FOURIER ANALYSIS O F 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 h ( t ) = T5----at
T5wc sin w,t
=
-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 T p , sin w,(t - k c )
cc
~ ( t=) If w,
=wJ2,
Setting t
then T,w,/a
= mT,
=
C x(kT,)k = - cc
77
w,(t - kT,)
1 and we have
( m = integer) and using the fact that w,T, sin ~
rn
Y ( ~ T=)
k=
-a
X (kTS)
=
2 ~we, get
( -m k)
77(m - k )
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, = 2 a / T 5 is greater than twice the highest frequency present in x ( t ) and w , = w J 2 , then y ( t ) = x ( t ) . If this condition on the bandwidth of x ( t ) is not satisfied, then y ( t ) z x ( t ) . However, if w, = 0 , / 2 , then y(mT,) = x(mT5) for any integer value of m .