Page 316 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS 303

When a sinusoid cos Rn is obtained by sampling a continuous-time sinusoid cos wt
with sampling interval T,, that is,

cos Rn = cos w t = cos wT,n (6.80)
all the results developed in this section apply if we substitute wT, for R:

R = oT, (6.81)
For a continuous-time sinusoid cos wt there is a unique waveform for every value of o in
the range 0 to w. Increasing w results in a sinusoid of ever-increasing frequency. On the
other hand, the discrete-time sinusoid cos Rn has a unique waveform only for values of R
in the range 0 to 27r because

COS[(R + 27rm)nI = cos(Rn + 27rmn) = cos Rn m = integer (6.82)
This range is further restricted by the fact that

cos(7r f R)n = cos .rrn cos Rn T sin 7rn sin Rn

Therefore,

Equation (6.84) shows that a sinusoid of frequency (7r + R) has the same waveform as one
with frequency (.rr - R). Therefore, a sinusoid with any value of R outside the range 0 to
7r is identical to a sinusoid with R in the range 0 to 7r. Thus, we conclude that every
discrete-time sinusoid with a frequency in the range 0 I R < .rr has a distinct waveform,
and we need observe only the frequency response of a system over the frequency range
OsR<7r.

B. Sampling Rate:
Let w, (= 27rfM) be the highest frequency of the continuous-time sinusoid. Then
from Eq. (6.81) the condition for a sampled discrete-time sinusoid to have a unique
waveform is 7r
wMTs <7r+ Ts< - or f,> 2fM (6.85)
WM
where f, = l/T, is the sampling rate (or frequency). Equation (6.85) indicates that to
process a continuous-time sinusoid by a discrete-time system, the sampling rate must not
be less than twice the frequency (in hertz) of the sinusoid. This result is a special case of
the sampling theorem we discussed in Prob. 5.59.

6.7 SIMULATION
Consider a continuous-time LTI system with input x(t) and output y(t). We wish to
find a discrete-time LTI system with input x[n] and output y[n] such that

if x[n] =x(nT,) then y[n] = y(nT,) (6.86)
where T, is the sampling interval.