fourier_transform_1_fft.ipynb

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  "metadata": {
    "colab": {
      "name": "fourier_transform_1_fft",
      "provenance": [],
      "authorship_tag": "ABX9TyPas7S5FEqxNavb4j+/Wvd3",
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/licciard/lu-phys3466/blob/main/fourier_transform_1_fft.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pZnLO279wi0y"
      },
      "source": [
        "## **The Fast Fourier Transform Algorithm**\n",
        "\n",
        "This notebook is dedicated to explain the optimal implementation of Fourier transform in numerical applications. The bottom line is that by exploiting a symmetry in the calculation of the transform, we can substantially reduce the number of calculations performed."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "RmiEnPjAxVyW"
      },
      "source": [
        "The **Discrete Fourier Transform** (DFT) of an array with $N$ data points can be calculated by: \n",
        "$$X[\\omega] = \\sum_{t=0}^{N-1} x[t] e^{-i \\frac{2\\pi t \\omega}{N}}$$\n",
        "using the same notations of our lectures, but with a reversed sign of $i$ for the direct FT because this is the standard in computer implementations."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "3q9Z-iCA9yEH"
      },
      "source": [
        "Let's first create a simple **signal**."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "D_EaS30s9-RC"
      },
      "source": [
        "# modules for the notebook\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "import scipy "
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wncLICDv-qhi"
      },
      "source": [
        "Here we start with a mono-frequency signal of 8 Hz, and sample it at 128 Hz. This means that while the signal oscillates with 10 cycles per second, we grab 128 data points per second. That gives a total of 16 samples per cycle. The choice of sampling is very important: too few samples won't give you a good idea of the signal. \n",
        "\n",
        "\n",
        "**Activity 1.** Play with the numbers of frequency and sampling rate to check this. For example, change the sampling rate to very low values to see that the signal seems to disappear. Then bring it back to 128 before proceeding to the next activity."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "yW6Qw5rc97jD"
      },
      "source": [
        "f = 8  # Frequency, in cycles per second\n",
        "f_s = 128  # Sampling rate = number of measurements per second\n",
        "\n",
        "t = np.linspace(0, 2, 2 * f_s, endpoint=False)\n",
        "print(\"We will need an array with an exact power of 2 entries in the following exercise. Our array has {} entries.\".format(len(t)))\n",
        "x = np.sin(f * 2 * np.pi * t)\n",
        "\n",
        "plt.figure(figsize=(10,5))\n",
        "fig, ax = plt.subplots()\n",
        "ax.plot(t, x, '-') # <-- change the option from '-' to '.' to see the actual points used in the plot\n",
        "ax.set_xlabel('Time [s]')\n",
        "ax.set_ylabel('Signal amplitude')\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-ko4VUBvN9bq"
      },
      "source": [
        "**Activity 2.** Complete the argument of the exponential function in our \"naive\" implementation of the DFT. Use the expression exactly like the formula above, including the notation (here we call $\\omega$ by $w$). Then run the next code block to print whether the implementation matches the result against a DFT implementation from python's module *scipy*."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "bMpMgUWTG37V"
      },
      "source": [
        "def our_DFT(xloc):\n",
        "    xloc = np.asarray(xloc, dtype=float)\n",
        "    N = xloc.shape[0]\n",
        "    w = np.arange(N)\n",
        "    t = w.reshape((N, 1))\n",
        "    M = np.exp(-2j * ???) # <-- complete this line\n",
        "    return np.dot(M, xloc)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "c9MGQeZWQW87",
        "outputId": "66f3e042-5912-432b-f589-57f2686c96de"
      },
      "source": [
        "our_X = our_DFT(x)\n",
        "X = scipy.fft.fft(x)\n",
        "correct = np.allclose(our_X,X)\n",
        "print(\"Does our DFT implementation match scipy's? {}\".format(correct))"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "Does our DFT implementation match scipy's? True\n"
          ],
          "name": "stdout"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uwDeBxtyQ4A-"
      },
      "source": [
        "**Activity 3.** Run the code below to compute the running times for our DFT and scipy's implementation. Take note of the times. (Disregard the number of loops.)"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "LO9f3ysLRN31"
      },
      "source": [
        "%timeit scipy.fft.fft(x)\n",
        "%timeit our_DFT(x)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "fX9AblbHTdFO"
      },
      "source": [
        "Introducing the **Fast Fourier Transform (FFT)** algorithm. Below is a very nice explanation taken from:\n",
        "*   https://jakevdp.github.io/blog/2013/08/28/understanding-the-fft/\n",
        "*   https://towardsdatascience.com/fast-fourier-transform-937926e591cb "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2C1sASpRR2Vf"
      },
      "source": [
        "Cooley and Tukey showed that it's possible to divide the DFT computation into two smaller parts.  From\n",
        "the definition of the DFT we have:\n",
        "$$\\begin{align}\n",
        "X_k &= \\sum_{n=0}^{N-1} x_n \\cdot e^{-i~2\\pi~k~n~/~N} \\\\\n",
        "    &= \\sum_{m=0}^{N/2 - 1} x_{2m} \\cdot e^{-i~2\\pi~k~(2m)~/~N} + \\sum_{m=0}^{N/2 - 1} x_{2m + 1} \\cdot e^{-i~2\\pi~k~(2m + 1)~/~N} \\\\\n",
        "    &= \\sum_{m=0}^{N/2 - 1} x_{2m} \\cdot e^{-i~2\\pi~k~m~/~(N/2)} + e^{-i~2\\pi~k~/~N} \\sum_{m=0}^{N/2 - 1} x_{2m + 1} \\cdot e^{-i~2\\pi~k~m~/~(N/2)}\n",
        "\\end{align}$$\n",
        "[*Note they use the variable names $\\omega \\leftrightarrow k$ and $t \\leftrightarrow n$.*]\n",
        "\n",
        "We've split the single Discrete Fourier transform into two terms which themselves look very similar to smaller Discrete Fourier Transforms, one on the odd-numbered values, and one on the even-numbered values. So far, however, we haven't saved any computational cycles.  Each term consists of $(N/2)*N$ computations, for a total of $N^2$.\n",
        "\n",
        "The trick comes in making use of symmetries in each of these terms.  Because the range of $k$ is $0 \\le k < N$, while the range of $n$ is $0 \\le n < M \\equiv N/2$, we see from the symmetry properties above that we need only perform half the computations for each sub-problem.  Our $\\mathcal{O}[N^2]$ computation has become $\\mathcal{O}[M^2]$, with $M$ half the size of $N$.\n",
        "\n",
        "[*Recall the big-O notation \"$\\mathcal{O}$\" from the lectures? That means a function that \"multiplies\" the factor in the argument, i.e. that is proportional to it.*]\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "CZ5DmBdyVbCj"
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      "source": [
        "![fft_1.png](data:image/png;base64,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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "pC4BLKx4TlmK"
      },
      "source": [
        "**Activity 4.** Compute the time required to run each of the three FT algorithms: our DFT, our FFT (below) and scipy's. Is our FFT time shorter than our DFT? By what factor? Is it consistent with the explanation above: $\\mathcal{O}[N^2] \\rightarrow \\mathcal{O}[N^2/2]$?"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "pKom9qRLzaJ9"
      },
      "source": [
        "def our_FFT(xloc):\n",
        "    xloc = np.asarray(xloc, dtype=float)\n",
        "    N = xloc.shape[0]\n",
        "    if N % 2 > 0:\n",
        "        raise ValueError(\"must be a power of 2\")\n",
        "    elif N <= 2:\n",
        "        return our_DFT(xloc)\n",
        "    else:\n",
        "        X_even = our_FFT(xloc[::2])\n",
        "        X_odd = our_FFT(xloc[1::2])\n",
        "        terms = np.exp(-2j * np.pi * np.arange(N) / N)\n",
        "        return np.concatenate([X_even + terms[:int(N/2)] * X_odd,\n",
        "                               X_even + terms[int(N/2):] * X_odd])"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "ZzcYmpxzzeGu"
      },
      "source": [
        "%timeit scipy.fft.fft(x)\n",
        "%timeit our_DFT(x)\n",
        "%timeit our_FFT(x)"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "onv6ijv1TnCr"
      },
      "source": [
        "There's no reason to stop there: as long as our smaller Fourier transforms have an even-valued $M$, we can reapply this divide-and-conquer approach, halving the computational cost each time, until our arrays are small enough that the strategy is no longer beneficial.  In the asymptotic limit, this recursive approach scales as $\\mathcal{O}[N\\log N]$."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jxgeFSklWutu"
      },
      "source": [
        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)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "uFVaeXkTWubR"
      },
      "source": [
        "So long as $N$ is a power of 2, the maximum number of times you can split into two equal halves is given by $p = \\ln(N)$.\n",
        "\n",
        "This recursive algorithm can be implemented very quickly in Python, falling-back on our slow DFT code when the size of the sub-problem becomes suitably small."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8PC7J7QwW0Mx"
      },
      "source": [
        "**Activity 5.** Run the code below to plot the DFT of your signal. Do you understand what is happening? Why do we have two peaks at $\\pm 8$ Hz? Vary the definition of your signal for different frequencies and then different samples to see if they make sense for you."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "eWo7M5J6Ay1k"
      },
      "source": [
        "freqs = scipy.fft.fftfreq(len(x)) * f_s\n",
        "fig, ax = plt.subplots()\n",
        "#ax.stem(freqs, np.abs(our_X), use_line_collection=True)\n",
        "ax.stem(freqs, np.abs(scipy.fft.fft(x)), use_line_collection=True)\n",
        "ax.set_xlabel('Frequency in Hertz [Hz]')\n",
        "ax.set_ylabel('Frequency Domain (Spectrum) Magnitude')\n",
        "ax.set_xlim(-f_s / 5, f_s / 5)\n",
        "ax.set_ylim(-5, 150)"
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}