Digital Signal Processing Reference
In-Depth Information
TABLE 3.1
DTFT Theorems
Property
ω
f(n)
F(
)
Periodicity
x
(
n
)
X
(
ω
) =
X
(
ω
+ 2
m
π
), for integer
m
Convolution
x
(
n
)
∗
h
(
n
)
X
(
ω
)
H
(
ω
)
Time Shift
x
(
n
-
n
)
X
(
ω
)
e
-
j
ω
n
0
0
Frequency Shift
e
j
ω
n
0
x
(
n
)
X
(
ω
-
ω
)
0
Time Reversal
x
(-
n
)
X
(-
ω
)
The function
X
(
e
j
ω
)
or
X
(
ω
)
is also called the
Discrete-Time Fourier Transform
(
DTFT
) of the discrete-time signal
x
(
n
). The inverse DTFT is defined by the
following integral:
π
1
2
πω ω
∫
jn
ω
xn
()
=
X
( )
e
d
(3.2)
−
π
for all values of
n
. The significance of the integration operation in Equation
3.2 will be clear after discussing the
periodicity
property of the DTFT in the
next section.
Properties of Discrete-Time Fourier Transform
A concise list of DTFT properties is given in Table 3.1.
Analog frequency and digital frequency
The fundamental relation between the analog frequency,
Ω
, and the digital
frequency,
ω
, is given by the following relation:
ω
=
Ω
T
(3.3a)
or alternately,
ω
=
Ω
/f
s
(3.3b)
where
T
is the sampling period, in sec., and
f
s
=
1
/T
is the sampling frequency
in Hz.
This important transformation will be discussed more thoroughly in
Chap-
ter 5
. Note, however, the following interesting points:
•
The unit of
Ω
is radian/sec., whereas the unit of
ω
is just radians.
•
The analog frequency,
Ω
, represents the
actual physical frequency of
the basic analog signal
, for example, an audio signal (0 to 4 kHz) or a
video signal (0 to 4 MHz). The digital frequency,
, is the transformed
frequency from Equation 3.3a or Equation 3.3b and can be considered
as a mathematical frequency, corresponding to the digital signal.
ω
