Signals and Systems - Electrical Engineering

564 C H A P T E R 9: The Z-Transform

9.6 What Have We Accomplished? Where Do We Go from Here?....................

Although the history of the Z-transform is originally connected with probability theory, for discrete-

time signals and systems it can be connected with the Laplace transform. The periodicity in the

frequency domain and the possibility of an infinite number of poles and zeros makes this connec-

tion not very useful. Defining a new complex variable in polar form provides the definition of the

Z-transform and thez-plane. As with the Laplace transform, poles of the Z-transform characterize

discrete-time signals by means of frequency and attenuation. One- and two-sided Z-transforms are

possible, although the one-sided version can be used to obtain the two-sided one. The region of con-

vergence makes the Z-transform have a unique relationship with the signal, and it will be useful in

obtaining the discrete Fourier representations in Chapter 10.

Dynamic systems represented by difference equations use the Z-transform for representation by

means of the transfer function. The one-sided Z-transform is useful in the solution of difference

equations with nonzero initial conditions. As in the continuous-time case, filters can be represented

by difference equations. However, discrete filters represented by polynomials are also possible. These

nonrecursive filters give significance to the convolution sum, and will motivate us to develop methods

that efficiently compute it.

You will see that the reason to present the Z-transform before the Fourier representation of discrete-

time signals and systems is to use their connection, thereby simplifying calculations.

Problems............................................................................................

9.1. Mapping ofs-plane into thez-plane

The poles of the Laplace transformX(s)of an analog signalx(t)are

p1,2=− 1 ±j 1

p 3 = 0

p4,5=±j 1

There are no zeros. If we use the transformationz=esTswithTs= 1 :

(a) Determine where the given poles are mapped into thez-plane.

(b) How would you determine if these poles are mapped inside, on, or outside the unit circle in thez-

plane? Explain.

(c) Carefully plot the poles and the zeros of the analog and the discrete-time signals in the Laplace and

thez-planes.

9.2. Mapping ofz-plane into thes-plane

Consider the inverse relation given byz=esTs—that is, how to map thez-plane into thes-plane.

(a) Find an expression forsin terms ofzfrom the relationz=esTs.

(b) Consider the mapping of the unit circle (i.e.,z= 1 ejω,−π≤ω < π). Obtain the segment in thes-plane

resulting from the mapping.

(c) Consider the mapping of the inside and the outside of the unit circle. Determine the regions in the

s-plane resulting from the mappings.