Digital Signal Processing Reference
In-Depth Information
TABLE 12.2
Properties of the Discrete-Time
Fourier Transform
Signal
Transform
X(Æ) =
q
n=-
q
x[n]e
-jnÆ
x[n]
x
[
n
]
X(Æ) = X(Æ+2p)
a
1
x
1
[n] + a
2
x
2
[n]
a
1
X
1
(Æ) + a
2
X
2
(Æ)
e
-jÆn
0
X(Æ)
x[n - n
0
]
e
jnÆ
0
x[n]
X(Æ-Æ
0
)
Re[X(Æ)] is even
Im[X(Æ)] is odd
ƒX(Æ) ƒ is even
argX(Æ) is odd
d
x
[
n
] real
x[-n]
X(-Æ)
x
[
n
]
*
y
[
n
]
X(Æ)Y(Æ)
1
2p
X(Æ)
*
Y(Æ)
x
[
n
]
y
[
n
]
j
dX(
Æ
)
dÆ
nx
[
n
]
q
n=-
q
ƒ x(n) ƒ
1
2p
L
2p
ƒX(Æ) ƒ
2
2
dÆ
Parseval's theorem:
a
=
12.3
DISCRETE-TIME FOURIER TRANSFORM
OF PERIODIC SEQUENCES
In this section, we consider the discrete-time Fourier transform of periodic se-
quences. The resulting development leads us to the discrete Fourier transform and
the fast Fourier transform.
Consider a periodic sequence with period
N
, such that
Of course,
N
must be an integer. We define
x[n]
x[n] = x[n + N].
x
0
[n]
to be the values of
x[n]
over the
period beginning at
n = 0,
such that
x[n],
0 F n F N - 1
x
0
[n] =
c
.
(12.17)
0,
otherwise
An example of a periodic sequence is shown in Figure 12.4, with
N = 3.
In this
case,
x
0
[n]
is the sequence composed of
x
0
[0] = 0, x
0
[1] = 1,
and
x
0
[2] = 1,
with
x
0
[n] = 0
for all other
n
.
