Fundamentals - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 20

One of the reasons for the success of the kernel e -j 2π ft (or its equivalent with f
complex, the Laplace transform) is that this family of special functions is one of the
eigenfunctions of invariant linear systems. When we analyze such systems, it is
legitimate to project (break down) the signals on this basis, the linearity of the
system ensuring individual processing of each component. Fundamentally, this
property is at the root of spectral equations that are as useful as the Wiener-Lee
relations.
Outside the invariant linear framework, the Fourier transform with the
exponential kernel is not justified as much, and other kernels may be better suited:
the linear transforms on Gallois fields in base 2 justify the use of binary kernel
transforms, the search for polynomial modulation laws requires higher order
Wigner-Ville representations, the highlighting of translations and time dilations
induces wavelet representations, etc.
The following chapters deal almost exclusively with the exponential kernel, but
the informed reader could easily generalize most of the concepts and results to other
kernels, by rather simple transpositions of concepts of frequency resolution,
estimation variance, bias-variance trade off, etc.
However, the importance of the exponential kernel for applications is practically
a fact of life, the other representations sharing a small portion of applications. This
justifies the bias of the current topic which is to center the discussion on spectral
analysis with the Fourier kernel , rather than make a presentation with a more
general class of kernels ψ ( t, u ) and to deduce from this the properties of Fourier
analysis: as a result the topic loses elegance, but no doubt gains in clarity and
pedagogy.
In addition, the majority of chapters is devoted to 2 nd order spectral analysis, for
the same reasons of extent of the fields of application. However, the extension to
multispectra is quite obvious, and the topics already listed show that multispectral
analysis borrows everything from 2 nd order analysis.
In the very large set of spectral analysis methods, we discern non-parametric
methods, the subject of Chapter 5, which make very few hypotheses on the signal:
these hypotheses are essentially the existence of Fourier transforms of the analyzed
quantities (the signal itself in the deterministic context, 2 nd order moments in random
context), and most often the stationarity (strict or local) of moments in random
context.
Parametric methods, the subject of Chapters 4 and 6, are based on a more
restrictive a priori assumption: we suppose that we know a behavioral model of the
signal, general model that will be adjusted to the physical signal using a set of