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pluto_fourier_transform.jl
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pluto_fourier_transform.jl
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### A Pluto.jl notebook ###
# v0.19.36
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 1bd03510-d1d2-11ec-04c2-fbbd23b59e29
begin
using PlutoUI
using Colors
using ColorVectorSpace
using Images
using ImageShow
using FileIO
using ImageIO
using Unitful
using ImageFiltering
using OffsetArrays
using TestImages
using Plots
using Markdown
using InteractiveUtils
using HypertextLiteral
using CalculusWithJulia
using SymPy
using LaTeXStrings
# Small patch to make images look more crisp:
# https://github.com/JuliaImages/ImageShow.jl/pull/50
Base.showable(::MIME"text/html", ::AbstractMatrix{<:Colorant}) = false
end
# ╔═╡ dcb9b0ed-8338-4eb0-839f-cf7611c8f3b3
md"""
#### Fourier transform
The Fourier transform of an intergrable function $f(x)$ on $\mathbb{R}$ is defined by
$\hat{f}(\xi):=\int_{\mathbb{R}}e^{-2\pi{i}x\xi}f(x)d\xi, \quad \xi\in\mathbb{R}.$
The Fourier transform of a Gaussian function is a Gaussian function:
$\int_{\mathbb{R}}e^{-2\pi{i}x\xi-\pi{x^2}}d\xi=e^{-\pi\xi^2}$
Julia Programming Language computes this integral:
"""
# ╔═╡ 30d870a3-1b06-4a89-8772-f3fdc6bc2bd9
begin
@syms x::real xi::real
integrate(exp(-2*PI*im*x*xi-PI*x^2),(x,-oo,oo)).simplify()
end
# ╔═╡ d08f5ee2-c7c2-4a59-a9fb-7dc2d88518aa
md"""
#### The Nyquist–Shannon Sampling Theorem and Aliasing
We state the Nyquist-Shannon sampling theorem and observe an example. This is basically concerned with one-variable integrable functions whose Fourier transforms are compactly supported. Such functions are automatically real analytic and can be extended to entire functions on $\mathbb{C}$. We employ the definition of Fourier transform as follows. For an appropriate one-variable function $f(x)$, its Fourier transform is defined by
$\hat{f}(\xi)
:=
\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^\infty
e^{-ix\xi}
f(x)
dx,
\quad
\xi\in\mathbb{R}.$
We use the normalized sinc function defined by
$\operatorname{sinc}(x)
:=
\frac{\sin(\pi x)}{\pi x}
\quad
(x\ne0),
\quad
\operatorname{sinc}(0)
:=1.$
"""
# ╔═╡ 8ac97a72-05de-483e-b21c-dbd533632d99
md"""
#### Theorem (the Nyquist-Shannon sampling theorem)
Let $B>0$. Suppose that $f(x)$ is an integrable smooth functions on $\mathbb{R}$ and $\hat{f}(\xi)=0$ for $\lvert\xi\rvert \geqq B$. Then for any $s\in(0,\pi/B]$, we have
$f(x)
=
\sum_{n=-\infty}^\infty
f(sn)\cdot\operatorname{sinc}\left(\frac{x}{s}-n\right),$
$\int_{-\infty}^\infty
\lvert{f(x)}\rvert^2
dx
=
s
\sum_{n=-\infty}^\infty
\lvert{f(sn)}\rvert^2.$
The parameter $s>0$ is said to be the sampling period.
"""
# ╔═╡ b5f40261-b208-4786-9a5f-0756784b3099
md"""
#### Observation: sampling and aliasing
Let $f(x)$ be a function defined by
$f(x)
:=
\frac{2\bigl(1-\cos(\pi x)\bigr)}{(\pi x)^2}
\quad
(x\ne0),
\qquad
f(0):=1.$
Then
$\hat{f}(\xi)
=
\frac{\sqrt{2}(\pi-\lvert\xi\rvert)}{\pi^{3/2}}
\quad
(\lvert\xi\rvert<\pi),
\qquad
\hat{f}(\xi)=0
\quad
(\lvert\xi\rvert\geqq\pi).$
If we set
$F(x;s):=
\sum_{-\infty}^\infty
f(sn)
\cdot
\operatorname{sinc}\left(\frac{x}{s}-n\right),
\quad
s>0,$
then the Nyquist–Shannon Sampling Theorem implies that
$f(\cdot)=F(\cdot;s), \quad s\in(0,1],$
$f(\cdot){\ne}F(\cdot;s), \quad s>1,$
The latter one is said to be aliasing.
"""
# ╔═╡ 6c3a6a78-86fa-4bc2-9236-92dfbc0fc0e9
md"""
s = $(@bind s Slider(0.8:0.1:2, show_value=true))
"""
# ╔═╡ 2d6010ab-b1c6-4a36-8c8a-d940b771f7f1
begin
x1 = range(-10, 10, length = 201);
F=sinc.(x1/s);
f=ones(201);
for k=1:100
f[k]=2*(1-cos(pi*x1[k]))/(pi*x1[k])^2;
f[k+101]=2*(1-cos(pi*x1[k+101]))/(pi*x1[k+101])^2;
end
for n=1:20
a=2*(1-cos(pi*n*s))/(pi*n*s)^2;
b=sinc.(x1/s-n*ones(201));
c=sinc.(x1/s+n*ones(201));
F=F+a*b+a*c;
end
end
# ╔═╡ 1d325d98-2ea3-4577-a442-79766c114c32
begin
plot([F,f],
grid=false,
linewidth=2,
ylim=(-0.3,1.2),
title="Example: Nyquist-Shannon sampling and aliasing",
xticks = ([0 51 101 151 201;], [-10,-5,0,5,10]),
xlabel="x",
yticks = ([0 0.5 1;], [0,0.5,1]),
label=[L"F(x;s)" L"f(x)"],
linecolor=["blue" "magenta"],
legend=:topright,
legendfont=font(12))
end
# ╔═╡ f5cc5a49-1c0a-4428-a647-d8f96edd8e83
md"""
#### Japanese seismic scale
Let $w_j(t)$ ($j=1,2,3$) be the mesurements at the time $t$ of seismic waves of the three directions respectively.
The Japanese seismic scale is obtained from the modified amplitude defined by
$A(t)
:=
\left\lvert
\sum_{j=1}^3
\int_{-\infty}^\infty
K(t-s)w_j(s)
ds
\right\rvert,$
where
$K(t)
:=
\int_{-\infty}^\infty
e^{2\pi{it\tau}}
\frac{\lvert\tau\rvert^3}{(1+\tau^2)^{9/2}}
d\tau
=
2\int_0^\infty
\cos(2\pi{t\tau})
\frac{\lvert\tau\rvert^3}{(1+\tau^2)^{9/2}}
d\tau.$
"""
# ╔═╡ 61ef730a-274d-4fd6-ac9d-34893792d08a
begin
N5=100
KK5=zeros(N5+1,N5+1);
h5=10/N5;
for j=1:N5+1
for k=1:N5+1
KK5[j,k]=2*cos(2*pi*(j-1)*(k-1)*h5^2)*((k-1)*h5)^3/(1+((k-1)*h5)^2)^(9/2);
end
end
K5=zeros(2*N5+1);
for k=1:N5+1
K5[N5+1]=K5[N5+1]+KK5[1,k];
end
for j=1:N5
for k=1:N5+1
K5[j]=K5[j]+KK5[N5+2-j,k];
K5[N5+1+j]=K5[N5+1+j]+KK5[j+1,k]
end
end
end
# ╔═╡ 4de3cbc1-0b58-453c-8957-336a6520dc68
begin
P5=plot(K5,
title="Convolution kernel for the Japanese seismic scaling",
grid=:false,
xlim=(21,181),
xticks=([51 101 151;],[-5,0,5]),
label=L"K(t)",
linewidth=2,
xtickfont=font(12),
ytickfont=font(12),
legendfont=font(12))
savefig("./pictures/seismicscaling.png")
P5
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
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