Discrete Time Signals and Systems

Abstract

All the signals considered so far are continuous-time signals in the sense that they are defined for every possible value of time; in other words a function x(t)is defined for all real values of time. However, in order to subject signals to numerical analysis we must have finite lists of numbers, which can be obtained by sampling the continuous-time signal at a finite number of points in time. This means that the value of x(t) at discrete points in time is obtained. The resulting discrete-time signal, x[n], can be stored as a sequence of numbers in a computer and analyzed. In order to store x[n]as a sequence of numbers a finite resolution of representation must necessarily be chosen; this is the process of quantization. In practice sampling as well as quantization is done by electronic analogue-to-digital converter circuits. The two main considerations in analogue to digital (A/D) conversion are (i) the rate of data collection or the sampling frequency, and (ii) the resolution of data representation or quantization. During the theoretical analysis of discrete-time signals it is convenient to separate the issues of sampling and quantization. Since the effects of sampling are usually more critical we will primarily deal with the sampled signal, x[n], assuming that the effects of quantization are absent. Quantization, which on a digital computer depends on the number of digital bits used to represent the numbers, will be discussed briefly; but for most physiological signals it is found that 8 bits, 12 bits or 16 bits of data resolution is adequate for representing the signals.