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Spatial timefrequency distribution of cross termbased directionofarrival estimation for weak nonstationary signal
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 239 (2019)
Abstract
In the radar array signal processing direction of arrival (DOA), the estimation of weak nonstationary signal is an important and difficult problem when both strong and weak signals are coexisting particularly because the weak nonstationary signals are often submerged in noise. In this paper, we proposed a novelty method to estimate the direction of arrival (DOA) of weak nonstationary signal in scenario for strong nonstationary interference signals and Gaussian white noise. The method utilizes spatial timefrequency distribution (STFD) of cross terms rather than suppressing cross terms in timefrequency analysis. The STFD of cross terms are introduced as an alternative matrix, which is similar to data covariance matrix in multiple signal classification (MUSIC), for the DOA estimation of a weak nonstationary signal. The crossterm amplitude of the strong and weak signals is usually above the noise and is easier to use than the autoterm of the weak signal. In the cross term, the information of the weak signal is included, and the autoterm of these weak signals is difficult to extract directly. The ability to incorporate the STFD of cross terms empowers information about a weak nonstationary signal for DOA estimation, leading to improved signal estimates for direction finding. The method based on the STFD of cross terms for DOA estimation of the weak nonstationary signal is revealed to outperform the timefrequency MUSIC and traditional MUSIC algorithm by simulation, respectively. This method has the advantages of the timefrequency direction finding method and also deals with the situation of weak signals. When the strong and weak signals exist at the same time and the two angles are similar, the crossterms can be used to perform DOA estimation on the weak signal.
Introduction
Among numerous nonstationary signals that arise in many radars [1] and communication [2, 3], instantaneous frequency (IF) signals, for instance, linear frequency modulated (LFM) signals, have obvious timefrequency characteristics which are continuous and decided the location. Similar to the timefrequency signatures, the spatial signature of the signal source also includes significant information about the signal source [4]. It ensures signal source identification due to the respective angle position received from a receiver antenna array, which is the directionsofarrival. The characterization of DOA is viewed as steering vectors in which the source signal demonstrates a difference of the signal phase over every sensor antennas when electromagnetic wave covers the receiver antenna array [5]. Timefrequency (TF) analysis plays important roles in the DOA estimation of the nonstationary signal [6].
Timefrequency analysis enables to process nonstationary signals overlying in both frequency and time domains in which windowing and filtratingbased means cannot separate the different signal source components [7, 8]. For analyzing the nonstationary signals, such as LFM signals, timefrequency signal representations and analyses are necessary [9]. We engage in the class of signals where the instantaneous frequency particularly or basically determines the timefrequency signatures of the signal source. A successful application of timefrequency distribution desires prior knowledge for the signal source in order that the most advisable distribution is preferred [10]. In this article, WignerVille distribution (WVD) is considered as the timefrequency distribution representation since it provides the most energy concentration in timefrequency domains, displays the nonstationary properties of the signal, and satisfies the marginal conditions [11]. Combining the timefrequency and spatial characteristic is accomplished into a framework named STFDs in timefrequency MUSIC (TFMUSIC) [12]. This structure applies the signal location characteristic and energy concentration for increasing signaltonoise ratio (SNR) and source signal identification before achieving the highresolution directionofarrival estimation [13]. The framework desires the calculations of the WVD from the data obtained at each sensor, for instance, autoterms of WVD and the cross terms of WVD between sensor antennas.
When the analyzed signal includes more than one signal source component, WVD performs signal autoterms as positive magnitudes at their instantaneous frequency areas and the cross terms as oscillatory magnitudes along the geometrical middle point of autoterms [14]. The crossterm problem of the WVD was first indicated in [15]. The suppression of cross terms is the core problem of bilinear timefrequency transformation. The references are abundant studies to suppress the cross terms and increase the timefrequency resolution. A method combined of the Hough transform (HT) and the WignerVille distribution (WVD) performs cross term suppression [14, 16]. Another study researched the blind source separation approach based on the use of timefrequency analysis to eliminate cross terms in WVD [17]. The main purpose of research in [18] is to accomplish the highresolution and the maximal crossterm contraction with the desirable diagonal or offdiagonal peculiarity of timefrequency distribution matrices in blind source separation applications. Researchers proposed a method named standardization of the pseudquadratic form to suppress the cross terms in [19]. In order to suppress the cross term, Zuo et al. further propose a smoothed highresolution timefrequency rate representation (SHRTFRR) via utilizing an FR window to the highresolutiontime frequency rate representation, which is expressed in the convolution form [20, 21]. A pure idea is offered by Aiordachioaie and Popescu, as starting initial point to compose an approach to suppress the cross terms and to attain an exact image of WVD, including only the autoterms [22]. A new method is offered by Wu and Li to suppress cross terms in the WVD of linear frequency modulation signals with multicomponent [23].
The existence of cross terms is difficult to be avoided. There are some researches in the references that take full use of cross terms in WVD in different research fields. Bird song syllable classification is realized using cross terms of WignerVille ambiguity function in [24]. The Wigner distribution achieves artifacts well known as cross terms that are contemplated and undesirable in some situations; nevertheless, it can be utilized to ascertain the existence of greatly little signal source terms, in this case, VLPs terms [25]. Using the ambiguity domain interpretation, Jeong and Williams explored the theory of the cross terms (or interferences) in spectrograms [26]. A blind source separation technology applying both cross terms and autoterms in the timefrequency distributions of the source signals was considered by Belouchrani et al. in [27]. As an application, Fadaili et al. showed that the source separation can be accomplished via exploiting one of these algorithms to a set of spatial quadratic timefrequency distribution matrices corresponding merely to the named cross terms and/or to the named autoterms [28]. When the cross term is considered from different perspectives, the cross term accurately reflects the relationship between multiple signal components [29]. When the energy difference between nonstationary signals is large, specifically in the situation of low signaltonoise ratio, a weak nonstationary signal may be buried in the noise. At this point, the weak nonstationary signal is the desired signal. It is difficult or almost impossible to extract the autoterm of the weak nonstationary signal. The crossterm amplitude of strong and weak nonstationary signal did not decrease significantly. In general, the cross terms contain sufficient information of weak nonstationary signal for its DOA estimation. Therefore, in the case that both strong and weak nonstationary signals existing simultaneously, the cross terms of STFDs are used to obtain DOA of the desired weak nonstationary signal in this paper.
This paper includes some sections. Section 2 reviews spatial timefrequency distribution in TFMUSIC. In Section 3, crossterm selection procedures of STFDs are introduced. The analytical results are used in this section in order to examine the proposed method performance. Several simulations are offered in Section 4. Section 5 gives conclusions.
Related works
In narrowband signal array processing, because n source signals access on a melement uniform linear array, the received data formula
is frequently presumed, where the m × n spatial matrix A(θ) = [a(θ_{1}), a(θ_{2}), …, a(θ_{n})] implies the steering matrix. In the directionofarrival estimate situations, we expect A to be a wellknown characteristic, and each column of matrix A associated with a single signal direction drives a distinct orientation. The analytical operation in this article does not rely upon any special matrix A structure characteristic. Because of the synthesis of the source signals at every antenna, the parts of the m × 1 data vector x(t) are multicomponent source signals; nevertheless, every source signal d_{i}(t) of the source signal vector d(t) is often a single signal component. n(t) is an additive zeromean white complex noise vector in which elements are presumed as temporal and spatial stationary random processes which are independent from the source signals.
In (1), it is presumed that the quantity m of sensors is more than the quantity n of source signals. Furthermore, matrix A has a column full rank because it involves that the steering vectors associated with n different directions of arrival have linear independence relationship. We furthermore presumed that the data covariance matrix
is not singular and that the processing duration includes N snapshots (N > m), where superscript H implies conjugate transpose, and E(·) implies the statistical expectation operator. From the above presumptions, the data correlation matrix is provided by
where R_{dd} = E[d(t)d^{H}(t)] is the signal correlation matrix, σ^{2} is the noise power at every sensor antenna, and I implies the identity matrix. Let λ_{1} > λ_{2} > … > λ_{n} > λ_{n + 1} = λ_{n + 2} = … = λ_{m} = σ^{2} imply the characteristic values of R_{xx}. The λ_{i}, i = 1, …, n are individual. The eigenvectors corresponding to λ_{1}, …, λ_{n} make up the columns of the matrix S = [s_{1}, …, s_{n}], and those associated with λ_{n + 1}, …, λ_{m} constitute the matrix G = [g_{1}, …, g_{m − n}]. Because the column vectors of A and S constitute their subspace, A^{H}G = 0.
In effect, R_{xx} is not known and could be approximated via the applicable data snapshots x(i), i = 1, …, N. The approximated data covariance matrix is provided from
Let \( \left\{{\hat{\mathbf{s}}}_1,\dots, {\hat{\mathbf{s}}}_n,{\hat{\mathbf{g}}}_1,\dots, {\hat{\mathbf{g}}}_{mn}\right\} \) imply the unitnorm eigenvectors of \( {\hat{\mathbf{R}}}_{\mathbf{xx}} \) that are arranged according to a descending order of the corresponding eigenvalues and make \( \hat{\mathbf{S}} \) and \( \hat{\mathbf{G}} \) imply the matrices determined by the set of vectors \( \left\{{\hat{\mathbf{s}}}_i\right\} \) and \( \left\{{\hat{\mathbf{g}}}_i\right\} \), severally. We retrospect that the DOAs can be evaluated by the traditional MUSIC approach via resolving the n values of θ for which the subsequent spatial spectrum is performed by maximization operation [30]:
where a(θ) is the steering vector associated with θ.
We then survey the concept and fundamental peculiarities of the STFDs. In this article, we straightforwardly examine a kind of Cohen’s class, specially, the WignerVille distribution (WVD) as well as its characteristic. The discrete formula of WVD of a signal source x(t), via an odd length L rectangular window, is given by
where * implies complex conjugate. The integral of the product of the conjugate of the signal reflects the energy distribution of the signal in the time and frequency dimensions. Formula (6) reflects the details of the signal energy distribution. The spatial WVD matrix is attained via changing x(t) by the data snapshot vector x(t)
Replacing (1) into (7), we attain
We note that D_{yy} (k_{t},k_{f}), D_{yn} (k_{t},k_{f}), D_{ny} (k_{t},k_{f}), and D_{nn} (k_{t},k_{f}) are matrices of m × m dimension. Due to the uncorrelated noise and signal presumption and the zeromean noise property, the mathematic expectation of the crossterm STFD matrices between the noise and signal vectors equal to zero, such as E[D_{yn}(k_{t}, k_{f})] = 0 and E[D_{ny}(k_{t}, k_{f})] = 0, and it pursues that
where the signal source timefrequency distribution matrix
is of n × n dimension matrix. For signal array processing researches, the hybrid matrix A includes the azimuth and projects the autoterms and cross terms in timefrequency distributions of the source signals to the autoterms and cross terms in timefrequency distributions of the received data.
Formula (9) is comparable to the other formula that has been generally utilized for direction finding problems, involving the source signal covariance matrix to the data spatial covariance matrix. From the above production, the covariance matrices are reestablished via the STFD matrices. The reestablished formulas for traditional array signal processing could be used, and core problems for various situations of array processing, especially those addressing nonstationary signal situations, can be addressed by bilinear transformations. It is prominent that (9) suits for every point. For reducing the noise influence and ensure the column full rank character for the involved matrix, many timefrequency points are used, contrary to a single one. Joint diagonalization [31] and the timefrequency averaging approach become the two key technologies that have been utilized for the objective [7, 13]. In this article, we merely use averaging operation for many timefrequency points.
The timefrequency distribution transforms the onedimensional time domain source signals into the twodimensional timefrequency domain source signals. The timefrequency distribution characteristic of accumulating the incoming signal while widening the noise to the integrated timefrequency domain enhances the efficient SNR. Next, we calculate the signal subspace and noise subspace projection through a finite snapshot quantity of data. In the situation where the STFD matrices are averaged for the timefrequency signatures, we consider those N–L + 1 timefrequency distribution points. The result is provided via
The unitnorm eigenvectors corresponding to \( {\hat{\lambda}}_1^{\mathrm{tf}},\dots, {\hat{\lambda}}_{n_0}^{\mathrm{tf}} \) are implied by the columns of \( {\hat{\mathbf{S}}}^{\mathrm{tf}}=\left[{\hat{\mathbf{s}}}_1^{\mathrm{tf}},\dots, {\hat{\mathbf{s}}}_{n_0}^{\mathrm{tf}}\right] \), and those associated with \( {\hat{\lambda}}_{n_0+1}^{\mathrm{tf}},\dots, {\hat{\lambda}}_m^{\mathrm{tf}} \) are implied by the columns of \( {\hat{\mathbf{G}}}^{\mathrm{tf}}=\left[{\hat{\mathbf{g}}}_1^{\mathrm{tf}},\dots, {\hat{\mathbf{g}}}_{m{n}_0}^{\mathrm{tf}}\right] \). The superscript tf implies that the associated term is inferred from the matrix \( \hat{\mathbf{D}} \). Parallelly, for timefrequency MUSIC with n_{0} source signals considered, the DOAs are confirmed via confirming the n_{0} peaks of the spatial spectrum which are determined from the signals’ timefrequency domain:
Method
The virtues of timefrequencybased DOA estimation approach may merely be realized when suitable timefrequency points are considered in the STFD matrices. The key point of this kind of method is how to choose the suitable timefrequency points. The purpose of this paper is to estimate the DOA of weak nonstationary source signals based on cross terms of spatial timefrequency distribution in the presence of strong nonstationary signal interference. In the STFD framework, the source signal timefrequency characteristics should not be highly overlapping. The source STFD matrix is
The two source signals, d_{1}(k_{t}) and d_{2}(k_{t}), are LFM signals. The following four types of timefrequency points are discussed. The firsttype timefrequency points are associated with signal autoterms merely. For those points, the source signal timefrequency distribution matrix has a rankone diagonal mathematics structure. The secondtype timefrequency points are associated with source signal crossterms merely. For the points, the signal timefrequency distribution matrix is offdiagonal matrix. That is that the matrix is considered to be offdiagonal because their diagonal elements equal to zeros. The thirdtype timefrequency points are associated with both source signal autoterms and cross terms. For the points, the signal timefrequency distribution matrix has no obvious algebraic specific structure which can be immediately used. The source signal cross terms and source signal autoterms are inexistence in the fourth timefrequency points.
The diagonal and offdiagonal mathematics structures of the first and secondtype TF points are frequently destroyed when the source signals are mixed. The first, second, and thirdtype TF points are significant to the DOA estimation problem. The fourth should be abandoned because they do not have any effect in this situation. In this paper, we exploit cross terms of spatial timefrequency distribution to perform DOA estimation of weak nonstationary signals when there are both strong and weak nonstationary signals.
Because of the fact that in the firsttype timefrequency points, D_{dd}(k_{t}, k_{f}) have a high outstanding algebraic structure which is a rankone diagonal matrix, an outstanding mean to solve the directionofarrival estimation problem will become to utilize matrix decomposition technology, which is a traditional technology for DOA estimation. Yet, the procedure of the automatic timefrequency point’s selection, in the general situation, is difficult. Complicated timefrequency point selection technologies will be usually needed, as studied in the following.
Under the above condition, the STFDs have the structure as follows:
Consider an n × m matrix W, named a whitening matrix, in order that WA implies a unitary matrix and implies U. It is
Pre and postmultiplying D_{xx}(k_{t}, k_{f}) by W results in the whitened matrix, defined as
where the second equation results from the W concept and (14). Distinctly, the whitening process results in a linear model in which a unitary mixed matrix is structured. In a whitened situation, some technologies use trace invariance in the matrix for unitary transform, taking it likely to judge the existence of signal cross terms. One technology [27] introduces that for the secondtype TF points, consider matrices that verify
where trace {·} implies matrix mathematics trace, · implies mathematics Frobenius norm, and ε is a userdetermined positive small value. For a noise condition, the selection procedure of timefrequency points of peak power (the first and secondtype timefrequency points) may develop a severe problem when the source signals are nearly submerged in noise. This is achieved by averaging the STFDs of cross terms to solve this problem.
Results and discussion
Assume an eightsensor uniform linear array with a half wavelength separation between each element and an observation duration of 512 samples. The two linear frequency modulation signal components are transmitted from two source signals located at angles θ_{1} and θ_{2}. The initiated and finished normalized frequencies of the source signal from θ_{1} = 30° are f_{s1} = 0 and f_{e1} = 0.3, whereas the homologous second frequencies for another source from θ_{2} = 40°are f_{s2} = 0.2 and f_{e2} = 0.5, severally. WVD is utilized to calculate the timefrequency distribution, and timefrequency averaging is utilized to build the noise subspace. The input SNR of θ_{1} is 5 dB, whereas the incoming SNR of θ_{2} is − 5 dB. Figure 1 shows the timefrequency spectrum of two LFM signals. From Fig. 1, the autoterm of the weak signal cannot be obtained on the timefrequency spectrum because of the submergence in noise. However, the cross terms in timefrequency plane can observe the normalized frequencies of which is approximately from 0.1 to 0.4. As shown in Fig. 2, the cross terms are extracted by the method mentioned above for the DOA estimation of the weak nonstationary signal. The DOA of strong and weak signals is estimated by cross terms in Fig. 3, which are approximately 30° and 40°, respectively.
If the SNR difference of the strong and weak signals increases further, the DOA of the weak signal cannot be directly obtained from the cross terms. The input SNR of θ_{1} is 10 dB, whereas the input SNR of θ_{2} is − 10 dB. The cross term is utilized from another perspective. Firstly, the position of the autoterms of the strong signal and cross terms in the timefrequency domain is estimated. Then the position of the autoterms of the weak signal is fitted according to the two positions in order to improve the SNR of the weak nonstationary signals. The spatial timefrequency distribution matrixes are extracted from the position of the weak signal for the DOA estimation. Figure 4 shows the TFspectrum of two linear frequency modulation signal components. Although the cross terms are no longer obvious on the timefrequency plane, they can also be extracted according to the characteristics of the matrix. In Fig. 5, the autoterms of the weak nonstationary signal are fitted by the method mentioned above for the DOA estimation. The DOA of the strong and weak signals is simultaneously estimated in Fig. 6, which are approximately 30° and 40°, respectively. The timefrequencyMUSIC technology is realized respectively for two sets of timefrequency points, each one from one source signal.
When the incoming SNR of θ_{1} is 5 dB, the input SNR of θ_{2} is from − 8 dB to 0 dB. The results were calculated via averaging 100 Monte Carlo runs. Figure 7 shows the DOA estimation rootmeansquare error with SNR for traditional MUSIC and timefrequencyMUSIC based on crossterms. The RMSE of TFMUSIC based on cross terms is less than that of the conventional MUSIC overall. The advantages of TFMUSIC based on cross terms in poor SNR conditions become obvious from the figure.
Next, the influence of snapshots on the algorithm is analyzed. With the other simulation conditions mentioned above unchanged, the number of snapshot is 256, 512, and 1024, respectively. From Fig. 8, the increase in the number of snapshots reduces the RMSE of DOA estimation. Then, the increase in that will also increase the calculation time. The impact of a small userdefined positive scalar ε on RMSE is also simulated. With the 512 snapshots, ε is 0.3, 0.5, and 0.7, respectively. From Fig. 9, different values of the same order have little influence on RMSE.
Figures 10 and 11 show the estimated spatial spectrum of TFMUSIC based on cross terms and the traditional MUSIC where the direction separation is close (θ_{1} = 30°, θ_{2} = 33°). It is obvious that the source signals can be separated via the timefrequencyMUSIC based on cross terms whereas the conventional MUSIC fails. This is attributed to the combination of timefrequency analysis method and MUSIC algorithm. In the timefrequency MUSIC algorithm, for each signal energy distribution, the MUSIC algorithm is calculated separately, and only a single signal is included in the data covariance matrix. Therefore, two curves are generated and the target with an angle approaching can be distinguished.
Conclusions
When the desired weak nonstationary signal may be buried in noise, especially in the condition of low signaltonoise ratio, it is difficult or almost impossible to extract the autoterm of the weak nonstationary signal. However, the cross terms of the strong and weak nonstationary signal did not decrease significantly, which contain sufficient information of weak nonstationary signal for its DOA estimation. Therefore, in the case that both strong and weak nonstationary signals exist, meanwhile, the cross terms of STFDs are used to obtain DOA of the desired weak nonstationary signal in this paper. The DOA estimation rootmeansquare error of TFMUSIC based on cross terms is less than that of conventional MUSIC. The DOA of two closely spaced signals is resolved by the TFMUSIC based on cross terms.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 DOA:

Direction of arrival
 HT:

Hough transform
 IF:

Instantaneous frequency
 LFM:

Linear frequency modulation
 MUSIC:

Multiple signal classification
 RMSE:

Rootmeansquare error
 SHRTFRR:

Smoothed highresolution timefrequency rate representation
 SNR:

Signaltonoise ratio
 STFD:

Spatial timefrequency distribution
 TF:

Timefrequency
 WVD:

WignerVille distribution
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Acknowledgements
This work was supported in part by the National Key R&D Program of China under Grant No. 2017YFC1405202, in part by the National Natural Science Foundation of China under Grant No. 61571157 and Grant No. 61571159, in part by the Public Science and Technology Research Funds Projects of Ocean under Grant No. 201505002, in part by the Natural Science Foundation of Shandong Province under Grant No. ZR2018PF001, and in part by the Foundation of Science and Technology on Communication Networks Key Laboratory.
Funding
National Key R&D Program of China under Grant No. 2017YFC1405202 is supporting the data acquisition devices and materials; National Natural Science Foundation of China under Grant No. 61571157 and Grant No. 61571159 are supporting the simulations; the Public Science and Technology Research Funds Projects of Ocean under Grant No. 201505002, the Natural Science Foundation of Shandong Province under Grant No. ZR2018PF001, and the Foundation of Science and Technology on Communication Networks Key Laboratory are supporting the data analyses.
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SS and AL conceived and designed the experiments; SS and HY performed the experiments; BL contributed simulation tools; and SS and HY wrote the paper. All authors have read and approved the final manuscript.
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Shao, S., Liu, A., Yu, C. et al. Spatial timefrequency distribution of cross termbased directionofarrival estimation for weak nonstationary signal. J Wireless Com Network 2019, 239 (2019). https://doi.org/10.1186/s1363801915555
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Keywords
 DOA estimation
 Timefrequency analysis
 Spatial timefrequency distribution (SFTD)
 Cross terms