Signals and Systems - Electrical Engineering

462 C H A P T E R 8: Discrete-Time Signals and Systems

Likewise, forx[n+M] thex[0] appears advanced byMsamples, or shifted to the left (i.e., when

n=−M). Negating the variablenflips over the signal with respect to the origin.

nExample 8.9

A triangular discrete pulse is defined as

x[n]=

{

n 0 ≤n≤ 10

0 otherwise

Find an expression fory[n]=x[n+3]+x[n−3] andz[n]=x[−n]+x[n] in terms ofn and

carefully plot them.

Solution

Replacingnbyn+3 andn−3 in the definition ofx[n], we get the advanced and delayed signals

x[n+3]=

{

n+ 3 − 3 ≤n≤ 7

0 otherwise

and

x[n−3]=

{

n−3 3≤n≤ 13

0 otherwise

so that when added, we get

y[n]=x[n+3]+x[n−3]=











n+ 3 − 3 ≤n≤ 2

2 n 3 ≤n≤ 7

n−3 8≤n≤ 13

0 otherwise

Likewise, we have that

z[n]=x[n]+x[−n]=











n 1 ≤n≤ 10

0 n= 0

−n − 10 ≤n≤− 1

0 otherwise

The results are shown in Figure 8.2. n

nExample 8.10

We will see that in the convolution sum we need to figure out how a signalx[n−k] behaves as a

function ofkfor different values ofn. Consider the signal

x[k]=

{

k 0 ≤k≤ 3

0 otherwise