Digital Signal Processing Reference
In-Depth Information
5.5.7 Duality
The CTFTs of a constant signal
x
(
t
)
=
1 and of an impulse function
x
(
t
)
= δ
(
t
)
are given by the following CTFT pairs (see Table 5.2):
and
CTFT
←−−→
CTFT
←−−→
1
2
πδ
(
ω
) and
δ
(
t
)
1
.
For the above examples, the CTFT exhibits symmetry across the time and
frequency domains in the sense that the CTFT of a constant
x
(
t
)
=
1 is an impulse
function, while the CTFT of an impulse function
x
(
t
)
= δ
(
t
) is a constant. This
symmetry extends to the CTFT of any arbitrary signal and is referred to as the
duality property. We formally define the duality property below.
CTFT
←−−→
If x
(
t
)
X
(
ω
)
, then
CTFT
←−−→
X
(
t
)
2
π
x
(
−ω
)
(5.54)
is also a CTFT pair.
Proof
By the definition of the inverse CTFT, Eq. (5.9), we know that
∞
1
2
π
X
(
r
)e
j
rt
d
r
,
x
(
t
)
=
−∞
where the dummy variable
r
is used instead of
ω
. Substituting
t
=−ω
in the
above equation yields
∞
−
j
ω
r
d
r
2
π
x
(
−ω
)
=
=ℑ
X
(
t
)
.
X
(
r
)e
−∞
To illustrate the application of the duality property, consider the CTFT pair
CTFT
←−−→
δ
(
t
)
1
,
with
x
(
t
)
= δ
(
t
) and
X
(
ω
)
=
1. By interchanging the role of the independent
variables
t
and
ω
, we obtain
X
(
t
)
=
1 and
x
(
ω
)
= δ
(
ω
). Using the duality
property, the converse CTFT pair is given by
CTFT
←−−→
1
2
πδ
(
−ω
)
=
2
πδ
(
ω
)
,
which is indeed the CTFT of the constant signal
x
(
t
)
=
1.
Example 5.20
As stated in Eq. (5.22), the following is a CTFT pair (see Example 5.6):
t
τ
ωτ
2
π
CTFT
←−−→
rect
τ
sinc
.
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