PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 473

c. [H, W] = freqz(G, F, W), over the range −π ≤ W ≤ π

d. [H, W] = freqz(G, F, W, Fs/2), where 0 ≤ W ≤ Fs/2 and Fs denotes the sampling

frequency

e. [H, W] = freqz(G, F, K), where K is the number of equally spaced points over the

domain 0 ≤ W ≤ π

Observe that if H(ejW) represents the transfer function of a given system, then

G(ejW) and F(ejW) represents the DTFT of its output and input, respectively.

R.5.43 Recall that DTFT was defi ned by

FejW f nejWn

n

()
()

 



∑

And replacing e−jw by the (new) variable z, DTFT is transformed into what is

referred to as the two-sided or bilateral ZT. The ZT of a discrete sequence f(n) is

then defi ned by

Fz f nzn

n

()
( )

 



∑

R.5.44 The ZT is the preferred tool in the analysis and synthesis of discrete-time systems,

defi ned by a linear constant coeffi cient difference equation. The ZT is for a discrete

system what LT is for a continuous system.

The drawback of DTFT is that the convergence and existence conditions may not

exist for many sequences, and therefore no frequency representation is possible

for these systems. The ZT, however, may exist for many sequences for which DTFT

does not exist.

R.5.45 Note that the equation that defi nes the ZT is in the form of an infi nite power series

of the complex variable z, and its coeffi cients are the discrete samples of f(n).

R.5.46 Observe that the resulting series expansion of ZT is a Laurent series*.

The notation used to denote the ZT of f(n) is as follows:

Zfn Fz fnzn

n

[()]

() ()

 



∑

where Z[f(n)] denotes the ZT of f(n).

R.5.47 Since z is a complex variable, z can be represented as a point on the complex plane

and expressed in rectangular and polar forms as

z = real(z) + jimag(z) (rectangular form)

z = |z|ejW = rejW (e x p o n e n t i a l f o r m )

where |z| = r = 1.

Observe that the complex variable z = ejW maps into a unit radius circle centered

at the origin of the complex plane.

* Since most transforms are named after a scientist such as Laplace, Fourier, or Hilbert, it would be logical to

call the ZT the Laurent transform.