{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"nbsphinx": "hidden"
},
"source": [
"# Characterization of Discrete 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 linear time-invariant (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 $H(e^{j \\Omega})$ the phase delay in samples is defined as follows\n",
"\n",
"\\begin{equation}\n",
"t_p(\\Omega) = - \\frac{\\varphi(e^{j \\Omega})}{\\Omega}\n",
"\\end{equation}\n",
"\n",
"where $\\varphi(e^{j \\Omega}) = \\arg \\{ H(e^{j \\Omega}) \\}$ denotes the phase of the transfer function. The phase delay quantifies the delay of a single harmonic exponential signal $e^{j \\Omega k}$ with normalized 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, the phase delay may not be defined for $\\Omega = 0$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Example - Phase delay of second-order recursive system**\n",
"\n",
"The phase delay $t_p(\\Omega)$ for the before introduced [second-order recursive LTI system](difference_equation.ipynb#Second-Order-System) with transfer function\n",
"\n",
"\\begin{equation}\n",
"H(z) = \\frac{1}{1 - z^{-1} + \\frac{1}{2} z^{-2}}\n",
"\\end{equation}\n",
"\n",
"is computed. First the transfer function is defined"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\frac{1}{1 - \\frac{1}{z} + \\frac{1}{2 z^{2}}}$"
],
"text/plain": [
" 1 \n",
"────────────\n",
" 1 1 \n",
"1 - ─ + ────\n",
" z 2\n",
" 2⋅z "
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import sympy as sym\n",
"sym.init_printing()\n",
"%matplotlib inline\n",
"\n",
"z = sym.symbols('z', complex=True)\n",
"W = sym.symbols('Omega', real=True)\n",
"H = 1 / (1 - z**(-1) + sym.Rational(1, 2)*z**(-2))\n",
"H"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now the phase delay $t_p(\\Omega)$ is computed and plotted for illustration"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
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"text/plain": [
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"source": [
"phi = sym.arg(H.subs(z, sym.exp(sym.I*W)))\n",
"tp = -phi/W\n",
"\n",
"sym.plot(tp, (W, -sym.pi, sym.pi), xlabel='$\\Omega$', ylabel='$t_p(\\Omega)$ in samples')"
]
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"source": [
"### Group Delay\n",
"\n",
"The group delay is defined as the derivative of the phase with respect to the normalized frequency\n",
"\n",
"\\begin{equation}\n",
"t_g(\\Omega) = - \\frac{\\mathrm{d} \\varphi(e^{j \\Omega})}{\\mathrm{d} \\Omega}\n",
"\\end{equation}\n",
"\n",
"given in samples.\n",
"\n",
"The group delay quantifies the delay the amplitude envelope of a group of exponential signals observes when passing through a system. The negative sign in above definition results in a positive group delay for a system imposing a delay onto the input signal. Note that the [phase](https://en.wikipedia.org/wiki/Instantaneous_phase) $\\varphi(e^{j \\Omega})$ is in general only unique for $- \\pi < \\varphi(e^{j \\Omega}) \\leq \\pi$. If the phase exceeds this range it is wrapped back. For meaningful results it is required to unwrap the phase before computing the group delay."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Example - Phase delay of second-order recursive system**\n",
"\n",
"The group delay $t_g(\\Omega)$ of above second-order recursive system is computed and plotted"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
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"source": [
"tg = - sym.diff(phi, W)\n",
"sym.plot(tg, (W, -sym.pi, sym.pi), xlabel='$\\Omega$', ylabel='$t_g(\\Omega)$ in samples')"
]
},
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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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