{ "cells": [ { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "source": [ "# Periodic Signals\n", "\n", "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Comunications 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": [ "## Relation between Spectrum and Fourier Series\n", "\n", "The Fourier transform $X(j \\omega) = \\mathcal{F} \\{ x(t) \\}$ of a periodic signal $x(t)$, [as derived before](spectrum.ipynb#Fourier-Transform), is a line spectrum. It consists of a weighted series of Dirac impulses. Periodic functions can be represented alternatively by a [Fourier series](https://en.wikipedia.org/wiki/Fourier_series). The relation between the spectrum $X(j \\omega)$ of a periodic signal and its Fourier series coefficients is derived in the following.\n", "\n", "The complex Fourier series of a periodic signal $x(t)$ is defined as\n", "\n", "\\begin{equation}\n", "x(t) = \\sum_{n = - \\infty}^{\\infty} X_n \\, e^{j n \\frac{2 \\pi}{T_\\text{p}} t}\n", "\\end{equation}\n", "\n", "where $T_\\text{p} > 0$ denotes the period of the signal and $X_n$ the Fourier series coefficients of $x(t)$. The Fourier series represents the signal as weighted superposition of complex exponential signals. The weights (expansion coefficients) $X_n$ are given as\n", "\n", "\\begin{equation}\n", "X_n = \\frac{1}{T_\\text{p}} \\int_{0}^{T_\\text{p}} x(t) \\, e^{- j n \\frac{2 \\pi}{T_\\text{p}} t} \\; dt\n", "\\end{equation}\n", "\n", "Introducing the [Fourier transform $X(j \\omega)$ of a periodic signal](spectrum.ipynb#Fourier-Transform) into the [inverse Fourier transform](../fourier_transform/definition.ipynb#Definition) yields\n", "\n", "\\begin{align}\n", "x(t) &= \\frac{1}{2 \\pi} \\int_{-\\infty}^{\\infty} X_0(j \\omega) \\cdot {\\bot \\!\\! \\bot \\!\\! \\bot} \\left( \\frac{\\omega T_\\text{p}}{2 \\pi} \\right) \\, e^{j \\omega t} \\; d \\omega \\\\\n", "&= \\frac{1}{T_\\text{p}} \\sum_{\\mu = -\\infty}^{\\infty} X_0 \\left( j \\, \\mu \\frac{2 \\pi}{T_\\text{p}} \\right) \\, e^{j \\, \\mu \\frac{2 \\pi}{T_\\text{p}} t}\n", "\\end{align}\n", "\n", "where $X_0(j \\omega) = \\mathcal{F} \\{ x_0(t) \\}$ denotes the Fourier transform of one period $x_0(t)$ of the periodic signal. Note, the [definition of the Dirac comb](spectrum.ipynb#The-Dirac-Comb) and the multiplication property of the Dirac impulse was used to derive the last equality. Comparing this result with the definition of the Fourier series reveals that both are equal for\n", "\n", "\\begin{equation}\n", "X_n = X_0 \\left( j \\, n \\frac{2 \\pi}{T_\\text{p}} \\right)\n", "\\end{equation}\n", "\n", "The Fourier series coefficients $X_n$ of a periodic signal are equal to the Fourier transform $X_0(j \\omega)$ of one period of the signal at the frequencies $\\omega = n \\frac{2 \\pi}{T_\\text{p}}$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Example**\n", "\n", "The Fourier series coefficients of the pulse train can be derived from the [Fourier transform of the pulse train](spectrum.ipynb#Fourier-Transform-of-the-Pulse-Train) as\n", "\n", "\\begin{equation}\n", "X_n = T \\, e^{-j \\omega \\frac{T}{2}} \\cdot \\text{sinc} \\left( \\frac{\\omega T}{2} \\right) \\bigg\\vert_{\\omega = n \\frac{2 \\pi}{T_\\text{p}}} = T \\, e^{-j n \\pi \\frac{T}{T_\\text{p}}} \\cdot \\text{sinc} \\left( n \\pi \\frac{T}{T_\\text{p}} \\right)\n", "\\end{equation}\n", "\n", "With these coefficients the pulse train can be represented by the Fourier series\n", "\n", "\\begin{equation}\n", "x(t) = \\sum_{n = -\\infty}^{\\infty} T \\, e^{-j n \\pi \\frac{T}{T_\\text{p}}} \\cdot \\text{sinc} \\left( n \\pi \\frac{T}{T_\\text{p}} \\right) \\, e^{j n \\frac{2 \\pi}{T_\\text{p}} t}\n", "\\end{equation}\n", "\n", "This series cannot be evaluated numerically due to its infinite limits. The series has to be truncated to a finite number of summands in a practical implementation. The consequences of truncating the series are illustrated in the following. First the weights $X_n$ of the Fourier series are defined" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle 2 e^{- \\frac{2 i \\pi n}{5}} \\operatorname{sinc}{\\left(\\frac{2 \\pi n}{5} \\right)}$" ], "text/plain": [ " -2⋅ⅈ⋅π⋅n \n", " ───────── \n", " 5 ⎛2⋅π⋅n⎞\n", "2⋅ℯ ⋅sinc⎜─────⎟\n", " ⎝ 5 ⎠" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import sympy as sym\n", "%matplotlib inline\n", "sym.init_printing()\n", "\n", "n = sym.symbols('n', integer=True)\n", "t = sym.symbols('t', real=True)\n", "\n", "T = 2\n", "Tp = 5\n", "\n", "Xn = T * sym.exp(-sym.I * n * sym.pi * T/Tp) * sym.sinc(n * sym.pi * T/Tp)\n", "Xn" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now the Fourier series is evaluated for a finite upper and lower limit $N$\n", "\n", "\\begin{equation}\n", "x_N(t) = \\sum_{n = -N}^{N} X_n \\, e^{j n \\frac{2 \\pi}{T_\\text{p}} t}\n", "\\end{equation}" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "application/pdf": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 15\n", "\n", "x = sym.Sum(Xn * sym.exp(sym.I*n*2*sym.pi/Tp*t), (n, -N, N)).doit()\n", "sym.plot(x, (t, 0, 10), xlabel='$t$', ylabel='$x_N(t)$')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Overshoots can be observed at the discontinuities of the pulse train. The relative magnitude of these overshoots remains at constantly 9% even when increasing the limits of the truncated Fourier series expansion. This effect is known as [*Gibbs phenomenon*](https://en.wikipedia.org/wiki/Gibbs_phenomenon). Truncated Fourier series are therefore not very well suited for the approximation of signals with discontinuities." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Exercise**\n", "\n", "* Examine the properties of the truncated Fourier series when you increase the limit $N$ in above example. Note: The evaluation of the Fourier series may take a while due to involved numerical complexity." ] }, { "cell_type": "markdown", "metadata": { "nbsphinx": "hidden" }, "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." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 1 }