 By Jean M. Firth (auth.)

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Example text

The analogue-digital conversion process induces errors (and other errors occur in subsequent computation, such as involve numerical integration, and the truncation of infinite series). In this chapter we shall first consider some of these matters, and the effects of sampling, and then proceed to introduce the Z-transform, which in effect is a discrete version ofthe Laplace transform. No prior knowledge of the Z-transform will be assumed, although some readers might have encountered it (possibly defined as a series rather than as a derivation from the Laplace transform), and might have seen some applications - as in the description of system transfer functions.

E. in terms of known functions of t), but it is, however, a tabulated function. 2. 3 at the end of the chapter, but we will discuss the implications and application of FTlO. 7 Show that ~ L=~oo b(t - mTs)} = fs m=~oo b(f - mfs) in which 1 fs= T s Solution We begin with some discussion. The function to be transformed is a train of unit impulses at points t = mTs on - 00 < t < 00, equally spaced if Ts is a constant. It can therefore be regarded as a periodic impulse function, and we can regard Ts as a sampling interval inasmuch as multiplying a function x(t) by this sum is equivalent to generating a sequence of sampled values of x(t) at discrete points t = mTs' illustrated in Fig.

In Chapter 2 we shall examine convolution and multiplication properties in more detail in the context of the Fourier transform and, more generally, consider how the evaluation of convolution integrals can be assisted by graphical considerations - especially when dealing with functions defined in a piecewise-continuous manner. Subsequently we shall X(I) ----t~---__t H(I) t--_~-- Y(I) Fig. 6 Transform of a linear system. 21 Problems consider convolution in the context of discrete signals and discrete transforms, as required for digital computation. 