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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "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)"
+   ]
+  }
+ ],
+ "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": "3.6.10"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}