Normalized Autocorrelation Function Research Articles

A modified hybrid sign estimator for the normalized autocorrelation function of a Gaussian stationary process

Fast digital estimators based mainly on additions are of great interest for computing the normalized a.c.f. of Gaussian stationary discrete processes.

Resolution bias errors in spectral density, frequency response and coherence function measurements, IV: Time delay bias errors

The problem of time delay bias errors is treated in this part of the present series of papers as a special case of a general theory of resolution bias errors in spectral estimation presented in a previous part. It is shown that the main effect of an additional time delay between the output and the input of a physical system is to multiply the frequency response and coherence estimators which would be obtained without delay with the normalized autocorrelation function of the window function and its square respectively. Experimental results for the rectangular and Hanning windows are in accord with the theoretical predictions. The theoretical results given in the literature previously are valid for the rectangular window only but not for an arbitrary window function.

Resolution bias errors in spectral density, frequency response and coherence function measurements, I: General theory

A general theory of resolution bias errors inherent in the usual ensemble (segment) averaging estimation procedure for spectral density, frequency response and coherence functions has been developed. It is shown that the estimation procedure is mathematically equivalent to a situation where the true correlation functions would be modified in a specified manner: that is, they would be multiplied with the normalized autocorrelation function of the window function and transformed into the frequency domain to yield the spectral density estimates. The theory is worked out for an arbitrary ideal, linear, and time-invariant system which is driven by a stationary stochastic input and for an arbitrary window function. Specifically, the rectangular and Hanning windows are compared with each other and the theoretical results are verified experimentally to great accuracy by using electrical RC- and LRC-circuits and a true random noise generator (see later parts of this series of papers). Furthermore, as an application of the theory, approximate formulas for the bias errors of spectral estimators are derived and discussed, thus extending some older results given in the literature.

Statistical Properties of the Dynamic Polychromatic Laser Speckle

This paper investigates theoretically the normalized autocorrelation function of the time-varying polychromatic laser speckle intensity fluctuation produced in the Fraunhofer diffraction field by the in-plane movement of a diffuse object. The variation of the correlation time in the radial direction indicates the spatial non-stationarity of an elongated speckle pattern while that in the angular direction indicates the three different spatial structures, randomly distributed, fibrous radial and radially separated, of elongated speckles in the angular direction. The theoretical results obtained are confirmed by experiments.

Statistical properties of the dynamic dichromatic laser speckle

The statistical properties of dynamic elongated speckle patterns produced in the far-field diffraction region by a moving diffuse object under illumination of the dichromatic laser beam are experimentally investigated by means of a correlation method. The experimental results show that the correlation length of the normalized autocorrelation function of time-varying dichromatic speckle intensity fluctuations depends strongly on the fibrous radial structure of elongated speckle patterns. The qualitative discussions are given for the experimental results.

Statistical Properties of Radar Echo Patterns and the Radar Echo Process

Abstract A stochastic process for explaining the statistical properties of radar echo patterns in a given direction is defined and investigated. We call this process the radar echo process. Earlier work by Kessler, Russo and Cole, who used the random telegraph process as a model for the statistical properties of radar echo patterns, led to certain discrepancies. The radar echo process is a much more satisfactory model for these statistical properties and completely explains the discrepancies due to the random telegraph assumption. The new process, being coverage dependent, reduces to the random telegraph process for fifty per cent coverage, is wide sense stationary and ergodic, has an exponential normalized autocorrelation function, implies realistic distributions of lengths, and is formulated in terms of conditional probabilities which are explicitly expressible in terms of elementary functions.

Frequency Correlation of Sound-Transmission Functions in Rooms

Steady-state frequency responses between pairs of points in a room can be considered samples of a random process. (This holds for frequencies above a certain critical value, typically several tens of cycles per second.) The autocorrelation functions of frequency responses of a variety of rooms are computed analyitcally and compared with experimental data and Monte Carlo results. For rooms whose normal modes have uniform decay rates (exponential reverberation), the normalized autocorrelation function of the squared modulus of the frequency response depends only on the reverberation time T60 and equals [1+(0.455⋅T60⋅Δf)2]−1. The autocorrelation of the modulus itself has a more involved analytical form but, surprisingly, is almost equal numerically to that of the modulus squared. The autocorrelation and crosscorrelation functions of the real and imaginary parts of frequency responses are also computed. The practical significance of these frequency-correlation functions for room acoustics is discussed. The analytical methods employed can be extended to the computation of spatial sound-pressure correlation functions, including those of nondiffuse sound fields.