

(College of Communication Engineering, Jilin University, Changchun 130012, China)

0　Introduction

In a variety of the sensor-based studies such as radar and sonar detection, speech and image processing, model analysis, and condition monitoring of engineering structures and systems, the information of source number/model order is essential for parameter estimation and information extraction of signal field. However, in practice, source number has been usually unknown, therefore source enumeration has been a crucial preprocessing step in many sensor-based studies, especially for the high-resolution subspace method in the DoAs (Direction-of-Arrivals) estimation its performance relies on the prior perfect information of source number. As the consequence, in the last three decades the various approaches have been developed to tackle this problem in array processing.

The property of source signals in sensor array has wide diversity and governs the applicability of the existing various enumeration methods. The two conventional classifications are extensively mentioned: one is independent and coherent, the other is narrow-band and wide-band. Employing the distribution characteristic of the eigenvalues of the spatial covariance matrix, information criterion methods[1,2]and hypothesis testing[3,4]are effective for the narrow-band independent signals and break down for the coherent case on account of the reduced rank of the signal covariance matrix. With the summation of the overlapping subarrays partitioned from the spatial covariance matrix, the SFBS(Spatial Forward/Backward Smoothing)[5,6]method has been regarded as a competitive tool of signal decorrelation. Because the original distribution characteristic of the eigenvalues still remain well after the smoothing operation, SFBS method has been suitable for the enumeration of the coherent source signals and the independent ones. However, the distinction or threshold between the signal eigenvalues and the unknown noise variance need to be well considered, especially for sensor array with the multi-noise-variance and the low SNR(Signal-to-Noise Ratio). Because phase differences of sensor outputs are no longer just dependent on the DoAs alone, but also depend on the frequency, the realization of the wide-band processing[7-9]has been performed in the frequency domain by dividing the wide-band signals into the non-overlapping narrow bands with the DFT(Discrete Fourier Transform). Since the number of snapshots in the time domain has no countpart parameter in the frequency domain, many narrow-band enumeration methods involve the number of snapshots fail to be applied to the wide-band case.

Furthermore, the properties of source signals received by sensor array are prior unknown to the processor in practice, enumeration methods or the subsequent modifications rely based on the unknown properties of source signals have undoubtedly weighed on their applications and potentials. An enumeration method that still works without knowing these properties is what the practice wants. Therefore, a more adaptable enumeration method has been proposed. Besides increasing practical value greatly when properties of source signals are prior unknown, it provides intense research in array signal processing with a new way of thinking.

It has been recognized that the SVD of Hankel matrix is an extensively-used robust method for determination of source number/model order in the variety of the practical engineerings. Likewise, Hankel matrix has been the promising tool to investigate the problems or topics related to the sensor-based studies[10-18]. To improve the detection capability, Gelli had introduced Hankel matrix of the cyclic conjugate cross correlation functions for the minimum-redundancy linear array[10]. Similar to the staggered operation of the subarrays in ESPRIT method, two ‘staggered’ Hankel matrices were developed into the noted MP(Matrix Pencil) method[11-13]. Hankel matrices consisting of the correlations of the outputs of the selected sensors had been derived for source enumeration[14]and DOA estimation in the SUMWE(Subspace-based Method Without Eigendecomposition)[15]. Utilizing Hankel matrix of the exponential sinusoidal model, Badeau had investigated the eigenvalue property and error bound of two cases of the overestimation and underestimation of model order[16]. Recently Hankel matrix was also exploited for the multidimensional array processing[17,18]. However, the forms of the constructed Hankel matrices had been confused and far from definite, in which the processing of noise term were different, noise term was not considered under the special noiseless condition[10], or excluded from Hankel matrix[15], or still included[11-14,16,17], or both discussed[18]. For the enumeration method based on Hankel matrix, due to the form variety, the adaptabilities of Hankel matrix to the different properties of source signals and enumeration behaviors and capabilities have not been clear. Hence formation mechanisms of the different forms of Hankel matrix and the effects on source enumeration and DoAs estimation have been investigated. It is significant to make the application of HANKEL matrix more specific in structure and exclude interference of the term of the unknown noise variance for source enumeration and DoAs estimation.

The paper is structured as follows: In Section 1 Hankel matrix of the correlations of sensor outputs has been constructed for the ULA. The proposed SVD-based enumeration method of Hankel matrix is described after the analytical factorized form is derived in Section 2. Analyzed with computer simulations are in Section 3. The conclusions are put into Section 4.

1　The Construction of Hankel Matrix of ULA

1.1　Array Signal Model

Considering an ULA composing of theMidentical sensors that receive theNfar-field narrow-band signalssi(t)(i=1,…,N) from the directions ofθ1,θ2,…,θN. Using the complex data representation, theM×1 column vector of sensor outputX(t) is expressed as

X(t)=A(θ)S(t)+N(t)

(1)

whereX(t)=[x1(t),…,xM(t)]T, source signal vectorS(t)=[s1(t),…,sN(t)]Tand noise vectorN(t)=[n1(t),…,nM(t)]T. Array steering vector or array reponseA(θ)=[a(θ1),…,a(θN)], and

(2)

where (·)Tdenotes the transpose operation,vi∈Cis phase delay or difference of source signalsi(t) between the two adjacent sensors,λand Δdare carrier wavelength and sensor spacing. The two conventional signal assumptions are following:

1) Source signalssi(t)(i=1,…,N) are the temporally complex stationary Gaussian random processes with zero mean and satisfy

E{S(t)SH(t)}=P

(3)

2) The additive noises are the temporally and spatially stationary complex white Gaussian random processes with zero mean and equal varianceσ2and independent of source signals.

E{N(t)NH(t)}=σ2I

(4)

The spatial covariance matrixRof sensor outputs is expressed as

(5)

where (·)Hdenotes the conjugate transpose,λiandui(i=1,2,…,M) are the eigenvalues and eigenvectors ofR, respectively. The eigenvalues ofRare ordered as

λ1≥λ2≥…≥λN>λN+1=…=λM=σ2

(6)

in (6) the eigenvalues are partitioned two groups, the firstNeigenvalues whose eigenvectors span the signal subspace of the observed space are larger thanσ2. The lastM-Neigenvalues whose eigenvectors span the noise subspace are equal to noise varianceσ2. The distribution property of the eigenvalues of the spatial covariance matrixRhas been the ground of the most existing enumeration methods including information criteria methods and hypothesis testing.

1.2　Construction of Hankel Matrix of the Noise-contaminated ULA

For the noise-contaminated ULA, we construct a specific Hankel matrix of the spatial correlations of sensor outputs in which the term of the unknown noise variance is not contained.

(7)

(8)

wherePis the number of snapshots.

2　The Factorization of Hankel Matrix

2.1　The Factorization of Hankel Matrix

The general factorized form of Hankel matrixHn×nis known as[19]

Hn×n=VDVT

(9)

(10)

(11)

(12)

(13)

1) The form of the diagonal matrixDbecome known and the referenced index is independent ofV1(θ) andV2(θ), all of which help to simplify rank analysis of Hankel matrix.

2.2　Rank Property of Hankel Matrix

(14)

2.3　The SVD-based Enumeration Method of Hankel Matrix

(15)

whereUis anm×mleft singular vector matrix andVis ann×nright singular vector matrix.Σis anm×nsingular value matrix and

(16)

where the non-zero singular valuesσ1≥σ2≥…≥σN>0. Then the SVD-based enumeration can be transformed into the rank determination of Hankel matrix. Source number is equivalent to that of the non-zero singular values.

The same enumeration capability is obtained as the referenced indexj=M. For the other referenced indexes, enumeration capabilities are less than int[M/2].

It is shown that Hankel matrix has the same enumeration capability concluded by the forward smoothing scheme as the referenced indexj=1 orM. However, since the term of the unknown noise variance is excluded from the consturcted Hankel matrix, theoretically, the effect of noise variance on the SVD-based enumeration method is removed, which is a feasible way to solve enumeration problem for the low SNR case.In addition, the conventional gap or distinction between signal eigenvalues and the unknown noise variance is converted into that between the non-zero singular values and zero, besides sensor array with one noise variance, the proposed SVD-based enumeration method is a promising tool for the more complicated array with the changeable noise variance or with the multi-noise-variance.

The implementation of the SVD-based source enumeration of Hankel matrix is summarized as follows.

Step2A feasible judgment thresholdλTis preset, and the number of the normalized singular values exceedλTis counted and let be equal to the estimated source number.

3　Numerical Simulation and Discussion

The proposed SVD-based enumeration methods of the single Hankel matrix have been tested through the computer simulations considering the uniform linear array of the half of wavelength separation. The Gaussian white random processes are employed to simulate the three far-field narrow-band source signals with the directions of [-30°,-5°,10°] and sensor noises in the following three scenarios. Statistical performance is evaluated by performing 200 Monte Carlo runs for each SNR. For comparison, two forms of Hankel matrices are constructed: the proposed one, and the other that contains the term of noise variance. It can be predicted that the SVD-based enumeration methods with two forms asymptotically approach the same performance as SNR tends to infinite. Therefore the performance comparison of two forms of Hankel matrices has been focused on the low and middle SNR regions.

表1　HANKEL矩阵的NSVs

a　矩阵　矩阵　矩阵图1　信号独立时矩阵和的基于SVD的估计方法的PCEs与SNRs和门限值之间关系Fig.1　The 2D contours of the SVD-based PCEs of the matrices (a) (b), and (c) versus SNRs and threshold values for the independent signals

In summary, because of the larger threshold scope and more stable distribution property of the symmetical ‘arch’ form, it is worthwhile to utilize the proposed Hankel matrix to perform the SVD-based enumeration, especially in the low SNR region. In addtion, it is noted that the addition of the term of noise variance into Hankel matrix and the corresponding number have the changeable and unforeseeable effect on distribution property of the PCEs and lead to the difficulty of threshold setting and source enumeration.

4　Conclusions

A SVD-based enumeration method of Hankel matrix of the correlations of sensor outputs has been developed for the ULA. Under the analytical factorized form, the form of Hankel matrix of sensor array has been derived by the exclusion of the term of the unknown noise variance. It is found by rank analysis that the proposed enumeration method of Hankel matrix is valid to the independent, mixed and coherent source signals and enumeration capability is up to the half of sensor number.

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