Introduction

A signal is the medium carrying information, transmitted by a source to a receiver. In other words, a signal is the vehicle of intelligence in systems. It transports commands in control and remote-control equipment; it carries data such as information, spoken words, or images across networks. It is particularly fragile and needs to be handled with a great deal of care. Signal processing is applied in order to extract information, alter the message being carried, or adapt the signal to the transmission techniques being used. It is here that digital techniques come into play. Indeed, if we imagine substituting the signal with a set of numbers, representing its value or amplitude at carefully chosen times, then its processing, even in the most elaborate of forms, boils down to a sequence of logical and arithmetical operations on that set of numbers, committing the results to memory.

A continuous analog signal is converted into a digital signal by sensors which act on readings, or directly in the devices producing or receiving the signal. The operations taking place in the wake of that conversion are carried out by digital computers, tasked or programmed to perform the sequence of operations by which the desired processing is defined.

Before introducing the content of each chapter of this book, it is wise to precisely define the processing of which we speak here.

Digital signal processing refers to the set of operations, arithmetic calculations, and number manipulations, which are applied to a signal to be processed, represented by a series or a set of numbers, to produce another series or set of numbers, which represent the processed signal. In this way, an immense variety of functions can be performed, such as spectral analysis, linear or nonlinear filtration, transcoding, modulation, detection, estimation, and parameter extraction. The machines used are digital computers.

The systems corresponding to this processing obey the laws of discrete systems. In certain cases, the numbers to which the processing is applied may be derived from a discrete process. However, they often represent the amplitude of samples taken from a continuous signal, and, in that case, the computer must be downstream of an analog-to-digital converter and possibly upstream of a digital-to-analog converter. In designing such systems, and in studying how they work, signal digitization is fundamentally important, and the operations of sampling and encoding must be analyzed in terms of their principles and their consequences. The theory of distributions is a concise, simple, and effective means of such analysis. Following the presentation of certain fundamental aspects concerning Fourier analysis, distributions, and signal representation, Chapter 1 contains the most important and most useful results for sampling and encoding of a signal.

The advent of digital processing dates from the discovery of fast computational algorithms of the discrete Fourier transform. Indeed, this transform is the basis for the study of discrete systems. In digital processing, it is the equivalent of the Fourier transform in analog processing, enabling us to transition from the discrete-time domain to the discretefrequency domain. It lends itself very well to spectral analysis, with a frequency step dividing the sampling frequency of the signals being analyzed.

Fast computation algorithms offer gains, as they enable operations to be performed in real time, provided certain elementary conditions are met. Thus, the discrete Fourier transform is not only a fundamental tool in determining the processing characteristics and in the study of the impacts of those characteristics on the signal, but it is also used in the production of popular devices, such as mobile radio and digital television. Chapters 2 and 3 are dedicated to these algorithms. To begin with, they present the elementary properties and the mechanism of fast computation algorithms and their applications before moving on to a set of variants associated with practical situations.

A significant portion of this book is devoted to the study of one-dimensional invariant linear time-discrete systems, which are easily accessible and highly useful. Multi-dimensional systems, and, in particular, two- and three-dimensional systems, are experiencing significant development. For example, they are applied to images. However, their properties are generally deduced from those of one-dimensional systems, of which they are often merely simplified extensions. Nonlinear or time-variable systems either contain a significant subset, retaining the properties of linearity and time-invariance, or can be analyzed with the same techniques as systems that have those properties.

Linearity and time-invariance lead to the existence of a convolution relation, which governs the operation of the system or filter having those properties. This convolution relation is defined on the basis of the system’s response to the elementary signal which represents a pulse – the impulse response – by an integral in the case of analog signals. Thus, if x(t) denotes the signal to be filtered, and h(t) is the filter impulse response, the filtered signal y(t) is given by the equation:

In these conditions, such a relation, which directly expresses the filter’s real operation, offers limited practical interest. To begin with, it is not particularly easy to determine the impulse response on the basis of criteria that define the filter’s intended operation. In addition, an equation that contains an integral cannot easily be used to recognize and check the filter’s behavior. Design is much easier to address in the frequency domain because the Laplace transform or Fourier transform can be used to move to a transformed plane where the convolution relations from the amplitude–time plane become simple products of functions. The Fourier transform matches the system’s frequency response to the impulse response, and the filtration is then the product of that frequency response by the Fourier transform, or spectrum, of the signal to be filtered.

In discrete digital systems, the convolution is expressed by a sum. The filter is defined by a series of numbers, representing its impulse response. Thus, if the series to be filtered is written as x(n), the filtered series y(n) is expressed by the following sum, where n and m are integers:

Two scenarios then arise. Firstly, the sum may pertain to a finite number of terms – i.e. the h(m) values are zero, except for a finite number of values of the integer variable m. The filter is known as a finite impulse-response filter. In reference to its realization, it is also referred to as non-recursive, because it does not require a feedback loop from output to input in its implementation. It occupies finite memory space because it only retains the memory of an elementary signal – an impulse, for example – for a limited time. The numbers h(m) are called the coefficients of the filter, which they define completely. They can be calculated directly, in a very simple way – for instance, by means of a Fourier series development of the frequency response. This type of filter exhibits highly interesting original features (for example, the possibility of a rigorously linear phase response – i.e., a constant group delay); the signals whose components are within the filter’s passband are not deformed as they pass through the filter. This possibility is exploited in data transmission systems, or spectral analysis, for example.

Alternatively, the sum may pertain to an infinite number of terms, and the h(m) may have an infinite number of nonzero values; the filter is called an infinite impulse-response filter, or recursive, because its memory must be set up as a feedback loop from output to input. Its operation is governed by an equation whereby an element in the output series y(n) is calculated by the weighted sum of a number of elements of the input series x(n), and a certain number of elements of the previous output series. For example, if L and K are integers, the filter’s operation may be defined by the following equation:

The al(l = 0, 1, …, L) and bk(k = 1, 2, …, K) are the coefficients. As is the case with analog filters, this type of filter generally cannot easily be studied directly; it is necessary to go through a transformed plane. The Laplace or Fourier transforms could be used for this purpose. However, there is a transform that is much more suitable – the Z transform, which is the equivalent for discrete systems. A filter is characterized by its Z-transfer function, generally written as H(Z), which involves the coefficients in the following equation:

To obtain the filter’s frequency response, in H(Z), we simply need to replace the variable Z with the following expression, where f denotes the frequency variable, and T the time step between the signal samples:

In this operation, the imaginary axis in the Laplacian plane corresponds to the circle with unit radius, centered at the origin in the plane of the variable Z. It is plain that the frequency response of the filter defined by H(Z) is a periodic function whose period is the sampling frequency. Another representation of the function H(Z), which is useful in...