Page 339 - Schaum's Outline of Theory and Problems of Signals and Systems

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

and

FREQUENCY RESPONSE

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

y[n] - +y[n - 11 + $y[n - 21 =x[n] (6.143)

where x[n] and y[n] are the input and output of the system, respectively (Prob. 4.32).
(a) Determine the frequency response H(n) of the system.
(b) Find the impulse response h[n] of the system.
(a) Taking the Fourier transform of Eq. (6.1431, we obtain

Y(R) - ie-'"~(fl) + ;e-j2'y (a) =X(R)
or
(1 - ie-ifl + Le -j 2n)Y(R) = X(R)

Thus,

(6) Using partial-fraction expansions, we have
1 - 2 - 1
-
H(R) = (1 - - I 1 - Ie-in 1 - Le-in
2
Taking the inverse Fourier transform of H(fl), we obtain
h[n] = [2(iln - (f)"]u[n]

which is the same result obtained in Prob. 4.32(6).

6.32. Consider a discrete-time LTI system described by

y[n] - ;y[n - 11 =x[n] + ix[n - 11

(a) Determine the frequency response H(n) of the system.
(b) Find the impulse response h[n] of the system.
(c) Determine its response y[n] to the input
iT
~[n] cos-n
=
2
(a) Taking the Fourier transform of Eq. (6.1441, we obtain

Y(R) - ie-jnY(R) =X(R) + ;e-jnx(R)