1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

368 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS

which can be written as

(

y[n] = e• ..._)n + ( e-• k)" = e i.!!..!!. • +e-i.!!..!!. • =2cos(7r
4

n)

7r

y[nJ = 2cos(

4

n).

R emark 9.11
The solution can also .be obtained by applying the z-transform identity with
a = % that was given in Example 9.5 of Section 9.1 to get

[

7r ] z^2 - cos(%)z

(^3) cos(4n) = z2- 2cos(i)z+l'
then we have
9.2.4 Convolution for Solving a Nonhomogeneous
Equation
(i) Solve the homogeneous equation y,.[n + 2) -2ay,.[n + 1) + by,.[n) = 0
and get y,.[n].

(ii) Use the transfer function H(z) = 1 _ 2 a.-+oz , and the unit-sample

response h [nJ.

(iii) Construct the particular solution using convolution

yp[n) = r^1 [H(z)X(z)], or

"

Yv[n) = L h[i)x[n - i].

i=O

(iv) The general solution to the nonhomogeneous difference equation is

y[n] = Yh[n) + Yp[nJ.

• EXAMPLE 9.20

(a) Find the general solution to y[n + 2) - ~./2y[n + 1] + ~y[n] = 0.

(b) Find the general solut ion to y[n+ 2) - ~J2y(n+ lj + ~y[n) = cos(ifn).