Signals and Systems - Electrical Engineering

530 C H A P T E R 9: The Z-Transform

x[k]

h[− 1 −k]

h[2−k]

h[5−k]

x[k]

x[k]

045123

k

k

k

− 3 − 2 − 1

− 3 − 2 − 1

− 3 − 2 − 1

0

0

1

12

23

3

4

4

5

5

n=− 1

n= 2

n= 5

y[−1]= 0

y[2]=x[0]h[2]+x[1]h[1]+x[2]h[0]

y[5]=x[3]h[2]

(a)

(b)

(c)

FIGURE 9.4

Graphical approach: convolution sum for (a)n=− 1 , (b)n= 2 , and (c)n= 5 with corresponding outputsy[−1],

y[2], andy[5]. Bothx[k]andh[n−k]are plotted as functions ofkfor a given value ofn. The signalx[k]remains

stationary, whileh[n−k]moves linearly from left to right. Thus, the convolution sum is also called a linear

convolution.

ish[−k] shifted to the right one sample. Multiplyingx[k] byh[1−k] gives two values different

from zero, which when added givesy[1]=1, and so on. For increasing values ofnwe shift to

the right one sample to geth[n−k], multiply it byx[k], and then add the nonzero values to

obtain the outputy[n]. Figure 9.4 displays the graphical computation of the convolution sum

forn=−1,n=2 andn=5.

Convolution sum property:We have

X(z)= 1 +z−^1 +z−^2 +z−^3

H(z)=

1

2

[

1 +z−^1 +z−^2

]