Signal_Correlation.ipynb

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    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
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    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cvZwEuCPM73x",
        "colab_type": "text"
      },
      "source": [
        "## Signal Correlation\n",
        "Using Matlab / Octave, the correlation coefficients can be found by implementing the same concept introduced in the lectures.\n",
        "The next example is a solution for Q5 in HW1:\n",
        "\n",
        "Find the correlation coefficient $C_n$ of signal $x(t)$ and each of the four pulses $g_1(t)$, $g_2(t)$, $g_3(t)$, and $g_4(t)$.\n"
      ]
    },
    {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "YdHyrpA3M1Qk"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c0 =  1\n",
      "c1 =   -1.4558e-16\n",
      "c2 = -1\n",
      "c3 =    1.0618e-16\n",
      "c4 =  0.90002\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "% (file name: ustep.m)\n",
    "% The unitstep function is a function of time 't'.\n",
    "% Usage y = ustep(t)\n",
    "%\n",
    "% ustep(t) = 0, if t < 0\n",
    "% ustep(t) = 1, if t > = 1 \n",
    "%\n",
    "% t - must be real-valued and can be a vector or a matrix\n",
    "%\n",
    "function y = ustep(t)\n",
    "y = (t >= 0);\n",
    "end \n",
    "\n",
    "clear\n",
    "\n",
    "% To generate 5 signals x(t), g1(t), ... g4(t);\n",
    "% of this Example\n",
    "% we can first decide the signal within the period of 'T' for g(t)\n",
    "\n",
    "Dt = 0.01; % time increment Dt\n",
    "T = 1.5;   % time duration = T\n",
    "t = [-0.5: Dt: T]; % \"t\" goes between [-1, T] in Dt increment\n",
    "\n",
    "% Then evaluate the signal over the range of \"t\" to plot\n",
    "x = sin(2*pi*t) .* (ustep(t) - ustep(t-1));\n",
    "g1 = sin(4*pi*t) .* (ustep(t) - ustep(t-1)) ;\n",
    "g2 = -sin(2*pi*t) .* (ustep(t) - ustep(t-1));\n",
    "g3 = 0.707 * (ustep(t) - ustep(t-1));\n",
    "g4 = 0.707 * (ustep(t) - 2*ustep(t-0.5) + ustep(t-1));\n",
    "\n",
    "% Figures\n",
    "subplot(231); sig1 = plot(t, x, 'k');\n",
    "xlabel('\\it t'); ylabel(' {\\it x} ({\\it t }) '); % Label axis\n",
    "set( sig1, 'Linewidth', 2);  % change linewidth\n",
    "axis( [-.5 T -1.2 1.2] ); grid % set plot range\n",
    "\n",
    "subplot(232); sig2 = plot(t, g1, 'k');\n",
    "xlabel('\\it t'); ylabel(' {\\it g}_l ({\\it t })');\n",
    "set(sig2, 'Linewidth', 2);\n",
    "axis([-.5 T -1.2 1.2]); grid\n",
    "\n",
    "subplot(233); sig3 = plot(t, g2, 'k');\n",
    "xlabel('\\it t'); ylabel('{\\it g}_2 ({\\it t})');\n",
    "set(sig3, 'Linewidth', 2);\n",
    "axis([-.5 T -1.2 1.2]); grid\n",
    "\n",
    "subplot(234); sig4 = plot(t, g3, 'k');\n",
    "xlabel('\\it t'); ylabel('{\\it g}_3 ({\\it t})');\n",
    "set(sig4, 'Linewidth', 2);\n",
    "axis([-.5 T -1.2 1.2]); %grid\n",
    "\n",
    "subplot(235); sig5 = plot(t, g4, 'k');\n",
    "xlabel('\\it t'); ylabel('{\\it g}_4 ({\\it t})');\n",
    "set(sig5, 'Linewidth', 2); %grid\n",
    "axis([-.5 T -1.2 1.2]);\n",
    "\n",
    "%Computing signal energies\n",
    "E0 = sum(x .* conj(x)) * Dt;\n",
    "E1 = sum(g1 .* conj(g1)) * Dt;\n",
    "E2 = sum(g2 .* conj(g2)) * Dt;\n",
    "E3 = sum(g3 .* conj(g3)) * Dt;\n",
    "E4 = sum(g4 .* conj(g4)) * Dt;\n",
    "\n",
    "c0 = sum(x .* conj(x)) * Dt / (sqrt(E0 * E0))\n",
    "c1 = sum(x .* conj(g1)) * Dt / (sqrt(E0 * E1))\n",
    "c2 = sum(x .* conj(g2)) * Dt / (sqrt(E0 * E2))\n",
    "c3 = sum(x .* conj(g3)) * Dt / (sqrt(E0 * E3))\n",
    "c4 = sum(x .* conj(g4)) * Dt / (sqrt(E0 * E4))"
   ]
  },
   {
      "cell_type": "markdown",
      "metadata": {
        "id": "cvZwEuCPM73x",
        "colab_type": "text"
      },
      "source": [
    "**References**:\n",
    "\n",
    " - Modern Digital and Analog Communication systems, 4th Edition, B.P. Lathi, Oxford University Press, 2009"
   ]
}

  ]
}