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(e
jW ) = F[f(n)]
(denotes the discrete time Fourier transform [DTFT])
f(n) = F
–1
[e
jW ]
(denotes the inverse discrete Fourier transform [IDFT])
f(n)↔F(e
jW )
(denotes the transform equivalent time–frequency pair)
R.5.29 The DTFT F(e
jW ) 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
F e
f n e
jW
jnW
n
( )
( )
ϭ
Ϫ
ϭϪϱ
ϩϱ
∑
and its expansion is given by
F e
f e
f e
f e
f n e
f
e
f
jW
jW
j W
jnW
jW
(
)
(0)
(1)
(2)
( )
( 1)
( 2
0
2
ϭ
ϩ
ϩ
ϩ ϩ
ϩ Ϫ
ϩ Ϫ
Ϫ
Ϫ
Ϫ
) )
( )
e
f ne
j2W
jnW
ϩ ϩ Ϫ
consisting of a sum of exponentials of the form e
−jnW
, for n = 0, ±1, ±2, …, ±∞.
Observe also that the function e
−j2πn 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[e
jW ] is a complex function of the real variable W, then F[e
JW ] can be represented 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[e
JW ] = real {F[e
jW ]} + jimag {F[e
jW ]}
(rectangular)
F[e
JW ] = |F[e
jW ]|e
−jθ(n)
(polar)
R.5.32 Let |F[e
JW ]| represent the magnitude, whereas θ(W) the phase of DTFT of f(n), then
the plot |F[e
JW ]|versus 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[e
jW ]} = F[e
jW ]cos[θ(W)]
imag {F[e
jW ]} = F[e
jW ]sin[θ(W)]
where
"
"
F e
real F e
imag F e
jW
jW
jW
( )
[ ( )]
[ ( )]
ϭ
ϩ
2
2
CRC_47760_CH005.indd 469
CRC_47760_CH005.indd 469
7/23/2008 1:47:18 PM
7/23/2008 1:47:18 PM