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Chapter 25
Visualizing the Fourier transform of some basic functions.
In the periodic case we saw that the Fourier transform passes between -periodic functions of a real variable
and functions of integers . In the non-periodic setting, a new version of
the Fourier transform will pass between functions
of a real variable and functions of a real variable . Our focus on the 1-dimensional Fourier transform
is sufficient for many signal processing applications and
for PDE's with 1 spatial and 1 time dimension (a good
warm-up before tackling the full complexities of nature).
Let be a function for which decays quickly enough as (so
converges). The Fourier transform is the function defined by
(The integrand involves variables and , but we "integrate over ", leaving a dependence on .)
The Fourier transform can also be reversed: For as above, if decays quickly enough
as
so that converges then the following formula
(with suitable interpretation at jump discontinuities of ) recovers from :
We also introduce convenient notation for expressing the effect of the Fourier transform on functions when it is unwieldy to use
the "" symbol. For a function , its Fourier transform is also denoted as :
For a function , its Fourier transform is also denoted as :
For a function , we define its inverse Fourier transform to be
Below we visualize the Fourier transform of some basic functions. You can adjust the parameters using the sliders.
Example 1: Indicator Function
−15 −10 −5 0 5 10 15 −0.2 0 0.2 0.4 0.6 0.8 1
original function Fourier transform
Example 2: Triangular Bump
−15 −10 −5 0 5 10 15 0 0.2 0.4 0.6 0.8 1
original function Fourier transform
Example 3: Exponential Decay
−15 −10 −5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 4
original function Fourier transform
Example 4: Gaussian
−15 −10 −5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5
original function Fourier transform