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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 L-periodic functions f(x) of a real variable x and functions F(n) of integers n. In the non-periodic setting, a new version of the Fourier transform will pass between functions f(x) of a real variable x and functions F(λ) of a real variable λ. 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 f:RC be a function for which |f(x)| decays quickly enough as |x|+ (so R|f(x)|dx converges). The Fourier transform f^:RC is the function defined by f^(λ)=Rf(x)eiλxdx (The integrand involves variables λ and x, but we "integrate over x", leaving a dependence on λ.)

The Fourier transform can also be reversed: For f(x) as above, if |f^(λ)| decays quickly enough as |λ|+ so that R|f^(λ)|dλ converges then the following formula (with suitable interpretation at jump discontinuities of f(x)) recovers f from f^: f(x)=12πRf^(λ)eiλxdλ.

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 f(x), its Fourier transform f^ is also denoted as Ff: For a function f(x), its Fourier transform f^ is also denoted as Ff: (Ff)(λ)=Rf(x)eiλxdx. For a function g(λ), we define its inverse Fourier transform to be (F1g)(x)=12πRg(λ)eiλxdλ.

Below we visualize the Fourier transform of some basic functions. You can adjust the parameters using the sliders.

Example 1: Indicator Function
Function

a(x)=I[a,a](x).

a^(λ)=2sin(λa)λ.

Parameter
0.5

−15−10−5051015−0.200.20.40.60.81
original functionFourier transform
Example 2: Triangular Bump
Function

a(x)={1|x|/a,|x|a,0,|x|>a.

a^(λ)=a(sin(λa/2)λa/2)2.

Parameter
0.5

−15−10−505101500.20.40.60.81
original functionFourier transform

Example 3: Exponential Decay
Function

fa(x)=ea|x|.

fa^(λ)=2aa2+λ2.

Parameter
0.5

−15−10−505101500.511.522.533.54
original functionFourier transform
Example 4: Gaussian
Function

fa(x)=eax2/2.

fa^(λ)=2πaeλ2/2a.

Parameter
0.5

−15−10−505101500.511.522.533.5
original functionFourier transform