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Double criterion-based estimator for signal number estimation for the colored noise with unknown covariance matrix

The subspace-based techniques are widely utilized to estimate the parameters of sums of complex sinusoids corrupted by noise, and the zoom ESPRIT algorithm utilizes the zoom technique to apply the ESPRIT to a narrow frequency band to improve the accuracy of frequency estimation. However, the Gaussian noise becomes non-Gaussian in the zoomed baseband after being filtered by a low-pass filter, and thus has an unknown covariance matrix. However, most exiting algorithms for model order estimation performs poorly for the case of colored noise with unknown covariance matrix. In order to accurately estimate the dimension of the signal subspace for the zoom ESPRIT algorithm, this paper proposes a novel strategy to estimate the number of signals for the case of colored noise with unknown covariance matrix. The proposed strategy is based on the analysis of the behavior of information theoretic criteria utilized in model order selection. Firstly, a first criterion is defined as the ratio of the current eigenvalue and the mean of the next ones, and its properties is analyzed with respect to the over-modeling and under-modeling. Secondly, a novel second criterion is designed as the ratio of the current value and the next value of the first criterion, and its properties is also analyzed with respect to the over-modeling and under-modeling. Then, a novel signal number estimation method is proposed by combining the second criterion with the first criterion to check whether the eigenvalue being tested is arising from a signal or from noise. The resulted signal number estimation method is called as the double criterion-based estimator as it utilizes two criteria to separate the signal eigenvalues from the noise eigenvalues. Finally, simulation results are presented to illustrate the performance of the proposed double criterion-based estimator and compare it with the existing methods.

Spectral analysis of signals is a major problem in statistical signal processing, and has wide applications in communications, radar, sonar, seismology, astronomy and so on [1]- [3]. The fast Fourier transform (FFT) processing is a well-known frequency estimation method due to its processing gain and low computational complexity [4].
However, the frequency resolution of FFT-based methods is limited to the number of samples. Based on FFT, spectral refinement methods such as the chirp z-transform (CZT) algorithm in [5] and the zoom FFT (ZFFT) techniques [6]- [7] have been proposed to increase the frequency resolution. The ZFFT algorithm is utilized to detect incipient rotor fault in a three-phase induction machine by using both stator current and stray flux sensors in [8], and a modified ZFFT algorithm is employed for GPR (ground penetrating radar) signal analysis in [9]. However, the ZFFT techniques for the highfrequency resolution require a long acquisition time.
Subspace methods such as multiple signal classification (MUSIC) algorithm [10] and an estimation of signal parameters via rotational invariance techniques (ESPRIT) [11] have been applied to estimate the frequencies while overcoming the frequency resolution limitation of the FFT method in low signal-to-noise ratios (SNRs). However, the subspace methods require long computation time to find frequency estimates when the autocorrelation matrix is large. In addition, in many real-world applications the number of samples is finite, and thus it is hard to increase the dimension of the autocorrelation matrix. The subspace methods are based on the eigen-structure of the autocorrelation matrix, and their performances are completely deteriorated by choosing a wrong dimension of the signal subspace or the noise subspace. Therefore, the subspace methods need accurate estimation of the signal subspace dimension, that is, of the number of harmonic components or signals that are superimposed and corrupted by noise. This estimation is particularly difficult when the signal-to-noise ratio (SNR) is low and the statistical properties of the noise are unknown. There are many methods to estimate the number of signals (or the dimension of the signal subspace). The traditional methods are based on the information theoretic criteria (ITC) such as AIC, MDL, etc. [12]- [14]. Utilizing the random matrix theory (RMT), a RMT estimator is proposed to estimate the number of signals via sequentially detecting the largest noise eigenvalues as arising from a signal or noise for a given over-estimation probability [15]. In [16], the linear shrinkage (LS) technique is employed to obtain more accurate estimates for the noise eigenvalues, and the LS-MDL estimator is proposed for signal number estimation. In [17], an improved AIC estimator is proposed for signal number estimation. In [18], a RMT estimator with adaptive decision criterion based on linear shrinkage (LS-RMT-ADC estimator) is proposed for signal number estimation when the number of samples is relatively small. However, these algorithms [12]- [18] assume that the noise is white Gaussian and the signals are uncorrelated, and will perform poorly when the signals are correlated or the noise is not white. In order to cope with this problem, some methods have been developed for determining the number of sources when the noise observations have an unknown covariance matrix [19]- [24].
The signal number estimation method in [20] makes use of two discriminant functions for separating the eigenvalues into two classes, whose separation limit can be found using two discriminant functions. However, this method may be suboptimal because of improper choice of the discriminant functions. The Gerschgörin disk estimator (GDE) criterion in [21] performs better than AIC and MDL for the case of colored noise, but it is less effective for white noise. In [22], the extreme value theory (EVT) was utilized to address the problem of model order selection associated with the detection of complex-valued sinusoids in the presence of unknown correlated noise, and a loglikelihood penalty term for EVT-based model selection was derived. In [23], the source detection was modelled as a multivariate regression problem and the model order was inferred from the covariance matrix of the residual error. In [24], a threshold-based model-order selection method is proposed without assuming precise distribution knowledge of the measurements. Specifically, a Hermitian Toeplitz data matrix is formulated and the signal number is estimated by thresholding the corresponding singular values. Although this method allows an unknown noise covariance matrix, it requires that the noise is sufficiently weaker than the signal. Therefore, its detection performance will degrade when the SNR becomes low.
As it was pointed out in [25,26], the estimation performance of high-resolution spectrum estimation methods degrades for closely spaced complex exponentials, and the estimates are vulnerable to large deviate from their true values. This severely limits the resolution of subspace methods for frequency estimation. In order to improve the frequency estimation performance of the subspace methods in the case of closely spaced frequencies, the frequency separation was artificially increased by decimating the signal before applying subspace-based methods [26]- [27]. In [26], an interleaving technique is proposed to improve the estimation performance of subspace-based methods for closely spaced sinusoidal frequencies. The idea is to artificially increase the frequency separation by using down sampling before estimating the data covariance.
In [27], a decimation approach is applied, and a total least squares problem is solved to obtain the frequency estimates of closely spaced signals. However, the down-sampling factor is chosen to make sure that no aliasing is introduced in [26,27]. Therefore, the original signal should be oversampled and thus will increases the memory requirement and computation complexity. In order to deal with this problem, a zoom MUSIC algorithm (ZMUSIC) that combined the zoom technique and the MUSIC algorithm was proposed to detect fault sensitive frequencies based on a frequency analysis of the stator current [28]. The ZMUSIC uses the zoom technique to focus on a narrow frequency band, and reduces the computational complexity by focusing on frequencies close to the fundamental frequency. However, the ZMUSIC obtains the frequency estimates from searching the pseudospectrum, and thus has a relatively high computation complexity. In order to overcome the drawbacks of the ZMUSIC algorithm, a zoom ESPRIT (ZESPRIR) algorithm that combined the zoom technique and the ESPRIT algorithm was proposed to estimate both the frequencies and amplitudes of the fault harmonics in [29]. The ZESPRIT algorithm has the advantages of providing more accurate estimates and being more computationally efficient than the ZMUSIC algorithm. However, the columns of the data matrix in [26]- [27] will have correlation to some degree due to the oversampling of the original signals, and thus the autocorrelation matrix may become ill-conditioned. In the ZMUSIC and ZESPRIT algorithms in [28]- [29], the noise becomes non-Gaussian in the zoomed baseband after being filtered by a low-pass filter, and thus will have an unknown covariance matrix.
In order to correctly estimate the dimension of signal subspace for the algorithms in [26]- [29], this paper considers the signal number estimation for the case of colored noise with unknown covariance matrix. To this aim, we design a new strategy to detect the number of sources without assuming precise distribution knowledge about the noise and the signals. The proposed strategy is inspired by the behavior of information theoretic criteria for model order selection in [30]- [31]. Firstly, a first criterion is defined as the ratio of the current eigenvalue and the mean of the next ones, and its properties is analyzed with respect to the over-modeling and under-modeling. Secondly, a novel second criterion is designed as the ratio of the current value and the next value of the first criterion, and its properties is also analyzed with respect to the overmodeling and under-modeling. Then, a novel signal number estimation method is proposed by utilizing both the first criterion and the first criterion to check whether the eigenvalue being tested is arising from a signal or from noise. The resulted signal number estimation method is called as the double criterion-based estimator as it utilizes two criteria to separate the signal eigenvalues from the noise eigenvalues. Finally, simulation results are presented to illustrate the estimation performance of the proposed double criterion-based estimator and compare it with the existing methods.
The remainder of this paper is organized as follows. Section 2 is the problem formulation and the related works that motivate the current work. In Section 3, proposed double criterion-based estimator for signal number estimation is presented. Simulation results and comparison of the proposed method with other approaches are provided in Section 4, followed by conclusions in Section 5.

Problem formulation and related works
In this Section, data model and problem formulation are firstly described. Then, the related works that motivate the current work are introduced.

Data model
It is assumed that the discrete-time signal ( ) y n can be represented by K complex sinusoids corrupted by noise [26], i.e., where N is the number of sample data, K is the number of pure sinusoids, ( ) w n is additive white Gaussian noise with zero mean and variance 2 σ , and ( ) i x n is the th i complex sinusoid wave (2 ) ( ) where s F is the sampling frequency. It is assumed that the amplitudes { } The subspace methods, such as the popular MUSIC [10], ESPRIT [11] can be utilized to estimate the frequencies { } i f , and the first step is the formation of a finite sample estimate of the data covariance matrix. Let Also, let in (4) can be rewritten as The vector ( ) n y is of dimension 1 M × . From (8), the following data matrix of dimension M L × is formed: The subspace methods are constructed on the basis of the sample covariance of ( ) n y in (8) Then, the eigen-decomposition of ˆy R can be written as  is a diagonal matrices containing eigenvalues of the signal subspaces arranged in descending order. The dimension K of the signal subspace (i.e., the number of sinusoids) can be estimated by using methods in [12]- [18].

Related works and motivation
Based on the eigen-decomposition of ˆy R in (13), the frequencies of signals can be estimated by the subspace methods. In the following, two subspace techniques are reviewed: ESPRIT estimator [11], and ZESPRIT estimator [29].

1) ESPRIT estimator:
The ESPRIT estimator exploits an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure [11]. Here, the least squared ESPRIT is described.
Let k v be the eigenvalues of LS Φ , 1 , 2, , k K =  . Then, the signal frequencies can be estimated using the following expression: In many applications, there are numerous sinusoids in a given frequency range and are affected by the signal-to-noise ratio. Therefore, the MUSIC method has to increase the dimension of the data matrix Y in (9), and thus takes a long computation time to find more frequencies.
As is illustrated in [32], it has been empirically found that gives a good performance.

2) ZESPRIR algorithm:
The ZMUSIC algorithm in [28] as well as the ZESPRIT algorithm in [29] is based on frequency shifting and decimating a discrete-time signal to determine the spectrum in a specified frequency bandwidth 1 1 [ , ] p f f F + , and the flowchart of the ZMUSIC (ZESPRIT) algorithm [28,29] is shown in Fig. 1. In the following, the ZESPRIT algorithm in [29] is briefly described.
Sampling for points where shift 1 0.5 p f f F = + , with 0.5 p F the cutoff frequency of the low-pass filter to avoid frequency aliasing. According to the translation of the discrete Fourier transform (DFT), the following relationships are obtained: Then, the autocorrelation of A Y is calculated as Then, the eigen-decomposition of A Y R can be written as where A E is the matrix of eigenvectors, and A Λ is a As the number of signals in the frequency range 1 (24) can be significantly reduced. As is suggested in [28], a reasonable estimate of the autocorrelation matrix dimension is obtained with . Therefore, the computational complexity of the ZESPRIT algorithm is significantly less that of the ESPRIT algorithm.
In practice, the number 0 K of the signals in the frequency range 1 1 [ , ] p f f F + cannot be known in advance, and thus has to be estimated. However, after being filtered by a low-pass filter, the noise becomes non-Gaussian in the zoomed baseband [ 2,2] p p F F − , and thus has an unknown covariance matrix. Therefore, the existing signal number estimation algorithms [12]- [24] perform poorly when utilizing the eigenvalues ,1 , 2 , (26). In order to solve this problem, in this paper a novel method called as double criterion-based estimator for signal number estimation is proposed for signal number estimation by analyzing the behavior of information theoretic criteria of the eigenvalues A Λ in (26), and will be presented in next Section.

Double criterion-based estimator for signal number estimation
As is illustrated in [30], the asymptotic estimation bounds of BIC (Bayesian Based on above observations (a) and (b) and with the aim to make a clear discrimination between the smallest signal eigenvalue and the largest noise eigenvalue, a novel decision criterion is proposed and is defined as follows: As the index k of the maximum value of , ( ) Step 1: Estimate the index of the maximum value of , ( ) Step 2: Estimate the number of signals as follows: (1) If ˆ1 k = , then the number of signals is estimated as 0 1 K = . This is because whether there are signals in the specific frequency band [ 2,2]  λ is arising from a signal, and thus the signal number is estimated as 0K k = .
Based on above discussions, the proposed method for signal number estimation is summarized in Algorithm 1.

Algorithm 1. Proposed method for signal number estimation
Step

D. Simulation results
In this section, simulations are carried out to evaluate the detection performance of the proposed double criterion-based estimator for signal number estimation and compare it with the existing methods including the conventional AIC estimator [12], the conventional MDL estimator [13], the LS-MDL estimator in [16], the MAIC estimator in [17], the discriminant functions-base method in [20], and the source enumeration method for the case of colored noise in [23]. To simplify the definition of signal-to-noise ratio (SNR), the scenario where K narrow-band and unit-power signals in a specific frequency band (34) It should be noted similar results can be obtained for the scenario of unequal-power signals. In addition, the performance measure for the signal number estimation is the correct estimation probability defined as P r( ) ( The parameters utilized in the simulations are set as follows.  Fig. 2 shows the probability of correct estimation of various algorithms versus SNR when the zoom factor 32 A = , and Fig. 3 shows corresponding results when the zoom factor 64 A = .
As can be seen from Fig. 2 and Fig. 3, the correct estimation probability of the proposed method is less than other methods when the SNR is relatively small. However, the correct estimation probability of the proposed method becomes higher than other methods when the SNR becomes larger. Moreover, when the SNR is sufficiently large, the correct estimation probability of the proposed method is 100%, while the correct estimation probability of the AIC estimator, MDL estimator, LS-MDL method, the MAIC method and the discriminant function-based method in [20] is 0%, and the correct estimation probability of the method in [23] is about 68%. Therefore, the proposed method is a consistent estimator, while the other algorithms are not consistent estimators for signal number estimation in this case. corresponding results when the zoom factor 64 A = . As can be seen from Fig. 4, the correct estimation probability of the proposed method is less than other methods when the SNR is relatively small. However, the correct estimation probability of the proposed method becomes higher than other methods when the SNR becomes larger. Moreover, when the SNR is sufficiently large, the correct estimation probability of the proposed method is 100%, while the correct estimation probability of the AIC estimator, MDL estimator, LS-MDL method, the MAIC method and the discriminant function-based method in [20] is 0%, and the correct estimation probability of the method in [23] is about 68%. Therefore, the proposed method is a consistent estimator, while the other algorithms are not consistent estimators for signal number estimation when the zoom factor 32 A = . As can be seen from Fig. 5, the probability of correct estimation of the proposed method is 43% when the SNR is -12dB and approaches about 96% when the SNR becomes large, while the other methods don't work properly when the zoom factor 64 A = .

Conclusions
In this paper, the model order estimation for the zoom subspace techniques such as the ZMUSIC/ZESPRIT algorithm was investigated, and a novel strategy called as the double criterion-based estimator has been proposed to estimate the number of signals for the case of colored noise with unknown covariance matrix. The proposed strategy is based on the analysis of the behavior of information theoretic criteria utilized in model order selection. Firstly, a first criterion is defined as the ratio of the current eigenvalue and the mean of the next ones, and its properties is analyzed with respect to the over-modeling and under-modeling. Secondly, a novel second criterion is designed as the ratio of the current value and the next value of the first criterion, and its properties is also analyzed with respect to the over-modeling and under-modeling. Then, the first criterion and the second criterion are combined to check whether the eigenvalue being tested is arising from a signal or from noise. Finally, simulation results have been presented to show that the proposed double criterion-based estimator has better detection performance than the existing methods.