IAA Spectral Estimation: Fast Implementation Using the Gohberg–Semencul Factorization

Dept. of Electr. & Comput. Eng., Univ. of Florida, Gainesville, FL, USA
IEEE Transactions on Signal Processing (Impact Factor: 2.79). 08/2011; 59(7):3251 - 3261. DOI: 10.1109/TSP.2011.2131136
Source: DBLP


We consider fast implementations of the weighted least-squares based iterative adaptive approach (IAA) for one-dimensional (1-D) and two-dimensional (2-D) spectral estimation of uniformly sampled data. IAA is a robust, user parameter-free and nonparametric adaptive algorithm that can work with a single data sequence or snapshot. Compared to the conventional periodogram, IAA can be used to significantly increase the resolution and suppress the sidelobe levels. However, due to its high computational complexity, IAA can only be used in applications involving small-sized data. We present herein novel fast implementations of IAA using the Gohberg-Semencul (G-S)-type factorization of the IAA covariance matrices. By exploiting the Toeplitz structure of the said matrices, we are able to reduce the computational cost by at least two orders of magnitudes even for moderate data sizes.

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    • "Literatures have also presence of certain unique attempts to perform spectral estimation. Adoption of Gohberg factorization technique was seen in the literature of Xue et al. [8], where the authors have used weighted least squared approach of iterative nature. The outcome of the study was analyzed using computational time as well as signal power, but the outcome of the study was not found to analyze with respect to the power spectral density, which is also one of the significant performance parameters to showcase the effectiveness of spectral estimation techniques. "
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    • "The basic idea of the recently developed IAA algorithm tries to solve (5) by minimizing the following weighted least-squares (LS) cost function [30] [31] [32] [33] [34] [35] [36] [37] [38] min∥x−γ k;i vðf d;k ; f s;i Þ∥ "

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    • "First, though, the covariance matrix˜R N is estimated using the IAA as described by (16) and (17), which can efficiently implemented without the need of direct estimation of the covariance matrix and its inverse. This can be accomplished using the celebrated Levinson-Durbin (LD) algorithm and some fast techniques for the evaluation of trigonometric polynomials related to structured matrices as detailed in [16], [17]. "
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    ABSTRACT: Recently, optimal linearly constrained minimum variance (LCMV) filtering methods have been applied to fundamental frequency estimation. Such estimators often yield preferable performance but suffer from being computationally cumbersome as the resulting cost functions are multimodal with narrow peaks and require matrix inversions for each point in the search grid. In this paper, we therefore consider fast implementations of LCMV-based fundamental frequency estimators, exploiting the estimators' inherently low displacement rank of the used Toeplitz-like data covariance matrices, using as such either the classic time domain averaging covariance matrix estimator, or, if aiming for an increased spectral resolution, the covariance matrix resulting from the application of the recent iterative adaptive approach (IAA). The proposed exact implementations reduce the required computational complexity with several orders of magnitude, but, as we show, further computational savings can be obtained by the adoption of an approximative IAA-based data covariance matrix estimator, reminiscent of the recently proposed Quasi-Newton IAA technique. Furthermore, it is shown how the considered pitch estimators can be efficiently updated when new observations become available. The resulting time-recursive updating can reduce the computational complexity even further. The experimental results show that the performances of the proposed methods are comparable or better than that of other competing methods in terms of spectral resolution. Finally, it is shown that the time-recursive implementations are able to track pitch fluctuations of synthetic as well as real-life signals.
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