Emanuel Parzen

The Estimation of Coherence, Frequency Response, and Envelope Delay

The inappropriateness of standard cross-spectrum estimntes when the envelope delay is nonzero is discussed. Appropriate methods for estimating cross-spectrum quantities, including the delay, are discussed. The techniques are illustrated by revisiting a chemical blender example.

Comments on the Discussions of Messrs. Tukey and Goodman

We are all grateful to Professor Tukey and Dr. Goodman for their comments and constructive criticism. Since Dr. Goodman has not chosen to discuss any of the issues raised in the two papers, I shall confine my reply to Professor Tukeys lengthy contribution which contains a wealth of knowledge and a depth of understanding of the subject. He has based much of his discussion on the analogy between spectral analysis and the components of variance aspect of the analysis of variance. There is clearly some justification for making this analogy but I would have thought that the differences are sufficiently great to make it unwise to pursue it too far. In general, the variances which appear in the expected values of the mean squares in a components of variance analysis will correspond to populations which we will have deliberately sampled and for the estimation of which we will have designed a specific experimental arrangement. When it comes to spectrum analysis, we have in theory an infinite number of components of variance from a sample, no matter how small. In general we have no a priori preference for particular components of variance but we are interested in as accurate a picture of the spectral curve as is possible. In other situations, we may not want all this detail; for example in designing an experiment to estimate the slope of a response surface when the errors are autocorrelated, we will only be interested in those frequencies for which the spectral density is smallest; in many other physical applications it is sufficient to give the average spectral density over fairly wide bands, presenting finally a picture looking exactly like a histogram. In all these situations, the choice of components of variance is much more arbitrary than in the classical components of variance situation. However, spectral analysis is usually a part of a components of variance analysis in the usual sense. This is due to the fact that one is not usually interested in single spectra; in many cases, spectrum analysis will form part of a much larger experimental programme. We might for example, be investigating the effect of atmospheric turbulence on aircraft structures and an experiment may have been planned using different pilots and several aircraft flown at each of several heights on a number of different days. Instead of one response in each cell of the experiment, we now have a time-series which we may replace by its spectrum. The variation in the estimated spectra may then be compared from one set of experimental conditions to the other and we might isolate components of variance for pilots, aircraft and days. There is a much greater degree of flexibility here than in the usual set-up since components of variance may be isolated for a number of frequency bands.