Digital Signal Processing Reference
In-Depth Information
EXAMPLE 12.1
Discrete-time Fourier transform (DTFT) of a signal
x[n] = a
n
u[n],
We now find the discrete-time Fourier transform of the function
where
a
n
,
n G 0
a
n
u[n] =
c
n 6 0
.
0,
Recall from Section 9.4 that this function is exponential in nature. From (12.1),
X(Æ) =
q
n=-
q
x[n]e
-jnÆ
=
q
n= 0
a
n
e
-jnÆ
= 1 + ae
-jÆ
+ a
2
e
-j2Æ
+
Á
= 1 + ae
-jÆ
+ (ae
-jÆ
)
2
+
Á
.
We now find
X(Æ)
in closed form. From Appendix C, we have the geometric series
q
n= 0
1
1 - b
;
Á
b
n
= 1 + b + b
2
ƒ
b
ƒ
6 1.
+
=
ae
-jÆ
= b,
In
X(Æ),
we let
resulting in the transform
1
1 - ae
-jÆ
;
ƒ ae
-jÆ
ƒ 6 1.
X(Æ) =
ƒ e
-jÆ
ƒ = ƒ cos Æ-j sin Æ ƒ = 1,
Because
this transform exists for
ƒ a ƒ 6 1;
we then have the
transform pair
1
1 - ae
-jÆ
,
a
n
u[n]
Î
"
ƒ a ƒ 6 1.
This transform is valid for either real or complex values of
a
. The discrete-time Fourier
transform of
a
n
u[n], ƒ a ƒ 7 1,
does not exist.
■
The linearity property of the DTFT
EXAMPLE 12.2
x[n] = a
ƒnƒ
.
Consider the discrete-time function
This function is plotted in Figure 12.1 for
a
real and
0 6 a 6 1
and can be expressed as
x[n] = a
n
u[n] + a
-n
u[-n - 1] = x
1
[n] + x
2
[n].
x
[
n
]
1
a
•••
•••
A plot of a
ƒnƒ
.
3
2
1
0
1
2
3
n
Figure 12.1
