m-thesis-introduction/01_fourier.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fourier Transforms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "import numpy as np\n",
    "import scipy.fft as ft\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.gridspec as gridspec\n",
    "import matplotlib.ticker as tck\n",
    "rng = np.random.default_rng()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Fourier transform (FT) of a quantity $t$ is defined as\n",
    "\n",
    "$$\n",
    "\\hat{f}(\\omega) = F(f(t)) = \\frac{1}{\\sqrt{2\\pi}} \\int_{-\\infty}^{\\infty} \\mathrm{d}t\\, f(t)\\, \\exp{ \\left(-i \\omega t \\right)}\n",
    ".\n",
    "$$\n",
    "\n",
    "In signal processing, taking the integral from $t = -\\infty$ to $t = \\infty$ is not practical. Together with a finite sample space $t$, the integral can be turned into a (discrete) sum, giving a Discrete Fourier Transform (DFT)\n",
    "\n",
    "$$\n",
    "F(t) = \\frac{1}{\\sqrt{2\\pi}} \\sum_{k=0}^{N-1} f(t_k)\\, \\exp{ \\left(-i \\omega t_k \\right)}\n",
    "\\label{eq:DFT} \\tag{1}\n",
    ",\n",
    "$$\n",
    "where the sum iterates over each sample $t_k$ up to the end ($k=N-1$) of the signal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interdependence $\\Delta t$, $T$, $\\Delta f$, $f_{min}$, $f_{max}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The highest frequency $f_\\mathrm{max}$ (also known as the Nyquist frequency) that can be (reliably) measured within a signal is dependent only on the sampling rate ($1/\\Delta t$) of the signal with\n",
    "$$\n",
    "    f_\\mathrm{max} = \\frac{1}{2} f_\\mathrm{sampling} = \\frac{1}{2} \\frac{1}{\\Delta t}\n",
    "    .\n",
    "$$\n",
    "Higher frequencies will fall in between two adjacent samples. This means that these frequencies are then 'measured\\` with differing phases in such a way that they are 'folded' around the Nyquist frequency and appear as aliased frequencies below it.\n",
    "\n",
    "Likewise, the lowest frequency depends on the length of the signal $T$, such that over the whole signal one oscillation can be found\n",
    "$$\n",
    "    f_\\mathrm{min} \\sim \\frac{1}{T}\n",
    "    .\n",
    "$$\n",
    "Even lower frequencies will no longer be detected as a frequency. Instead, they will introduce a bias for the $0 Hz$ component in the spectrum."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# FTs of ideal signals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ft_spectrum( signal, sample_rate, fft=None, freq=None, mask_bias=False):\n",
    "    \"\"\"Return a FT of $signal$, with corresponding frequencies\"\"\"\n",
    "    n_samples = len(signal)\n",
    "    real_signal = np.isrealobj(signal)\n",
    "    \n",
    "    if fft is None:\n",
    "        if real_signal:\n",
    "            fft = ft.rfft\n",
    "            freq = ft.rfftfreq\n",
    "        else:\n",
    "            fft = ft.fft\n",
    "            freq = ft.fftfreq\n",
    "\n",
    "    if freq is None:\n",
    "        freq = ft.fftfreq\n",
    "    \n",
    "    spectrum = fft(signal) / sample_rate\n",
    "    freqs = freq(n_samples, 1/sample_rate)\n",
    "    \n",
    "    if not mask_bias:\n",
    "        return spectrum, freqs\n",
    "    else:\n",
    "        return spectrum[1:], freqs[1:]\n",
    "\n",
    "    \n",
    "def plot_spectrum( ax, spectrum, freqs, plot_complex=False, plot_power=False, plot_amplitude=None):\n",
    "    \"\"\" Plot a signal's spectrum on an Axis object\"\"\"\n",
    "    plot_amplitude = plot_amplitude or (not plot_power and not plot_complex)\n",
    "    alpha = 1\n",
    "    \n",
    "    ax.set_title(\"Spectrum\")\n",
    "    ax.set_xlabel(\"f (Hz)\")\n",
    "    ylabel = \"\"\n",
    "    if plot_amplitude or plot_complex:\n",
    "        ylabel = \"Amplitude\"\n",
    "    if plot_power:\n",
    "        if ylabel:\n",
    "            ylabel += \"|\"\n",
    "        ylabel += \"Power\"\n",
    "    ax.set_ylabel(ylabel)\n",
    "\n",
    "    if plot_complex:\n",
    "        alpha = 0.5\n",
    "        ax.plot(freqs, np.real(spectrum), '.-', label='Real', alpha=alpha)\n",
    "        ax.plot(freqs, np.imag(spectrum), '.-', label='Imag', alpha=alpha)\n",
    "\n",
    "    if plot_power:\n",
    "        ax.plot(freqs, np.abs(spectrum)**2, '.-', label='Power', alpha=alpha)\n",
    "    \n",
    "    if plot_amplitude:\n",
    "        ax.plot(freqs, np.abs(spectrum), '.-', label='Abs', alpha=alpha)\n",
    "\n",
    "    ax.legend()\n",
    "\n",
    "    return ax\n",
    "\n",
    "\n",
    "def plot_phase( ax, spectrum, freqs):\n",
    "    ax.set_ylabel(\"Phase\")\n",
    "    ax.set_xlabel(\"f (Hz)\")\n",
    "\n",
    "    ax.plot(freqs, np.angle(spectrum), '.-')\n",
    "    ax.set_ylim(-1*np.pi, np.pi)\n",
    "    \n",
    "    return ax\n",
    "\n",
    "\n",
    "def plot_combined_spectrum(spectrum, freqs, \n",
    "                           spectrum_kwargs={}, fig=None, gs=None):\n",
    "    \"\"\"Plot both the frequencies and phase in one figure.\"\"\"\n",
    "    \n",
    "    # configure plotting layout\n",
    "    if fig is None:\n",
    "        fig = plt.figure(figsize=(8, 16))\n",
    "\n",
    "    if gs is None:\n",
    "        gs = gridspec.GridSpec(2, 1, figure=fig, height_ratios=[2,1])\n",
    "\n",
    "    ax1 = fig.add_subplot(gs[:-1, -1])\n",
    "    ax2 = fig.add_subplot(gs[-1, -1], sharex=ax1)\n",
    "\n",
    "    axes = np.array([ax1, ax2])\n",
    "    \n",
    "    # plot the spectrum \n",
    "    plot_spectrum(ax1, spectrum, freqs, **spectrum_kwargs)\n",
    "    \n",
    "    # plot the phase\n",
    "    plot_phase(ax2, spectrum, freqs)\n",
    "    \n",
    "    return fig, axes\n",
    "    \n",
    "\n",
    "def plot_signal( ax, signal, sample_rate = 1):\n",
    "    ax.set_title(\"Signal\")\n",
    "    ax.set_xlabel(\"t (s)\")\n",
    "    ax.set_ylabel(\"A(t)\")\n",
    "\n",
    "    ax.plot(np.arange(len(signal)) / sample_rate, signal)\n",
    "    \n",
    "    return ax\n",
    "\n",
    "\n",
    "def plot_signal_and_spectrum( signal, sample_rate, title=None,\n",
    "                             ft_kwargs = {}, spectrum_kwargs = {}):\n",
    "    \"\"\"Plot a signal and its spectrum in one go.\"\"\"\n",
    "    fig = plt.figure(figsize=(16, 4))\n",
    "    \n",
    "    if title:\n",
    "        fig.suptitle(title)\n",
    "\n",
    "    # setup plot layout\n",
    "    gs0 = gridspec.GridSpec(1, 2, figure=fig)\n",
    "    gs00 = gs0[0].subgridspec(1, 1)\n",
    "    gs01 = gs0[1].subgridspec(2, 1)\n",
    "      \n",
    "    # plot the signal\n",
    "    ax1 = fig.add_subplot(gs00[0, 0])\n",
    "    plot_signal(ax1, signal)\n",
    "    \n",
    "    # plot spectrum\n",
    "    signal_fft, freqs = ft_spectrum(signal, sample_rate, **ft_kwargs)\n",
    "    _, (ax2, ax3) = plot_combined_spectrum(signal_fft, freqs, spectrum_kwargs = spectrum_kwargs, fig=fig, gs=gs01)\n",
    "\n",
    "    # return the axes\n",
    "    axes = np.array([ax1, ax2, ax3])\n",
    "    \n",
    "    return (fig, axes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "sample_rate = 1/1e-3 # s\n",
    "n_samples = int(1e4)\n",
    "t = np.arange(n_samples) / sample_rate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### FTs: constant signal\n",
    "\n",
    "A constant signal (also termed offset or bias) results in only a single component in the Fourier transform, at the zero frequency.\n",
    "\n",
    "Since often such a component is included in a signal (for example when a signal is not on average zero), it can help to mask out the zero frequency when plotting the spectrum.\n",
    "Doing this for a constant signal, we get a totally flat spectrum."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
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    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "signal_constant = np.ones(int(n_samples))\n",
    "\n",
    "fig, axes = plot_signal_and_spectrum(signal_constant, sample_rate, \"Constant signal\")\n",
    "axes.flat[1].set_xlim(0,10);\n",
    "\n",
    "fig, axes = plot_signal_and_spectrum(signal_constant, sample_rate, \"Constant signal (masked zero)\", ft_kwargs={'mask_bias':True})\n",
    "axes.flat[1].set_xlim(0,10);\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### FTs: noise (normal distributed)\n",
    "\n",
    "A FT of normal distributed noise will not have easily identifiable peaks in the spectrum.\n",
    "Instead, the spectrum is on average a bit flat, with only low power in any individual peak."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "signal_normal_noise = rng.normal(size=n_samples)\n",
    "\n",
    "plot_signal_and_spectrum(signal_normal_noise, sample_rate, \"Noise (normal distributed)\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### FTs: sine wave"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = 1e2 # Hz\n",
    "if f > sample_rate/2:\n",
    "    print(\"Sampling a frequency above the sample_rate/2 gives problems\")\n",
    "\n",
    "signal_sine = np.sin( 2*np.pi*f*t )\n",
    "\n",
    "if False:\n",
    "    # aliased peak\n",
    "    f_alias = (1.1* sample_rate/2)\n",
    "    signal_sine += 1/2*np.sin( 2*np.pi* f_alias *t)\n",
    "\n",
    "fig, axes = plot_signal_and_spectrum(signal_sine, sample_rate, \"Single frequency at $f = {:g}$MHz\".format(f/10**6))\n",
    "axes.flat[1].axvline(sample_rate/2, color='r', label='Nyquist Frequency');\n",
    "axes.flat[1].legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A sine wave will give very clear peaks at the frequency of the sine wave."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA80AAAHgCAYAAACfG480AAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjMsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+AADFEAAAgAElEQVR4nOzde1jV1533/ffanFVAUYEgCgoGiagbBQ+JGG2DZoySpslEo03SaKtt7NOZZOaZmbtNM537SSdpx0xzkGYak/ueTjTRpjO9U++JxsRoY2yjgGBU8BRPSBQ8cZCDwN7r+QPZkQAGhc126+d1XVzh99trfdcXtlf0u9dav2WstYiIiIiIiIhIew5fJyAiIiIiIiJyvVLRLCIiIiIiItIJFc0iIiIiIiIinVDRLCIiIiIiItIJFc0iIiIiIiIinVDRLCIiIiIiItKJQF8n0JsGDRpkExMTfZ2GiIiI19U1NgPQJ/im+qteRERucgUFBWestYN7MuZN9TdpYmIi+fn5vk5DRETE6wqOnQNgQkKUjzMRERHpPcaYYz0dU8uzRURERERERDqhollERERERESkEyqaRURERERERDqhollERERERESkEyqaRURERERERDpxUz09W0RE5GZxqGoP+yqLCAibhjPa6et0RERE/JaKZhERkRtMUUURP9/1BE3uRtYd/w9em/maCmcREZFrpOXZIiIiN5j88nya3E0ANLoayS/P93FGIiIi/ktFs4iIyA0mIyYDc+l7YwwZMRk+zUdERMSfqWgWERG5wYweNJoAEwTAwNCBWpotIiLSDSqaRUREbjB7z+yl2TZya+QYTtef5lTtKV+nJCIi4rdUNIuIiNxgWvcw5yQ8CkDeqTxfpiMiIuLX9PRsERGRG0z+qXyG9BnObQPGEx4cTn55PnOT5vo6LRERv9XU1MSJEydoaGjwdSpySWhoKPHx8QQFBXl9LBXNIiIiN5AmdxM7K3Zye/TdOEwAE2ImaKZZRKSbTpw4QXh4OImJiRhjvrqDeJW1lrNnz3LixAmGDx/u9fG0PFtEROQGUnK2hPrmelL6jwNanqRdWlOqfc0iIt3Q0NDAwIEDVTBfJ4wxDBw4sNdm/lU0i4iI3EBaZ5VH9W95YnZmbCaAzmoWEekmFczXl958P1Q0i4iI3EDyy/NJikwiIngAACkDUggPCif/lIpmERF/9tJLL5GamsrChQu7FWfFihUkJydjjOHMmTOe+9ZafvjDH5KcnMzYsWPZuXOn57Xf/OY3jBw5kpEjR/Kb3/zGcz8xMbFNjC1btjBnzpxu5Xc90p5mERGRG0Szu5md5TvbPPQrwBHA+JjxmmkWEfFzv/rVr1i/fn239/DecccdzJkzh+nTp7e5v379eg4ePMjBgwfZvn073//+99m+fTvnzp3jn/7pn8jPz8cYw4QJE8jJyWHAgAHdysOfaKZZRETkBlFytoS65joyYjPa3M+MzeRY9TEq6ip8lJmIiHTH9773PQ4fPkxOTg6//OUvuxUrPT2dxMTEdvffeecdHnnkEYwxTJ48mcrKSk6ePMl7771HdnY2UVFRDBgwgOzsbDZs2PCV48yePRun04nT6SQyMrLNDLW/0UyziIjIDaJ1NjkjJoNjl9XHGTEtRXT+qXxmj5jti9RERG4YB8prqGlo6tGY4aFB3BoT3unr//Zv/8aGDRvYvHkzgwYNavPa/v37mTdvXof9tmzZQv/+/buUQ1lZGUOHDvVcx8fHU1ZW1un9VjNmzCAgIACACxcuMGrUKADeffddAAoKCnjsscf4xje+0aU8rkcqmkVERG4QeafyGB45nEFhgzjGOc/9lKgU+gX1I688T0WziMgNJiUlhaKiom7Hsda2u2eM6fR+q8sL+S1btrB8+XLPa2fOnOHhhx/mt7/9LZGRkd3O0Vf8vmg2xgQA+UCZtfbG23UuIiLSBc3uZgorCpk9vH1RHOgIbNnXrIeBiYh025VmhH2hp2aa4+PjKS0t9VyfOHGCuLg44uPj2bJlS5v7X94P3RGXy8X8+fN5+umnSUtL61IO1yu/L5qBvwJKgAhfJyIiIuIr+8/t50LThXb7mVtlxGTw0YmPOF13msF9BvdydiIi4i09NdOck5PDihUrmD9/Ptu3bycyMpJbbrmFWbNm8aMf/Yjz588DsHHjRp599tmvjPcP//APjB07lvnz53c7N1/z6weBGWPigXuA13ydi4iIiC+1ns/cun/5y1rPay4oL+i1nERE5Prz0ksvER8fz4kTJxg7dizf+c53gJYHd40YMYLk5GS++93v8qtf/QqAqKgofvKTn5CZmUlmZiZPP/00UVFRXznO8uXL2bhxo+dhYH/4wx+8+nN5k+lojbq/MMb8DngWCAf+9quWZ2dkZNj8fC1NExGRG88PNv2AY9XHWHffOgAKjrXsaZ6Q0PIPm2Z3M1PXTOWe4ffwkyk/8VmeIiL+qKSkhNTUVF+nIV/S0ftijCmw1nb8CfI18tuZZmPMHKDCWnvFj8yNMUuMMfnGmPzTp0/3UnYiIiK9x+V2UVBe0OnSbGjZ15wenU5eeV4vZiYiIuL//LZoBu4AcowxR4E1wNeMMau+3Mha+6q1NsNamzF4sPZwiYjIjWf/+Uv7mTtZmt0qMzaTI1VHOFN/ppcyExER8X9+WzRba/+HtTbeWpsIzAc+tNZ+y8dpiYiI9Lqv2s/cynNec7m2KomIiHSV3xbNIiIi0iL/VD4JEQnE9I25YrvUgan0Ceyjo6dERESuwg1RNFtrt+iMZhERuRm53C4KKgq+cpYZIMgRRHp0uopmERGRq3BDFM0iIiI3qwPnD1DTWHPFh4BdLiM2g8+qPuNs/VkvZyYiInJjUNEsIiLix1r3J395prnPtv9kwAuPU/fem23u67xmERGRq6OiWURExI/lncpjaPhQYvvGeu7Vvfcm5mfPc3F9Ecef/J9tCufbBt5GWGCY5+FhIiIicmUqmkVERPyU27opKC/wzB63qtuyHusCMFhXy3Urz75mPUFbRMTvBAQE4HQ6SUtLY+7cuVRWVvoslw0bNpCSkkJycjLPPffcNbfxByqaRURE/NTB8wepbqxuvzR7+l98cWG+dE3LUu5DlYc433C+N9IUEZEeEhYWRlFREXv27CEqKorc3Fyf5OFyuVi2bBnr16+nuLiYt956i+Li4qtu4y9UNIuIiPipzs5nDhgYCxhMoBuAEOcdbV7XvmYREe8rOHae3M2HKDjmnQ8op0yZQllZGQCrVq1i4sSJOJ1Oli5disvl6rBPRUUFOTk5rFy5kqSkJHJycnC73Vc99o4dO0hOTmbEiBEEBwczf/583nnnnS63OXToEIMHDyYxMRGn00lUVBRJSUlUV1dfdS69IdDXCYiIiMi1yS/PZ0i/IdzS75Y296vf/DVgcXz/YVwvr6b6jRcZ8Lf/6nl99MDRnn3NdyXc1ctZi4j4t39at5fiz69c3NU0NLHvVA1uCw4Do2LDCQ8N6rT9bXER/OPc0V3OweVysWnTJhYvXkxJSQlr165l27ZtBAUF8fjjj7N69WoeeeSRdv0ee+wxFi5cyNtvv82qVatYtmwZeXl5TJo0CYCsrCxqamra9Vu+fDl33fXF3xdlZWUMHTrUcx0fH8/27dvb9LlSm+TkZKZOncqTTz5JVlYW06dP5+WXXyYiIqLLv4PepKJZRETED7mtm/zyfGYMndHmvnW7qdq2m5ChYZyf/T36v/Em1Rs3M+Bvv2gTFBDEuMHjyCvXw8BERLyhuqEZt2353m1brq9UNHdVfX09TqeTo0ePMmHCBLKzs3nllVcoKCggMzPT0yY6Orpd37KyMo4fP86CBQt46qmnSE9PZ9iwYTQ2NnrabN26tUt5WGvb3TPGXFWbvXv3kpaWBsC+fftISUnp0ti+oKJZRETEDx08f5Cqi1XtlmY3bP4dTVWWoHumgsNBxNSxnPm/RTQd2kVQ8jhPu8zYTF4ufJnKhkr6h/bv7fRFRPxWV2aEC46dZ+Frn9DU7CYo0MGL89OZkDCg22O37mmuqqpizpw55ObmYoz
      "text/plain": [
       "<Figure size 1152x576 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "with_spectrum = True\n",
    "phase_offsets = np.linspace(0, 2*np.pi, 8, endpoint=False) # rad\n",
    "f = 1e2 # Hz\n",
    "\n",
    "if f > sample_rate/2:\n",
    "    print(\"Sampling a frequency above the sample_rate/2 gives problems\")\n",
    "\n",
    "if with_spectrum:\n",
    "    fig, axes = plt.subplots(2,1, sharex=True, figsize=(16,8))\n",
    "else:\n",
    "    fig, ax = plt.subplots(figsize=(16,4))\n",
    "    axes = np.array([None, ax])\n",
    "    \n",
    "if with_spectrum:\n",
    "    axes[0].set_ylabel(\"Amplitude\")\n",
    "    axes[0].axvline(f, alpha=0.3, label='f = {:g}Hz'.format(f))\n",
    "\n",
    "axes[1].axvline(f, alpha=0.3, label='f = {:g}Hz'.format(f))\n",
    "axes[1].set_ylabel(\"Phase\")\n",
    "axes[1].set_xlabel(\"f\")\n",
    "axes[1].set_xlim( f-2, f+2)\n",
    "axes[1].set_ylim( -1 - 0.2, 1+ 0.2)\n",
    "axes[1].grid()\n",
    "axes[1].yaxis.set_major_formatter(tck.FormatStrFormatter('%g $\\pi$'))\n",
    "axes[1].yaxis.set_major_locator(tck.MultipleLocator(base=0.5))\n",
    "\n",
    "for phase in phase_offsets:\n",
    "    signal = np.sin( 2*np.pi*f*t + phase )\n",
    "    \n",
    "    ft_signal, freqs = ft_spectrum(signal, sample_rate)\n",
    "    \n",
    "    if with_spectrum:\n",
    "        axes[0].plot(freqs, np.real(ft_signal),  '.-', label='Re $\\\\varphi = {}\\\\pi$'.format(phase/np.pi))\n",
    "    \n",
    "    axes[1].plot(freqs, np.angle(ft_signal)/np.pi, '.-', label='$\\\\varphi = {}\\\\pi$'.format(phase/np.pi))\n",
    "\n",
    "# trigger legends\n",
    "if with_spectrum:\n",
    "    axes[0].legend(loc='center right');\n",
    "axes[1].legend(loc='center right');\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a sine wave, the phase of the spectrum at it's frequency is $\\theta( \\varphi = 0 ) = - \\dfrac{\\pi}{2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### FTs: step function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t_0 = t[n_samples//4]\n",
    "signal_step = np.heaviside(t - t_0, 1)\n",
    "fig, axes = plot_signal_and_spectrum(signal_step, sample_rate, \"Step function $t_0 = {:g}ns$\".format(t_0 * 10**9), ft_kwargs={'mask_bias':True});\n",
    "axes.flat[1].set_xlim(0, 20);\n",
    "\n",
    "t_0 = t[n_samples//2]\n",
    "signal_step = np.heaviside(t - t_0, 1)\n",
    "fig, axes = plot_signal_and_spectrum(signal_step, sample_rate, \"Step function $t_0 = {:g}ns$\".format(t_0 * 10**9), ft_kwargs={'mask_bias':True});\n",
    "axes.flat[1].set_xlim(0, 20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### FTs: delta peak\n",
    "\n",
    "All frequencies are accounted for with the same amplitude, but the phase will change depending on the position of the delta peak within the signal.\n",
    "\n",
    "Still, the individual phases are linearly related to their corresponding frequencies."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "for n in [4, 2, -4]:\n",
    "    index_t0 = n_samples//n\n",
    "\n",
    "    t_0 = t[index_t0]\n",
    "    signal_step = np.zeros(n_samples)\n",
    "    signal_step[index_t0] = 1\n",
    "\n",
    "    fig, axes = plot_signal_and_spectrum(signal_step, sample_rate, ft_kwargs={'mask_bias':True}, spectrum_kwargs={'plot_complex':True, 'plot_amplitude':True});\n",
    "    fig.suptitle(\"Delta function at ${}/{}$th of the sample\".format(np.sign(n), np.abs(n)))\n",
    "    axes.flat[1].set_xlim(0, 20);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Peculiarities $F^{-1}( F(A(t)) ) \\sim A(t)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# DFT\n",
    "\n",
    "A naive implementation of a DFT is simple to setup by implementing Eq. (1)\n",
    "\n",
    "$$\n",
    "\\hat{f}(\\omega) = F(t) = \\frac{1}{\\sqrt{2\\pi}} \\sum_{k=0}^{N-1} f(t_k)\\, \\exp{ \\left(-i \\omega t_k \\right)}\n",
    ".\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def naive_DFT( signal ):\n",
    "    \"\"\"\n",
    "    Calculate DFT in a naive way, returns a complex DFT.\n",
    "    \"\"\"\n",
    "\n",
    "    N = np.size(signal)\n",
    "\n",
    "    x_k = 2*np.pi/N*np.arange(N)\n",
    "    ft = np.zeros(N, dtype=np.complex128)\n",
    "    for j in range(N):\n",
    "        ft[j] = np.dot(signal, np.exp(- 1j * j * x_k ))\n",
    "\n",
    "    return ft"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### DFT: Cooley and Tukey (or Divide and Conquer)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A smarter DFT (Cooley and Tukey) can be implemented by splitting the sum in Eq. (1) into two parts: $k$ even and $k$ odd.\n",
    "\n",
    "With a uniform sample space $t$, the label $t_k$ can be rewritten to $t_k = k \\Delta t = $.\n",
    "\n",
    "This gives\n",
    "$$\n",
    "F(t) = \n",
    "    \\frac{1}{\\sqrt{2\\pi}} \\sum_{k=0}^{N/2-1} f(t_{2k})\\, \\exp{ \\left(-i \\omega \\, t_{2k} \\right)}\n",
    "    +\n",
    "    \\frac{1}{\\sqrt{2\\pi}} \\sum_{k=0}^{N/2-1} f(t_{2k+1})\\, \\exp{ \\left(-i \\omega \\, t_{2k+1} \\right)}\n",
    "    \\\\\n",
    "    =\n",
    "    \\frac{1}{\\sqrt{2\\pi}} \\sum_{k=0}^{N/2-1} f(t_{2k})\\, \\exp{ \\left(-i \\omega \\, 2k \\Delta t \\right)}\n",
    "    +\n",
    "    \\frac{1}{\\sqrt{2\\pi}} \\sum_{k=0}^{N/2-1} f(t_{2k+1})\\, \\exp{ \\left(-i \\omega \\, 2k \\Delta t \\right)}\\exp{ \\left(-i \\omega \\, \\Delta t \\right)}\n",
    "    \\\\\n",
    "    =\n",
    "    \\frac{1}{\\sqrt{2\\pi}} \\sum_{k=0}^{N/2-1} f(t_{2k})\\, \\exp{ \\left(-i \\omega \\, 2k \\Delta t \\right)}\n",
    "    +\n",
    "    \\frac{\\exp{ \\left(-i \\omega \\, \\Delta t \\right)} }{\\sqrt{2\\pi}} \\sum_{k=0}^{N/2-1} f(t_{2k+1})\\, \\exp{ \\left(-i \\omega \\, 2k \\Delta t \\right)}\n",
    "    \\\\\n",
    "    =\n",
    "    F(t_{even}) + F(t_{odd}) \\exp{ \\left(-i \\omega \\, \\Delta t \\right)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cooley_tukey_DFT( signal ):\n",
    "    \"\"\"\n",
    "    Calculate a DFT according to the Cooley and Tukey algorithm (radix-2)\n",
    "    also known as Divide and Conquer.\n",
    "    \n",
    "    It divides the terms into 2 streams with odd and even index.\n",
    "    \"\"\"\n",
    "    radix = 2\n",
    "\n",
    "    N = np.size(signal)\n",
    "    \n",
    "    if N <= radix:\n",
    "        return naive_DFT(signal)\n",
    "    \n",
    "    if N % radix > 0:\n",
    "        raise ValueError(\"Provide a signal with  N%{radix} == 0 values.\".format(radix=radix))\n",
    "    \n",
    "    # divide and conquer\n",
    "    DFT_even = cooley_tukey_DFT(signal[::radix])\n",
    "    DFT_odd  = cooley_tukey_DFT(signal[1::radix])\n",
    "    \n",
    "    phase_terms = np.exp( -2j * np.pi * np.arange(0, N) / N )\n",
    "    \n",
    "    return np.tile(DFT_even, 2) + np.tile(DFT_odd,2) * phase_terms\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 1152x288 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = 1e2 # Hz\n",
    "if f > sample_rate/2:\n",
    "    print(\"Sampling a frequency above the sample_rate/2 gives problems\")\n",
    "\n",
    "signal_sine = np.sin( 2*np.pi*f*t )\n",
    "\n",
    "fig, axes = plot_signal_and_spectrum(signal_sine, sample_rate, \"Single frequency at $f = {:g}$MHz\".format(f/10**6), ft_kwargs={'fft': cooley_tukey_DFT})\n",
    "axes.flat[1].axvline(sample_rate/2, color='r', label='Nyquist Frequency');"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}