The continuous signal is a piecewise continuous function f(x) of a continuous variable x that mainly has the physical meaning of time but can also be a space distance or other physical quantity. The physical quantity expressed by the signal values (e.g. electrical voltage) has for the most part no significance in our discussions. Note: often such signals are called continuous-time signals but obviously this is less general, although perhaps more precise in the particu lar sense of common signals. The discrete signal is an ordered sequence of values f =f(i) which is a function of the integer-valued variable index, i. From a theoretical point of view, the origin of the sequence is irrelevant; most frequently it is a series of some measurement values.

The spectrum as defined by the integral Fourier transform for continuous signals does not exist for discrete signals. This chapter introduced the notion of quasicontinuous signals, in Section 2.1, which enable us to generalise the definition of the transform even to the discrete signals interpreted as limit cases of certain continuous signals. The relation (2.7) then gives the sampled-signal spectrum as an infinite sum of mutually shifted replicas of the original continuous signal spectrum.

Filtering is usually formulated as a type of process that selects certain components out of a mixture of signals and suppresses other components. More generally, filtering can be understood as a means of changing the properties of individual components, e.g. their relative magnitudes and the mutual time or phase relations in the resulting signal. The components of the processed signals are commonly understood or formulated in the frequency domain they are then harmonic components the amplitudes and initial phases of which are modified by the filtering. The effect of a filter is thus expressed by two frequency characteristics: amplitude frequency response, and phase frequency response. As the filters under discussion are discrete, their characteristics are periodic and it is sufficient to define them only in the frequency range (0, ωs/2) where ωs is the angular sampling frequency. A special type of filters are the band filters which ideally have unit amplitude transfer in frequency passbands and zero transfer in stopbands; the transitions between both types of band should ideally be of zero width. Ideal characteristics of basic kinds of band filters are depicted in Figure 5.1 (the missing type band stop would be the reverse of a bandpass filter). The responses of realisable filters only approximately approach the ideal responses. The flatness of the characteristics inside the bands and the steepness of the continuous transitions between the stop and passbands serve as quality criteria of the filter. Apart from the design method, they can be influenced primarily by the degree of complexity of the filter.

Signals in their classical analogue form, represented by a suitable physical quantity, e.g. by electrical voltage at a certain point of a circuit, naturally acquire only real values in every time instant. Two such voltages would be needed to represent a signal, the values of which could be complex; nevertheless, to construct reliable circuits maintaining the proper relationship between the component signals would be extremely difficult. On the other hand, digital representation enables us to represent simply signals with samples of complex value such signals will be called complex signals. Subsequently, advanced methods using complex signals can be conceived that would not be feasible with analogue signal representation.

Spectral (or frequency) analysis generally serves to describe a signal in terms of its components in the frequency domain. We shall deal in this chapter only with spectra in the sense of the Fourier transform; thus, we will have in mind harmonic components of different frequencies and unlimited duration. If we can find the description of a signal in the frequency domain, i.e. magnitudes and possibly even mutual phase (or time) shifts of its harmonic components, we can make certain conclusions about the character of the signal. This may enable us to classify or to recognise the signal, to use the information to design a suitable communication channel or archiving medium for such a signal or to consider restricting information about less important components in order to compress the data describing the signal. Applications of spectral analysis are very broad; we shall meet some illustrative examples in this and further chapters. More detailed information on concrete applications, such as in the area of speech recognition, nondestructive testing in different technical fields, biomedical diagnostic applications, multimedia signal processing, telecommunications etc. can be found in specialised literature.

In the previous chapter, when designing restoration filters which should provide optimal estimates of original signals based on their observed noisy and distorted versions, we used explicit information obtained by a priori identification of signal-source properties. Alternatively, the stationary signal-source parameters could also be estimated from the signal itself providing that a suitable signal-generation model has been introduced. Consequently, it is possible to design an optimal or suboptimal restoration system; in the case of stationary processes, the system is, or aims at, a time-invariant filter, such as the classical Wiener filter. Nevertheless, if the filter is to work in an unknown environment (either because the identification is impossible or the environment is time varying in an unpredictable way), it must be capable of adapting to such a situation. We shall therefore deal in this chapter with adaptive filters that are able to learn from a given environment, i.e. they are capable of providing the necessary information estimates of the needed quantities in the course of their work, with out any a priori information. It is then possible to expect that such filters will be able to react (with a certain delay) even to changes in environmental properties and thus to also process signals generated by nonstationary processes. Adaptive filters can be designed as filters with infinite or finite impulse response. General recursive filters (ARMA-type filters) promise naturally, in principle, better estimates. Their basic disadvantage is that they may become unstable in the course of adaptation; thus, complicated precautions in the adaptation mechanisms are needed to prevent parameter-adjustment leading to instability. We shall therefore limit ourselves in this book to common nonrecursive FIR adaptive filters (e.g. MA type) which are always stable.

Neural networks are a relatively new means of data processing. Interpreted in this way, it can be said that a neural network realises a mapping from an input vector space into the output vector space.In this chapter attention is given primarily on neural-network applications as signal processors, when the input and output vectors represent the relevant signals. Another, perhaps more frequent use is their application as classifiers then the input vector represents a section of a signal or image and the output indicates to which class the section belongs. Further, it is possible to utilise the ability of some networks to solve optimisation problems; they can be used for signal restoration with respect to chosen criteria.