{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Characterization of Systems in the Time Domain\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Impulse Response\n",
    "\n",
    "The response $y(t)$ of a linear time-invariant (LTI) system $\\mathcal{H}$ to an arbitrary input signal $x(t)$ is derived in the following. The input signal can be represented as an integral when applying the [sifting-property of the Dirac impulse](../continuous_signals/standard_signals.ipynb#Dirac-Impulse)\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = \\int_{-\\infty}^{\\infty} x(\\tau) \\cdot \\delta(t-\\tau) \\; d \\tau\n",
    "\\end{equation}\n",
    "\n",
    "Introducing above relation for the the input signal $x(t)$ into the output signal $y(t) = \\mathcal{H} \\{ x(t) \\}$ of the system yields\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = \\mathcal{H} \\left\\{ \\int_{-\\infty}^{\\infty} x(\\tau) \\cdot \\delta(t-\\tau) \\; d \\tau \\right\\}\n",
    "\\end{equation}\n",
    "\n",
    "where $\\mathcal{H} \\{ \\cdot \\}$ denotes the system response operator. The integration and system response operator can be exchanged under the assumption that the system is linear\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = \\int_{-\\infty}^{\\infty}  x(\\tau) \\cdot \\mathcal{H} \\left\\{ \\delta(t-\\tau) \\right\\} \\; d \\tau \n",
    "\\end{equation}\n",
    "\n",
    "where $\\mathcal{H} \\{\\cdot\\}$ was only applied to the Dirac impulse, since $x(\\tau)$ can be regarded as constant factor with respect to the time $t$. It becomes evident that the response of a system to a Dirac impulse plays an important role in the calculation of the output signal for arbitrary input signals. \n",
    "\n",
    "The response of a system to a Dirac impulse as input signal is denoted as [*impulse response*](https://en.wikipedia.org/wiki/Impulse_response). It is defined as\n",
    "\n",
    "\\begin{equation}\n",
    "h(t) = \\mathcal{H} \\left\\{ \\delta(t) \\right\\}\n",
    "\\end{equation}\n",
    "\n",
    "If the system is time-invariant, the response to a shifted Dirac impulse is $\\mathcal{H} \\left\\{ \\delta(t-\\tau) \\right\\} = h(t-\\tau)$. Hence, for an LTI system we finally get\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = \\int_{-\\infty}^{\\infty}  x(\\tau) \\cdot h(t-\\tau) \\; d \\tau \n",
    "\\end{equation}\n",
    "\n",
    "Due to its relevance in the theory of LTI systems, this operation is explicitly termed as [*convolution*](https://en.wikipedia.org/wiki/Convolution). It is commonly abbreviated by $*$, hence for above integral we get $y(t) = x(t) * h(t)$. In some books the mathematically more precise nomenclature $y(t) = (x*h)(t)$ is used, since $*$ is the operator acting on the two signals $x$ and $h$ with regard to time $t$.\n",
    "\n",
    "In can be concluded that the properties of an LTI system are entirely characterized by its impulse response. The response $y(t)$ of a system to an arbitrary input signal $x(t)$ is given by the convolution of the input signal $x(t)$ with its impulse response $h(t)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following example considers an LTI system whose relation between input $x(t)$ and output $y(t)$ is given by an ordinary differential equation (ODE) with constant coefficients\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) + \\frac{d}{dt} y(t) = x(t)\n",
    "\\end{equation}\n",
    "\n",
    "The system response is computed for the input signal $x(t) = e^{- 2 t} \\cdot \\epsilon(t)$ by \n",
    "\n",
    "1. explicitly solving the ODE and by \n",
    "2. computing the impulse response $h(t)$ and convolution with the input signal.\n",
    "\n",
    "The solution should fulfill the initial conditions $y(t)\\big\\vert_{t = 0-} = 0$ and $\\frac{d}{dt}y(t)\\big\\vert_{t = 0-} = 0$ due to causality.\n",
    "\n",
    "First the ODE is defined in `SymPy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$y{\\left (t \\right )} + \\frac{d}{d t} y{\\left (t \\right )} = x{\\left (t \\right )}$$"
      ],
      "text/plain": [
       "       d              \n",
       "y(t) + ──(y(t)) = x(t)\n",
       "       dt             "
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "%matplotlib inline\n",
    "sym.init_printing()\n",
    "\n",
    "t = sym.symbols('t', real=True)\n",
    "x = sym.Function('x')(t)\n",
    "y = sym.Function('y')(t)\n",
    "\n",
    "ode = sym.Eq(y + y.diff(t), x)\n",
    "ode"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ODE is solved for the given input signal in order to calculate the output signal. The integration constant is calculated such that the solution fulfills the initial conditions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$y{\\left (t \\right )} = \\left(1 - e^{- t}\\right) e^{- t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "       ⎛     -t⎞  -t             \n",
       "y(t) = ⎝1 - ℯ  ⎠⋅ℯ  ⋅Heaviside(t)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution = sym.dsolve(ode.subs(x, sym.exp(-2*t)*sym.Heaviside(t)))\n",
    "integration_constants = sym.solve(\n",
    "    (solution.rhs.limit(t, 0, '-'), solution.rhs.diff(t).limit(t, 0, '-')), 'C1')\n",
    "y1 = solution.subs(integration_constants)\n",
    "y1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets plot the output signal derived by explicit solution of the ODE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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       "<Figure size 614.4x460.8 with 1 Axes>"
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     "metadata": {},
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   ],
   "source": [
    "sym.plot(y1.rhs, (t,-1,10), ylabel=r'$y(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The impulse response $h(t)$ is computed by solving the ODE for a Dirac impulse as input signal, $x(t) = \\delta(t)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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      "text/latex": [
       "$$h{\\left (t \\right )} = e^{- t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "        -t             \n",
       "h(t) = ℯ  ⋅Heaviside(t)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = sym.Function('h')(t)\n",
    "solution2 = sym.dsolve(ode.subs(x, sym.DiracDelta(t)).subs(y, h))\n",
    "integration_constants = sym.solve((solution2.rhs.limit(\n",
    "    t, 0, '-'), solution2.rhs.diff(t).limit(t, 0, '-')), 'C1')\n",
    "h = solution2.subs(integration_constants)\n",
    "h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets plot the impulse response $h(t)$ of the LTI system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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      "text/plain": [
       "<Figure size 614.4x460.8 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sym.plot(h.rhs, (t,-1,10), ylabel=r'$h(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As alternative to the explicit solution of the ODE, the system response is computed by evaluating the convolution $y(t) = x(t) * h(t)$. Since `SymPy` cannot handle the Heaviside function properly in integrands, the convolution integral is first simplified. Both the input signal $x(t)$ and the impulse response $h(t)$ are causal signals. Hence, the convolution integral degenerates to\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = \\int_{0}^{t} x(\\tau) \\cdot h(t - \\tau) \\; d\\tau\n",
    "\\end{equation}\n",
    "\n",
    "for $t \\geq 0$. Note that $y(t) = 0$ for $t<0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\left(- e^{- t} + e^{- 2 t}\\right) \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       " ⎛   -t    -2⋅t⎞             \n",
       "-⎝- ℯ   + ℯ    ⎠⋅Heaviside(t)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tau = sym.symbols('tau', real=True)\n",
    "\n",
    "y2 = sym.integrate(sym.exp(-2*tau) * h.rhs.subs(t, t-tau), (tau, 0, t))\n",
    "y2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets plot the output signal derived by evaluation of the convolution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   "source": [
    "sym.plot(y2, (t,-1,10), ylabel=r'$y(t)$');"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Compare the output signal derived by explicit solution of the ODE with the signal derived by convolution. Are both equal?\n",
    "* Check if the impulse response $h(t)$ is a solution of the ODE by manual calculation. Hint $\\frac{d}{dt} \\epsilon(t) = \\delta(t)$.\n",
    "* Check the solution of the convolution integral by manual calculation including the Heaviside functions."
   ]
  },
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   "source": [
    "**Copyright**\n",
    "\n",
    "The notebooks are provided as [Open Educational Resource](https://de.wikipedia.org/wiki/Open_Educational_Resources). Feel free to use the notebooks for your own educational purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Lecture Notes on Signals and Systems* by Sascha Spors."
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