UploadFile_6417

118 Chapter 4 THE z-TRANSFORM
From these calculations
0.5 − j
X(z) =
1 − 0.8e +j π 4 z + 0.5+j
, |z| > 0.8
−1 1 − 0.8e −j π 4 z−1 and from Table 4.1, we have
x(n) =(0.5 − j)0.8 n e +j π 4 n u(n)+(0.5+j)0.8 n e −j π 4 n u(n)
=0.8 n [0.5{e +j π 4 n + e −j π 4 n }−j{e +j π 4 n − e −j π 4 n }]u(n)
( ) ( )]
=0.8
[cos
n πn πn
+2sin u(n)
4
4
MATLAB verification:
>> [delta, n] = impseq(0,0,6);
x = filter(b,a,delta) % check sequence
x =
Columns 1 through 4
1.00000000000000 1.69705627484771 1.28000000000000 0.36203867196751
Columns 5 through 8
-0.40960000000000 -0.69511425017762 -0.52428800000000 -0.14829104003789
>> x = ((0.8).^n).*(cos(pi*n/4)+2*sin(pi*n/4))
x =
Columns 1 through 4
1.00000000000000 1.69705627484771 1.28000000000000 0.36203867196751
Columns 5 through 8
-0.40960000000000 -0.69511425017762 -0.52428800000000 -0.14829104003789 □
4.4 SYSTEM REPRESENTATION IN THE z-DOMAIN
Similar to the frequency response function H(e jω ), we can define the
z-domain function, H(z), called the system function. However, unlike
H(e jω ), H(z) exists for systems that may not be BIBO stable.
DEFINITION 1
[The System Function] The system function H(z) isgiven by
∞∑
H(z) = △ Z [h(n)] = h(n)z −n ; R h− < |z|