Page 353 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

and the system function of the discrete-time system is given by

Thus, the frequency response Hd(fl) of the discrete-time system is
1 - 1
Hd(fl) = Hd(z)l,,,,~~ 1 - e-nTs e-~n 1 - e-2n~, ,-in (6.179)
=
Note that if the system function of a continuous-time LTI system is given by

then the impulse-invariance method yields the corresponding discrete-time system with the
system function H,( z given by

6.45. A differentiator is a continuous-time LTI system with the system function [Eq. (3.2011

A discrete-time LTI system is constructed by replacing s in Hc(s) by the following
transformation known as the bilinear transformation:

to simulate the differentiator. Again T, in Eq. (6.183) is a positive number to be
chosen as part of the design procedure.

(a) Draw a diagram for the discrete-time system.
(b) Find the frequency response Hd(fl) of the discrete-time system and plot its
magnitude and phase responses.

(a) Let H,(z) be the system function of the discrete-time system. Then, from Eqs. (6.182)
and (6.183) we have

Writing Hd( z) as

then, from Probs. (6.35) and (6.38) the discrete-time system can be constructed as a
cascade connection of two systems as shown in Fig. 6-3Ma). From Fig. 6-3Ma) it is seen
that we can replace two unit-delay elements by one unit-delay element as shown in Fig.
6-30( 6).