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Measurement of Matrix Frequency Response Functions and Multiple Coherence Functions

Report Number: AFFDL TR 65-56
Author(s): N. R. Goodman
Corporate Author: Measurement Analysis Corporation
Laboratory: Air Force Flight Dynamics Laboratory
Date of Publication: 1965-06
Pages: 36
Contract: AF 33(615)-1418
Project: 1370
Task: 137005
AD Number: AD0619993

Fundamental concepts involved in the statistical analysis of multiple-input single-output time-invariant linear systems are described. The definitions of a matrix frequency response function and a multiple coherence function are presented. Formulas for computing simultaneous confidence bands for all elements of the matrix frequency response function are obtained, using the standard 'F' distribution. Expressions for the confidence bands are given both as a function of the various types of coherences and of the elements of the spectral density matrix. The effect of the various quantities on the width of the confidence bands is discussed in detail. Confidence bands for the gains and phases of the frequency response functions are also developed. The interpretation of linear system computational results in terms of a time invariant nonlinear system model is described. It is shown how the linear system results provide what may be thought of as a 'best' linear fit to the nonlinear model. The multiple coherence function then gives a quantitative measure of goodness of this fit. In this sense the coherence function may be used to provide a test for system linearity.

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