PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

472 Practical MATLAB® Applications for Engineers

R.5.40 Observe that the FT (analog case) and DTFT (discrete case) present similarities, as
well as differences. The contrasting characteristics are summarized as follows:
a. The DTFT is periodic with period equal to 2π, whereas FT is nonperiodic.
b. The frequency W of DTFT is given in radians, whereas the frequency w of the FT
is given in radians/seconds.
c. The magnitude spectrums in either case are symmetric.
d. The phase spectrums in either case are asymmetric.
e. The plots of DTFT are continuous and periodic functions in the frequency
domain, whereas discrete in the time domain.

FT (and FS) of a periodic continuous time function results in a discrete-frequency
spectrum.

R.5.41 The process of computing DTFT by hand (or the Inverse Discrete Time Fourier Trans-
form [IDTFT]) can be accomplished by using Table 5.1, a process that may involve
some algebraic manipulations as well as applications of some properties, discussed in
R.5.38. Let us illustrate the transform process by presenting the following example.
Evaluate by hand the DTFT of the following discrete function:

f(n) = (0.7)nu(n − 3)

ANALYTICAL Solution

a. The transform of (from the Table 5.1)

(.)07nun jW

e

()^1

1 0.7

↔ −



b. The shift in time of the step function by three time units returns the following in

frequency (from the time-shift property of R.5.38):

1

107

3

 



.e

jWejW

c. The coeffi cient of f(n) must also be shifted by (0.7)^3 , resulting in the following

transform:

(.) ( ) (.)

.

07 3 07

107

n^33 jW

un jW

e

e







↔ −

R.5.42 Let the discrete-system transfer function be H(eJW), given by

He

Ge

Fe

gge ge ge

ffe f

jW

jW

jW

jW j W

n

jnW

() jW

()

()

 



 



01 2

2

01

22

efejW^2

m


 jmW

Recall that the MATLAB signal processing toolbox includes the function freqz,
which can be used to evaluate DTFT of H(ejW). Some of the most common forms
of the freqz function are listed as follows (they were introduced in Chapter 7 of
Practical MATLAB® Basics for Engineers).
a. [H, W] = freqz(G, F), where G and F are row vectors defi ned by their coeffi cients
(g and f) arranged in descending power of ejnW and ejmW
b. H = freqz(G, F)