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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Homework \\#3 - 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": [
+    "## DTFT,DFT, and numerical computations\n",
+    "\n",
+    "Consider the following infinite non-periodic discrete time signal, where $M\\in\\mathbb{N}$:\n",
+    "$$\n",
+    "x[n]=\\begin{cases}\n",
+    "      0 & n<0,\\\\\n",
+    "      1 & 0\\leq n< M,\\\\\n",
+    "      0 & n\\geq M.\n",
+    "      \\end{cases}\n",
+    "$$\n",
+    "\n",
+    "  The goal is to plot the magnitude of $X(e^{j\\omega})$ using a numerical package. Using a numerical package implies that we will only obtain an approximation of the true value.\n",
+    "* Compute $X(e^{j\\omega})$ analytically\n",
+    "* Using a numerical package plot $|X(e^{j\\omega})|$ over $[-\\pi, \\pi]$ for $M=20$ using 10,000 points.\n",
+    "* Now generate a finite sequence $\\hat{x}[n]$ of length $N=30$ such that $\\hat{x}[n]=x[n]$ for $n=0,1, \\ldots, N-1$. Compute its DFT using the numerical package's FFT algorithm and plot its magnitude. Compare it with the plot obtained previously.\n",
+    "* Repeat now for different values of $N = 50, 100, 1000$. What can you conclude?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Solution\n",
+    "The analytical expression for the DTFT of the signal is \n",
+    "$$\n",
+    "     X(e^{j\\omega}) = \\ldots\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# we want to plot the analytic DTFT over 10,000 points: create the values for \\omega\n",
+    "w = ...\n",
+    "\n",
+    "# analytic expression for the DTFT\n",
+    "M = 20\n",
+    "X = ...\n",
+    "\n",
+    "# plotting\n",
+    "plt.plot(w, X);\n",
+    "plt.xlim([-np.pi, np.pi]);  # fit the x-axis"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# now compute the FFT-based approximations\n",
+    "N = [30, 50, 100, 1000]\n",
+    "for pts in N:\n",
+    "    # build the signal\n",
+    "    x = ...\n",
+    "    \n",
+    "    # compute the DFT and shift it to align it with the DTFT\n",
+    "    X_a = ...\n",
+    "    plt.plot(np.abs(X_a));\n",
+    "    plt.show();"
+   ]
+  }
+ ],
+ "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": "3.7.7"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}