Jens Steinwandt

Deterministic Cramér-Rao Bound for Strictly Non-Circular Sources and Analytical Analysis of the Achievable Gains

Recently, several high-resolution parameter estimation algorithms have been developed to exploit the structure of strictly second-order (SO) non-circular (NC) signals. They achieve a higher estimation accuracy and can resolve up to twice as many signal sources compared to the traditional methods for arbitrary signals. As a benchmark for these NC methods, we derive the closed-form deterministic R-D NC Cramér-Rao bound (NC CRB) for the multi-dimensional parameter estimation of strictly non-circular (rectilinear) signal sources in this paper. Assuming a separable centro-symmetric R-D array, we show that in some special cases, the deterministic R-D NC CRB reduces to the existing deterministic R-D CRB for arbitrary signals. This suggests that no gain from strictly non-circular sources (NC gain) can be achieved under the deterministic data assumption in these cases. For more general scenarios, finding an analytical expression of the NC gain for an arbitrary number of sources is very challenging. Thus, in this paper, we simplify the derived NC CRB and the existing CRB for the special case of two closely-spaced strictly non-circular sources captured by a uniform linear array (ULA). Subsequently, we use these simplified CRB expressions to analytically compute the maximum achievable asymptotic NC gain for the considered two source case. The resulting expression only depends on the various physical parameters and we find the conditions that provide the largest NC gain. Our analysis is supported by extensive simulation results.

Beamspace direction finding based on the conjugate gradient and the auxiliary vector filtering algorithms

Motivated by the performance of the direction finding algorithms based on the auxiliary vector filtering (AVF) method and the conjugate gradient (CG) method as well as the advantages of operating in beamspace (BS), we develop two novel direction finding algorithms for uniform linear arrays (ULAs) in the beamspace domain, which we refer to as the BS AVF and the BS CG methods. The recently proposed Krylov subspace-based CG and AVF algorithms for the direction of arrival (DOA) estimation utilize a non-eigenvector basis to generate the signal subspace and yield a superior resolution performance for closely spaced sources under severe conditions. However, their computational complexity is similar to the eigenvector-based methods. In order to save computational resources, we perform a dimension reduction through the linear transformation into the beamspace domain, which additionally leads to significant improvements in terms of the resolution capability and the estimation accuracy. A comprehensive complexity analysis and simulation results demonstrate the excellent performance of the proposed algorithms and show their computational requirements. As examples, we investigate the efficacy of the developed methods for the discrete Fourier transform (DFT) and the discrete prolate spheroidal sequences (DPSS) beamspace designs.

Esprit-type algorithms for a received mixture of circular and strictly non-circular signals

Recently, ESPRIT-based parameter estimation algorithms have been developed to exploit the structure of signals from strictly second-order (SO) non-circular (NC) sources. They achieve a higher estimation accuracy and can resolve up to twice as many sources. However, these NC methods assume that all the received signals are strictly non-circular. In this paper, we present the C-NC Standard ESPRIT and the C-NC Unitary ESPRIT algorithms designed for the more practical scenario of a received mixture of circular and strictly non-circular signals. Assuming that the number of circular and strictly non-circular signals is known, the two proposed methods yield closed-form estimates and C-NC Unitary ESPRIT also enables an entirely real-valued implementation. As a main result, it is shown that the estimation accuracy of the presented algorithms improves with an increasing number of strictly non-circular signals among a fixed number of sources. Thereby, not only the estimation accuracy of the strictly non-circular signals themselves is improved, but also the estimation accuracy of the circular signals. These results are validated by simulations.

Performance analysis of ESPRIT-type algorithms for non-circular sources

High-resolution parameter estimation algorithms designed to benefit from the presence of non-circular (NC) source signals allow for an increased identifiability and a lower estimation error. In this paper, we present a 1-D first-order performance analysis of the NC standard ESPRIT and NC Unitary ESPRIT estimation schemes for strictly second-order (SO) non-circular sources, where NC Unitary ESPRIT has a lower complexity and a better performance in the low signal-to-noise ratio (SNR) regime. Our derived expressions are asymptotic in the effective SNR and explicit in the noise realizations, i.e., no assumptions about the noise statistics are necessary. As a main result, we show that the asymptotic performance of both NC ESPRIT-type algorithms is identical in the effective SNR and that NC Unitary ESPRIT is even applicable to array geometries without a centro-symmetric structure as required for Unitary ESPRIT.

Non-data-aided adaptive beamforming algorithm based on the Widely Linear Auxiliary Vector Filter

We propose a non-data-aided adaptive beamforming algorithm based on Widely Linear (WL) processing techniques and the Auxiliary Vector Filtering (AVF) algorithm for non-circular signals, where only the steering vector of the desired user is known. The proposed Widely Linear Auxiliary Vector Filtering (WL-AVF) algorithm recursively updates the filter weights by a sequence of auxiliary vectors that are designed according to the Widely Linearly Constrained Minimum Variance (WLCMV) criterion. It takes full advantage of the second-order statistics of the non-circular data, achieving a higher maximum signal-to-interference-plus-noise ratio (SINR) than the linear AVF. Key properties of the proposed WL-AVF are analyzed. Simulation results show that the WL-AVF beamforming algorithm performs the best among the existing adaptive algorithms.

Widely linear adaptive beamforming algorithm based on the conjugate gradient method

A new widely linear (WL) adaptive beamforming algorithm for non-circular sources based on the conjugate gradient (CG) method is proposed, which we refer to as the widely linear conjugate gradient (WL-CG) algorithm. It is designed according to the widely linearly constrained minimum variance (WL-CMV) criterion and takes full advantage of the second-order statistics of the non-circular data. Since only the knowledge of the steering vector of the desired user is necessary, the proposed method is non-data-aided. The CG method is a powerful algorithm used to design an adaptive beamformer and has an attractive trade-off between performance and complexity. Simulation results show that the proposed WL-CG algorithm provides an excellent performance, while requiring a low complexity compared to existing widely linear beamforming techniques.

SECRECY RATE MAXIMIZATION FOR MIMO GAUSSIAN WIRETAP CHANNELS WITH MULTIPLE EAVESDROPPERS VIA ALTERNATING MATRIX POTDC

In this paper, we consider the problem of optimizing the transmit co-variance matrix for a multiple-input multiple-output (MIMO) Gaussian wiretap channel. The scenario of interest consists of a transmitter, a legitimate receiver, and multiple non-cooperating eavesdroppers that are all equipped with multiple antennas. Specifically, we design the transmit covariance matrix by maximizing the secrecy rate under a total power constraint, which is a non-convex difference of convex functions (DC) programming problem. We develop an algorithm, termed alternating matrix POTDC algorithm, based on alternating optimization of the eigenvalues and the eigenvectors of the transmit covariance matrix. The proposed alternating matrix POTDC method provides insights into the non-convex nature of the problem and is very general, i.e., additional constraints on the co-variance matrix can easily be incorporated. The secrecy rate performance of the proposed algorithm is demonstrated by simulations.

Knowledge-aided direction finding based on Unitary ESPRIT

In certain applications involving direction finding, a priori knowledge of a subset of the directions to be estimated is sometimes available. Existing knowledge-aided (KA) methods apply projection and polynomial rooting techniques to exploit this information in order to improve the estimation accuracy of the unknown signal directions. In this paper, a new strategy for incorporating prior knowledge is developed for situations with a low signal-to-noise ratio (SNR) and a limited data record based on the Unitary ESPRIT algorithm. The proposed KA-Unitary ESPRIT algorithm processes an enhanced covariance matrix estimate obtained by applying a shrinkage covariance estimator, which linearly combines the sample covariance matrix and an a priori known covariance matrix in an automatic fashion. Simulations show that the derived algorithm achieves significant performance gains in estimating the unknown sources and additionally provides a high robustness in the case of inaccurate prior knowledge.

Sparsity-aware direction finding for strictly non-circular sources based on rank minimization

Exploiting the statistical properties of strictly non-circular (NC) signals in direction of arrival (DOA) estimation has long been an active area of research due to its associated performance improvements. Recently, this concept has been introduced to DOA estimation via sparse signal recovery (SSR), where similar benefits from processing NC signals are achieved. However, the standard approach to NC SSR requires solving a two-dimensional (2-D) SSR problem in the spatial and the phase rotation domain, which is not only associated with a high computational complexity itself but also with a 2-D off-grid problem. In this paper, we propose an entirely new NC SSR approach based on nuclear norm (rank) minimization after lifting the original bilinear optimization problem to a linear optimization problem in a higher-dimensional space. Thereby, the SSR-based 2-D estimation problem is reduced to a 1-D estimation problem only in the sampled spatial domain, which automatically provides gridless estimates of the rotation phases. In our second contribution, we present a simple closed-form grid offset estimator for a single NC source and a numerical joint grid offset estimation procedure for two closely-spaced NC sources assuming a uniform linear array (ULA). Simulations validate the effectiveness of the new approach.

ASYMPTOTIC PERFORMANCE ANALYSIS OF ESPRIT-TYPE ALGORITHMS FOR CIRCULAR AND STRICTLY NON-CIRCULAR SOURCES WITH SPATIAL SMOOTHING

Spatial smoothing is a widely used preprocessing scheme to improve the performance of high-resolution parameter estimation algorithms in case of coherent signals or a small number of available snapshots. In this paper, we present a first-order performance analysis of Standard and Unitary ESPRIT as well as NC Standard and NC Unitary ESPRIT for strictly second-order (SO) non-circular (NC) sources when spatial smoothing is applied. The derived expressions are asymptotic in the effective signal-to-noise ratio (SNR), i.e., the approximations become exact for either high SNRs or a large sample size. Moreover, they are explicit in the noise realizations, i.e., only a zero-mean and finite SO moments of the noise are required. We show that both NC ESPRIT-type algorithms with spatial smoothing perform asymptotically identical in the high effective SNR. Also, for the special case of a single source, we analytically derive the optimal number of subarrays for spatial smoothing and show that no gain from strictly non-circular sources is achieved in this case.