Page 345 - Schaum's Outline of Theory and Problems of Signals and Systems

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

6.38. Convert the discrete-time low-pass filter shown in Fig. 6-18 (Prob. 6.34) to a high-pass
filter.
From Prob. 6.34 the discrete-time low-pass filter shown in Fig. 6-18 is described by [Eq.
( 6.145 )I

Using Eq. (6.154), the converted high-pass filter is described by

which leads to the circuit diagram in Fig. 6-23. Taking the Fourier transform of Eq. (6.157) and
by Eq. (6.77 ), we have

From Eq. (6.158)

and

which are sketched in Fig. 6-24. We see that the system is a discrete-time high-pass FIR filter.

Fig. 6-23

6.39. The system function H(z) of a causal discrete-time LTI system is given by

where a is real and la1 < 1. Find the value of b so that the frequency response H(R)
of the system satisfies the condition
IH(n)l= 1 all R (6.160)
Such a system is called an all-pass filter.

By Eq. (6.34) the frequency response of the system is