diff --git a/discrete_fourier_transform.html b/discrete_fourier_transform.html new file mode 100644 index 0000000..f654a69 --- /dev/null +++ b/discrete_fourier_transform.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/discrete_fourier_transform.jl b/discrete_fourier_transform.jl new file mode 100644 index 0000000..c1a4cce --- /dev/null +++ b/discrete_fourier_transform.jl @@ -0,0 +1,1338 @@ +### A Pluto.jl notebook ### +# v0.19.40 + +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 + +# ╔═╡ a22d543c-6a50-11ed-002d-7f2fe5bd7276 +using Plots, PlutoUI, LaTeXStrings, ShortCodes + +# ╔═╡ 4de22c75-0d00-4838-8ba4-52bab4dab89e +md""" +# Discrete Fourier Transform + +In this notebook, we compute the Discrete Fourier Transform (DFT) of a periodic function ``f(x): \mathbb{R} \rightarrow \mathbb{R}`` with a period ``L=2 \pi``, i.e., ``f(x+L) = f(x)``. + +The discrete fourier transform with ``N`` equidistant points is given by +```math +\hat y_k = \frac{1}{N} \sum_{j=0}^{N-1} y_j e^{-i j k\frac{2\pi}{N}} ~~~~ (k=0, \ldots, N-1), +``` +where ``y_j`` is the function at the sampling points ``y_j = f(x_j)`` with ``x_j = 2 \pi j/N``. We can rewrite the DFT as +```math +\hat y_k = \frac{1}{N} \sum_{j=0}^{N-1} y_j e^{-i k x_j} = \frac{1}{N} \left( \sum_{j=0}^{N-1} y_j \cos(k x_j) - i +\sum_{j=0}^{N-1} y_j \sin(k x_j) \right) ~~~~ (k=0, \ldots, N-1). +``` + +The top-left graph presents the function ``f(x)``, which will undergo a Discrete Fourier Transform (DFT). If selected, this graph also includes the functions ``\cos(kx)`` and/or ``\sin(kx)`` that ``f(x)`` will be correlated with. The bottom-left graph displays the product of ``f(x)`` with ``\cos(kx)`` and/or ``\sin(kx)`` at the chosen sample points. On the top-right, the graph illustrates the real component of the DFT results, whereas the bottom-right graph visualizes the imaginary component of these results. +""" + +# ╔═╡ 6e4a15eb-8c68-4837-81dd-caac7c81ba7e +md""" +##### How to add a new function +To add your own function (and show the corresponding DFT), you need to do the following things: +* Implement a new function of a single variable `x` (let's call it `myfun`). +* Add `myfun` to the dropdown menu above (under "Interpolation settings"). +One example, for which this has already been done in this notebook, is `cos_plus_1`. To get started, simply copy `cos_plus_1`, rename it to something unique, and implement away 🙂 +""" + +# ╔═╡ 6b49c667-b16e-44f3-9f0d-2b8a62f5efda +md""" +## Auxiliary functions and internals +If you are only interested in *using* the interactive plotting features in this notebook, there is **no need to read any further**. The remainder of this document contains auxiliary functions for computing the Taylor polynomial. Furthermore, implementation details that are required for this workbook to execute correctly are included here. +""" + +# ╔═╡ b8f81a3f-d773-4e67-b8d7-e1734b04abaa +md""" +### Load required packages +This noteboook requires some additional packages to work: +* Plots.jl is used for plotting the results +* PlutoUI.jl is used to create the interactive GUI elements +* LaTeXStrings.jl is used to make some mathematical notations look nicer +* ShortCodes.jl allows to embed references to papers etc. more easily +""" + +# ╔═╡ 94c7aca0-1f23-4b2a-94ae-1aef90fed8d9 +md""" +### Definitions for additional functions to interpolate +""" + +# ╔═╡ 9e81ff03-1e93-4664-a030-32c593b2be0a +function _2cosx_2sin3x(x) + return 2*cos(x) + 2*sin(3*x) +end + +# ╔═╡ 889fb26d-338c-45f8-b621-640ad42c44ae +md""" +##### Fourier transform settings +Function to take DFT: ``f = ``$(@bind selected_function Select([cos, sin, _2cosx_2sin3x])) ``(\omega_0 (x - \phi))`` + +Frequency ``\omega_0``: $(@bind omega0 Slider(1:0.05:15, default=1, show_value=true)) + +Shift ``\phi``: $(@bind phi Slider(0:0.05:2 * pi, default=0, show_value=true)) + +Number of nodes ``N``: $(@bind N Slider(1:100, default=5, show_value=true)) +""" + +# ╔═╡ c03bb676-d1d5-48fc-b966-b680a3088d52 +md""" +##### +Current frequency ``k``: $(@bind k Slider(0:N-1, default=0, show_value=true)) + +Plot cos: $(@bind plot_cos Select(["no", "yes"])) + +Plot sin: $(@bind plot_sin Select(["no", "yes"])) +""" + +# ╔═╡ 8156b2cc-bd11-4714-82c5-00f97eceec8c +md""" +### Definitions for functions to compute node locations +The following function compute the node locations with an equidistant distribution. +""" + +# ╔═╡ e560e608-ac7a-427d-9e2f-715ad01283da +""" + equidistant_nodes(a, b, N) + +Compute ``N`` equidistant node locations (including end points) on the interval +``[a, b]``. +""" +function equidistant_nodes(N) + + # Initialize nodes to zero and compute spacing `h` + nodes = zeros(N) + h = 2 * pi / N + + # Compute node locations + for i in 1:N + nodes[i] = (i - 1) * h + end + + return nodes +end + +# ╔═╡ ac216c60-8288-465b-ab2b-2ba3e7701d02 +begin + using FFTW + # Auxiliary variables + n = N - 1 + x_min = 0.0 + x_max = 2 * pi + xvalues = range(x_min, x_max, 1001) + + # Compute exact function values + exact_values = selected_function.(omega0 * (xvalues .- phi)) + y_min = min(-2, minimum(exact_values) - 0.1) + y_max = max(2, maximum(exact_values) + 0.1) + + # Compute nodes + nodes = equidistant_nodes(N) + yvalues = selected_function.(omega0 * (nodes .- phi)) + + # Compute Fourier frequencies + fourier_real_cont = cos.(k * xvalues) + fourier_imag_cont = sin.(k * xvalues) + fourier_real_disc = cos.(k * nodes) + fourier_imag_disc = sin.(k * nodes) + + # Plot canvas 1 and set some default options + p1 = plot( + xlims=(x_min - 0.1, x_max), + ylims=(y_min, y_max), + legend=:bottomright, + legendfontsize=10, + tickfontsize=10, + gridlinewidth=1.5, + title=L"\textrm{Function \, Values}", + ) + + # Plot exact function + label_exact = L"f = %$(nameof(selected_function))$\,(%$(omega0) x)$" + plot!(p1, xvalues, exact_values, label=label_exact, linewidth=3, linecolor=:blue) + + # Plot markers at node locations + scatter!(p1, nodes, yvalues, label=nothing, markersize=5, markercolor=:white, markershape = :rect, markerstrokecolor=:blue, markerstrokewidth = 2) #"nodes" + + # Plot Fourier functions + if plot_cos == "yes" + plot!(p1, xvalues, fourier_real_cont, label=L"\cos(k x)", linewidth=3, linestyle = :dash, linecolor = :green) + scatter!(p1, nodes, fourier_real_disc, label=nothing, markersize=5, markercolor=:green, markershape = :diamond) + end + if plot_sin == "yes" + plot!(p1, xvalues, fourier_imag_cont, label=L"\sin(k x)", linewidth=3, linestyle = :dash, linecolor = :red) + scatter!(p1, nodes, fourier_imag_disc, label=nothing, markersize=5, markercolor=:red) + end + + real_part = (fourier_real_disc .* yvalues) + imag_part = (fourier_imag_disc .* yvalues) + + + # Plot canvas 4 and set some default options + p4 = plot( + xlims=(x_min - 0.1, x_max), + ylims=(y_min, y_max), + legend=:bottomright, + legendfontsize=10, + tickfontsize=10, + gridlinewidth=1.5, + title=L"f \cdot e^{i\,k\,x}", + ) + + if plot_cos == "yes" + for i in 1:N + plot!(p4, [nodes[i], nodes[i]], [0, real_part[i]], label=nothing, linecolor=:green, linewidth = 3) #, markersize=5 + end + scatter!(p4, nodes, real_part, label=L"y_j \cos(k x_j)", markershape = :diamond, markercolor=:green) + end + if plot_sin == "yes" + for i in 1:N + plot!(p4, [nodes[i], nodes[i]], [0, imag_part[i]], label=nothing, linecolor=:red, linewidth = 3, markercolor=:red) #, markersize=5 + end + scatter!(p4, nodes, imag_part, label=L"y_j \sin(k x_j)", markercolor=:red) + end + + # Compute FFT (FFTW does not divide by N) + freq = fft(yvalues) / N + + # Plot canvas 2 and set some default options + y_min = min(-1, -maximum(abs.(real(freq)))) - 0.1 + y_max = max(1, maximum(abs.(real(freq)))) + 0.1 + p2 = plot( + xlims=(0, N), + ylims=(y_min, y_max), + legend=:bottomright, + legendfontsize=10, + tickfontsize=10, + gridlinewidth=1.5, + title=L"\textrm{Re}\, (\hat{y}_k) = \frac{1}{N}\sum_j y_j \cos(k x_j)", + ) + + # Plot markers at node locations on the x-axis + scatter!(p2, 0:k, real(freq[1:k + 1]), label=nothing, markersize=5, markercolor=:red) + plot!(p2, [(N/2), (N/2)], [y_min, y_max], linecolor = :blue, label = nothing) + annotate!(p2, (N/2), y_min + 0.25, text(L"\textrm{Wavelength} = 2 \Delta x", :blue, :left, 12)) + + # Plot canvas 2 and set some default options + y_min = min(-1, -maximum(abs.(imag(freq)))) - 0.1 + y_max = max(1, maximum(abs.(imag(freq)))) + 0.1 + p3 = plot( + xlims=(0, N), + ylims=(y_min, y_max), + legend=:bottomright, + legendfontsize=10, + tickfontsize=10, + gridlinewidth=1.5, + title=L"\textrm{Im}\, (\hat{y}_k) = -\frac{1}{N}\sum_j y_j \sin(k x_j)", + ) + # Plot markers at node locations on the x-axis + scatter!(p3, 0:k, imag(freq[1:k + 1]), label=nothing, markersize=5, markercolor=:red) + 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["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +version = "17.4.0+0" + +[[deps.x264_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" +uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "2021.5.5+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.5.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+0" +""" + +# ╔═╡ Cell order: +# ╟─4de22c75-0d00-4838-8ba4-52bab4dab89e +# ╟─ac216c60-8288-465b-ab2b-2ba3e7701d02 +# ╟─889fb26d-338c-45f8-b621-640ad42c44ae +# ╟─c03bb676-d1d5-48fc-b966-b680a3088d52 +# 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\ No newline at end of file diff --git a/finite_difference_method.jl b/finite_difference_method.jl new file mode 100644 index 0000000..0979e7f --- /dev/null +++ b/finite_difference_method.jl @@ -0,0 +1,1236 @@ +### A Pluto.jl notebook ### +# v0.19.19 + +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 + +# ╔═╡ a22d543c-6a50-11ed-002d-7f2fe5bd7276 +using Plots, PlutoUI, LaTeXStrings, ShortCodes, LinearAlgebra + +# ╔═╡ f87f000f-e0c9-46eb-a242-89a4290bd84f +md""" +### Numerical solution +In the discretized equation $(*)$ above, each temperature value $u_{i,j}$ depends on its neighboring values. Thus, we end up with a linear system, +```math +A u = b +``` +which we need to solve to obtain the temperature values at each inner node. Here, $A$ is the coefficient matrix, $u$ contains our unknown temperature values $u_{i,j}$ that we want to solve for, and $b$ holds the right-hand side of the system, i.e., $-h^2 f_{i,j}$. + +We will solve the system using one of Julia's built-in linear solvers from the `LinearAlgebra` module. For simplicity, we will use $f(x,y) = 2$ everywhere: +""" + +# ╔═╡ a7e57f8e-2032-419f-b957-12e90c3b9773 +md""" +In the plot below, you can see a surface plot of the numerical solution, where the height indicates the value of the temperature at a given point. Increasing the number of intervals $N$, and thus decreasing the spacing between the nodes $h$, will lead to longer computing times but also a more accurate solution. +""" + +# ╔═╡ 25a2120b-e9cd-48bc-901e-5611a2330f99 +md""" +Number of intervals ``N``: $(@bind N Slider(4:30, default=5, show_value=true)) | +Camera angle: $(@bind angle Slider(1:90, default=45, show_value=false)) +""" + +# ╔═╡ 4de22c75-0d00-4838-8ba4-52bab4dab89e +md""" +# Finite difference method + +In this notebook, we will approximate the solution to the steady state heat equation in two dimensions, +```math +\begin{equation} +\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = -\frac{1}{\kappa} f, +\end{equation} +``` +using a **finite difference method**. Here, ``u(x, y)`` is the temperature, ``x`` and ``y`` are the spatial directions, ``\kappa`` is the thermal conductivity, and ``f(x, y)`` is a known heat source. For simplicity, we will assume $\kappa = 1$ in the following sections and drop it from the equations. The steady state heat equation is a prototypical example for a so-called *Poisson problem*. + +Our computational domain is $\Omega = [0, 1]^2$, and we assume $u$ to be zero on all boundaries, i.e., $u(x,y) = 0$ if $x \in \{0,1\}$ or $y \in \{0, 1\}$. We discretize our domain with $N + 1$ nodes in each spatial direction on a rectangular grid. The nodes $(x_i, y_j)$ with $i, j = 0, 1, \ldots, N$ are equidistantly placed and their positions are computed as +```math +\begin{align} +x_i &= ih,\\ +y_j &= jh, +\end{align} +``` +where the $h = 1/N$ is the distance between the nodes in each axis direction. + +To discretize the heat equation, we use second-order central finite differences to approximate the partial derivative in the $x$-direction, +```math +\frac{\partial^2 u(x, y)}{\partial x^2} \approx \frac{u(x-h, y) - 2 u(x, y) + u(x+h, y)}{h^2} +``` +and likewise for the derivative in the $y$-direction. Inserted in the heat equation above, we obtain +```math +\begin{equation} +\frac{u_{i-1,j} - 2 u_{i,j} + u_{i+1,j}}{h^2} + \frac{u_{i,j-1} - 2 u_{i,j} + u_{i,j+1}}{h^2} = -f_{i,j},\quad i,j = 1, \ldots, N-1, +\end{equation} +``` +where $u_{i,j} = u(x_i, y_j)$ is the value of the temperature at node $(i,j$), and $f_{i,j} = f(x_i, y_j)$. We can further simplify this equation to +```math +\begin{equation} +u_{i-1,j} + u_{i,j-1} - 4 u_{i,j} + u_{i+1,j} + u_{i,j+1} = -h^2 f_{i,j},\quad i,j = 1, \ldots, N-1, \quad (*) +\end{equation} +``` +In the plot below, you can see the domain $\Omega$ and the node locations for the chosen $N$ (violet circles for inner nodes, red diamonds for boundary nodes). The green shape denotes the numerical stencil for a user-selected $(i, j)$ pair, i.e., it includes all neighbor nodes that are used to compute approximate the solution at the central node. + +Index ``i``: $(@bind idx_i Slider(1:(N-1), default=3, show_value=true)) | Index ``j``: $(@bind idx_j Slider(1:(N-1), default=1, show_value=true)) +""" + +# ╔═╡ c5689d78-a0ae-432d-b7cd-a3de894d4443 +begin + h = 1 / N + plot([0.0, 1.0, 1.0, 0.0, 0.0], [0.0, 0.0, 1.0, 1.0, 0.0], + aspect_ratio=:equal, xlims=(-0.1, 1.1), ylims=(-0.1, 1.1), label=L"\Omega", color=:black, lw=3) + plot!([0.0, 1.0], [1.0, 1.0], fillrange=0, fillcolor=:black, fillalpha=0.1, label="") + scatter!([(x, y) for x in (0.0+h):h:(1.0-h) for y in (0.0+h):h:(1.0-h)], + color=:violet, label="inner nodes") + scatter!([(x, y) for x in 0.0:h:1.0 for y in 0:1], + color=:red, label="boundary nodes", markershape=:diamond) + scatter!([(x, y) for x in 0:1 for y in (0.0+h):h:(1.0-h)], + color=:red, label="", markershape=:diamond) + + cx, cy = (idx_i*h, idx_j*h) + plot!([ + (cx-0.2h, cy-1.2h), + (cx+0.2h, cy-1.2h), + (cx+0.2h, cy-0.2h), + (cx+0.2h, cy-0.2h), + (cx+1.2h, cy-0.2h), + (cx+1.2h, cy+0.2h), + (cx+0.2h, cy+0.2h), + (cx+0.2h, cy+1.2h), + (cx-0.2h, cy+1.2h), + (cx-0.2h, cy+0.2h), + (cx-1.2h, cy+0.2h), + (cx-1.2h, cy-0.2h), + (cx-0.2h, cy-0.2h), + (cx-0.2h, cy-1.2h), + ], + lw=2, label="stencil", color=:green) +end + +# ╔═╡ 138b0481-7d42-4197-8ca2-b3fb073366a0 +md""" +### Assembling the linear system +From equation $(*)$ above, we can assemble our coefficient matrix $A$ that represents our linear system of equations (see class notes): +""" + +# ╔═╡ 8e2710f6-9e2c-47a9-8a1e-33c6df4386cf +begin + m = (N - 1)^2 # Matrix size + + # Banded matrix with full bands + A = diagm( 0 => fill(-4, m), + 1 => fill( 1, m - 1), + N-1 => fill( 1, m - (N - 1)), + -1 => fill( 1, m - 1), + -(N-1) => fill( 1, m - (N - 1)), + ) + + # Reset values to obtain block structure + for i in (N - 1):(N - 1):(m - 1) + A[i, i+1] = 0 + A[i+1, i] = 0 + end + + nothing # hide output +end + +# ╔═╡ f185aacd-0334-4217-8fa9-9fd0776c12c6 +begin + b = -h^2 * 2 * ones(m) # build right-hand side + u = A \ b # solve linear system + nothing # hide output +end + +# ╔═╡ eb184f17-09c4-426e-80bb-f7d5867a01a4 +begin + surface(reshape(u, (N - 1, N - 1)), camera=(angle, 45), zlims=(0.0, 0.15)) # Plot +end + +# ╔═╡ 52e5b8cc-d18a-4e20-bc66-54c010e2cc1f +md""" +The following plot shows the sparsity pattern of the matrix $A$ that represents the discretization with a finite difference method as a linear system. +""" + +# ╔═╡ ac216c60-8288-465b-ab2b-2ba3e7701d02 +begin + markersize = if N == 4 + 15 + elseif N == 5 + 11 + elseif N == 6 + 7 + elseif N == 7 + 4 + elseif N == 8 + 3 + else + 2 + end + markersizes_small = Dict(5=>11, 6=>7, 7=>4, 8=>3) + + spy(A + UniformScaling(5), markersize=markersize, markershape=:square) +end + +# ╔═╡ 4376970c-ea81-415e-bdb8-c35d36751536 +md""" +Below, you can also see the numerical values of the matrix A: +""" + +# ╔═╡ 625a431f-9187-4c17-9411-3e9078499c38 +A + +# ╔═╡ 95fdd1b4-3e76-4dbd-976b-2243aea1e8b5 +md""" +Note that for simplicity we use a classical or *dense* matrix representation here, where each entry of the matrix is stored explicitly. In a real application, one would use a *sparse* matrix representation that only stores the non-zero elements explicitly. +""" + +# ╔═╡ 6b49c667-b16e-44f3-9f0d-2b8a62f5efda +md""" +## Auxiliary functions and internals +If you are only interested in *using* the interactive plotting features in this notebook, there is **no need to read any further**. The remainder of this document contains auxiliary functions. Furthermore, implementation details that are required for this workbook to execute correctly are included here. +""" + +# ╔═╡ b8f81a3f-d773-4e67-b8d7-e1734b04abaa +md""" +### Load required packages +This noteboook requires some additional packages to work: +* Plots.jl is used for plotting the results +* PlutoUI.jl is used to create the interactive GUI elements +* LaTeXStrings.jl is used to make some mathematical notations look nicer +* ShortCodes.jl allows to embed references to papers etc. more easily +* `LinearAlgebra` is a Julia standard library module for linear algebra operations +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +ShortCodes = "f62ebe17-55c5-4640-972f-b59c0dd11ccf" + +[compat] +LaTeXStrings = "~1.3.0" +Plots = 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+[[deps.libvorbis_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] +git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" +uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" +version = "1.3.7+1" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" +version = "1.48.0+0" + +[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +version = "17.4.0+0" + +[[deps.x264_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" +uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "2021.5.5+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "3.5.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.4.1+0" +""" + +# ╔═╡ Cell order: +# ╟─4de22c75-0d00-4838-8ba4-52bab4dab89e +# ╟─c5689d78-a0ae-432d-b7cd-a3de894d4443 +# ╟─f87f000f-e0c9-46eb-a242-89a4290bd84f +# ╠═f185aacd-0334-4217-8fa9-9fd0776c12c6 +# ╟─a7e57f8e-2032-419f-b957-12e90c3b9773 +# ╟─eb184f17-09c4-426e-80bb-f7d5867a01a4 +# ╟─25a2120b-e9cd-48bc-901e-5611a2330f99 +# ╟─138b0481-7d42-4197-8ca2-b3fb073366a0 +# ╠═8e2710f6-9e2c-47a9-8a1e-33c6df4386cf +# ╟─52e5b8cc-d18a-4e20-bc66-54c010e2cc1f +# ╟─ac216c60-8288-465b-ab2b-2ba3e7701d02 +# ╟─4376970c-ea81-415e-bdb8-c35d36751536 +# ╟─625a431f-9187-4c17-9411-3e9078499c38 +# ╟─95fdd1b4-3e76-4dbd-976b-2243aea1e8b5 +# ╟─6b49c667-b16e-44f3-9f0d-2b8a62f5efda +# ╟─b8f81a3f-d773-4e67-b8d7-e1734b04abaa +# ╠═a22d543c-6a50-11ed-002d-7f2fe5bd7276 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/finite_difference_method.plutostate b/finite_difference_method.plutostate new file mode 100644 index 0000000..e717adc Binary files /dev/null and b/finite_difference_method.plutostate differ diff --git a/html/finite_difference_method.jl.html b/html/finite_difference_method.jl.html new file mode 100644 index 0000000..73c3d45 --- /dev/null +++ b/html/finite_difference_method.jl.html @@ -0,0 +1,15 @@ + + + + + +
\ No newline at end of file diff --git a/index.html b/index.html new file mode 100644 index 0000000..6dab3d2 --- /dev/null +++ b/index.html @@ -0,0 +1,23 @@ + + + + + + + + + + + + +

Notebooks

+ + + + diff --git a/play_sounds.html b/play_sounds.html new file mode 100644 index 0000000..55094f7 --- /dev/null +++ b/play_sounds.html @@ -0,0 +1,20 @@ + + + + + + + + + + +
\ No newline at end of file diff --git a/play_sounds.jl b/play_sounds.jl new file mode 100644 index 0000000..f1b7fa0 --- /dev/null +++ b/play_sounds.jl @@ -0,0 +1,1255 @@ +### A Pluto.jl notebook ### +# v0.19.40 + +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 + +# ╔═╡ a22d543c-6a50-11ed-002d-7f2fe5bd7276 +using Plots, LaTeXStrings, WAV, Measures, PlutoUI + +# ╔═╡ 4de22c75-0d00-4838-8ba4-52bab4dab89e +md""" +# Discrete Fourier Transform - Introduction + +In this notebook, we explore the superposition of acoustic (sinusoidal) signals to create sound. + +A sinusoidal acoustic wave with amplitude ``A_1`` and frequency ``f_1`` is given by +```math +y_1(t) = A_1 \sin(2 \pi f_1 t). +``` +""" + +# ╔═╡ 5319474d-bfbd-4bef-a4cb-2b827214f998 +md""" +Let's start by defining the amplitude and frequency of a musical note: + +Amplitude ``A_1``: $(@bind A1 Slider(0:0.05:1, default=1, show_value=true)) + +Frequency ``f_1``: $(@bind f1 Slider(0:10:1000, default=440, show_value=true)) [Hz] + +Sampling Frequency: $(@bind fs Slider(0:1:1e4, default=1e4, show_value=true)) [Hz] + +(the musical note "A" has a frequency of 440 Hz) +""" + +# ╔═╡ c72389b8-9db4-493c-847f-144185ebdded +md""" +Let's visualize the note 1 +""" + +# ╔═╡ ac216c60-8288-465b-ab2b-2ba3e7701d02 +begin + t = 0.0:1/fs:prevfloat(1.0) # Time sampling for one second with frequency fs + y1 = A1 * sin.(2pi * f1 * t) + # Plot canvas 2 and set some default options + y_min = -2.1 + y_max = 2.1 + tf = 0.08 + p1 = plot( + xlims=(0, tf), + ylims=(y_min, y_max), + legend=:bottomright, + legendfontsize=10, + tickfontsize=10, + gridlinewidth=1.5, + title="Musical Note 1", + xlabel=L"x", ylabel="Pressure", + size=(1200, 450), + margin = 5mm + ) + plot!(p1, t, y1, label=L"y_{1}",) + +end + +# ╔═╡ 61b8fecf-ef4f-4d7d-be6a-fe6985b264c4 +begin + y1 + md""" + And we can also listen to this signal! + + $(@bind play1 CounterButton("Play Note 1")) + """ +end + +# ╔═╡ 865db857-1a01-47b6-9a07-245a88806170 +begin + if play1 > 0 + wavplay(y1, fs) + else + nothing + end +end + +# ╔═╡ fbcf9b4c-2dfc-4dc1-9a61-d3638c3782a9 +md""" +Let's now define another musical note: + +```math +y_2(t) = A_2 \sin(2 \pi f_2 t). +``` +""" + +# ╔═╡ 96b56aa9-0295-4f7e-8507-5e4e5b389751 +md""" +Amplitude ``A_2``: $(@bind A2 Slider(0:0.05:1, default=1, show_value=true)) + +Frequency ``f_2``: $(@bind f2 Slider(0:10:1000, default=260, show_value=true)) [Hz] + +(the musical note "C" has a frequency of approximately 260 Hz) +""" + +# ╔═╡ 3e7ed6c8-d064-4043-9b67-b4d92d82eb5a +begin + y2 = A2 * sin.(2pi * f2 * t) + # Plot canvas 2 and set some default options + p2 = plot( + xlims=(0, tf), + ylims=(y_min, y_max), + legend=:bottomright, + legendfontsize=10, + tickfontsize=10, + gridlinewidth=1.5, + title="Musical Note 2", + size=(1200, 450), + xlabel=L"x", + ylabel="Pressure", + margin = 5mm + ) + plot!(p2, t, y2, label=L"y_2") +end + +# ╔═╡ c03df22b-e757-4004-9788-95ec286bd95c +begin + y2 + md""" + Let's listen to the note 2! + + $(@bind play2 CounterButton("Play Note 2")) + """ +end + +# ╔═╡ 7b4f1aa0-b3df-4aa8-82ff-ee7148452e58 +begin + if play2 > 0 + wavplay(y2, fs) + else + nothing + end +end + +# ╔═╡ 490a6892-1e52-4aa4-948a-21288d7e34bb +md""" +What happens when we sum up the two signals? + +```math +y = y_{1} + y_{2} +``` +""" + +# ╔═╡ 33552e73-68dd-4a04-b33f-0c3dc6adf308 +begin + y = y1 + y2 + # Plot canvas 3 and set some default options + p3 = plot( + xlims=(0, tf), + ylims=(y_min, y_max), + legend=:bottomright, + legendfontsize=10, + tickfontsize=10, + gridlinewidth=1.5, + title="Musical Note 3", + size=(1200, 450), + xlabel=L"x", + ylabel="Pressure", + margin = 5mm + ) + plot!(p3, t, y, label=L"y = y_{1} + y_{2}") +end + +# ╔═╡ 26ea1c0f-c757-41fd-a894-7ccf9c22676f +begin + y + md""" + $(@bind playsum CounterButton("Play Sum")) + """ +end + +# ╔═╡ c7fde4ce-0ca0-4654-aaad-152ab0faa87f +begin + if playsum > 0 + wavplay(y, fs) + else + nothing + end +end + +# ╔═╡ 6b49c667-b16e-44f3-9f0d-2b8a62f5efda +md""" +## Auxiliary functions and internals +If you are only interested in *using* the interactive plotting features in this notebook, there is **no need to read any further**. The remainder of this document contains auxiliary functions for computing the Taylor polynomial. Furthermore, implementation details that are required for this workbook to execute correctly are included here. +""" + +# ╔═╡ b8f81a3f-d773-4e67-b8d7-e1734b04abaa +md""" +### Load required packages +This noteboook requires some additional packages to work: +* Plots.jl is used for plotting the results +* LaTeXStrings.jl is used to make some mathematical notations look nicer +* ShortCodes.jl allows to embed references to papers etc. more easily +* WAV.jl is used to play the sounds +* PlutoUI.jl is used to create the interactive GUI elements +""" + +# ╔═╡ bf205421-a919-4090-acb1-9b2f2555b056 +md""" +### Notebook style adjustments +Make notebook wider to show plots side by side: +""" + +# ╔═╡ 282da2f2-c0da-4310-9bc4-54b5addc6b11 +html""" + +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" +Measures = "442fdcdd-2543-5da2-b0f3-8c86c306513e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" 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+# ╠═282da2f2-c0da-4310-9bc4-54b5addc6b11 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 diff --git a/play_sounds.plutostate b/play_sounds.plutostate new file mode 100644 index 0000000..78b4943 Binary files /dev/null and b/play_sounds.plutostate differ diff --git a/pluto_export.json b/pluto_export.json new file mode 100644 index 0000000..63ee712 --- /dev/null +++ b/pluto_export.json @@ -0,0 +1 @@ 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\ No newline at end of file