melvincabatuan/Convolution.ipynb

## Convolution.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Convolution Integral"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A *convolution*, $(h*x)(t)$, is a mathematical operator which manipulates two functions $x$ and $h$ and produces an output that represents the amount of overlap between $x$ and a reversed and translated version of $h$. </br>\n",
    "\n",
    "### In signal processing, one of the functions ($h$) is taken to be a fixed filter impulse response, and is known as a *kernel*.\n",
    "\n",
    "$$  (x \\ast h)(t) = \\int_{-\\infty}^{\\infty}   x(\\tau)h(t - \\tau) \\, d\\tau$$\n",
    "\n",
    "or\n",
    "\n",
    "$$ (h \\ast x)(t) = \\int_{-\\infty}^{\\infty}   h(\\tau)x(t - \\tau) \\, d\\tau $$\n",
    "\n",
    "Illustration: [1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe src=http://mathworld.wolfram.com/images/gifs/convrect.gif width=300 height=260></iframe>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 188,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import HTML\n",
    "HTML('<iframe src=http://mathworld.wolfram.com/images/gifs/convrect.gif width=300 height=260></iframe>')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe src=http://mathworld.wolfram.com/images/gifs/convgaus.gif width=300 height=260></iframe>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 189,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import HTML\n",
    "HTML('<iframe src=http://mathworld.wolfram.com/images/gifs/convgaus.gif width=300 height=260></iframe>')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Steps </br>\n",
    "\n",
    "### 1. The impulse response $h(\\tau)$ is flipped or time-reversed (that is, reflected about the origin) to obtain $h(-\\tau)$ and then shifted by to form $h(t - \\tau) = h[-(\\tau - t)]$, which is a function of $\\tau$ with parameter $t$.\n",
    "### 2. The signal $x(\\tau)$ and $h(t - \\tau)$ are multiplied together for all values of $\\tau$ with $t$ fixed at some value.\n",
    "### 3. The product $x(\\tau)h(t - \\tau)$ is integrated over all $\\tau$ to produce a single output value $y(t)$. \n",
    "\n",
    "</br>\n",
    "Ex."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def step(t, t_0):\n",
    "    return 1 * ((t - t_0) >= 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline  \n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Basic Example:\n",
    "\n",
    "## Calculate the convolution of a unit step input with a unit step response system.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "AABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaf9c544c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "x = step(t,0) \n",
    "h = step(t,0)\n",
    "\n",
    "fig = plt.figure(figsize=(12,4))\n",
    "\n",
    "plt.subplot(1, 2, 1)\n",
    "plt.plot(t,x,'g-',linewidth=6);\n",
    "plt.title('Input signal, $x(t)$')\n",
    "plt.xlabel('t')\n",
    "\n",
    "plt.subplot(1, 2, 2)\n",
    "plt.plot(t,h,'r-',linewidth=6);\n",
    "plt.title('Impulse response, $h(t)$')\n",
    "plt.xlabel('t');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$  (x \\ast h)(t) = \\int_{-\\infty}^{\\infty}   u(\\tau)u(t - \\tau) \\, d\\tau$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 1: The impulse response $h(\\tau)$ is flipped or time-reversed (that is, reflected about the origin) to obtain $h(-\\tau)$ and then shifted by to form $h(t - \\tau) = h[-(\\tau - t)]$, which is a function of $\\tau$ with parameter $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "SZLmjhEsSZKkufP/AemYlWYYTrtSAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaf76af8c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "h = step(-t, 0)\n",
    "fig = plt.figure(figsize=(12,4))\n",
    "plt.plot(t,h,'r-',linewidth=6);\n",
    "plt.title('Impulse response, $h(-\\u03c4)$ at t = 0')\n",
    "plt.xlabel('T');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 2-3 The signal $x(\\tau)$ and $h(t - \\tau)$ are multiplied together for all values of $\\tau$ with $t$ fixed at some value. Then, the product $x(\\tau)h(t - \\tau)$ is integrated over all $\\tau$ to produce a single output value $y(t)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Annotation at 0xb00f372c>"
      ]
     },
     "execution_count": 194,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xafe919ac>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "h = step(-t, -0.5)\n",
    "x = step(t, 0)\n",
    "fig = plt.figure(figsize=(12,4))\n",
    "ax1 = fig.add_axes([0, 0, 1, 1])   # [left, bottom, width, height]\n",
    "\n",
    "plt.plot(t,h,'r-',linewidth=6);\n",
    "plt.plot(t,x,'g-',linewidth=6);\n",
    "plt.title('Impulse response, $h(-\\u03c4)$ at t = 0.5 with the input signal, $u(t)$')\n",
    "plt.xlabel('T');\n",
    "ax1.fill_between(t,0,h,where=x==h,color='black');\n",
    "ax1.annotate('$t$', xy=(0.5, 0), xytext=(0.6, 0.1),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that the time reversed Impulse response, $u(-\\tau)$, starts intersecting the signal, which is also a unit step, $u(\\tau)$, at $\\tau = 0$ until $\\tau = t$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, \n",
    "\n",
    "$$  (x \\ast h)(t) = \\int_{0}^{t}   x(\\tau)h(t - \\tau) \\, d\\tau$$  \n",
    "\n",
    "$$  (x \\ast h)(t) = \\int_{0}^{t}   u(\\tau)u(t - \\tau) \\, d\\tau$$  \n",
    "\n",
    "$$  (x \\ast h)(t) = \\int_{0}^{t}   (1)(1) \\, d\\tau$$ \n",
    "\n",
    "$$  (x \\ast h)(t) = \\left. \\tau \\right |_0^t  $$\n",
    "\n",
    "$$  (x \\ast h)(t) = t \\, u(t) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$  (x \\ast h)(t) = \\begin{cases} \n",
    " 0                 & \\mbox{if } -\\infty < t < 0 \\\\ \n",
    " t                 & \\mbox{if }  \\, 0 < t < \\infty \\\\ \n",
    "\\end{cases} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "ExagkiRJkqQ18f8Df5qL3XOPjgMAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xafbd25cc>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "\n",
    "# input\n",
    "x = step(t,0)  \n",
    "\n",
    "# impulse response\n",
    "h = step(t,0) \n",
    "\n",
    "# result\n",
    "result = np.zeros(len(t))\n",
    "result = t * (step(t,0))\n",
    "\n",
    "fig = plt.figure(figsize=(16,4))\n",
    "\n",
    "plt.plot(t,x,'go-',linewidth=5, label='Input signal, $x(t)$');\n",
    "plt.plot(t,h,'rd-',linewidth=1, label='Impulse response, $h(t)$');\n",
    "plt.plot(t,result,'b-',linewidth=3, label='Output signal, $y(t)$');\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "legend = plt.legend(loc='upper left');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaf9006cc>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "x = step(t,1) - step(t,3)\n",
    "h = (t + 2)*(step(t,-2) - step(t,-1))\n",
    "\n",
    "fig = plt.figure(figsize=(12,4))\n",
    "\n",
    "plt.subplot(1, 2, 1)\n",
    "plt.plot(t,x,'g-',linewidth=5);\n",
    "plt.title('Input signal, $x(t)$')\n",
    "plt.xlabel('t')\n",
    "\n",
    "plt.subplot(1, 2, 2)\n",
    "plt.plot(t,h,'r-',linewidth=3);\n",
    "plt.title('Impulse response, $h(t)$')\n",
    "plt.xlabel('t');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Drill Problem 1: Compute the convolution, $(x \\ast h)(t)$, of the signals represented in the figure above:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Step 1: The impulse response $h(\\tau)$ is flipped or time-reversed (that is, reflected about the origin) to obtain $h(-\\tau)$ and then shifted by to form $h(t - \\tau) = h[-(\\tau - t)]$, which is a function of $\\tau$ with parameter $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaf7dcc0c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "h = (-t + 2)*(step(t,1) - step(t,2))\n",
    "fig = plt.figure(figsize=(12,4))\n",
    "plt.plot(t,h,'r-',linewidth=3);\n",
    "plt.title('Impulse response, $h(-\\u03c4)$ at t = 0')\n",
    "plt.xlabel('T');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: This is the $h(t - \\tau)$ at $t = 0$ and the signal starts at $\\tau = t + 1$ and ends at $\\tau = t + 2$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 2-3 The signal $x(\\tau)$ and $h(t - \\tau)$ are multiplied together for all values of $\\tau$ with $t$ fixed at some value. Then, the product $x(\\tau)h(t - \\tau)$ is integrated over all $\\tau$ to produce a single output value $y(t)$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case I: NO Overlap, $y(t) = 0$ at $-\\infty \\to t_1$, where $t_1 = -1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{-\\infty}^{-1} x(\\tau)h(t - \\tau) \\, d\\tau  = 0 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb03550ac>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "h_4 = (-t - 2)*(step(t,-3) - step(t,-2))\n",
    "h_3 = (-t - 1)*(step(t,-2) - step(t,-1))\n",
    "h_2 = (-t - 0)*(step(t,-1) - step(t,-0))\n",
    "h_1 = (-t + 1)*(step(t,0) - step(t,1))\n",
    "fig = plt.figure(figsize=(16,4))\n",
    "plt.plot(t,h_4,'r-',linewidth=3, label=\"$t=-4$\");\n",
    "plt.plot(t,h_3,'c-',linewidth=3, label=\"$t=-3$\");\n",
    "plt.plot(t,h_2,'m-',linewidth=3, label=\"$t=-2$\");\n",
    "plt.plot(t,h_1,'y-',linewidth=3, label=\"$t=-1$\");\n",
    "plt.title('Impulse response, $h(-(\\u03c4 - t))$ at t = -4,-3,-2,-1, with the input signal, $x(t)$')\n",
    "plt.plot(t,x,'g-',linewidth=5, label=\"$x(t)$\");\n",
    "plt.xlabel('T');\n",
    "\n",
    "legend = plt.legend() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case II: Leading-edge overlap, at $t_1 \\to t_2$ or at $ -1 < t < 0 $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{a}^{b} x(\\tau)h(t - \\tau) \\, d\\tau  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "ORk0MzMzMzMrof8PCxMcn5JTJjUAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb0685b0c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "h_half = (-t + 1.5)*(step(t,0.5) - step(t,1.5))\n",
    "fig = plt.figure(figsize=(12,2.5))\n",
    "ax1 = fig.add_axes([0, 0, 1, 1])   # [left, bottom, width, height]\n",
    "plt.plot(t,h_half,'r-',linewidth=3, label=\"$t=0.5$\");\n",
    "plt.title('Impulse response, $h(-(\\u03c4 - t))$ at t = -0.5, with the input signal, $x(t)$')\n",
    "plt.plot(t,x,'g-',linewidth=5, label=\"$x(t)$\");\n",
    "plt.xlabel('T');\n",
    "ax1.fill_between(t,0,h_half,where=x>=h_half,color='black')\n",
    "ax1.annotate('$t + 2$', xy=(1.5, 0), xytext=(1.6, 0.1),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            )\n",
    "ax1.annotate('same \\n length \\n as \\n base', xy=(1, 0.2), xytext=(0.55, 0.2),\n",
    "            arrowprops=dict(facecolor='yellow'),\n",
    "            )\n",
    "ax1.annotate('t+2-1', xy=(1.25, 0), xytext=(1.1, -0.15),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            )\n",
    "legend = plt.legend() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The overlap starts at $\\tau = 1$ and ends at $\\tau = t + 2$. Recall that the impulse response ends at $t+2$. Thus,  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{1}^{t+2} x(\\tau)h(t - \\tau) \\, d\\tau  $$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the above interval, $x(\\tau) = 1$ and $h(t - \\tau) = - \\tau + t + 2  $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{1}^{t+2} (1)(- \\tau + t + 2) \\, d\\tau  $$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\left. (t + 2) \\tau - \\dfrac{\\tau^2}{2} \\right |_1^{t+2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\left((t + 2)(t + 2) - \\dfrac{(t+2)^2}{2}\\right) - \\left((t + 2) - \\dfrac{1^2}{2}\\right)  $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\dfrac{(t+2)^2}{2} - t - \\dfrac{3}{2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\dfrac{t^2}{2} + \\dfrac{4t}{2} + \\dfrac{4}{2} -t - \\dfrac{3}{2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) =  \\dfrac{t^2}{2} + t + \\dfrac{1}{2} $$ or $$  y(t) = \\dfrac{1}{2} (t + 1)^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, this can be evaluated by the triangle area:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ A = \\dfrac{1}{2} bh $$\n",
    "\n",
    "$$ A = \\dfrac{1}{2} (t+2-1)(t+1) $$\n",
    "\n",
    "$$ A = \\dfrac{1}{2}(t+1)^2 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case III: Full overlap, at $t = 0$ to $t = 1$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{a}^{b} x(\\tau)h(t - \\tau) \\, d\\tau  $$  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb014088c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "h_half = (-t + 2.5)*(step(t,1.5) - step(t,2.5))\n",
    "fig = plt.figure(figsize=(12,2.5))\n",
    "ax1 = fig.add_axes([0, 0, 1, 1])   # [left, bottom, width, height]\n",
    "plt.plot(t,h_half,'r-',linewidth=3, label=\"$h$ at $t=0.5$\");\n",
    "plt.title('Impulse response, $h(-(\\u03c4 - t))$ at t = -0.5, with the input signal, $x(t)$')\n",
    "plt.plot(t,x,'g-',linewidth=5, label=\"$x(t)$\");\n",
    "plt.xlabel('T');\n",
    "ax1.fill_between(t,0,h_half,where=x>=h_half,color='black')\n",
    "legend = plt.legend(loc='center') \n",
    "ax1.annotate('$t + 2$', xy=(2.5, 0), xytext=(2.6, 0.1),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            )\n",
    "ax1.annotate('$t + 1$', xy=(1.5, 0), xytext=(1.2, 0.1),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The overlap starts at $\\tau = t + 1$ and ends at $\\tau = t + 2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{t+1}^{t+2} x(\\tau)h(t - \\tau) \\, d\\tau  $$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the above interval, $x(\\tau) = 1$ and $h(t - \\tau) = - \\tau + t + 2  $. Thus,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{t+1}^{t+2} (1) (- \\tau + t + 2) \\, d\\tau  $$\n",
    "\n",
    "$$ y(t) = \\left. (t + 2) \\tau - \\dfrac{\\tau^2}{2} \\right |_{t+1}^{t+2} $$\n",
    "\n",
    "$$ y(t) = \\left((t + 2)(t + 2) - \\dfrac{(t+2)^2}{2}\\right) - \\left((t + 2)(t+1) - \\dfrac{(t+1)^2}{2}\\right)  $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\dfrac{(t+2)^2}{2} - (t + 1)(t+2 - \\dfrac{t+1}{2}) $$\n",
    "\n",
    "$$ y(t) =  \\dfrac{(t^2 + 4t + 4)}{2} - (t + 1)(\\dfrac{t}{2} + \\dfrac{3}{2})$$\n",
    "\n",
    "$$ y(t) = \\dfrac{t^2}{2} + \\dfrac{4t}{2} + \\dfrac{4}{2} - \\dfrac{t^2}{2} - \\dfrac{4t}{2} - \\dfrac{3}{2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\dfrac{1}{2} $$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, this can be evaluated by the triangle area:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ A = \\dfrac{1}{2} bh $$\n",
    "\n",
    "$$ A = \\dfrac{1}{2} (1)(1) $$\n",
    "\n",
    "$$ A = \\dfrac{1}{2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case IV: Trailing edge overlap, at $1 < t < 2$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{a}^{b} x(\\tau)h(t - \\tau) \\, d\\tau  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "UQAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaf13950c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "h_half = (-t + 3.5)*(step(t,2.5) - step(t,3.5))\n",
    "fig = plt.figure(figsize=(12,2.5))\n",
    "ax1 = fig.add_axes([0, 0, 1, 1])   # [left, bottom, width, height]\n",
    "plt.plot(t,h_half,'r-',linewidth=3, label=\"$h$ at $t=0.5$\");\n",
    "plt.title('Impulse response, $h(-(\\u03c4 - t))$ at t = -0.5, with the input signal, $x(t)$')\n",
    "plt.plot(t,x,'g-',linewidth=5, label=\"$x(t)$\");\n",
    "plt.xlabel('T');\n",
    "ax1.fill_between(t,0,h_half,where=x>=h_half,color='black')\n",
    "legend = plt.legend(loc='center') \n",
    "ax1.annotate('$t + 1$', xy=(2.5, 0), xytext=(2.2, 0.1),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The overlap starts at $\\tau = t + 1$ and ends at $\\tau = 3$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{t+1}^{3} x(\\tau)h(t - \\tau) \\, d\\tau  $$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the above interval, $x(\\tau) = 1$ and $h(t - \\tau) = - \\tau + t + 2  $. Thus,\n",
    "\n",
    "$$ y(t) = \\int_{t+1}^{3} (1) (- \\tau + t + 2) \\, d\\tau  $$\n",
    "\n",
    "$$ y(t) = \\left. (t + 2) \\tau - \\dfrac{\\tau^2}{2} \\right |_{t+1}^{3} $$\n",
    "\n",
    "$$ y(t) =  \\left((t + 2)(3) - \\dfrac{(3)^2}{2}\\right) - \\left((t + 2)(t + 1) - \\dfrac{(t+1)^2}{2}\\right) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\left(3t + 6 - \\dfrac{9}{2}\\right) - \\left((t + 1)((t+2)- \\dfrac{(t+1)}{2} )\\right) $$ \n",
    "\n",
    "$$ y(t) = (3t + \\dfrac{3}{2}) - (t + 1)(\\dfrac{t}{2} + \\dfrac{3}{2}) $$ \n",
    "\n",
    "$$ y(t) = 3t + \\dfrac{3}{2} - \\dfrac{t^2}{2} - \\dfrac{4t}{2} -  \\dfrac{3}{2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = t - \\dfrac{t^2}{2} $$ \n",
    "\n",
    "or\n",
    "\n",
    "$$ y(t) = t (1 - \\dfrac{t}{2}) $$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Alternatively, this can be evaluated by subtracting the smaller triangle area, $A_t$, from the bigger triangle, $A_T$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ A = A_T - A_t$$\n",
    "\n",
    "$$ A = \\dfrac{1}{2} - \\dfrac{1}{2} bh $$\n",
    "\n",
    "$$ A = \\dfrac{1}{2} -  \\dfrac{1}{2} (t+2-3)(t+2-3) $$\n",
    "\n",
    "$$ A = \\dfrac{1}{2} -  \\dfrac{1}{2}(t-1)^2 $$\n",
    "\n",
    "$$ A = t - \\dfrac{t^2}{2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case V: No overlap, at $ t > 2$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{2}^{\\infty} x(\\tau)h(t - \\tau) \\, d\\tau  = 0 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xafec55ec>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "h_half = (-t + 4)*(step(t,3) - step(t,4))\n",
    "fig = plt.figure(figsize=(12,2.5))\n",
    "ax1 = fig.add_axes([0, 0, 1, 1])   # [left, bottom, width, height]\n",
    "plt.plot(t,h_half,'r-',linewidth=3, label=\"$h$ at $t=2$\");\n",
    "plt.title('Impulse response, $h(-(\\u03c4 - t))$ at t = -0.5, with the input signal, $x(t)$')\n",
    "plt.plot(t,x,'g-',linewidth=5, label=\"$x(t)$\");\n",
    "plt.xlabel('T');\n",
    "ax1.fill_between(t,0,h_half,where=x>=h_half,color='black')\n",
    "legend = plt.legend(loc='center') \n",
    "ax1.annotate('No Overlap', xy=(3, 0.4), xytext=(2.2, 0.4),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Final Answer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore, the convolution result is:\n",
    "\n",
    "\n",
    "\n",
    "$$ y(t) = \\begin{cases} \n",
    "0                  & \\mbox{if } -\\infty < t < -1 \\\\ \n",
    "\\frac{1}{2}(t+1)^2 & \\mbox{if } -1 < t < 0 \\\\  \n",
    "\\frac{1}{2}        & \\mbox{if }  \\, 0 < t < 1 \\\\ \n",
    "t - \\frac{t^2}{2}  & \\mbox{if }  \\, 1 < t < 2 \\\\\n",
    " 0                 & \\mbox{if }  \\, 2 < t < \\infty \\\\ \n",
    "\\end{cases} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "ms8AAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaf8df16c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "\n",
    "# input\n",
    "x = step(t,1) - step(t,3)\n",
    "\n",
    "# impulse response\n",
    "h = (t + 2)*(step(t,-2) - step(t,-1))\n",
    "\n",
    "# result\n",
    "result = np.zeros(len(t))\n",
    "result = 0.5*(t+1)**2 * (step(t,-1)-step(t,0)) + 0.5* (step(t,0)-step(t,1)) + t * (1 - 0.5*t) * (step(t,1)-step(t,2))\n",
    "\n",
    "fig = plt.figure(figsize=(14,4))\n",
    "\n",
    "plt.plot(t,x,'g-',linewidth=5, label='Input signal, $x(t)$');\n",
    "plt.plot(t,h,'r-',linewidth=3, label='Impulse response, $h(t)$');\n",
    "plt.plot(t,result,'b-',linewidth=3, label='Output signal, $y(t)$');\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "legend = plt.legend(loc='upper left');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "ZQAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaf87a24c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-2.0, 2.0, 0.01)\n",
    "x = step(t,0) - step(t,1)\n",
    "h = (-t + 1)*(step(t,0) - step(t,1))\n",
    "\n",
    "fig = plt.figure(figsize=(15,3))\n",
    "\n",
    "plt.subplot(1, 2, 1)\n",
    "plt.plot(t,x,'g-',linewidth=5);\n",
    "plt.title('Input signal, $x(t)$')\n",
    "plt.xlabel('t')\n",
    "\n",
    "plt.subplot(1, 2, 2)\n",
    "plt.plot(t,h,'r-',linewidth=3);\n",
    "plt.title('Impulse response, $h(t)$')\n",
    "plt.xlabel('t');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Drill Problem 2: Compute the convolution, $(h \\ast x)(t)$, of the signals represented in the figure above:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = (h \\ast x)(t) = \\int_{-\\infty}^{\\infty}   h(\\tau)x(t - \\tau) \\, d\\tau $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: In this exercise, the signal is time-reversed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Step 1: The impulse response $h(\\tau)$ is flipped or time-reversed (that is, reflected about the origin) to obtain $h(-\\tau)$ and then shifted by to form $h(t - \\tau) = h[-(\\tau - t)]$, which is a function of $\\tau$ with parameter $t$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "SZIkAP4/masSIuj+mawAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb01256ac>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-2.0, 2.0, 0.01)\n",
    "x = step(t,-1) - step(t,0)\n",
    "\n",
    "fig = plt.figure(figsize=(12,3))\n",
    "ax1 = fig.add_axes([0, 0, 1, 1])   # [left, bottom, width, height]\n",
    "\n",
    "plt.plot(t,x,'g-',linewidth=5);\n",
    "plt.title('Time-reversed input signal, $x(t-\\u03c4)$')\n",
    "plt.xlabel('T');\n",
    "ax1.annotate('$t$', xy=(0, 0), xytext=(0.1, 0.1),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );\n",
    "ax1.annotate('$t-1$', xy=(-1, 0), xytext=(-1.2, 0.1),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: This rectangular pulse begins at $\\tau = t-1$ and ends at $\\tau = t$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step 2-3 The signal $x(\\tau)$ and $h(t - \\tau)$ are multiplied together for all values of $\\tau$ with $t$ fixed at some value. Then, the product $x(\\tau)h(t - \\tau)$ is integrated over all $\\tau$ to produce a single output value $y(t)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaff2328c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-2.0, 2.0, 0.01)\n",
    "x_0 = step(t,-1) - step(t,0)\n",
    "x__5 = step(t,-1.5) - step(t,-0.5)\n",
    "x_1 = step(t,-2) - step(t,-1)\n",
    "h = (-t + 1)*(step(t,0) - step(t,1))\n",
    "\n",
    "fig = plt.figure(figsize=(12,3))\n",
    "ax1 = fig.add_axes([1, 0, 1, 1])   # [left, bottom, width, height]\n",
    "\n",
    "plt.plot(t,x_1,'cd-.',linewidth=5, label=\"x at $t=-1$\");\n",
    "plt.plot(t,x__5,'m--',linewidth=5, label=\"x at $t=-0.5$\");\n",
    "plt.plot(t,x_0,'g-',linewidth=3, label=\"x at $t=0$\");\n",
    "plt.title('Time-reversed input signal, $x(t-\\tau)$ with the Impulse response, $h(t)$')\n",
    "plt.plot(t,h,'r-',linewidth=3, label=\"h(t)\");\n",
    "plt.xlabel('T');\n",
    "\n",
    "legend = plt.legend(loc='upper right');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case I: NO Overlap, $y(t) = 0$ at $-\\infty \\to t_1$, where $t_1 = 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ (h \\ast x)(t) = \\int_{-\\infty}^{0}   h(\\tau)x(t - \\tau) \\, d\\tau = 0 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case II: Leading-edge overlap, at $t_1 \\to t_2$ or at $ 0 < t < 1 $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{a}^{b} x(\\tau)h(t - \\tau) \\, d\\tau  $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "q+dORYcAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb03abb4c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-2.0, 2.0, 0.01)\n",
    "x = step(t,-0.5) - step(t,0.5)\n",
    "h = (-t + 1)*(step(t,0) - step(t,1))\n",
    "\n",
    "fig = plt.figure(figsize=(12,3))\n",
    "ax1 = fig.add_axes([1, 0, 1, 1])   # [left, bottom, width, height]\n",
    "\n",
    "plt.plot(t,x,'g-',linewidth=5, label=\"x at $t=0.5$\");\n",
    "plt.title('Time-reversed input signal, $x(t-T)$ with the Impulse response, $h(t)$')\n",
    "plt.plot(t,h,'r-',linewidth=3, label=\"h\");\n",
    "plt.xlabel('T');\n",
    "\n",
    "ax1.annotate('$t$', xy=(0.5, 0), xytext=(0.6, 0.15),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );\n",
    "\n",
    "ax1.fill_between(t,0,h,where=x>=h,color='black')\n",
    "legend = plt.legend(loc='upper right');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The overlap starts at $\\tau = 0$ and ends at $\\tau = t$. Thus,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{0}^{t}   h(\\tau)x(t - \\tau) \\, d\\tau = 0 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the above interval, $x(t -\\tau) = 1$ and $h(\\tau) = -\\tau + 1  $."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\int_{0}^{t}  (-\\tau + 1) (1) \\, d\\tau $$\n",
    "\n",
    "$$ y(t) = \\left. \\dfrac{-\\tau^2}{2} + \\tau \\right |_{0}^{t} $$\n",
    "\n",
    "$$ y(t) = \\left( \\dfrac{-t^2}{2} + t \\right) - 0 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = t - \\dfrac{t^2}{2} $$ or $$ y(t) = t  ( 1 - \\dfrac{t}{2}) $$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case III: Full overlap, at $t = 1$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "kgD4/00pFNgGpBy/AAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb052a86c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-2.0, 2.0, 0.01)\n",
    "x = step(t,0) - step(t,1)\n",
    "h = (-t + 1)*(step(t,0) - step(t,1))\n",
    "\n",
    "fig = plt.figure(figsize=(12,3))\n",
    "ax1 = fig.add_axes([1, 0, 1, 1])   # [left, bottom, width, height]\n",
    "\n",
    "plt.plot(t,x,'g-',linewidth=5, label=\"x at $t=0.5$\");\n",
    "plt.title('Time-reversed input signal, $x(t-T)$ with the Impulse response, $h(t)$')\n",
    "plt.plot(t,h,'r-',linewidth=3, label=\"h\");\n",
    "plt.xlabel('T');\n",
    "\n",
    "ax1.annotate('$t$', xy=(1, 0), xytext=(1.1, 0.15),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );\n",
    "\n",
    "ax1.annotate('$t-1$', xy=(0, 0), xytext=(-0.2, 0.15),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );\n",
    "\n",
    "ax1.fill_between(t,0,h,where=x>=h,color='black')\n",
    "legend = plt.legend(loc='upper right');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = (h \\ast x)(t) = \\int_{t-1}^{t}   h(\\tau)x(t - \\tau) \\, d\\tau $$\n",
    "\n",
    "$$ y(t) = \\int_{t-1}^{t}  (-\\tau + 1) (1) \\, d\\tau $$\n",
    "\n",
    "$$ y(t) = \\left. \\dfrac{-\\tau^2}{2} + \\tau \\right |_{t-1}^{t} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\left( \\dfrac{-t^2}{2} + t \\right) - \\left( \\dfrac{-(t-1)^2}{2} + (t-1) \\right) $$\n",
    "\n",
    "$$ y(t) = \\dfrac{-t^2}{2} + t - (t-1)(\\dfrac{-(t-1)}{2} + 1) $$\n",
    "\n",
    "$$ y(t) = \\dfrac{-t^2}{2} + t - (t-1)(\\dfrac{-t}{2} + \\dfrac{1}{2} + 1) $$\n",
    "\n",
    "$$ y(t) = \\dfrac{-t^2}{2} + t - ( \\dfrac{-t^2}{2} + \\dfrac{4t}{2} - \\dfrac{3}{2} ) $$\n",
    "\n",
    "$$ y(t) = \\dfrac{3}{2} - t $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "at $t = 1$, $$ y(t) = (h \\ast x)(t) = \\int_{0}^{1}   h(\\tau)x(t - \\tau) \\, d\\tau = \\dfrac{3}{2} - 1 = \\dfrac{1}{2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case IV: Trailing edge overlap, at $1 < t < 2$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xafec372c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-2.0, 2.0, 0.01)\n",
    "x = step(t,0.5) - step(t,1.5)\n",
    "h = (-t + 1)*(step(t,0) - step(t,1))\n",
    "\n",
    "fig = plt.figure(figsize=(12,3))\n",
    "ax1 = fig.add_axes([1, 0, 1, 1])   # [left, bottom, width, height]\n",
    "\n",
    "plt.plot(t,x,'g-',linewidth=5, label=\"x at $t=1.5$\");\n",
    "plt.title('Time-reversed input signal, $x(t-T)$ with the Impulse response, $h(t)$')\n",
    "plt.plot(t,h,'r-',linewidth=3, label=\"h\");\n",
    "plt.xlabel('T');\n",
    "\n",
    "ax1.annotate('$t-1$', xy=(0.5, 0), xytext=(0.3, 0.15),\n",
    "            arrowprops=dict(facecolor='yellow', shrink=0.05),\n",
    "            );\n",
    "\n",
    "ax1.fill_between(t,0,h,where=x>=h,color='black')\n",
    "legend = plt.legend(loc='upper right');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = (h \\ast x)(t) = \\int_{t-1}^{1}   h(\\tau)x(t - \\tau) \\, d\\tau $$\n",
    "\n",
    "$$ y(t) = \\int_{t-1}^{1}  (-\\tau + 1) (1) \\, d\\tau $$\n",
    "\n",
    "$$ y(t) = \\left. \\dfrac{-\\tau^2}{2} + \\tau \\right |_{t-1}^{1} $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = \\left( \\dfrac{-1^2}{2} +1 \\right) -  \\left( \\dfrac{-(t - 1)^2}{2} + (t - 1) \\right) $$\n",
    "\n",
    "$$ y(t) = \\dfrac{1}{2} - (t - 1)(\\dfrac{-(t -1)}{2} + 1)$$\n",
    "\n",
    "$$ y(t) = \\dfrac{1}{2} - (t - 1)(\\dfrac{-t}{2} + \\dfrac{1}{2} + 1)$$\n",
    "\n",
    "$$ y(t) = \\dfrac{1}{2} - ( \\dfrac{-t^2}{2} + \\dfrac{4t}{2} - \\dfrac{3}{2} ) $$\n",
    "\n",
    "$$ y(t) = \\dfrac{t^2}{2} - 2t + 2 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Case V: No overlap, at $ t > 2$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb057760c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-2.0, 2.0, 0.01)\n",
    "x = step(t,1) - step(t,2)\n",
    "h = (-t + 1)*(step(t,0) - step(t,1))\n",
    "\n",
    "fig = plt.figure(figsize=(12,3))\n",
    "ax1 = fig.add_axes([1, 0, 1, 1])   # [left, bottom, width, height]\n",
    "\n",
    "plt.plot(t,x,'g-',linewidth=3, label=\"x at $t=2$\");\n",
    "plt.title('Time-reversed input signal, $x(t-T)$ with the Impulse response, $h(t)$')\n",
    "plt.plot(t,h,'r-',linewidth=3, label=\"h\");\n",
    "plt.xlabel('T');\n",
    "legend = plt.legend(loc='upper right');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ y(t) = (h \\ast x)(t) = \\int_{2}^{\\infty}   h(\\tau)x(t - \\tau) \\, d\\tau  = 0 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Final Answer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore, the convolution result is:\n",
    "\n",
    "\n",
    "\n",
    "$$ y(t) = \\begin{cases} \n",
    "0                  & \\mbox{if } -\\infty < t < 0 \\\\ \n",
    "t ( 1 - \\frac{t}{2}) & \\mbox{if } 0 < t < 1 \\\\  \n",
    "\\frac{1}{2}        & \\mbox{if }  \\, t = 1 \\\\ \n",
    "\\frac{t^2}{2} - 2t + 2  & \\mbox{if }  \\, 1 < t < 2 \\\\\n",
    " 0                 & \\mbox{if }  \\, 2 < t < \\infty \\\\ \n",
    "\\end{cases} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb052734c>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(-4.0, 4.0, 0.01)\n",
    "\n",
    "# input\n",
    "x = step(t,0) - step(t,1)\n",
    "\n",
    "# impulse response\n",
    "h = (-t + 1)*(step(t,0) - step(t,1))\n",
    "\n",
    "# result\n",
    "result = np.zeros(len(t))\n",
    "result = t * (1 - 0.5*t) * (step(t,0) - step(t,1)) + (0.5*t*t - 2*t + 2)*(step(t,1) - step(t,2))\n",
    "\n",
    "fig = plt.figure(figsize=(16,4))\n",
    "\n",
    "plt.plot(t,x,'g-',linewidth=5, label='Input signal, $x(t)$');\n",
    "plt.plot(t,h,'r-',linewidth=3, label='Impulse response, $h(t)$');\n",
    "plt.plot(t,result,'b-',linewidth=3, label='Output signal, $y(t)$');\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "legend = plt.legend(loc='upper left');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By: Melvin Cabatuan melvincabatuan@gmail.com\n",
    "\n",
    "This ipython notebook is licensed under the CC-BY-NC-SA license: http://creativecommons.org/licenses/by-nc-sa/4.0/\n",
    "\n",
    "![http://i.creativecommons.org/l/by-nc-sa/3.0/88x31.png](http://i.creativecommons.org/l/by-nc-sa/3.0/88x31.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "References:\n",
    "    \n",
    "    [1] Weisstein, Eric W. \"Convolution.\" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Convolution.html \n",
    "    [2] Notes from  Prof. Alan Oppenheim. http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/"
   ]
  }
 ],
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   "language": "python",
   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
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