PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 469

R.5.28 Observe that DTFT closely resembles the analog FT presented in Chapter 4 of this

book.

The syntaxes (notation) used to indicate the direct and inverse FT pairs are as

follows:

F(ejW) = F[f(n)] (denotes the discrete time Fourier transform [DTFT])

f(n) = F–1[ejW] (denotes the inverse discrete Fourier transform [IDFT])

f(n)↔F(ejW) (denotes the transform equivalent time–frequency pair)

R.5.29 The DTFT F(ejW) is a periodic function of the real variable W, with a period equal

to 2π. This is the reason why the integral of the inverse transform is evaluated over

the range −π ≤ W ≤ +π. Also observe that

FejW f nejnW

n

()
()

 



∑

and its expansion is given by

Fe fe fe fe fne

fe f

jW jW j W jnW

jW

( ) (0) (1) (2) ( )

(1) (2

 ^02 

 




))()efnej2W
 jnW

consisting of a sum of exponentials of the form e−jnW, for n = 0, ±1, ±2, ..., ±∞.

Observe also that the function e−j^2 πn repeats in frequency of every 2π rad.

R.5.30 Recall that the FT of an analog signal is a function of w, with radian/second as unit,

whereas W is used to denote discrete frequencies, with radian as unit.

Some authors differentiate the analog case from the discrete case by using the

variable Ω; in this book, W (uppercase) is used to denote discrete frequencies.

R.5.31 Since F[ejW] is a complex function of the real variable W, then F[eJW] can be repre-

sented by two parts, real and imaginary in rectangular form, or with a magnitude

and phase in polar form (similar to the continuous case) as follows:

F[eJW] = real {F[ejW]} + jimag {F[ejW]} (rectangular)

F[eJW] = |F[ejW]|e−jθ(n) (polar)

R.5.32 Let |F[eJW]| represent the magnitude, whereas θ(W) the phase of DTFT of f(n), then

the plot |F[eJW]|v e r s u s W is referred to as the magnitude spectrum plot of f(n),

whereas θ(W) versus W is referred to as the phase spectrum plot of f(n).

R.5.33 Recall that the relation between the polar and rectangular forms is

real {F[ejW]} = F[ejW]cos[θ(W)]

imag {F[ejW]} = F[ejW]sin[θ(W)]

where

F e( )jW 
real F e^22 [( )]jW imag F e[( )]jW