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Rectangular Pulse and Its Fourier Transform

A
0.5
T
43
t
0
-9.6
This Demonstration illustrates the relationship between a rectangular pulse signal and its Fourier transform. There are three parameters that define a rectangular pulse: its height
A
, width
T
in seconds, and center
t
0
. Mathematically, a rectangular pulse delayed by
t
0
seconds is defined as
g(t-
t
0
)=Arect
t-
t
0
T
=
A
t-
t
0
T
1
2
0
otherwise
and its Fourier transform or spectrum is defined as
G(f)=ATsinc(πfT)exp(-i2πf
t
0
)
.
This Demonstration illustrates how changing
g(t)
affects its spectrum. Both the magnitude and phase of the spectrum are displayed.

Details

This Demonstration illustrates the following relationship between a rectangular pulse and its spectrum:
1. As the pulse becomes flatter (i.e., the width
T
of the pulse increases), the magnitude spectrum loops become thinner and taller. In other words, the zeros (the crossings of the magnitude spectrum with the
x
axis) move closer to the origin. In the limit, as
T
becomes very large, the magnitude spectrum approaches a Dirac delta function located at the origin.
2. As the height of the pulse become larger and its width becomes smaller, it approaches a Dirac delta function and the magnitude spectrum flattens out and becomes a constant of magnitude 1 in the limit.
3. As
t
0
changes, the pulse shifts in time, the magnitude spectrum does not change, but the phase spectrum does.
4. We notice a
180°
phase shift at each frequency defined by
k
T
, where
k
is an integer other than zero, and
T
is the pulse duration. These frequencies are the zeros of the magnitude spectrum.

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