Page 371 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Let tn=nAt and wk=kAw. Then

Substituting Eq. (6.243) into Eqs. (6.237) and (6.239), we get
N- 1
X(k Ao) = C Atx(n At) e-j(2"/N)nk
n =O
(N/2)-1
k
and x(n At) = - C ~ ( bw) e(2"/N)nk
2a k= -N/Z
Rewrite Eq. (6.245) as
1
- 1
X(k Aw) e'(2"/N'nk + C X(k Aw) ei(2"/N)nk
k= -N/2
Then from Eq. (6.244) we note that X(k Aw) is periodic in k with period N. Thus, changing
the variable k = m - N in the second sum in the above expression, we get

Multiplying both sides of Eq. (6.246) by At and noting that Aw At = 2,rr/N, we have
1 N-1
x(n At) At = - C X(k Aw) ei(2"/N)nk
k=O
Now if we define
x[n] = Atx(n At) = T,x(nT,) (6.248)
X[k] = X(k Aw)
then Eqs. (6.244) and (6.247) reduce to the DFT pair, that is,

6.61. (a) Using the DFT, estimate the Fourier spectrum X(w) of the continuous-time
signal

Assume that the total recording time of x(t) is T, = 10 s and the highest
frequency of x(t) is w, = 100 rad/s.
(b) Let X[k] be the DFT of the sampled sequence of x(t). Compare the values of
X[O], X[l], and X[10] with the values of X(O), X(Aw), and X(10Aw).