Computationally efficient sparsity-inducing coherence spectrum estimation of complete and non-complete data sets

Abstract

The magnitude squared coherence (MSC) spectrum is an often used frequency-dependent measure for the linear dependency between two stationary processes, and the recent literature contain several contributions on how to form high-resolution data-dependent and adaptive MSC estimators, and on the efficient implementation of such estimators. In this work, we further this development with the presentation of computationally efficient implementations of the recent iterative adaptive approach (IAA) estimator, present a novel sparse learning via iterative minimization (SLIM) algorithm, discuss extensions to two-dimensional data sets, examining both the case of complete data sets and when some of the observations are missing. The algorithms further the recent development of exploiting the estimators' inherently low displacement rank of the necessary products of Toeplitz-like matrices, extending these formulations to the coherence estimation using IAA and SLIM formulations. The performance of the proposed algorithms and implementations are illustrated both with theoretical complexity measures and with numerical simulations.

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“…So the complexity can be reduced in this way. However, these results (19), (20) are under the assumption that W 4 was known. Actually, W 4 remains to be calculated.…”

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“…With these results, we are now ready to introduce the displacement of R À2 N 1 N 2 , noting that this particular form is a special case of more general matrix products studied recently in the context of the efficient implementation of 1-D and 2-D Magnitude Squared Coherence estimators [29][30][31]. Using (41),…”