Properties of the Toeplitz Covariance Matrix Numerical Likelihood Ratio Maximization: Convex-Like Optimization Behavior

Abstract

Yuri Abramovich, Victor Abramovich and Tanit Pongsiri

Abstract In this paper, we continue to investigate the well-known problem of the numerical likelihood maximization of the positive definite Toeplitz covariance matrix of complex Gaussian data. In our recent papers, we demonstrated that direct LR maximization, using Vandermonde parameterization, applied to initial Toeplitz matrices distant from the true Toeplitz covariance matrix, predominantly yields an inappropriate solution with negative eigenvalues [1,2]. Yet in all cases where the MATLAB fmincon routine converges to a positive definite Toeplitz matrix, the process converges to the same solution as if initiated by a true Toeplitz covariance matrix, irrespective of the initialization one. We also demonstrated that the optimized likelihood ratio (LR) exceeded that of the true Toeplitz covariance matrix and that, by starting fmincon iterations from the true covariance matrix �?�?�, we converged to the same solution as when starting from practical initializations. These two properties of the globally optimal solutions are crucial for practical Toeplitz matrix estimation. In this paper, we continue to explore these convex-like properties of the MATLAB fmincon LR maximization. In particular, to avoid generating large numbers of non-positive-definite solutions, we propose using the Carathéodory Toeplitz matrix representation in the initial step of the LR maximization with the fmincon routines. We demonstrate that this optimization exceeds the LR value of the true Toeplitz covariance matrix. Still, in most cases, it does not exceed the LR maximum obtained by optimizing the covariance lags of the optimized Toeplitz matrix. Therefore, the second stage of LR optimization, which we performed in covariance lags, drove the optimized LR to the maximum. Yet, we demonstrate that this second-stage LR maximization improves the LR but, in fact, degrades the proximity of the optimized solution to the true Toeplitz covariance matrix used instead of the sample matrix in the LR. We demonstrate that the proposed LR maximization allows for the essential reduction of the minimal input SNR or inter-source separation of the properly estimated sources' DOAs, compared to the classical MUSIC algorithm, applied to the sample matrix R�?�?�.

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