Digital Signal Processing Reference
In-Depth Information
Symmetry of FT and IFT using the example of the rectangle function rect (t)
x(t)
X(
ω
)
1
2a
FT
IFT
⎯⎯→
←⎯⎯
2
a
π
π
π
2
a
π
−
−
ω
-a 0 +a
0
t
f
1
a
1
2
a
1
2
a
1
a
−
−
∞
∞
a
1
sin
ω
a
sin
ω
a
(
)
³
−
jt
ω
³
−
jt
ω
³
−
jt
ω
ja
ω
−
j
ω
a
X
()
ω
=
xt e t
()
=
t t e t
()
=
e t
=
e
−
e
=
2
=
2
a
j
ω
ω
ω
a
−∞
−∞
−
a
1
(
)
ja
ω
−
ja
ω
Here
j
si
n
ω
a
=
e
−
e
was used.
2
The structural symmetry of FT and IFT (up to a scale factor 2
π
) makes it possible,
to call the FT-signal
X
(
ω
) an "equally formed" time signal
=
placed by the time t. After this, the result of
X t
( )
x t
( ). Generally
with
X
(
ω
) the variable
ω
is simply re
FT is already determined: rect(
ω
) instead of rect(t)!
sin
ω
a
sin
at
X
(
ω
)
=
2
symmetry principle
X t
( )
==
x t
( )
2
↔↔
FT
2
π
rect
(
−
ω
)
a
a
ω
t
ω
<
a
-
°
1
sin
at
π
rect
rect
Both sides divided by 2
:
↔−
()
ω
=
()
ω
=
®
°
¯
a
a
π
t
ω
>
a
0
X(
ω
)
x(t)
a
π
1
FT
IFT
⎯⎯→
←⎯ ⎯
-a 0 +a
0
ω
π
t
π
2
a
π
2
a
π
−
−
Illustration 313:
Singular signal: FOURIER- transformation and symmetry principle
This example was already dealt with in llustration 91. Here, the mathematical background is added. For
practical reasons, the angular frequency Ȧ is taken as a variable instead of f. The angular frequency Ȧ
(Ȧ=2ʌ f) is here the complementary term to the time t and vice versa. It is only necessary to take a scale
factor of 2ʌ into consideration.
The reciprocity of the terms is immediately striking: Everything that is big in the one domain turns out to
be small in the complementary domain and vice versa (see Illustration 35). The faster the alteration in the
one domain, the bigger the expansion of the transformed signal in the other. Everything angular turns out
to be round in the complementary domain.
The time domain and frequency domain represent two “worlds”, where equal “figures”, projected into the
complementary world, create the same images. In addition, two- dimensional signals, such as pictures,
can undergo a FOURIER- transformation: all the horizontal information is then contained in the vertical
curve. They are perpendicular to each other.
