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## Über dieses Buch

This book covers the basics of processing and spectral analysis of monovariate discrete-time signals. The approach is practical, the aim being to acquaint the reader with the indications for and drawbacks of the various methods and to highlight possible misuses. The book is rich in original ideas, visualized in new and illuminating ways, and is structured so that parts can be skipped without loss of continuity. Many examples are included, based on synthetic data and real measurements from the fields of physics, biology, medicine, macroeconomics etc., and a complete set of MATLAB exercises requiring no previous experience of programming is provided. Prior advanced mathematical skills are not needed in order to understand the contents: a good command of basic mathematical analysis is sufficient. Where more advanced mathematical tools are necessary, they are included in an Appendix and presented in an easy-to-follow way. With this book, digital signal processing leaves the domain of engineering to address the needs of scientists and scholars in traditionally less quantitative disciplines, now facing increasing amounts of data.

## Inhaltsverzeichnis

### Chapter 1. Introduction

This introductory chapter discusses the meaning of the words that appear in the book’s title. It then gives a concise description of the historical background in which the techniques introduced in the book were developed. The structure of the book is also presented and commented on. Finally, some possible further readings are suggested.
Silvia Maria Alessio

### Chapter 2. Discrete-Time Signals and Systems

Discrete-time signals are, in general, infinite-length sequences of numerical values that may either arise from sampling of continuous-time signals, or be generated directly by inherently-discrete-time processes. Deterministic signals have a univocal mathematical description, so that future signal values are exactly predictable. Random signals do not allow for such a description: their treatment requires statistical tools since signal evolution cannot be exactly foreseen. This chapter introduces basic concepts related to discrete-time signals, as well as mathematical operators, called discrete-time systems, that are employed to process them. The main constraints imposed on discrete-time systems, namely linearity, time invariance, stability and causality, are introduced along with the quantities used to describe a system: the impulse response, the transfer function and the frequency response. Finite-impulse-response (FIR) and infinite-impulse-response (IIR) systems are defined. Linear convolution and the linear constant-coefficient-difference equation (LCCDE) are introduced to express the input-output relation of discrete-time systems.
Silvia Maria Alessio

### Chapter 3. 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.
Silvia Maria Alessio

### Chapter 4. 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.
Silvia Maria Alessio

### Chapter 5. Spectral Analysis of Deterministic Discrete-Time Signals

In this chapter we will describe the spectral representation of a discrete-time deterministic signal. Even if intrinsically the signal has infinite length, we can only observe it through a window of finite width, and we cannot compute its DTFT, provided it exists, but only the DFT of the segment framed by the window. The consequences of these limitations are investigated through examples using sinusoidal signals, and the issues of leakage and loss of resolution are examined. A description of the main windows used in spectral analysis follows. We then move to more conceptual topics. Deterministic bounded signals can be energy signals or power signals. For energy signals we can use the squared magnitude of the DTFT to define the energy spectrum, describing how the energy of the signal distributes over frequency. The energy spectrum can equivalently be defined as the DTFT of the autocovariance (AC) sequence, quantifying the signal’s self-similarity. On the other hand, the spectral representation of power signals, i.e., signals with infinite energy but finite average power, necessarily passes through the transform of the AC sequence of the power signal, provided that the AC has finite energy.
Silvia Maria Alessio

### Chapter 6. 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.
Silvia Maria Alessio

### Chapter 7. 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.
Silvia Maria Alessio

### Chapter 8. IIR Filter Design

Digital IIR filters have an infinitely long impulse response and therefore can be associated with analog filters that have the same characteristics. For this reason, the classical method for digital IIR filter design is based on the design of an analog filter, which is later transformed into an equivalent digital filter through a mapping in the complex plane. The advantage of such a technique lies in the fact that analog filter design and mapping methods for analog-to-digital (A/D) transformation are well known and have sound theoretical foundations. The process is based on a lowpass filter, and frequency transformation methods are later applied if a different type of frequency selectivity is desired. These frequency transformations are again well-known mapping procedures in the complex plane. In this chapter, we will first discuss the design of the main types of lowpass analog filters, namely Butterworth, Chebyshev and elliptic lowpass filters, all of which represent different ways of approximating the desired ideal frequency response. Then we will learn how to transform the analog lowpass filter into an equivalent IIR digital lowpass filter via bilinear transformation, and finally how to transform the IIR lowpass filter into a highpass, bandpass or bandstop filter, if needed. The appendix to this chapter provides deeper insight into the mathematical facets of elliptic design, discussing elliptic integrals of the first kind, Jacobi elliptic functions, and the elliptic rational function on which the transfer function of the analog elliptic filter is based. It must be explicitly noted that all classically-designed lowpass IIR filters are inadequate for specifications with cutoff frequency close to zero and narrow transition band. In such cases, it may be advisable to preliminarily downsample the signal and then design a filter with a less extreme cutoff frequency, allowing for a wider transition band.
Silvia Maria Alessio

### Chapter 9. Statistical Approach to Signal Analysis

Discrete-time signals often derive from periodically repeated measurements of some quantity over a finite time span, and we are interested in the characteristics of the quantity and of the process that generates it, rather than in those of the particular sequence we measured. The measured record, affected by random errors, is interpreted as a segment of a persistent discrete-time random power signal, conceptually resulting from sampling a continuous-time random signal (process) possibly varying with time. The value assumed by a random signal is not exactly specified at a given past or current instant, and future values are not predictable with certainty on the basis of past behavior. This chapter provides a brief introduction to the basic theory of discrete-time random processes, which can be described using theoretical average quantities. The latter could be calculated if the probabilistic laws associated with the random process were known, but usually they are unknown; the problem is then simplified by assuming stationarity, i.e., no dependence on time, and ergodicity, a property that allows for substituting theoretical averages with time averages performed on a single finite-length data record. This path leads to a spectral representation for the discrete-time random power signal, i.e., to the power spectrum. Also the representation of the common spectral content of two random signals is introduced.
Silvia Maria Alessio

### Chapter 10. 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.
Silvia Maria Alessio

### Chapter 11. 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.
Silvia Maria Alessio

### Chapter 12. 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.
Silvia Maria Alessio

### Chapter 13. Non-stationary Spectral Analysis

A modern approach to spectral analysis of non-stationary signals is provided by the continuous wavelet transform (CWT), in which the signal in its entirety is not compared with infinitely-long sinusoids, but with waveforms called wavelets, which are concentrated in time and frequency. In this method, the concept of period (inverse of a frequency) is replaced by the concept of scale. Using the language of continuous time-signals that allows for avoiding some mathematical difficulties, we will describe what a wavelet is and how signals can be analyzed in time and scale; we will then establish a relation between scale and frequency and investigate CWT resolution in time and frequency, thus introducing the concept of multiresolution analysis (MRA). We will also define the conditions under which a signal can be reconstructed from its CWT coefficients. For practical applications, the CWT must be made discrete in time and scale; we will discuss the most popular discretization scheme. Next we will show how an average power spectrum estimate, the global power spectrum (GWS), can be derived from CWT by averaging over time, and how significance tests for the spectral features detected in CWT analysis can be devised. Real-world application examples will be provided.
Silvia Maria Alessio

### Chapter 14. 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.
Silvia Maria Alessio

### Chapter 15. 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.
Silvia Maria Alessio

### Chapter 16. 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.
Silvia Maria Alessio

### Backmatter

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