Add julia notebook

diff --git a/DigitalSignalProcessing.jl b/DigitalSignalProcessing.jl
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+### A Pluto.jl notebook ###
+# v0.11.14
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ c8101fc0-e4a8-11ea-2924-f78341e62d7a
+begin
+       using Plots, LinearAlgebra
+       default(label = false, minorgrid = true)
+end
+
+# ╔═╡ b33ad050-e489-11ea-3b17-e598a406a091
+md"# Week 2 - Vector space"
+
+# ╔═╡ e0f11590-e489-11ea-0147-0928a6e40330
+x = [1, 2, 4]
+
+# ╔═╡ 35be38a0-e48a-11ea-3616-3f314782c3ad
+let
+       α = 10
+       α * x # Multiply by a scala.
+end
+
+# ╔═╡ 3db39320-e48a-11ea-02a9-6dc184ea3fdf
+let
+       y = [3, 4, 5]
+       x + y
+end
+
+# ╔═╡ 502b22a0-e48c-11ea-3a62-c7de5e345f02
+md"Addition in $L2[-1, 1]$"
+
+# ╔═╡ 79c736a0-e48a-11ea-2886-cf90c9c874db
+let    
+       t = -1:1/200:1  
+       f1(t) = sin(π * t)
+       f2(t) = 0.3 * sin(25 * π * t)
+       f3(t) = f1(t) + f2(t)
+       plot(t, f3, label = false)
+end
+
+# ╔═╡ 5c99778e-e48b-11ea-23cf-7b1537bf57d9
+md"## Approximation of a square function"
+
+# ╔═╡ 294b51f0-e55e-11ea-068d-2f6663453bbc
+md"## Inner product"
+
+# ╔═╡ f0112030-e48c-11ea-3d07-d383bd861e81
+let
+       v1 = [1, 2, 3]
+       v2 = [2, 3, 4]
+       
+       # Same two expressions for inner product.
+       dot(v1, v2),
+       transpose(v1) * v2
+end
+
+# ╔═╡ 5bd37650-e579-11ea-354b-65f651b7fc69
+let
+       A = [ 0 0 0 1
+                 1 0 0 0
+                 0 1 0 0
+                 0 0 1 0 ]
+
+       A^4
+end
+
+# ╔═╡ 5cde4c70-eaa4-11ea-214c-633b65f18a18
+md"# Week 3 & 4 - Fourier analysis"
+
+# ╔═╡ 78adff50-ed5a-11ea-3c3a-cf4de58dce71
+md"""
+### Analysis
+
+$X[k] = \sum_{n=0}^{N-1} x[n] e^{-j \frac{2 \pi}{N} \cdot n \cdot k}, k \in \mathbb{Z}$
+
+Produces an **N-periodic** signal in the time domain
+
+### Synthesis
+
+$x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j \frac{2 \pi}{N} \cdot n \cdot k}, n \in \mathbb{Z}$
+
+Produces an **N-point** signal in the frequency domain
+"""
+
+# ╔═╡ 0f4abce0-ed34-11ea-2245-0590b879023f
+begin
+       analysis(x, k, N) = sum(x(n) * exp(-im * 2π / N * n * k) for n ∈ 0:N-1)
+       synthesis(X, n, N) = 1 / N * sum(X(k) * exp(im * 2π / N * n * k) for k ∈ 0:N-1)
+       
+       # Input signal 'f'.
+       f(n) = sin(n * 2π / 100) + 0.2 * sin(10 * n * 2π / 100 + π / 4)
+       
+       # Analysis of 'f'.
+       N = 100
+       points_in_freq_domain = [analysis(f, k, N) for k ∈ 0:N-1]     
+       F(n) = points_in_freq_domain[n % length(points_in_freq_domain) + 1]
+       
+       # Synthesis of 'F'.
+       points_in_time_domain = [synthesis(F, n, N) for n ∈ 0:N-1] .|> real
+end
+
+# ╔═╡ c48cd7e0-e515-11ea-02ba-5999ac4f4c9a
+let
+       t = -1:1/200:1  
+       f(t, k) = sin((2k + 1) * π * t) / (2k + 1)
+       g(N) = t -> (f(t, k) for k ∈ 0:N) |> sum
+       plot(t, g(20), label = false)
+end
+
+# ╔═╡ 10a85110-ed34-11ea-36ff-175bceac4be1
+scatter(0:N-1, f, label = false, title = "Input: f(n)")
+
+# ╔═╡ 1609ba40-ed34-11ea-29d2-41e5742df891
+scatter(0:N-1, points_in_freq_domain |> imag, label = false, title = "Frequency domain: F(n): Imaginary")
+
+# ╔═╡ 1915e8d0-ed34-11ea-37b7-2f4938bcf27e
+scatter(0:N-1, points_in_freq_domain |> real, label = false, title = "Frequency domain: F(n): Real")
+
+# ╔═╡ 20bb8390-ed5e-11ea-2415-cbc53e60cc0a
+scatter(points_in_freq_domain, label = false, title = "Frequency domain: F(n)")
+
+# ╔═╡ 0f318a60-efb7-11ea-0416-3ded1c40b27a
+scatter(0:N-1, points_in_freq_domain .|> norm, label = false, title = "Frequency domain: F(n): Magnitude")
+
+# ╔═╡ 1badaab0-ed34-11ea-2709-df7b90e4e6a6
+scatter(0:N-1, points_in_time_domain |> real, label = false, title = "Back to time domain")
+
+# ╔═╡ e0ae4d60-f054-11ea-249c-41c05d1b7cf7
+md"## Example of a linear signal"
+
+# ╔═╡ 1e4174f0-ed34-11ea-0c3a-69ad2f7d6561
+let
+       N = 32
+       x = [ 2n / (N - 1) - 1 for n ∈ 0:N-1]
+       f(n) = x[(n % N) + 1]
+       points_in_freq_domain = [analysis(f, k, N) for k ∈ 0:N-1]
+       
+       plot_signal = scatter(x, label = false, title = "Time domain", )
+       plot_fft = scatter(0:N-1, points_in_freq_domain .|> norm, label = false, title = "Frequency domain: F(n): Magnitude")
+       plot(plot_signal, plot_fft, layout = (2, 1))
+end
+
+# ╔═╡ 4d5cabe2-f06a-11ea-349f-6f940c85692e
+md"## Example of exponation decaying sequence"
+
+# ╔═╡ 3dac19b0-f06a-11ea-34ee-291908925549
+let
+       α = 0.9
+       f(n) = if n < 0 0 else α^n end
+       scatter(-5:40, f)
+end
+
+# ╔═╡ Cell order:
+# ╟─b33ad050-e489-11ea-3b17-e598a406a091
+# ╠═c8101fc0-e4a8-11ea-2924-f78341e62d7a
+# ╠═e0f11590-e489-11ea-0147-0928a6e40330
+# ╠═35be38a0-e48a-11ea-3616-3f314782c3ad
+# ╠═3db39320-e48a-11ea-02a9-6dc184ea3fdf
+# ╟─502b22a0-e48c-11ea-3a62-c7de5e345f02
+# ╠═79c736a0-e48a-11ea-2886-cf90c9c874db
+# ╟─5c99778e-e48b-11ea-23cf-7b1537bf57d9
+# ╠═c48cd7e0-e515-11ea-02ba-5999ac4f4c9a
+# ╟─294b51f0-e55e-11ea-068d-2f6663453bbc
+# ╠═f0112030-e48c-11ea-3d07-d383bd861e81
+# ╠═5bd37650-e579-11ea-354b-65f651b7fc69
+# ╟─5cde4c70-eaa4-11ea-214c-633b65f18a18
+# ╟─78adff50-ed5a-11ea-3c3a-cf4de58dce71
+# ╠═0f4abce0-ed34-11ea-2245-0590b879023f
+# ╠═10a85110-ed34-11ea-36ff-175bceac4be1
+# ╠═1609ba40-ed34-11ea-29d2-41e5742df891
+# ╠═1915e8d0-ed34-11ea-37b7-2f4938bcf27e
+# ╠═20bb8390-ed5e-11ea-2415-cbc53e60cc0a
+# ╟─0f318a60-efb7-11ea-0416-3ded1c40b27a
+# ╠═1badaab0-ed34-11ea-2709-df7b90e4e6a6
+# ╟─e0ae4d60-f054-11ea-249c-41c05d1b7cf7
+# ╠═1e4174f0-ed34-11ea-0c3a-69ad2f7d6561
+# ╟─4d5cabe2-f06a-11ea-349f-6f940c85692e
+# ╠═3dac19b0-f06a-11ea-34ee-291908925549