PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 475

R.5.50 H(z) is called the discrete-system function or discrete-system transfer function.

Recall that in the continuous case, the notation used for the transfer function was

H(w) or H(s). Note that the discrete case is similar and given by

Hz

Gz

Fz

()

()

()

ZT of the output sequence

ZT of its input sequence

Hz

bz

az

k

n

n

k m

n

()

 






 



∑ −

∑

where a and b are the polynomial coeffi cients of the input and output sequences,

respectively, arranged in descending powers of z, assuming that the initial condi-

tions are set to zero.

For example, let the difference equation of a causal system be given by

g(n) − 7g (n − 1) + 3g(n − 2) = f(n) + 3f(n − 2) − 5f(n − 3)

Evaluate by hand, the expression of the system transfer function H(z).

ANALYTICAL Solution

G(z) − 7z−^1 G(z) + 3z−^2 G(z) = F(z) + 3z−^2 F(z) − 5z−^3 F(z)

G(z)[1 − 7z−^1 + 3z−^2 ] = F(z)[1 + 3z−^2 − 5z−^3 ]

Hz Gz

Fz

zz

zz

() ()

()









13 5

17 3

23

12

R.5.51 Since the ZT was obtained by a simple variable substitution in the DTFT, it is logical

to think that the properties of ZT and DTFT are similar. For completeness, the ZT

properties are summarized as follows:

a. Linearity

a 1 f 1 (n) + a 2 f 2 (n) ↔ a 1 F 1 (z) + a 2 F 2 (z)

b. Time shifting

f(n − n 0 ) ↔ z−noF(z)

c. Time reversal

f(−n) ↔ F(z−^1 )

d. Time multiplication by a power sequence

anf(n) ↔ F(z/a)

e. Conjugate of a sequence

f * (n) ↔ F * (z) (where * denotes conjugate)