Page 376 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS

6.75. Consider a continuous-time LTI system with the system function

Determine the frequency response Hd(R) of the discrete-time system designed from this
system based on the impulse invariance method.
-in
,
Am. H(n) = T, e-Ts , where T, is the sampling interval of hc(t).
(1 - e-T3 e-ia
)
6.76. Consider a continuous-time LTI system with the system function
1
HAS) = s+l

Determine the frequency response Hd(R) of the discrete-time system designed from this
system based on the step response invariance, that is,

where sc(t) and sd[n] are the step response of the continuous-time and the discrete-time
systems, respectively.

6.77. Let Hp(z) be the system function of a discrete-time prototype low-pass filter. Consider a new
discrete-time low-pass filter whose system function H(z) is obtained by replacing z in Hp(z)
with (z - a)/(l - az), where a is real.
(a) Show that

(b) Let R,, and R, be the specified frequencies (< T) of the prototype low-pass filter and
the new low-pass filter, respectively. Then show that

ein~ - a
Hint: Set einpl = and solve for a.
1 -a ein~

6.78. Consider a discrete-time prototype low-pass filter with system function

(a) Find the 3-dB bandwidth of the prototype filter.
(6) Design a discrete-time low-pass filter from this prototype filter so that the 3-dB bandwidth
of the new filter is 21~/3.