PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 477

R.5.53 For example, let an arbitrary discrete-time sequence be given by

f(n) = 3 
(n) + 5 
(n − 2) − 6 
(n − 5) + 2 
(n + 3)

Then its ZT[Z{f(n)}] can easily be evaluated by applying the relations presented in

R.5.52 yielding the following analytical solution:

ANALYTICAL Solution

Z[f(n)] = Z[3
(n)] + Z[5
(n − 2)] − Z[6
(n − 5)] + Z[2
(n + 3)]

Z[f(n)] = 3Z[
((n)] + 5Z[
(n − 2)] − 6Z[
(n − 5)] + 2Z[
(n + 3)]

Z[f(n)] = 3[1] + 5z−^2 − 6z−^5 + 2z^3

Z[f(n)] = 2z^3 + 3 + 5z−^2 − 6z−^5

R.5.54 Recall that it is possible in many cases to express the ZT of an infi nitely long

sequence in a close compact form by using the following relation:

a. aaan

n






0

∑^11 for ^1

b. aaa an

n

N

N




 

0

1

∑^11 for any

c.^ nan a a a

n




 

0

112

∑ ()for

R.5.55 By using the properties given in R.5.51 and R.5.52, any LTI system difference equa-

tion that relates the input sequence to its output sequence can easily be transformed

from the time domain to the z (frequency) domain.

For example, let us analyze the following system difference equation:

gn()
gn( ) f n()

1

2

 1







where f(n) and g(n) represent the input and output sequences of a given system.

A compact representation of the system transfer function H(z) = G(z)/F(z) can be

easily obtained and the condition for convergence can then be stated.

ANALYTICAL Solution

Gz
Gzz Fz() () ()

1

2

1

Gz() 1 z Fz()

1

2

 
 1







and

Hz

Gz

Fz z

()

()

() (.)

 

1

(^1051)
convergence occurs for |0.5z−^1 | < 1 or |z| > |0. 5|.