{ "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", "\$$\n", "x(t) = \\int_{-\\infty}^{\\infty} x(\\tau) \\cdot \\delta(t-\\tau) \\; d \\tau\n", "\$$\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", "\$$\n", "y(t) = \\mathcal{H} \\left\\{ \\int_{-\\infty}^{\\infty} x(\\tau) \\cdot \\delta(t-\\tau) \\; d \\tau \\right\\}\n", "\$$\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", "\$$\n", "y(t) = \\int_{-\\infty}^{\\infty} x(\\tau) \\cdot \\mathcal{H} \\left\\{ \\delta(t-\\tau) \\right\\} \\; d \\tau \n", "\$$\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", "\$$\n", "h(t) = \\mathcal{H} \\left\\{ \\delta(t) \\right\\}\n", "\$$\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", "\$$\n", "y(t) = \\int_{-\\infty}^{\\infty} x(\\tau) \\cdot h(t-\\tau) \\; d \\tau \n", "\$$\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", "\$$\n", "y(t) + \\frac{d}{dt} y(t) = x(t)\n", "\$$\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": [ "$\\displaystyle y{\\left(t \\right)} + \\frac{d}{d t} y{\\left(t \\right)} = x{\\left(t \\right)}$" ], "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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"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": { "application/pdf": 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\n", 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