{
"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": [
"## Phase and Group Delay\n",
"\n",
"The [phase and group delay](https://en.wikipedia.org/wiki/Group_delay_and_phase_delay) characterize the phase and delay properties of an LTI system. Both quantify the frequency dependent delay that is imprinted on a signal when passing through a system. In many applications the delay introduced by a system should be as small as possible or within reasonable limits."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Phase Delay\n",
"\n",
"For an LTI system with transfer function $H(j \\omega)$ the phase delay is defined as follows\n",
"\n",
"\\begin{equation}\n",
"t_p(\\omega) = - \\frac{\\varphi(j \\omega)}{\\omega}\n",
"\\end{equation}\n",
"\n",
"where $\\varphi(j \\omega) = \\arg \\{ H(j \\omega) \\}$ denotes the phase of the transfer function. The phase delay quantifies the delay of a single harmonic exponential signal $e^{j \\omega t}$ with frequency $\\omega$ when passing through the system. The negative sign in the definition of the phase delay results in a positive phase delay $t_p(\\omega) > 0$ when a signal is delayed by a system. Note that the phase delay may not be defined for $\\omega = 0$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Example**\n",
"\n",
"As example, the phase delay $t_p(\\omega)$ is computed for the [2nd order low-pass filter introduced before](../laplace_transform/network_analysis.ipynb#Example:-Second-Order-Low-Pass-Filter). First the transfer function $H(j \\omega)$ is defined in `SymPy`"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\frac{1}{- C L \\omega^{2} + i C R \\omega + 1}$"
],
"text/plain": [
" 1 \n",
"──────────────────────\n",
" 2 \n",
"- C⋅L⋅ω + ⅈ⋅C⋅R⋅ω + 1"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"%matplotlib inline\n",
"import sympy as sym\n",
"sym.init_printing()\n",
"\n",
"L, R, C = sym.symbols('L R C', positive=True)\n",
"w = sym.symbols('omega', real=True)\n",
"s = sym.I * w\n",
"\n",
"H = 1 / (C*L*s**2 + C*R*s + 1)\n",
"H"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now the phase delay $t_p(\\omega)$ is computed"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle - \\frac{\\arg{\\left(\\frac{1}{- C L \\omega^{2} + i C R \\omega + 1} \\right)}}{\\omega}$"
],
"text/plain": [
" ⎛ 1 ⎞ \n",
"-arg⎜──────────────────────⎟ \n",
" ⎜ 2 ⎟ \n",
" ⎝- C⋅L⋅ω + ⅈ⋅C⋅R⋅ω + 1⎠ \n",
"─────────────────────────────\n",
" ω "
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"phi = sym.arg(H)\n",
"tp = - phi/w\n",
"tp"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"and the result is visualized using the normalized values $R=1$, $L=0.5$ and $C=0.4$ for the elements of the low-pass filter"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
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\n",
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"