The Statistical Characteristics of Power-Spectrum Subband Energy Ratios under Additive Gaussian White Noise

TL;DR

This paper analyzes the statistical properties of power-spectrum subband energy ratios (PSER) under additive Gaussian white noise, revealing their distributional characteristics and independence from noise variance, with implications for signal detection.

Contribution

It develops a probability distribution model for PSER in noisy environments and provides an approximation method for identifying valid signals amidst noise.

Findings

PSER follows a beta distribution under pure noise.

In mixed signals, PSER follows a doubly non-central beta distribution.

The quantile of PSER is unaffected by noise variance, aiding signal detection.

Abstract

The power-spectrum subband energy ratio (PSER) has been applied in a variety of fields, but reports on its statistical properties have been limited. As such, this study investigates these characteristics in the presence of additive Gaussian white noise for both pure noise and mixed signals. By analyzing the probability and independence of power spectrum bins, and the relationship between the F and beta distributions, we develop a probability distribution for the PSER. Results showed that in the case of pure noise, the PSER follows a beta distribution. In addition, the probability density function and the quantile exhibited no relationship with the noise variance, only with the number of lines in the power spectrum, that is, PSER is not affected by noise. When Gaussian white noise was mixed with the known signal, the resulting PSER followed a doubly non-central beta distribution. In this case, it was difficult to identify the quantile, as the probability density and cumulative distribution functions were represented by infinite series. However, when a spectral bin did not contain the power spectrum of the known signal, an approximated quantile was found. This quantile is strictly proved to be in agreement with the quantile in the case of pure noise and offers a convenient methodology for identifying valid signals.