{
"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",
"It 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": {
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"text/latex": [
"$\\displaystyle y{\\left(t \\right)} + \\frac{d}{d t} y{\\left(t \\right)} = x{\\left(t \\right)}$"
],
"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",
"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": [
"$\\displaystyle y{\\left(t \\right)} = \\left(1 - e^{- t}\\right) e^{- t} \\theta\\left(t\\right)$"
],
"text/plain": [
" ⎛ -t⎞ -t \n",
"y(t) = ⎝1 - ℯ ⎠⋅ℯ ⋅θ(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": {
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\n",
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"