On The Positive Definiteness of Polarity Coincidence Correlation Coefficient Matrix
Farzan Haddadi, Mohammad Mahdi Nayebi, Mohammad Reza Aref
TL;DR
This paper investigates the conditions under which the polarity coincidence correlator (PCC) produces positive semi-definite covariance matrices, proving guarantees for low dimensions and providing counterexamples for higher dimensions.
Contribution
It provides a formal proof that PCC yields PSD estimates for real signals with up to 3 variables and complex signals with up to 2, and presents counterexamples in higher dimensions.
Findings
PCC guarantees PSD matrices for real signals when p<=3.
PCC guarantees PSD matrices for complex signals when p<=2.
Counterexamples show PCC may not be PSD in higher dimensions.
Abstract
Polarity coincidence correlator (PCC), when used to estimate the covariance matrix on an element-by-element basis, may not yield a positive semi-definite (PSD) estimate. Devlin et al. [1], claimed that element-wise PCC is not guaranteed to be PSD in dimensions p>3 for real signals. However, no justification or proof was available on this issue. In this letter, it is proved that for real signals with p<=3 and for complex signals with p<=2, a PSD estimate is guaranteed. Counterexamples are presented for higher dimensions which yield invalid covariance estimates.
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