Spectral Analysis

Parametric and Non-Parametric Digital Methods

Francis Castanié, Francis Castanié

• Editore: Wiley-ISTE
• Anno: 2013
• ISBN:
  9781118614273
• Edizione: 1
• Argomento: Technology & Engineering
• Categoria: Signals & Signal Processing

This book deals with these parametric methods, first discussing those based on time series models, Capon's method and its variants, and then estimators based on the notions of sub-spaces. However, the book also deals with the traditional "analog" methods, now called non-parametric methods, which are still the most widely used in practical spectral analysis.

PART I

Tools and Spectral Analysis

Chapter 1

Fundamentals ¹

1.1. Classes of signals

Every signal-processing tool is designed to be adapted to one or more signal classes and presents a degraded or even deceptive performance if applied outside this group of classes. Spectral analysis too does not escape this problem, and the various tools and methods for spectral analysis will be more or less adapted, depending on the class of signals to which they are applied.

We see that the choice of classifying properties is fundamental, because the definition of classes itself will affect the design of processing tools.

Traditionally, the first classifying property is the deterministic or non-deterministic nature of the signal.

1.1.1. Deterministic signals

The definitions of determinism are varied, but the simplest is the one that consists of calling any signal that is reproducible in the mathematical sense of the term as a deterministic signal, i.e. any new experiment for the generation of a continuous time signal x(t) (or discrete time x(k)) produces a mathematically identical signal. Another subtler definition, resulting from the theory of random signals, is based on the exactly predictive nature of x(t)t > t₀ from the moment that it is known for t < t₀ (singular term of the Wold decomposition for example; see Chapter 4 and [LAC 00]). We will discuss here only the definition based on the reproducibility of x(t), as it induces a specific strategy on the processing tools: as all information of the signal is contained in the function itself, any bijective transformation of x(t) will also contain all this information. Representations may thus be imagined, which, without loss of information, will demonstrate the characteristics of the signal better than the direct representation of the function x(t) itself.

The deterministic signals are usually separated into classes, representing integral properties of x(t), strongly linked to some quantities known by physicists.

Finite energy signals verify the integral properties [1.1] and [1.2] with continuous or discrete time

[1.1]

[1.2]

We recognize the membership of x(t) to standard function spaces (noted as L₂ or l₂ respectively), as well as the fact that this integral, to within some dimensional constant (an impedance in general), represents the energy E of the signal.

Signals of finite average power verify:

[1.3]

[1.4]

If we accept the idea that the sums of equation [1.1] or [1.2] represent “energies”, those of equation [1.3] or [1.4] then represent powers.

It is clear that these integral properties correspond to mathematical characteristics whose morphological behavior along the time axis is very different: the finite energy signals will be in practice “pulse shaped”, or “transient” signals such that |x(t)| → 0 for |t| → ∞. This asymptotic behavior is not at all necessary to ens...