Page 370 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Since x(t ) = 0 for t < 0, the Fourier transform X(w) of x(t ) is given by [Eq.

Let TI be the total recording time of x(t) required to evaluate X(w). Then the above integral
can be approximated by a finite series as
N-l
X(w) = At z x(tn) e-;"'"
n=O
where tn = n At and T, = NAt. Setting w = w, in the above expression, we have
N- 1
X(wk) = At x(t,) e-'"'kln (6.237)
n=O
is
Next, since the highest frequency of x(t) is w,, the inverse Fourier transform of ~(w) given
by [Eq. (5.3211

Dividing the frequency range -oM I w I w, into N (even) intervals of length Aw, the above
integral can be approximated by

where 2wM = NAw. Setting t = t, in the above expression, we have

Since the highest frequency in x(t) is w,, then from the sampling theorem (Prob. 5.59) we
should sample x(t) so that

where T, is the sampling interval. Since T, = At, selecting the largest value of At (the Nyquist
interval), we have
7
At = -
OM
and

Thus, N is a suitable even integer for which
T, ~ W M WM TI
-- -- =N and N2-
T, Aw T
From Eq. (6.240) the frequency resolution Aw is given by