Digital Signal Processing Reference
In-Depth Information
3.3.4 Time Reversal Property
Let us consider
x(n)
=
n
=
0
a
n
e
−
jωn
. Next, to find
a
n
u(n)
.ItsDTFT
X(e
jω
)
=
the DTFT of
x(
−
n)
, if we replace
n
by
−
n
, we would write the DTFT of
x(
−
n)
as
−∞
n
0
a
−
n
e
jωn
(but that is wrong), as illustrated by the following example:
=
∞
X(e
jω
)
a
n
e
−
jωn
ae
−
jω
a
2
e
−
j
2
ω
a
3
e
−
j
3
ω
=
=
1
+
+
+
+···
n
=
0
ae
jω
But the correct expression for the DTFT of
x(
−
n)
is of the form 1
+
+
a
2
e
j
2
ω
a
3
e
j
3
ω
+
+···
.
So the compact form for this series is
n
=−∞
a
−
n
e
−
jωn
. With this clarifica-
tion, we now prove the property that if
x(n)
X(e
jω
)
then
⇔
X(e
−
jω
)
x(
−
n)
⇔
(3.24)
=
n
=−∞
x(
n)e
−
jωn
. We substitute
(
Proof
:DTFTof
x(
−
n)
−
−
n)
=
m
,and
we get
n
=−∞
x(
=
m
=−∞
x(m)e
jωm
=
m
=−∞
x(m)e
−
j(
−
ω)m
n)e
−
jωn
−
=
X(e
−
jω
)
.
Example 3.3
Consider
x(n)
=
δ(n)
. Then, from the definition for DTFT, we see that
δ(n)
⇔
X(e
jω
)
1
for all
ω
.
From the time-shifting property, we get
=
e
−
jωk
δ(n
−
k)
⇔
(3.25)
The Fourier transform
e
−
jωk
has a magnitude of one at all frequencies but a
linear phase as a function of
ω
that yields a constant group delay of
k
samples. If
we extend this result by considering an infinite sequence of unit impulses, which
can be represented by
k
=−∞
δ(n
k)
, its DTFT would yield
k
=−∞
e
−
jωk
.
But this does not converge to any form of expression. Hence we resort to a
different approach, as described below, and derive the result (3.28).
−
Example 3.4
k)
. Its DTFT is given by
X(e
jω
)
e
jωk
We consider
x(n)
=
δ(n
+
k)
+
δ(n
−
=
+
e
−
jωk
2cos
(ωk)
. In this example, note that the DTFT is a function of the
continuous variable
ω
whereas
k
is a fixed number. It is a periodic function of
ω
with a period of 2
π
, because 2 cos
((ω
=
=
2cos
(ωk)
,where
r
is an
integer. In other words, the inverse DTFT of
X(e
jω
)
+
2
rπ)k)
=
2cos
(ωk)
is a pair of
impulse functions at
n
=
k
and
n
=−
k
, and this is given by
1
2
[
δ(n
cos
(ωk)
⇔
+
k)
+
δ(n
−
k)
]
(3.26)
Search WWH ::
Custom Search