{
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   "source": [
    "# Homework \\#7 - Python exercise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%pylab inline\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sampling and aliasing\n",
    "  \n",
    "Consider the linear chirp defined as\n",
    "\n",
    "$$\n",
    "    x(t) = \\cos(\\varphi(t)) = \\cos(\\pi at^2 + 2\\pi bt)\n",
    "$$\n",
    "\n",
    "where $a$ and  $b$ are real-valued parameters.\n",
    "\n",
    "* What can you say about the frequency of the signal $x(t)$? In particular, if we sample the signal with a given sampling rate, what do you think is going to happen?\n",
    "* Write a function `chirp(a, b, Fs, len)` that computes the sampled version of a chirp at a sampling frequency `Fs` for `len` seconds\n",
    "* Look at a typical shape of a chirp before aliasing by plotting `chirp(40, 4, 400, 1)`. Now plot `chirp(400, 4, 400, 1)` and explain what you observe\n",
    "* Consider the case where $a=50$ and $b=600$. What is the minimum sampling frequency so that aliasing does not occur before $t=1$ second? %\n",
    "* Play a chirp signal computed using $a=40$, $b=4$ and $F_s=900$ for a duration of 10 seconds. Repeat the operation with $F_s=700$. What do you observe?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solution\n",
    "\n",
    "The instantaneous angular frequency of the chirp at time $t$ is ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def chirp(a, b, sf, len=1):\n",
    "    ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the plots, it's best to use wider viewports:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.rcParams[\"figure.figsize\"] = (14,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(chirp(40, 4, 400, 1));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see from this plot that when $a=40$, a sampling frequency of ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(chirp(400, 4, 400, 1));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When $a=400$, the sampling frequency ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The minimum sampling frequency ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can play the chirp for a duration of 10 seconds by executing the following code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import IPython\n",
    "IPython.display.Audio(chirp(40, 4, 900, 10), rate=900)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "IPython.display.Audio(chirp(40, 4, 700, 10), rate=700)"
   ]
  }
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