Principles of System Identification

10

Fourier Transforms and Spectral

Analysis of Deterministic Signals

The chapter is devoted to the frequency-domain analysis of signals with the main objective

of introducing the concept of power spectrum. The focus topics of this chapter are Fourier

representations, power spectral density and the correlation theorem, all of these set up in the

context of deterministic signals. We shall also briefly introduce the notions of cross power

spectrum and coherence. The concepts learnt in this chapter lay the foundations for Chapter

11 concerning stochastic processes.

10.1 MOTIVATION

In Chapter 8 we learnt how to characterize random signals (processes) in terms of their time-domain

statistical properties, specifically, using correlation functions. Subsequently, Chapter 9 offered time-

domain parametric descriptions of linear stationary and integrating type processes.

A radically different route of describing and analyzing these signals takes us through the fre-

quency domain using the Fourier transform, popularly known as spectral representations. Compre-

hending the associated theory becomes considerably easier once we understand the machinery in the

framework of deterministic signals. With this idea, we devote this chapter to the Fourier transforms

and spectral analysis of deterministic processes. Spectral representations of random signals are then

taken up in Chapter 11.

Transforms are essentially change of basis of representation. The Fourier transform is based on

the use of sinusoidal basis functions, which are pure tone waveforms. A natural question is whether

this new representation of signals offers any theoretical and/or practical benefits over the regular

time-domain counterpart?

Fourier transform of signals brings with it several benefits to the fields of signal processing

and identification. Central to these applications are the concepts of spectral density function and

phase spectrum. The notion of spectral density is also vital to the milestone spectral factorization

result. For this reason, analysis in the Fourier domain is also known as spectral or frequency-domain

analysis.

We highlight the salient applications of frequency-domain analysis below.

1. Detection of oscillatory or periodic components in measurements: Determining the presence

of periodicities in a measurement is highly infeasible using a pure time-domain analysis due

to the presence of noise and multiple frequency components. This is one of the most widely

encountered problems in all spheres of engineering and science. In climatology, we are inter-

ested in knowing the periodicity of climatic phenomena; in health monitoring of control loops,

the first step is to detect loops with oscillatory outputs (Tangirala, Shah and Thornhill, 2005);

fault detection and diagnosis of vibration machinery and several other mechanical equipments

rely on monitoring of frequency content of measurements (Randall, 2011); biomedical signals

such as EEG and ECG are primarily characterized by their frequency content (Blinowska and

Zygierewicz, 2012) and so on. Among the different tools used in these applications, Fourier

transform occupies the most prominent and indispensable place.

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