Page 375 - Schaum's Outline of Theory and Problems of Signals and Systems

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

6.71. A causal discrete-time LTI system is described by

y[n] - iy[n - 11 + iy[n - 21 =x[n]
where x[n] and y[nl are the input and output of the system, respectively.
(a) Determine the frequency response H(R) of the system.
(b) Find the impulse response h[n] of the system.
(c) Find y[nl if x[nl = (i)"u[nl.

6.72. Consider a causal discrete-time LTI system with frequency response
H(R) = Re{ H(R)) + j Im{H(R)) = A(R) + jB(R)

(a) Show that the impulse response h[n] of the system can be obtained in terms of A(R) or
B(R) alone.
(b) Find H(R) and h[n] if

(a) Hint: Process in a manner similar to that for Prob. 5.49.
(b) Ans. H(R) = 1 + ePin, h[n] = ~[n] + S[n - 11

6.73. Find the impulse response h[n] of the ideal discrete-time HPF with cutoff frequency R,
(0 < R, < r) shown in Fig. 6-42.
sin R,n
Am. h[n] = S[n] - -
T n

Fig. 6-42

6.74. Show that if HLPF(z) is the system function of a discrete-time low-pass filter, then the
discrete-time system whose system function H(z) is given by H(z) = HLPF(-z) is a high-pass
filter.
Hint: Use Eq. (6.156) in Prob. 6.37.