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Coherence, Classical Correlations, and their Invariances

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Coherence

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Abstract

This chapter proceeds from a discussion of scalar-valued coherence to multiple coherence to matrix-valued coherence. Connections are established with principal angles and with canonical correlations. The study of factorizations of two-channel covariance matrices leads to filtering formulas for MMSE filters and their error covariances. When covariance matrices are estimated from measurements, then the filter and error covariance are random matrices. Their distribution is given. The multistage Wiener filter (MSWF), a conjugate gradient algorithm, is reviewed as a way to recursively update the order of the MMSE filter. Beamforming is offered as an illustration of these ideas. Half- and full-canonical coordinates are shown to be the correct bases for model order reduction. When three channels are admitted into the discussion, then the theory of partial coherence arises as a way to quantify the efficacy of a third channel when solving regression problems.

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Notes

  1. 1.

    In the original, and influential, work of Goldstein and Reed, this was termed the multistage Wiener filter [140].

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Authors and Affiliations

  1. Universidad Carlos III de Madrid, Madrid, Spain

    David Ramírez

  2. Universidad de Cantabria, Santander, Spain

    Ignacio Santamaría

  3. Colorado State University, Fort Collins, CO, USA

    Louis Scharf

Authors
  1. David Ramírez
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  2. Ignacio Santamaría
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  3. Louis Scharf
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Ramírez, D., Santamaría, I., Scharf, L. (2022). Coherence, Classical Correlations, and their Invariances. In: Coherence. Springer, Cham. https://doi.org/10.1007/978-3-031-13331-2_3

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