{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"nbsphinx": "hidden"
},
"source": [
"# Characterization of Systems in the Spectral 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": [
"## The Transfer Function\n",
"\n",
"Different time-domain representations are used to characterize linear time-invariant (LTI) systems. For instance, the constant coefficients of ordinary partial differential equations (ODEs), the [impulse response](../systems_time_domain/impulse_response.ipynb) and the [step response](../systems_time_domain/step_response.ipynb). The [transfer function](../systems_time_domain/eigenfunctions.ipynb#Transfer-Function) denotes the complex factor $H(s)$ an eigenfunction $e^{s t}$ at the input of an LTI system is weighted with when passing through the system. Since the Laplace transform decomposes signals with respect to these eigenfunctions, the transfer function constitutes a representation of an LTI system in the spectral (Laplace or Fourier) domain. The links between the transfer function and the time-domain representations of LTI systems are discussed in this section."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Relation to Impulse Response and In-/Output Signal\n",
"\n",
"It was already outlined in the context of the [convolution theorem of the Fourier](../fourier_transform/theorems.ipynb#Convolution-Theorem) and [Laplace transform](../laplace_transform/theorems.ipynb#Convolution-Theorem), that an LTI system can be represented equivalently in the temporal and spectral domain. In the temporal domain the output $y(t)$ of an LTI system is given by convolving the input signal $x(t)$ with the [impulse response](../systems_time_domain/impulse_response.ipynb) $h(t)$ of the system\n",
"\n",
"\\begin{equation}\n",
"y(t) = x(t) * h(t)\n",
"\\end{equation}\n",
"\n",
"Applying the convolution theorem of the Laplace transform yields\n",
"\n",
"\\begin{equation}\n",
"Y(s) = X(s) \\cdot H(s)\n",
"\\end{equation}\n",
"\n",
"where $X(s) = \\mathcal{L} \\{ x(t) \\}$ and $Y(s) = \\mathcal{L} \\{ y(t) \\}$ are the Laplace transforms of the input and output signal, respectively. The transfer function $H(s)$ is given as the Laplace transform of the impulse response $h(t)$\n",
"\n",
"\\begin{equation}\n",
"H(s) = \\mathcal{L} \\{ h(t) \\}\n",
"\\end{equation}\n",
"\n",
"This may also be concluded from the [previously derived link between the impulse response and the transfer function](../systems_time_domain/eigenfunctions.ipynb#Link-between-Transfer-Function-and-Impulse-Response). The derived results show that an LTI system can be fully characterized either in the temporal or spectral domain by its impulse response $h(t)$ or transfer function $H(s)$ or $H(j \\omega)$, respectively.\n",
"\n",
"![LTI system in the temporal and spectral domain](LTI_system_time_spectral_domain.png)\n",
"\n",
"It can furthermore be concluded that the transfer function is given as the quotient between output $Y(s)$ and input signal $X(s)$ in the Laplace domain\n",
"\n",
"\\begin{equation}\n",
"H(s) = \\frac{Y(s)}{X(s)}\n",
"\\end{equation}\n",
"\n",
"for $X(s) \\neq 0$. The same relations hold in the Fourier domain if the region of convergence (ROC) of the impulse response includes the imaginary axis $\\Re \\{ s \\} = 0$. In general, the transfer function can be derived by divison of the spectra of the output and input signal. This can be used to measure the transfer function of a system by specific input signals that fulfill $X(s) \\neq 0$ with $X(s) \\approx$ constant."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Example**\n",
"\n",
"The [transfer function $H(s)$ of the 2nd order low-pass filter](../systems_time_domain/network_analysis.ipynb#Transfer-Function) was derived as\n",
"\n",
"\\begin{equation}\n",
"H(s) = \\frac{1}{C L s^2 + C R s + 1}\n",
"\\end{equation}\n",
"\n",
"According to above findings, the impulse response $h(t)$ of the low-pass is given by inverse Laplace transform of the transfer function $H(s)$. First the transfer function is defined in `SymPy`"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\frac{1}{C L s^{2} + C R s + 1}$"
],
"text/plain": [
" 1 \n",
"──────────────────\n",
" 2 \n",
"C⋅L⋅s + C⋅R⋅s + 1"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import sympy as sym\n",
"%matplotlib inline\n",
"sym.init_printing()\n",
"\n",
"R, L, C = sym.symbols('R L C', positive=True)\n",
"s = sym.symbols('s', complex=True)\n",
"\n",
"H = 1/(C*L*s**2 + C*R*s + 1)\n",
"H"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then the inverse Laplace transform is computed for the specific normalized values $L = .5$, $R = 1$, $C = .4$ of the network elements"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\frac{5 e^{- t} \\sin{\\left(2 t \\right)} \\theta\\left(t\\right)}{2}$"
],
"text/plain": [
" -t \n",
"5⋅ℯ ⋅sin(2⋅t)⋅Heaviside(t)\n",
"───────────────────────────\n",
" 2 "
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = sym.symbols('t', real=True)\n",
"RLC = {R: 1, L: sym.Rational('.5'), C: sym.Rational('.4')}\n",
"\n",
"h = sym.inverse_laplace_transform(H.subs(RLC), s, t)\n",
"h"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This result is plotted for illustration"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
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\n",
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],
"text/plain": [
"