Page 295 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

From Eq. (5.137) the impulse response h(t) of the ideal low-pass filter is given by

sin w, t T5wc sin w,t
h(t) = T5----- = - -
at i~ w,t

From Eq. (5.182) we have

By Eq. (2.6) and using Eqs. (2.7) and (1.261, the output y(t) is given by

Using Eq. (5.1891, we get
cc Tp, sin w,(t -kc)
~(t) C x(kT,)-
=
k= - cc 77 w,(t - kT,)
If w, = wJ2, then T,w,/a = 1 and we have

Setting t = mT, (m = integer) and using the fact that w,T, = 2~, we get
mk)
rn sin ~ ( -
Y(~T) 77(m - k)
=
X( kTS)
k= -a
Since

we have

which shows that without any restriction on x(t), y(mT5) = x(mT,) for any integer value of m.
Note from the sampling theorem (Probs. 5.58 and 5.59) that if w, = 2a/T5 is greater than
twice the highest frequency present in x(t) and w, = wJ2, then y(t) = x(t). If this condition
on the bandwidth of x(t) is not satisfied, then y(t) zx(t). However, if w, = 0,/2, then
y(mT,) = x(mT5) for any integer value of m.