sp-continuous-time unit step and unit impulse-for github.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "60066f66",
   "metadata": {},
   "source": [
    "### Signal Processing\n",
    "#### Continuous-time unit-impulse $\\delta(t)$ and continuous-time unit step $u(t)$:\n",
    "$\\delta(t)=\\frac{du(t)}{dt},\\; u(t)=\\int_{-\\infty}^{t} \\delta(\\tau)d\\tau$\n",
    "<hr>\n",
    "https://github.com/ostad-ai/Signal-Processing\n",
    "<br> Explanation: https://www.pinterest.com/HamedShahHosseini/Signal-Processing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "868ded86",
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c2f7b7d5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#approximate continuous-time unit impulse\n",
    "def d(t,delta=1e-3):\n",
    "    if 0<=t<=delta:\n",
    "        return 1./delta\n",
    "    else:\n",
    "        return 0.\n",
    "    \n",
    "#approximate continuous-time unit step\n",
    "def u(t,delta=1e-3):\n",
    "    if t<0:\n",
    "        return 0\n",
    "    elif 0<=t<=delta:\n",
    "        return t/delta\n",
    "    else:\n",
    "        return 1.\n",
    "    \n",
    "#differentiation    \n",
    "def diff(f,t,h=1e-5):\n",
    "    return (f(t+h)-f(t))/h\n",
    "\n",
    "ts=np.arange(-1,1,1e-3)\n",
    "ds=[diff(u,t) for t in ts]\n",
    "ds=np.asarray(ds)\n",
    "plt.plot(ts,ds,label='$\\delta(t)$')\n",
    "plt.xlabel('$t$')\n",
    "plt.title(r'Approximate $\\delta(t)$ as $\\frac{du(t)}{dt}$')\n",
    "plt.legend(); plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8ed78786",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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