DTFT, DFT, ZT, and FFT
485
Evaluate the Z[f(n)] in the following ways:
a. Analytically
b. Symbolically by using MATLAB
ANALYTICAL Solution
F z
Z f n
f n z
z
z
n
n
( )
[ ( )]
( )
ϭ
ϭ
ϭ ϩ
ϩ
ϩ
Ϫ
ϭ
ϩϱ
Ϫ
Ϫ
0
1
2
1
1
3
1
9
1
27
∑

 

 

 

 


 

  z
z
n
n
Ϫ
Ϫ
ϩ ϩ
3
1
3
F z
z
z
z
z
z
( )
( / )
( / )
ϭ Ϫ
ϭ Ϫ
Ϫ
1 1 3
1 3
1
3
1
for
Ͼ
MATLAB Solution
>> syms z n
>> Fz = ztrans(1/3^n)
Fz =
3*z/(3*z-1)
>> pretty(Fz)
z
3 ------3 z - 1
R.5.77 Let us gain some experience by evaluating the ZT of the following functions by
using MATLAB:
a. f 1 (n) = u(n)
b. f 2 (n) = a
n u(n)
c. f 3 (n) = cos(an)u(n)
d. f 4 (n) = sin(an)u(n)
e. f 5 (n) = b
n sin(an)u(n)
MATLAB Solution
>> syms a b n
>> f1 _ n = 1^n;
% part(a)
>> F1 _ z = ztrans(f1 _ n)
F1 _ z =
z/(z-1)
>> pretty(F1 _ z)
z
----z - 1
>> f2 _ n = a^n;
% part(b)
>> F2 _ z =ztrans(f2 _ n)
CRC_47760_CH005.indd 485
CRC_47760_CH005.indd 485
7/23/2008 1:47:25 PM
7/23/2008 1:47:25 PM