Page 101 - Schaum's Outline of Theory and Problems of Signals and Systems

LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

Substituting r = m - k and interchanging the order of summation, we have

Now, since

we have

2.27. Show that
(a) x[n] * 6[n] = x[n]
(b) x[n]*6[n -n,]=x[n -no]
n
(el x[n]*u[n]= x[kl
k= -m
"-"11
(dl x[n]* u[n - no] = xlk]
k= - r,
(a) By Eq. (2.35)and property(1.46)of 6[n -&]we have

(b) Similarly, we have
m
*
~[n] S[n - no] = C x[k] S[n - k - n,,] =x[n -no]
k= -m
(c) By Eq. (2.35) and definition (1.44) of u[n - k] we have

(dl In a similar manner, we have
rn n -ntl
x[n]*u[n-no]= z x[k]u[n-k-no]= x[k]
k= -m k= -m