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   "source": [
    "# Homework \\#4 - Python exercise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Gibbs Phenomenon\n",
    "  \n",
    "In this exercise you will verify by yourself the existence of the Gibbs phenomenon using Python (or any other numerical package). The idea is to plot a zoomed-in version of the frequency response of a truncated ideal lowpass filter with cutoff frequency $\\pi/2$:\n",
    "$$\n",
    "      \\hat H(e^{j\\omega}) = \\sum_{n=-N}^{N} (1/2)\\mbox{sinc}(n/2) \\; e^{-j\\omega n}\n",
    "$$\n",
    "where we are interested in plotting the transform over a small interval around the cutoff frequency.\n",
    "  \n",
    "* Plot $\\hat H(e^{j\\omega})$ over 2000 points in the interval $1.4 \\leq \\omega \\leq 1.7$ for $N = 20$. \n",
    "* Repeat the above point for $N = 100$ and $N=200$ and verify that the peak of the magnitude is still approximately $9\\%$ of the value of the discontinuity.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solution\n",
    "\n",
    "First let's define a function that computes the DTFT $\\hat H(e^{j\\omega})$ over a given set of values $\\{\\omega_0, \\ldots, \\omega_M\\}$. To do so, we will define a matrix $\\mathbf{W}$ such that $W_{m,n} = e^{-j\\omega_m n}$ and, with this, we can obtain all $M$ DTFT values as the matrix-vector multiplication $\\mathbf{Wx = W}[h[-N], \\ldots, h[N]]^T$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# this function computes H(e^jw) for a given set of values of omega\n",
    "def H(omega: np.ndarray, N: int):\n",
    "    # Create the transform matrix W\n",
    "    ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "wlim = [1.4, 1.7]\n",
    "omega = np.linspace(wlim[0], wlim[1], 2000)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "for N in (20, 100, 200, 1000):\n",
    "    ax.plot(omega, H(omega, N), label=f\"$N={N}$\")\n",
    "ax.plot(wlim, [1.09, 1.09], 'r--')\n",
    "ax.set_xlim(wlim)\n",
    "ax.set_xlabel(r'$\\omega$', fontsize=12)\n",
    "ax.set_ylabel(r'$\\left|\\hat{H}(e^{j\\omega})\\right|$', fontsize=12)\n",
    "ax.grid()\n",
    "ax.legend()"
   ]
  }
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