Estimating signal spectra with a method of determining concealed periodicities in zero crossings

Abstract

We apply methods for determining concealed periodicities in zero crossings to spectral analysis of signals. The proposed methods increase confidence in the estimates obtained by using preliminary computation of "quasispectra" and nonlinear signal processing.In certain areas of science and technology, e.g., geophysics, optics, and audio-and video-signal processing, it is necessary to determine the parameters of quasiperiodic signals, i.e., signals that are additive mixtures of several sinusoidal waves that have time-varying parameters with white noise. The conventional approach to solution of this problem uses methods of detecting hidden periodicities, which has advantages over FFT methods in that there is no need to establish a period for the signal; the period can be determined during processing. Among the disadvantages of the method of detecting hidden periodicities is that it requires cumbersome computation and a large sample to achieve a given resolution [1].At the present time, considerable attention has been drawn to a method of detecting hidden periodicities by computing the number of zero crossings in a measurement signal, since the method of detecting hidden periodicities from zero crossings has an excellent resolution capability, has a lower computational cost (since it only requires addition operations), and inherits the properties of the ordinary method of detecting hidden periodicities. This makes it necessary to determine whether the method can be used for spectral analysis of signals.The fundamentals of the method of detecting hidden periodicities from zero crossings were presented by Rice in [2], where the statistical and probabilistic characteristics of signals taken to be mixtures of sinusoidal waves mixed with normally distributed white noise were determined. In particular, Rice derived a formula for determining the mathematical expectation of the number of zero crossings for such signals. Study of the problem was continued by White, Raynel, Bendat, and others. Bendat [3], for example, generalized Rice's results to the case of sinusoids with variable parameters and obtained a simpler formula for the mathematical expectation for the number of zeros, as well as a formula for the mathematical expectation of the mean square number of zero crossings by a process under investigation.The zeros problem was subsequently further studied by B. Kedem, who extended previous results to discrete time and concluded that the number of times a signal crosses zero provides an idea of the spectral properties of the signal, which makes it possible to use zeros of a signal to estimate its spectrum. In particular, it was shown in [4] that the mathematical expectation for the number of zeros D of a signal composed of white noise is connected to the spectral function of the signal by the following relation:where E(*) is the mathematical expectation operator, F(co) is the spectral function of the signal, N is the size of the sample, co is the relative angular frequency, i.e., the frequency normalized by the quantization...

Method for Evaluating Periodic Trends in Measured Signals Based on the Number of Zero Crossings

“…It can be used both to estimate the spectrum of the signal being studied, and for other practical tasks, in particular the detection of periodic signals [4]. There is a signifi cant limitation to the quasispectrum; it has low sensitivity at the edges of the frequency band, and this must be taken into account in choosing the sampling rate [3,5].The quasispectral component is the frequency at which NZC appear, i.e., it is used to determine periodicity in the initial sample at some frequency. To estimate the frequency of a harmonic trend in the resulting quasispectrum, the maximum value of the quasispectral component is found and the corresponding relative frequency is calculated.…”

“…Since NZC methods are associated with the spectral function of the analyzed signal, it is proposed that this fact be used for a rough spectral analysis [3]. To do this, repetitive-difference and -summation fi lters are applied to the signal sequentially in different combinations (the maximum order of fi lters is limited) and the number of zero crossings is calculated for each combination of these fi lters.…”

“…Relative errors δ ƒ in a frequency determination for a signal model with gaussian noise with S/N ratios of 3.0 (1), 1.0 (2), and 0.75(3).…”

“…It can be used both to estimate the spectrum of the signal being studied, and for other practical tasks, in particular the detection of periodic signals [4]. There is a signifi cant limitation to the quasispectrum; it has low sensitivity at the edges of the frequency band, and this must be taken into account in choosing the sampling rate [3,5].…”