Coherent Signal Parameter Estimation by Exploiting Decomposition of Tensors

Liu, Long; Wang, Ling; Wang, Yuexian; Xie, Jian; Zhang, Zhaolin

doi:https://doi.org/10.1155/2019/5794791

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 5794791 | https://doi.org/10.1155/2019/5794791

Coherent Signal Parameter Estimation by Exploiting Decomposition of Tensors

Long Liu,¹Ling Wang,¹Yuexian Wang,²Jian Xie,¹and Zhaolin Zhang¹

Academic Editor: Alessio Gizzi

Received15 Jul 2019

Accepted01 Nov 2019

Published13 Nov 2019

Abstract

The problem of parameter estimation of coherent signals impinging on an array with vector sensors is considered from a new perspective by means of the decomposition of tensors. Signal parameters to be estimated include the direction of arrival (DOA) and the state of polarization. In this paper, mild deterministic conditions are used for canonical polyadic decomposition (CPD) of the tensor-based signal model; i.e., the factor matrices can be recovered, as long as the matrices satisfy the requirement that at least one is full column rank. In conjoint with the estimation of signal parameters via the algebraic method, the DOAs and polarization parameters of coherent signals can be resolved by virtue of the first and second factor matrices. Hereinto, the key innovation of the proposed approach is that the proposed approach can effectively estimate the coherent signal parameters without sacrificing the array aperture. The superiority of the proposed algorithm is shown by comparing with the algorithms based on higher order singular value decomposition (HOSVD) and Toeplitz matrix. Theoretical and numerical simulations demonstrate the effectiveness of the proposed approach.

1. Introduction

The problem of parameter estimation for coherent signals is encountered in a variety of signal processing applications including wireless communication, satellite navigation, and radar. Generally, the signal covariance matrix must be full rank for the subspace-based high-resolution estimation approaches such as multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRITs). However, there is also a fact that cannot be ignored; that is, the presence of the coherent signals can cause the rank loss of the covariance matrix. Under such circumstances, the spatial smoothing technique and its derivative algorithms [1–5] are commonly used as a conventional solution to this problem. According to previous research studies, these approaches have been proved to play an important role in eliminating the rank loss of the covariance matrix, but they are not flawless because an unignorable large aperture loss appears under these approaches as well, degrading the accuracy of parameter estimation. Apart from these approaches, the approaches of parameter estimation for coherent signals based on the Toeplitz matrix signal model have received considerable attention and developments because the rank of the Toeplitz matrix is only related to the DOA of signals and cannot be affected by the coherency between them [6–10]. In addition, in the context of underwater environment and target imaging, the coherent signal analysis methods proposed in [11, 12] are also worthy of attention. For all the approaches mentioned above, they share a common feature that the process of parameter estimation is based on matrix operations.

However, the matrix can only reflect the two-dimensional characteristics of the signal. For multidimensional signals, stacking dimensions into one highly structured matrix can effectively improve the accuracy of parameter estimation. In [13], the measurement tensor has been defined, and the estimation of the subspace through higher order singular value decomposition (HOSVD) has been proposed. Similarly, the HOSVD-based algorithms have been studied to improve the performance of parameter estimation in [14–17]. However, in order to achieve the parameter estimation of coherent signals, smoothing for tensor is unavoidable in terms of the HOSVD-based algorithms, which will also lead to the sacrifice of array aperture.

Recently, parameter estimation methods by exploiting the canonical polyadic decomposition (CPD) of tensor, which is a minimal decomposition into a sum of rank-1 tensors, have been studied in [18–21]. One common feature of the existing tensor CPD-based parameter estimation methods is that they are all directed to noncoherent signal models. In this paper, we will study the CPD-based approach of parameter estimation under the condition that at least two signals satisfy the coherent premise.

Here, we present a signal model based on the third-order tensor, with three dimensions corresponding to the temporal, spatial, and polarized information of the signal. Furthermore, the mild deterministic conditions [22] are used for CPD of the tensor-based signal model; i.e., as long as the coherent signals comply with any one of the spatial and polarized diversities, the proposed approach can effectively estimate the parameters of the coherent signals. The parameters to be estimated mainly include the DOA and polarization parameters delivered by the signal. Computer results are reported as a function of the signal-to-noise ratio (SNR), in comparison to the advanced subspace-based approaches. In addition, the estimation accuracy of polarized parameters is also studied with respect to the different SNR values.

The rest of this paper is organized as follows: Section 2 derives the tensor-based signal model, followed by the proposed algorithm for extracting the signal parameters in Section 3. Meanwhile, numerical simulations are provided in Section 5. Finally, the last part of the paper will present our concluding remarks. Table 1 summarizes the algebra notations involved in this paper.

2. Signal Model

Let and serve as the indications of the azimuth angle and the elevation angle. Set the number of electromagnetic (EM) vector sensors in the array as L, and the components measured by an EM vector sensor are indexed as separately. The spatial steering vector for the array with DOA is given bywhere , , and indicates the spatial location of the lth sensor, . Let indicate the spatial-polarization parameter, where symbolizes the polarization auxiliary angle and symbolizes the polarization-phase difference. The polarization steering vector can be denoted aswhere and is defined as the polarization selection matrix for the EM vector sensor. In particular, for the complete EM vector sensor, the polarization selection matrix is represented as and belongs to the three dipoles, in which symbolizes the unit matrix. The spatial-polarization steering vector of the signal with the spatial-polarization parameter can be further given by

Hereby, the R signals with the complex amplitudes are assumed to be received by the array. Furthermore, assume that the first signals are coherent and the last signals are noncoherent, but all the signals are in cofrequency, where . Construct the following equation to represent the vector output of the array at the instant :where the additive prewhitening noise is assumed to have a Gaussian complex circular. In order to better distinguish the multidomain diversities of the signal, we perform tensorization on (3) so that the multidomain diversities of the signal caters to the multidimensional nature of the tensor, and then the following representation is obtained:

Furthermore, the tensor output of the consecutive K snapshots for the array can be modeled aswhere and . In addition, the noise tensor is yielded by tensorization with respect to the noise matrix .

3. The Proposed Approach

Let . Thereby, equation (6) can be expressed as

Considering that , , and are nonzero vectors, tensor is the rank-1 tensor. If the number R of the rank-1 terms in is minimal, then is called the CPD of and R is called the rank of (denoted by ). Since the addition of noise component , the structure of the tensor is significantly different from the structure of the tensor . Herein, we are not going to talk about the structure of the tensor , but instead regard it as the rank R tensor and decompose it. Furthermore, the estimation of the factor matrices can be obtained.

3.1. Construction of Intermediate Tensor

The CPD of can be expressed aswhere , , and . Let , , and . We can also write (8) as , where A, B, and C are called the first, second, and third factor matrix of , respectively. Again, we can write as , where , , and . Furthermore, we call that the CPD of the tensor is unique when it is the only subject to the trivial indeterminacies, in which the factor matrices can be arbitrarily permuted and the vectors belonging to any factor matrix can be arbitrarily scaled. Obviously, the factor matrices of and the factor matrices of do not satisfy the above indeterminacies due to the presence of the noise. Therefore, , , and are considered as the estimated values of , , and , respectively.

Herein, we only consider the algebraic algorithm for CPD under the condition that at least one factor matrix of has full column rank. That is to say, in the tensor , at least one factor matrix contains highly collinear vectors due to the presence of noise disturbances. In order to adapt to the analysis of the coherent signal tensor model, we extend Theorem 2 and Theorem 8 in [22]. The following theorem can be obtained.

Theorem 1. Let and , Assume thatThen, , and the CPD of is unique. In addition, the CPD of can be obtained algebraically. Under this premise, the proposed algorithm can achieve a unique decomposition of the tensor .

Matrix is defined in Appendix A. denotes the subspace spanned by all vectors of the form , where is repeated times.

We hereby execute the decomposition of the tensor-based signal model to obtain the factor matrices, which are then used to extract the parameters of the signals. Assume that . First of all, the matrix needs to be built, and the -th column of the matrix can be obtained as follows:where , represents the frontal slices of the tensor , and . An equation for the polarized compound matrix can be defined aswhere the m-th compound matrix of a given matrix can be obtained from Appendix B.

Subsequently, how to get the vectors from the matrix is the focus of this paper. In the absence of noise, for the tensor , as described in [23], vectors form a basis of . Combining the construction process of the matrix , the vector satisfying has the following structural features:where , , and the element is constant 1. Similarly, we can also write as , where the submatrices and contain the first a columns and the last columns of , respectively. Therefore, the following equation is established:where represents the ith column of the submatrix .

However, the vectors cannot be obtained by solving the basis of the matrix due to the presence of noise. Inspired by (12) and (13) summarized in this paper, an optimization problem model is established as follows:

Herein, the alternating least squares (ALS) method is used to solve the optimization problem (14), and then the vectors can be obtained according to (12).

Let denotes the canonical basis of , and then is the canonical basis of , . The symmetrizer on the space can be denoted as

In addition, the intermediate matrix needs to be built. The columns of the matrix can be defined as follows:

Furthermore, the matrix can be obtained. Reshape the matrix into an tensor . Afterwards, we need to compute the CPD of the tensor by generalized eigenvalue decomposition (GEVD). The tensor CPD procedure by GEVD is shown in [21].

3.2. Estimation of Factor Matrices

Once the tensor is constructed, we can obtain the matrix unfoldings:where , symbolizes the th frontal slice of , denotes the ith row of the , and is the diagonalization operator. According to (17), the following equation can be easily derived:where , , and denotes the Moore–Penrose inverse. Obviously, the columns of the factor matrix are the eigenvectors of . The corresponding eigenvalues are the first larger eigenvalues, which are arranged in a descending order. It is worth noting that when , there are multiple combinations that can be used for EVD. In order to fully exert the statistical characteristics of (17), possible combinations are selected for EVD to obtain , where , . Herein, we make the agreement that tuples are ordered lexicographically: the th tuple is preceding the th tuple if and only if and . According to the trivial indeterminacies aforementioned for CPD uniqueness, and coincide up to column permutation and scaling, . Therefore, we need to match multiple sets of results; i.e., all the eigenvectors need to be normalized and arranged in a descending order according to the corresponding eigenvalues. The results of the matching processing are recorded as , and the estimation of the factor matrix can be obtained by performing the following statistical operation:

Once the matrix is obtained, we can recover the matrix from . First, compute R submatrices of columns of that are linearly dependent. Then, compute as orthogonal complements to the submatrices aforementioned. It is worth noting that, in the presence of noise, the algorithm proposed in this paper only guarantees that . Furthermore, the factor matrices and can be recovered, thanks to .

3.3. Estimation of Signal Parameters

The construction of the factor matrices clearly shows that the factor matrix contains the spatial-polarization parameters of the signals, while the factor matrix includes only the spatial ones of them. Putting it in other way, and are the estimated counterparts of and , respectively. Thus, the following equations are going to be used to obtain the spatial DOA of the rth signal impinging on the array:

It is obvious that the DOA of the rth signal can be procured through any two equations from (20). Suppose is the representation of the nth subequations’ solution. Under such circumstances, the DOA can be estimated through the average of multiple sets’ solutions:

Once the DOA is estimated, a further step is taken to extract the polarization parameters of the rth signal from the following formula:

In order to simplify the decomposition of the tensor-based signal model, the value of the snapshots K is supposed to be as small as possible when is satisfied. Therefore, we take N-segment K snapshots to improve the estimation accuracy of signal parameters. That is, the signal parameters which are estimated through decomposing over N times are going to be averaged.

4. Simulation Results

Consider a uniform linear array (ULA) with 4 complete EM vector sensors whose spacing is half wavelength. Suppose there are three signals with the polarization parameters , , and impinging on the ULA from , , and , respectively. is independent of , and is the replica of , which means that there is a coherency between and . All of the signals have the same power. In addition, the noise component is assumed to be zero-mean additive white Gaussian noise. The root mean square error (RMSE) is defined aswhere M is the Monte Carlo trial number, denotes one of the parameters , and is the estimation of in the mth trial.

The DOA estimation of the signals at with 10 independent trials is illustrated in Figure 1. As expected, our proposed algorithm succeeds in differentiating the signals without paying special attention to their coherency.

For all the following simulations, their results derive from 300 Monte Carlo trials and have been compared with the Cramér–Rao lower bound (CRB) benchmark which is described in Appendix C.

In Figure 2, the proposed algorithm is compared with two advanced subspace-based algorithms in terms of the RMSE versus number of snapshots for . It can be seen from Figure 2 that the DOA estimation accuracy of the proposed algorithm outperforms the other two algorithms. Especially for the case of fewer number of snapshots, the advantages of the proposed algorithm are obvious.

Figure 3 illustrates the superior performance of the proposed algorithm as compared to the other two algorithms when the snapshots number is set as 200. It is worth noting that as SNR increases, the tensor-based methods, i.e., the proposed algorithm and the HOSVD-based algorithm, present a better accuracy of the DOA estimation than the Toeplitz-based algorithm. What is more, the estimation performance of the polarization parameters is examined in Figure 4, which is conducted under the same scenarios with those in Figure 3. According to the operation process mentioned above, we know that, in the proposed algorithm, polarization parameters cannot be estimated independently without obtaining the DOA estimation because the estimation of polarization parameters can only be achieved with the predicted DOA at hand at first. When comparing Figure 4 with Figure 3, we can see clearly that the estimation accuracy of polarization parameters is inferior to that of DOA parameters, and the reason lies in the accumulation of the estimation error during these two steps. Nevertheless, the estimation accuracy of polarization parameters in the proposed algorithm is still better than that of the other two algorithms.

5. Conclusions

With the purpose of surmounting the problem of parameter estimation for coherent signals, this paper proposes a new approach which features decomposition for tensors. Specifically, factor matrices can be effectively estimated at first and then be used to extract the parameters of signals irrespective of the coherency between them. In view of this, an obvious advantage of this approach lies in its effective estimation of factor matrices even with noises present. Besides, the problem of aperture loss, which commonly appears in most traditional algorithms, can also be avoided under this approach. Meanwhile, according to the simulation results, our proposed algorithm shows a higher accuracy of parameter estimation in comparison with the state-of-the-art ones.

Appendix

A. Construction of the Intermediate Matrix

Let serve as the indication of the set of all permutations of the set . It is important to note that the cardinality of is counted by considering multiplicities (e.g., is made up of six identical entries ) if some of the values coincide.

It is easy to know that any integer from can be uniquely denoted as and any integer from as , whereand , , and . These equations are of great use if there is going to have a switch between tensor, matrix, and vector indications. Hereinto, the th entry of the matrix can be defined as follows:

B. Construction of the Compound Matrix

The m-th compound matrix of a given matrix is derived from minors of that matrix. Let . The matrix whose -th entry is is called the m-th compound matrix of and is represented by , where . Let represent integers. The multi-index notation can be defined as

We assume that the elements of are ordered lexicographically. It is well known that . For example, let , where . Then,

It is well known that and is equal to the zero matrix if and only if .

C. Cramér–Rao Lower Bound for the Vector Sensor Array

The following matrix form is established after considering the situations described in (6):in which and . represents the unknown parameters vector, where serves as an indication of the unknown parameters vector of the rth source, . The matrix in (A.5) is assumed to be column full rank so is the Jacobian . On the basis of this, we further setwhere the number of elements in vector is represented by . Exploring the performance of estimating in (A.5) from is our main concern in this paper.

The purpose of constructing the following two intermediate matrices lies in simplifying the expression of the Cramér–Rao lower bound mentioned before:where represents the covariance of the signal matrix , symbolizes the noise variance, and serves as the denotation of a unit matrix. For , the Cramér–Rao lower bound of its unbiased estimation is where 1 is an indication of a matrix with all entries equal to one, . Assuming that the -th block entry of the matrix is represented by with dimension , the block operators , , , and are defined as follows.

Definition 1. Block trace operator:

Definition 2. Block transpose:

Definition 3. Block Kronecker product:

Definition 4. Block Hadamard product:

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant nos. 61771404 and 61601372.

References

S. U. Pillai and B. H. Kwon, “Forward/backward spatial smoothing techniques for coherent signal identification,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 1, pp. 8–15, 1989.
View at: Publisher Site | Google Scholar
W. Du and R. L. Kirlin, “Improved spatial smoothing techniques for DOA estimation of coherent signals,” IEEE Transactions on Signal Processing, vol. 39, no. 5, pp. 1208–1210, 1991.
View at: Publisher Site | Google Scholar
J. Dai and Z. Ye, “Spatial smoothing for direction of arrival estimation of coherent signals in the presence of unknown mutual coupling,” IET Signal Processing, vol. 5, no. 4, pp. 418–425, 2011.
View at: Publisher Site | Google Scholar
J. Wen, B. Liao, and C. Guo, “Spatial smoothing based methods for direction-of-arrival estimation of coherent signals in nonuniform noise,” Digital Signal Processing, vol. 67, pp. 116–122, 2017.
View at: Publisher Site | Google Scholar
J. Shi, G. Hu, and X. Zhang, “Direction of arrival estimation in low-grazing angle: a partial spatial-differencing approach,” IEEE Access, vol. 5, pp. 9973–9980, 2017.
View at: Publisher Site | Google Scholar
F. M. Han and X. D. Zhang, “An ESPRIT-like algorithm for coherent DOA estimation,” IEEE Antennas and Wireless Propaga Letters, vol. 4, pp. 443–446, 2005.
View at: Publisher Site | Google Scholar
X. Zhang and D. Xu, “Improved coherent DOA estimation algorithm for uniform linear arrays,” International Journal of Electronics, vol. 96, no. 2, pp. 213–222, 2009.
View at: Publisher Site | Google Scholar
Y.-H. Choi, “ESPRIT-Based coherent source localization with forward and backward vectors,” IEEE Transactions on Signal Processing, vol. 58, no. 12, pp. 6416–6420, 2010.
View at: Publisher Site | Google Scholar
C. Li, G. Liao, S. Zhu, and S. Wu, “An ESPRIT-like algorithm for coherent DOA estimation based on data matrix decomposition in MIMO radar,” Signal Processing, vol. 91, no. 8, pp. 1803–1811, 2011.
View at: Publisher Site | Google Scholar
A. Goian, M. I. AlHajri, R. M. Shubair et al., “Fast detection of coherent signals using pre-conditioned root-MUSIC based on Toeplitz matrix reconstruction,” in Proceedings of the IEEE 11th International Conference on Wireless and Mobile Computing, Networking and Communications, pp. 168–174, Abu Dhabi, UAE, October 2015.
View at: Publisher Site | Google Scholar
J. Li, Q.-H. Lin, C.-Y. Kang, K. Wang, and X.-T. Yang, “DOA estimation for underwater wideband weak targets based on coherent signal subspace and compressed sensing,” Sensors, vol. 18, no. 3, p. 902, 2018.
View at: Publisher Site | Google Scholar
D. Li, M. Zhan, J. Su, H. Liu, X. Zhang, and G. Liao, “Performances analysis of coherently integrated CPF for LFM signal under low SNR and its application to ground moving target imaging,” IEEE Transactions on Geoscience and Remote Sensing, vol. 55, no. 11, pp. 6402–6419, 2017.
View at: Publisher Site | Google Scholar
M. Haardt, F. Roemer, and G. Del Galdo, “Higher-order SVD-based subspace estimation to improve the parameter estimation accuracy in multidimensional harmonic retrieval problems,” IEEE Transactions on Signal Processing, vol. 56, no. 7, pp. 3198–3213, 2008.
View at: Publisher Site | Google Scholar
M. Boizard, G. Ginolhac, F. Pascal, S. Miron, and P. Forster, “Numerical performance of a tensor MUSIC algorithm based on HOSVD for a mixture of polarized sources,” in Proceedings of the 21st European Signal Processing Conference (EUSIPCO 2013), pp. 1–5, Marrakech, Morocco, September 2013.
View at: Google Scholar
K. Han and A. Nehorai, “Nested vector-sensor array processing via tensor modeling,” IEEE Transactions on Signal Processing, vol. 62, no. 10, pp. 2542–2553, 2014.
View at: Publisher Site | Google Scholar
P. Forster, G. Ginolhac, and M. Boizard, “Derivation of the theoretical performance of a tensor MUSIC algorithm,” Signal Processing, vol. 129, pp. 97–105, 2016.
View at: Publisher Site | Google Scholar
X. Lan, L. Wang, Y. Wang, C. Choi, and D. Choi, “Tensor 2-D DOA estimation for a cylindrical conformal antenna array in a massive MIMO system under unknown mutual coupling,” IEEE Access, vol. 6, pp. 7864–7871, 2018.
View at: Publisher Site | Google Scholar
X. J. Guo, S. Miron, D. Brie, S. H. Zhu, and X. W. Liao, “A CANDECOMP/PARAFAC perspective on uniqueness of DOA estimation using a vector sensor array,” IEEE Transactions on Signal Processing, vol. 59, no. 7, pp. 3475–3481, 2011.
View at: Publisher Site | Google Scholar
F. Raimondi, P. Comon, O. Michel, S. Sahnoun, and A. Helmstetter, “Tensor decomposition exploiting diversity of propagation velocities: application to localization of icequake events,” Signal Processing, vol. 118, pp. 75–88, 2016.
View at: Publisher Site | Google Scholar
L. Liu, L. Wang, and Z. Zhang, “Vector-sensor-based signal parameter estimation by exploiting CPD of tensors,” IEEE Sensors Letters, vol. 2, no. 3, pp. 1–4, 2018.
View at: Publisher Site | Google Scholar
L. Liu, L. Wang, J. Xie, and Z. L. Zhang, “DOA and polarization parameters estimation by exploiting canonical polyadic decomposition of tensors,” Wireless Communications and Mobile Computing, vol. 2019, Article ID 7389306, 12 pages, 2019.
View at: Publisher Site | Google Scholar
I. Domanov and L. De Lathauwer, “Canonical polyadic decomposition of third-order tensors: relaxed uniqueness conditions and algebraic algorithm,” Linear Algebra and its Applications, vol. 513, pp. 342–375, 2017.
View at: Publisher Site | Google Scholar
I. Domanov and L. D. Lathauwer, “Canonical polyadic decomposition of third-order tensors: reduction to generalized eigenvalue decomposition,” SIAM Journal on Matrix Analysis and Applications, vol. 35, no. 2, pp. 636–660, 2014.
View at: Publisher Site | Google Scholar

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Copyright © 2019 Long Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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