DOA estimation via shift-invariant matrix completion☆
Introduction
The need of large bandwidths in 5G networks motivates to operate in mm-Wave bands, which require large-scale antenna arrays to compensate for the path loss [1], [2]. Indeed, research in wireless communication systems has shifted towards the use of large antenna arrays as in massive multiple-input multiple-output (MIMO) systems [3]. This poses new challenges not only to antenna calibration and complexity issues associated with channel state information acquisition and precoding [4], but also to energy consumption. It is acknowledged that power consumption requirements in 5G networks increase by about 3 times over 4G, and that the signal processing in massive MIMO systems can represent up to 40 of the total power consumption for below-6 GHz bands, and even larger in mm-Wave bands [5].
A classical problem when processing multiple signals received by a uniform linear array (ULA) is that of estimating their directions-of-arrival (DOAs). DOA estimation has a long and rich history in array processing [6], and numerous high-resolution direction finding algorithms have been proposed over the last decades. As representative examples it is worth mentioning subspace-based methods such as the multiple signal classification (MUSIC) algorithm [7] and the estimation of signal parameters via rotational invariance technique (ESPRIT) [8], which provide high angular resolution. However, using MUSIC or ESPRIT with a large-scale fully-digital receive antenna array can be challenging due to their computational complexity and high energy consumption requirements. A possible solution is to reduce the number of radio frequency (RF) transceiver chains by performing antenna selection at the receiving array (cf. Fig. 1). At every time instant a random switch selects a subset of antennas whose RF signals are downconverted and further processed. Since the number of targets or sources is typically much smaller than the number of antennas, it is feasible to reconstruct (or at least to approximate) the low-rank signal data matrix using matrix completion (MC) algorithms as if it had been received by the full array, as long as we sample a sufficiently large fraction of the sensors [9].
Low-rank MC methods for DOA estimation are used in Pal and Vaidyanathan [10] for scenarios in which the number of sources exceeds the number of sensors, and in Ito et al. [11] in the presence of diffuse noise. An iterative reweighted nuclear norm minimization method is used in Tan and Feng [12] for DOA estimation with nested arrays. When a sparse coprime array is used, array interpolation techniques can be applied to improve the DOA estimation performace. In [13] the authors consider this scenario and apply MC techniques to reconstruct the Toeplitz virtual array covariance matrix. In [14], a DOA estimation algorithm based on virtual array interpolation and MC techniques is developed for coherent sources in coprime arrays. A Toeplitz reconstruction algorithm based on nuclear norm minimization is proposed in Wu et al. [15] for uniform and sparse linear arrays. A different approach is proposed in Liao et al. [16], where MC algorithms are used to reconstruct the entries of the sample covariance matrix (SCM) along its diagonal, which are deliberately set to zero. In [17] MC is used for order estimation in the presence of noise with a diagonal spatial covariance matrix. Whereas most of existing MC methods in array signal processing target the reconstruction of the signal covariance matrix, it is the data matrix itself that needs to be reconstructed when only a subset of sensors is sampled.
In this paper, an energy-efficient approach to DOA estimation is proposed based on the recovery of the data matrix by means of a MC method. We consider that only a randomly chosen subset of sensors are sampled at each time instant. By reducing the number of RF chains of the receiver, the overall hardware cost and energy consumption are reduced as well. In our approach, the matrix completion problem is tailored to enforce the shift-invariance property of ULAs by including an additional regularization term in the MC cost function. Then, the Optimal Subspace Estimation (OSE) technique proposed by Vaccaro and Ding in Vaccaro and Ding [18] is used to estimate the signal subspace and, finally, the DOAs are estimated using ESPRIT as a high-resolution technique. The simulations show that the number of RF chains can be largely reduced without significant performance loss in DOA estimation accuracy.
The rest of the paper is organized as follows. Section 2 presents the signal model assuming an array architecture with random antenna switching, and formulates the problem. The proposed Shift-Invariant Matrix Completion (SIMC) method is described in Section 3. A direct application of the Davis–Kahan theorem [19] allows us to analyze in Section 4 the chordal distance between the true signal subspace and the signal subspace of the sparse and reconstructed matrices. The simulation results are discussed in Section 5, and concluding remarks are provided in Section 6.
Notation. Bold lowercase letters denote vectors and bold uppercase matrices; is the entry in the th row and th column of matrix . Superscripts and denote transpose, complex conjugate and Hermitian, respectively. denotes the modulus of a complex number and is the -norm of vector . The trace, nuclear, spectral, Frobenius and infinity norms of a matrix are denoted, respectively, as and . The -th largest singular value is denoted as . Furthermore, denotes a proper Gaussian random vector in with zero mean and covariance .
Section snippets
Observed data matrix and problem statement
Let us consider narrowband signals impinging on a large half-wavelength ULA with antennas. For a fully digital receiver with RF-branches, the received signal at time instant or snapshot iswhere is the noise vector, is the signal vector with complex gains and is the complex array response to the th source with electrical angle which is unknown; and is the steering
Matrix completion
The problem of estimating the low-rank signal matrix from can be solved using MC techniques. According to [9], we can recover by solvingwhere denotes the nuclear norm of the set of observed entries of and is a tolerance parameter that limits the fitting error.
The main assumption for a successful recovery in low-rank MC is that of incoherence, which means that each singular vector of the unknown matrix must be
Perturbation analysis
The main factor impacting the performance of the random multi-switch sampling scheme is how well the signal subspace is preserved. The SIMC algorithm aims at estimating an improved signal subspace by leveraging its shift-invariant low-rank structure. This section analyzes how DOA estimation is impacted when performed after MC.
Since the DOA estimates are essentially determined by the singular vectors of the signal subspace, we want to assess how much the principal directions change after each
Simulation results
In this section we illustrate the performance of the proposed SIMC algorithm by means of Monte Carlo simulations. For comparison, we include the performance of the following methods:
- •
SCM: The sample covariance matrix without MC is estimated as .
- •
OSE: The shift-invariance property is enforced by applying OSE to (without MC).
- •
MC: The standard MC algorithm solution given by (6) is used to reconstruct from .
- •
MC-OSE: OSE is applied as a post-processing step to the previous method.
- •
SIMC:
Conclusion
The high hardware complexity and energy consumption of massive MIMO systems is a challenge for its fully-digital implementation. A solution is to reduce the number of RF chains by performing random antenna selection techniques, which result in a data matrix with multiple missing entries. In this paper we have proposed a matrix completion technique tailored to this array processing architecture. The reconstruction algorithm exploits both the low-rank structure of the partially observed matrix
CRediT authorship contribution statement
Vaibhav Garg: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing - original draft. Pere Giménez-Febrer: Validation, Formal analysis, Investigation, Writing - review & editing. Alba Pagès-Zamora: Validation, Writing - review & editing, Supervision. Ignacio Santamaria: Validation, Writing - review & editing, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (30)
- et al.
Millimeter Wave Wireless Communications
(2015) Millimeter wave mobile communications for 5G cellular: it will work!
IEEE Access
(2013)Scaling up MIMO: opportunities and challenges with very large arrays
IEEE Signal Process. Mag.
(2013)- et al.
Channel estimation and hybrid precoding for millimeter wave cellular systems
IEEE J. Sel. Top. Signal Process.
(2014) - et al.
A survey of energy-efficient techniques for 5G networks and challenges ahead
IEEE J. Sel. Areas Commun.
(2016) Detection, Estimation and Modulation Theory: Optimum Array Processing (Part IV)
(2002)Multiple emitter location and signal parameter estimation
IEEE Trans. Antennas Propag.
(1986)- et al.
ESPRIT- estimation of signal parameters via rotational invariance techniques
IEEE Trans. on Acoust. Speech Signal Process.
(1989) - et al.
Matrix completion with noise
Proc. IEEE
(2010) - et al.
A grid-less approach to underdetermined direction of arrival estimation via low rank matrix denoising
IEEE Signal Process. Lett.
(2014)
Covariance matrix reconstruction for direction finding with nested arrays using iterative reweighted nuclear norm minimization
Int. J. Antennas Propag.
Direction-of-arrival estimation for coprime array via virtual array interpolation
IEEE Trans. Signal Process.
Direction-of-arrival estimation of coherent signals via coprime array interpolation
IEEE Signal Process. Lett.
A Toeplitz covariance matrix reconstruction approach for direction-of-arrival estimation
IEEE Trans. Veh. Technol.
Cited by (5)
Truncated quadratic norm minimization for bilinear factorization based matrix completion
2024, Signal ProcessingLow-Complexity DOA Estimation for Uniform Circular Arrays with Directional Sensors Using Reconfigurable Steering Vectors
2023, Circuits, Systems, and Signal ProcessingAccessorial Locating for Internet of Vehicles Based on DOA Estimation in Industrial Transportation
2021, Wireless Communications and Mobile ComputingOrder Estimation via Matrix Completion for Multi-Switch Antenna Selection
2021, IEEE Signal Processing Letters
- ☆
This work was supported by the Ministerio de Ciencia e Innovación (MICINN) of Spain, and AEI/FEDER funds of the E.U., under grants TEC2016-75067-C4-4-R /2-R (CARMEN), PID2019-104958RB-C43/C41 (ADELE) and BES-2017-080542.