Silvia Maria Alessio

Singular Spectrum Analysis (SSA)

This chapter is devoted to an approach of extracting periodic or quasi-periodic components from a random signal. Singular Spectrum Analysis (SSA) is not, in a strict sense, a simple spectral method, since it is aimed at representing the signal as a linear combination of elementary variability modes that are not necessarily harmonic components, but can exhibit amplitude and frequency modulations in time, and are data-adaptive, i.e., modeled on the data. It does not provide a stationary spectral estimate but can separate auto-coherent from random features. SSA is a non-parametric method, since it does not assume any specific model for the generation of the signal. It can also be viewed as a powerful de-noising technique; finally, it can be exploited as a tool for filling gaps in data records that is soundly based from a theoretical point of view. Examples the real-world applications of SSA are provided.

Discrete Wavelet Transform (DWT)

The wavelet transform can be seen as a wavelet-based expansion (decomposition) of a finite-energy signal. In the discrete wavelet transform (DWT), economy in the representation of the signal and possibility of perfect signal reconstruction (PR) are crucial. The simplest formulation of the DWT problem includes two types of basis functions for the expansion: the scaling and wavelet functions. We will see how an ideal, infinite-length but finite-energy signal can be decomposed from the point of view function spaces, and how this decomposition can be obtained using a two-channel digital filter bank. The description of a fast wavelet decomposition/reconstruction algorithm will lead us to the practical implementation of the DWT in the real-life case of a finite-length, sampled input signal, as well as to the properties of PR filters, which are strictly related to the scaling and wavelet functions. After allowing for the necessary conditions that the filters of the bank must satisfy, primarily biorthogonality or orthogonality, a number of degrees of freedom remain available to design different wavelet systems suited for different purposes. A real-world example of signal DWT decomposition will be provided. The chapter ends with an appendix in which the various wavelet systems used for the CWT and/or DWT are reviewed.

Exercises with Matlab

This final chapter presents exercises on most of the techniques discussed in the book, including filter design and filtering implementation, stationary and non-stationary spectral analysis, etc.

De-noising and Compression by Wavelets

This chapter offers an introduction to DWT-based signal de-noising and compression. Real-life examples will be provided. The DWT outputs a series of approximation coefficients, representing the signal’s coarse features, and a number of detail-coefficient sequences, each belonging to a single level of decomposition, i.e., to a particular scale and level of time resolution with which we look at the signal. The basic idea consists of thresholding the detail coefficients of the noisy signal, preserving only those that are larger than the characteristic amplitude of the noise. The threshold can generally be a function of level and time, but usually it is function of level only, or even a scalar, and several variants of threshold selection methods exist. In cases in which the presence of non-white noise is suspected, the amplitude of the noise can be estimate level-by level. Signal compression is aimed at retaining only the information necessary to reconstruct significant features of the original signal, for reasons of storage saving. The relation with the de-noising issue is obvious, but in compression the focus is on the extent to which the number of DWT coefficients to be stored can be reduced with respect to the complete set, while preserving a substantial amount of the signal’s variability.

Sampling of Continuous-Time Signals

This chapter deals with the minimum sampling interval \(T_s\) needed to correctly represent an analog signal by samples extracted periodically from it, so as to be able to reconstruct the continuous-time signal from its discrete-time version. The sampling theorem prescribes this lower limit and highlights the fact that a representative sampling is possible if, and only if, the analog signal does not contain frequencies higher than the Nyquist frequency \(1/(2 T_s)\): no finite-rate sampling can capture the variations of an analog signal which is not bandlimited. Other issues related to analog signals, such as the signal’s concentrations in the time and frequency domains and their mutual inverse dependence (uncertainty principle), as well as the definition of bounded support in both domains, are also discussed. An appendix provides a summary of the relations among the variables used to express the concept of frequency in the continuous-time and discrete-time cases.

Non-Parametric Spectral Methods

This chapter deals with obtaining a good estimate of the power spectrum of a random signal on the basis of a finite number of samples of a typical realization of the underlying random process—one among the infinite sequences that the process can generate when we measure it. The simplest approach to spectral estimation, i.e., the periodogram, turns out to perform poorly: the variance of the estimate is high and does not decrease with increasing length of the data record—it is not a consistent estimate of the power spectrum. The search for a stable and consistent spectral estimate leads to the methods of Bartlett and Welch, and to the Blackman-Tukey method. We will also present statistical tests used judge the significance of any peak detected in a spectrum. A description of the multitaper method (MTM) and a brief account of the estimation of the cross-spectrum of two random signals will be followed by a discussion about the use of FFT for practical computation of spectral estimates and about the different normalization schemes adopted in literature for the power spectrum.

Digital Filter Properties and Filtering Implementation

The first part of this chapter examines the properties of frequency-selective filters that are LTI stable systems with a real and causal impulse response and a rational transfer function. Their frequency response is classified according to four prototypes: the lowpass, highpass, bandpass, and bandstop ideal filters. Ideal filters have real frequency response with jump discontinuities at band edges and are not computationally realizable: they must be approximated by continuous complex functions. The conditions for realizability are discussed in terms of magnitude and phase of the frequency response. The phase of a realizable filter presents jump discontinuities that are eliminated passing to a continuous-phase representation. Linear phase (LP) and generalized linear phase (GLP) filters are then studied, which do not cause phase distortion between input and output waveforms. Only FIR filters of four types can have exactly LP/GLP, and their impulse response must satisfy precise symmetry conditions. The second part of the chapter deals with implementing digital filtering, by arranging the difference equation into the most convenient structure. Finally, applications using downsampling before filtering are described.

FIR Filter Design

The beginning of this chapter contains general considerations about the design of a digital filter. The form in which the filter specifications must be expressed by the designer are illustrated, and the reasons why an IIR or an FIR filter might be preferred are listed. Then the discussion focuses on the design of linear phase (LP) or generalized linear phase (GLP) FIR filters, which exist in four types. Issues related to the selection of the design method and to the quantitative approximation criteria that may be established to judge the resemblance of the designed filter with the desired one are discussed. The properties of LP/GLP FIR filters are examined in detail, and a factorization of the zero-phase response, useful to unify the symmetry condition for the coefficients of the four filter types, is presented: the zero-phase response is split into a fixed factor, depending on the filter type but not on specifications, and an adjustable factor, with coefficients to be determined according to specifications. The most flexible and optimum design method for LP/GLP FIR filters is then described: this is the minimax method, which ensures the filter meets specifications with the minimum possible order. The properties of optimum FIR filters are finally studied.

Transforms of Discrete-Time Signals

In this chapter, the invertible transforms used to work on discrete-time signals are discussed. Given a complex variable z, the z-transform is defined as an infinite series in the z-plane that exists in the region(s) of the plane where the series exhibits absolute convergence to an analytic function. The corresponding infinite-length signal is required to be absolutely summable. Unit-amplitude z values identify the unit circle, on which the z-transform becomes a continuous function of frequency, called the discrete-time Fourier transform (DTFT). The DTFT representation can also be extended to sequences for which the z-transform does not exist, such as signals that are only square-summable, or periodic signals like sinusoids. If a sequence has finite length, it may be represented in the frequency domain by a finite number of values obtained by properly sampling the DTF, i.e., by the discrete Fourier transform (DFT). The properties of the DFT emerge clearly if this transform is introduced passing through the discrete Fourier series (DFS) of the signal’s periodic extension. The DFT can be efficiently computed via fast Fourier transform (FFT). Each inverse transform represents an expansion of the signal in an orthogonal basis. At the end of the chapter, an appendix provides an overview of the mathematical foundations of analog and discrete-time signal expansions.

Parametric Spectral Methods

In this chapter, parametric methods of spectral estimation are presented. They rely on fitting a proper stochastic model to the data record. The model is supposed to represent the persistence, i.e., autocorrelation, present in the process generating the observed signal. The signal’s spectral characteristics are then derived from the estimated model. This approach requires selecting model type and order (number of parameters), and then estimating the parameters. This can be done in several different ways, and the method of parameter estimation gives its name to the parametric spectral method: we thus have the Yule-Walker method, the covariance and modified covariance methods, Burg’s method and the maximum entropy method. These methods provide better resolution than non-parametric ones, especially when the record is short.