FFTs_and_probability.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "FFTs_and_probability.ipynb",
      "version": "0.3.2",
      "provenance": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/benjym/CIVL5999/blob/master/FFTs_and_probability.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "metadata": {
        "id": "W8LfQxmd-VGP",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Run this cell first\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "TZFAmmzrL8J8",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "# CIVL5999: FFTs and probability\n",
        "\n",
        "## 3.1 Fourier transform\n",
        "[First, read through this](https://pudding.cool/2018/02/waveforms/).\n",
        "\n",
        "[And then watch this](https://www.youtube.com/watch?v=spUNpyF58BY).\n",
        "\n",
        "If you are still not sure what the Fourier transform is, [then read this](https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/).\n",
        "\n",
        "In short, what the Fourier transform does is as simple as this analogy taken from the link above:\n",
        "- **What does the Fourier Transform do?** Given a smoothie, it finds the recipe.\n",
        "- **How?** Run the smoothie through filters to extract each ingredient.\n",
        "- **Why?** Recipes are easier to analyze, compare, and modify than the smoothie itself.\n",
        "- **How do we get the smoothie back?** Blend the ingredients.\n",
        "\n",
        "Smoothie = signal made out of many ingredients\n",
        "\n",
        "Ingredient = single sine wave with its own amplitude, frequency and phase\n",
        "\n",
        "The more sure we are what the ingredients are, the longer the signal lasts.  \n",
        "\n",
        "## Question 3.1.1\n",
        "Let's mix our own smoothie using Python! We can get different types of very interesting functions (patterns) just by adding sine waves of different amplitudes $a_1$, $a_2$, ..., frequencies $f_1$, $f_2$, ..., and phases $\\phi_1$, $\\phi_2$, ...\n",
        "\n",
        "Plot the function $y$ that will add 5 different sine waves:\n",
        "\n",
        "$$\n",
        "y = a_1\\sin(f_1x+\\phi_1) + a_2\\sin(f_2x+\\phi_2) + a_3\\sin(f_3x+\\phi_3) + a_4\\sin(f_4x+\\phi_4) + a_5\\sin(f_5x+\\phi_5)\n",
        "$$\n",
        "\n",
        "from $x=0$ to $x=10$. Play around with the values and make something interesting. Maybe a triangular wave? What is the secret recipe for triangular wave smoothie? Save it as `FFT_doodle.png`. To get started, use the following code:"
      ]
    },
    {
      "metadata": {
        "id": "rZ-6TaeL-RTw",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 364
        },
        "outputId": "775bd96f-f14a-4e03-ba02-2cb2e59f82b7"
      },
      "cell_type": "code",
      "source": [
        "n = 1001 # number of points, chosen arbitrarily\n",
        "x = np.linspace(0,10,n) # vector storing the x values\n",
        "a = [1,1,1,1,1] # amplitudes\n",
        "f = [1,2,3,4,5] # frequencies\n",
        "phi = [0,0,0,0,0] # phases\n",
        "y = np.zeros([n]) # empty array to store y in \n",
        "\n",
        "for i in range(5): # 5 times\n",
        "    y += a[i]*np.sin(f[i]*x + phi[i]) # add up the parts of y\n",
        "\n",
        "plt.plot(x,y) # make a graph"
      ],
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7fdb409bff60>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 7
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAd8AAAFKCAYAAABcq1WoAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n/wMcKXXXQVjrywAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 576x396 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "4Yu-Uuh-MtFQ",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 3.1.2\n",
        "Plot the fourier series you used to make `FFT_doodle.png` by plotting the $a$’s against the $f$’s. Save this as `FFT_doodle_frequency_domain.png`"
      ]
    },
    {
      "metadata": {
        "id": "7JcB-R3IMlES",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "CS0AQB3sMtMP",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 3.1.3\n",
        "Numpy has a built in functions `fft.fft` and `fft.fftfreq` for doing fourier transforms. Run the following code to see how they work."
      ]
    },
    {
      "metadata": {
        "id": "x5At7aPOMdzM",
        "colab_type": "code",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 398
        },
        "outputId": "7caf48ae-54e9-499b-fe86-ad19ea2bc9ef"
      },
      "cell_type": "code",
      "source": [
        "n = 1001 # number of time steps, arbitrary\n",
        "t = np.linspace(0,10,n) # vector of time\n",
        "dt = t[1] - t[0] # get the time interval between each time step \n",
        "x = np.sin(2*np.pi*10*t) # signal to process, here just a sine wav\n",
        "\n",
        "F = np.fft.fft(x) # fourier transform of x\n",
        "F_real = np.real(F) # just look at the real (not imaginary) part of the FFT\n",
        "\n",
        "freq = np.fft.fftfreq(n,dt) # get the frequencies that correspond to the FFT values\n",
        "\n",
        "F_real_positive = F[:n//2] # just get the positive part of F\n",
        "freq_positive = freq[:n//2] # just get the positive part of freq\n",
        "\n",
        "plt.plot(freq_positive, F_real_positive) # plot the real part"
      ],
      "execution_count": 16,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.6/dist-packages/numpy/core/numeric.py:492: ComplexWarning: Casting complex values to real discards the imaginary part\n",
            "  return array(a, dtype, copy=False, order=order)\n"
          ],
          "name": "stderr"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7fdb40728dd8>]"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 16
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAeQAAAFKCAYAAADMuCxnAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAEgBBDIAACmAQAYAIAUQyAAApID/DzxOFkMQJcDXAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 576x396 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "_H2JsX6MMtOy",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 3.1.4\n",
        "Calculate the FFT of simple sine wave whose length is 5 periods, and then do it once again for the same wave but make the length of the wave to be 500 periods. Plot and compare the results. What does it tell you?\n"
      ]
    },
    {
      "metadata": {
        "id": "6l46PgT7MN91",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "M9UjBia_MtUS",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 3.1.5\n",
        "[Now give this a go](https://musiclab.chromeexperiments.com/Spectrogram/)."
      ]
    },
    {
      "metadata": {
        "id": "uRQiPgWfai1E",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        ""
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "Ll0Lo4hnMx0A",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "_1QTkmJpMtXJ",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 3.1.6\n",
        "That website showed you a spectrogram (try [scipy.signal.spectrogram](https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.spectrogram.html). Save a spectrogram of the data in `Dataset_01/Signal_C_01` as `test_spectrogram.png`.\n",
        "\n",
        "Generally speaking, what is a spectrogram and what does it show? For the particular case of `test_spectrogram.png`, what is this spectrogram telling you?"
      ]
    },
    {
      "metadata": {
        "id": "3Y84fyIb9qoZ",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "from scipy.signal import spectrogram\n",
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "yvJmvFNg9yLQ",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "# 3.2 Probability\n",
        "\n",
        "## Question 3.2.1\n",
        "Generate random numbers by using two probability functions. First, generate 200 uniformly distributed numbers between 1-100 and assign those values to the variable `uniform`. Second, generate 200 normally distributed numbers with mean $\\mu=50$ and standard deviation $\\sigma=30$ , and store those values in the variable `normal`. Plot both arrays as scatter plots on two different charts. Let the values of the $x$ axis be randomly generated numbers and to each one of those numbers assign the $y$ value to be 1. You should plot something like this:\n",
        "\n",
        "# BM UP TO HERE"
      ]
    },
    {
      "metadata": {
        "colab_type": "code",
        "id": "dtBU_VMIAEPo",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "colab_type": "text",
        "id": "0s29RTVIAEoK"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 2.2.4\n",
        "Do the same for $y=sin(\\pi x) + x$ from $x=0$ to $x=6$ What does this tell you?"
      ]
    },
    {
      "metadata": {
        "colab_type": "code",
        "id": "BXuX_qXEAEqn",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "colab_type": "text",
        "id": "zETfBISTAEl2"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 2.2.5\n",
        "Do this again for `Dataset_01/Signal_C_01` and plot the autocorrelation. What does this tell you?"
      ]
    },
    {
      "metadata": {
        "colab_type": "code",
        "id": "EKugA-NpAEjH",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "colab_type": "text",
        "id": "lJvUOyaBAEjK"
      },
      "cell_type": "markdown",
      "source": [
        "# 2.3 Cross-correlation\n",
        "## Question 2.3.1\n",
        "Write a script to calculate the cross-correlation coefficient of two lists of values for all values of  $\\Delta t$. Recall the definition of the cross-correlation coefficient, $C$ for two functions   and $X(t)$ and $Y(t+\\Delta t)$ with means $\\mu_x$ and $\\mu_y$ and standard deviations $\\sigma_x$ and $\\sigma_y$ respectively:\n",
        "\n",
        "$$\n",
        "C(\\Delta t) = \\frac{()E[X(t) - \\mu_x(t))(Y(t+\\Delta t)-\\mu_y(t+\\Delta t))]}{\\sigma_x(t)\\sigma_y(t+\\Delta t)}\n",
        "$$"
      ]
    },
    {
      "metadata": {
        "colab_type": "code",
        "id": "miCm9pnbAEoH",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "colab_type": "text",
        "id": "kWhFQoq_AEgb"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 2.3.2\n",
        "Calculate the cross-correlation of `Dataset_01/Signal_C_01` and `Dataset_01/Signal_C_02`. What does this tell you?"
      ]
    },
    {
      "metadata": {
        "colab_type": "code",
        "id": "0R32O2lKAEgX",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "colab_type": "text",
        "id": "4WiJjGE8AEqq"
      },
      "cell_type": "markdown",
      "source": [
        "## Question 2.3.3\n",
        "Have a look at the `np.correlate` function. Compare it against your calculations in Questions 2.25 and 2.3.2. How did you go?"
      ]
    },
    {
      "metadata": {
        "colab_type": "code",
        "id": "V5GsfqgnAElx",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "# Type your code here"
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}