Page 309 - Schaum's Outline of Theory and Problems of Signals and Systems

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

G. Time Scaling:
In Sec. 5.4D the scaling property of a continuous-time Fourier transform is expressed
as [Eq. (5.5211

However, in the discrete-time case, x[an] is not a sequence if a is not an integer. On the
other hand, if a is an integer, say a = 2, then x[2n] consists of only the even samples
of x[n]. Thus, time scaling in discrete time takes on a form somewhat different from
Eq. (6.47).
Let m be a positive integer and define the sequence

x[n/m] =x[k] if n = km, k = integer
ifn#km

Then we have

Equation (6.49) is the discrete-time counterpart of Eq. (6.47). It states again the inverse
relationship between time and frequency. That is, as the signal spreads in time (m > I), its
Fourier transform is compressed (Prob. 6.22). Note that X(rnR) is periodic with period
27r/m since X(R) is periodic with period 27r.

H. Duality:

In Sec. 5.4F the duality property of a continuous-time Fourier transform is expressed
as [Eq. (5.5411

There is no discrete-time counterpart of this property. However, there is a duality between
the discrete-time Fourier transform and the continuous-time Fourier series. Let

From Eqs. (6.27) and (6.41)

Since fl is a continuous variable, letting R = t and n = -k in Eq. (6.51), we have