Digital Filters

A signal is defined as any physical quantity that varies with changes of one or more independent variables, and each can be any physical value, such as time, distance, position, temperature, or pressure (Elali, 2003; Smith, 2002). The independent variable is usually referred to as “time”. Examples of signals that we frequently encounter are speech, music, picture, and video signals. If the independent variable is continuous, the signal is called continuous-time signal or analog signal, and is mathematically denoted as x(t). For discrete-time signals, the independent variable is a discrete variable; therefore, a discrete-time signal is defined as a function of an independent variable n, where n is an integer. Consequently, x(n) represents a sequence of values, some of which can be zeros, for each value of integer n. The discrete–time signal is not defined at instants between integers, and it is incorrect to say that x(n) is zero at times between integers. The amplitude of both the continuous and discrete-time signals may be continuous or discrete. Digital signals are discrete-time signals for which the amplitude is discrete. Figure 1 illustrates the analog and the discrete-time signals. Most signals that we encounter are generated by natural means. However, a signal can also be generated synthetically or by computer simulation (Mitra, 2006). Signal carries information, and the objective of signal processing is to extract useful information carried by the signal. The method of information extraction depends on the type of signal and the nature of the information being carried by the signal. “Thus, roughly speaking, signal processing is concerned with the mathematical representation of the signal and algorithmic operation carried out on it to extract the information present,’’ (Mitra, 2006, pp. 1).