Latent Signal Analysis and the Analytic Signal

TL;DR

This paper redefines the complex extension problem in signal analysis, proposing alternative methods to the Hilbert Transform for more accurate instantaneous parameter estimation by relaxing harmonic constraints.

Contribution

It introduces a new approach to complex signal extension that relaxes harmonic correspondence, enabling potentially more accurate amplitude and frequency analysis.

Findings

Alternative complex extensions can be more accurate than Hilbert Transform-based methods.

Relaxing harmonic correspondence removes uniqueness but offers flexibility in analysis.

Solutions are provided for a practical latent signal analysis problem.

Abstract

In this paper we present the latent signal analysis problem as a recasting of the complex extension problem. Almost universally, the approach has been to use the Hilbert Transform (HT) to construct Gabor's analytic signal. This approach depends on harmonic correspondence and forces the use of simple harmonic components in the analysis, which may lead to incorrect Instantaneous Amplitude (IA) and Instantaneous Frequency (IF) parameters. We show that by relaxing the harmonic correspondence condition, the resulting complex extension can still be an analytic function and we can arrive at alternate IA/IF parameterizations which may be more accurate at describing the latent signal. However, in relaxing the harmonic correspondence condition there is no longer a unique rule for the complex extension. In addition, we discuss the problems associated with the use of the HT as an AM--FM demodulator for the latent signal. Finally we provide solutions to an example latent signal analysis problem.