Digital Signal Processing Reference
In-Depth Information
Use of the
z
-transform to find a DTFT
EXAMPLE 12.3
x[n] = na
n
u[n].
We wish to find the discrete-time Fourier transform of the function
From
Table 11.5,
az
(z - a)
2
;
z
[na
n
u[n]] =
ƒ z ƒ 7 ƒ a ƒ .
na
n
u[n]
ƒ e
jÆ
ƒ = 1 7 ƒ a ƒ ,
Hence, for the discrete-time Fourier transform of
to exist,
and we
have the transform
ae
jÆ
(e
jÆ
- a)
2
; ƒ a ƒ 6 1.
[na
n
u[n]] =
Consider next the discrete-time Fourier transform of cos(
bn
)
u
[
n
]. From Table 11.5,
z
2
- z cos b
z
[cos(bn)u[n]] =
- 2z cos b + 1
;
ƒ z ƒ 7 1.
z
2
TABLE 12.1
Discrete-Time Fourier Transforms
f
[
n
]
F
(
æ
)
1.
d[n]
1
q
2. 1
2p
a
d(Æ- 2pk)
k=-
q
1 - e
-jÆ
+
q
1
3.
u
[
n
]
pd(Æ- 2pk)
k=-
q
1
1 - ae
-jÆ
a
n
u[n];
4.
ƒ a ƒ 6 1
ae
jÆ
(e
jÆ
- a)
2
na
n
u[n];
5.
ƒ a ƒ 6 1
ae
jÆ
1 - ae
jÆ
a
-n
u[-n - 1];
6.
ƒ a ƒ 6 1
2p
q
k=-
q
e
jÆ
0
n
7.
d(Æ-Æ
0
- 2pk)
p
q
k=-
q
8.
cos[Æ
0
n]
[d(Æ-Æ
0
- 2pk) + d(Æ+Æ
0
- 2pk)]
j
q
k=-
q
p
9.
sin[Æ
0
n]
[d(Æ-Æ
0
- 2pk) - d(Æ+Æ
0
- 2pk)]
1
2
)
sin Æ(N +
10. rect[
n
/
N
]
sin(Æ/2)
sin [
Æ
1
n]
pn
11.
rect(Æ/Æ
1
)
2p
q
k=-
q
12.
x
[
n
] periodic
a
k
d(Æ- 2pk/N)
n
0
+N- 1
1
N
a
x[n]e
-j2pkn
with period
N
a
k
=
N
n=n
0
