Difference between revisions of "Fourier Series"
(→Common Exponential Fourier Series Pairs) |
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\mbox{Basic Signals} & | \mbox{Basic Signals} & | ||
x(t), y(t), z(t);~T_x=T_y=T & | x(t), y(t), z(t);~T_x=T_y=T & | ||
− | X[k], Y[k], Z[k];~\omega_0=\frac{2\pi}{T}\\ \hline | + | \mathbb{X}[k], \mathbb{Y}[k], \mathbb{Z}[k];~\omega_0=\frac{2\pi}{T}\\ \hline |
% | % | ||
\mbox{Linearity} & | \mbox{Linearity} & | ||
z(t)=Ax(t)+By(t) & | z(t)=Ax(t)+By(t) & | ||
− | Z[k]= | + | \mathbb{Z}[k]=A\mathbb{X}[k]+B\mathbb{Y}[k]\\ \hline |
% | % | ||
\mbox{Time Shifting} & | \mbox{Time Shifting} & | ||
z(t)=x\left(t-t_0\right) & | z(t)=x\left(t-t_0\right) & | ||
− | Z[k]=X[k]e^{-jk\omega_0t_0}\\ \hline | + | \mathbb{Z}[k]=\mathbb{X}[k]e^{-jk\omega_0t_0}\\ \hline |
% | % | ||
\mbox{Frequency Shifting} & | \mbox{Frequency Shifting} & | ||
z(t)=e^{jk_0\omega_0t}x(t) & | z(t)=e^{jk_0\omega_0t}x(t) & | ||
− | Z[k]=X[k-k_0]\\ \hline | + | \mathbb{Z}[k]=X[k-k_0]\\ \hline |
% | % | ||
\mbox{Conjugation} & | \mbox{Conjugation} & | ||
z(t)=x^*(t) & | z(t)=x^*(t) & | ||
− | Z[k]=X^*[-k]\\ \hline | + | \mathbb{Z}[k]=X^*[-k]\\ \hline |
% | % | ||
\mbox{Time Reversal} & | \mbox{Time Reversal} & | ||
z(t)=x(-t) & | z(t)=x(-t) & | ||
− | Z[k]=X[-k]\\ \hline | + | \mathbb{Z}[k]=X[-k]\\ \hline |
% | % | ||
\mbox{Time Scaling} & | \mbox{Time Scaling} & | ||
z(t)=x(\alpha t), \alpha>0 & | z(t)=x(\alpha t), \alpha>0 & | ||
− | Z[k]=X[k], T_z=\frac{T_x}{\alpha}\\ \hline | + | \mathbb{Z}[k]=\mathbb{X}[k], T_z=\frac{T_x}{\alpha}\\ \hline |
% | % | ||
\mbox{Periodic Convolution} & | \mbox{Periodic Convolution} & | ||
z(t)={\displaystyle \int_{T}x(\tau)y(t-\tau)d\tau} & | z(t)={\displaystyle \int_{T}x(\tau)y(t-\tau)d\tau} & | ||
− | Z[k]= | + | \mathbb{Z}[k]=T\mathbb{X}[k]\mathbb{Y}[k]\\ \hline |
% | % | ||
\mbox{Multiplication} & | \mbox{Multiplication} & | ||
z(t)=x(t)y(t) & | z(t)=x(t)y(t) & | ||
− | {\displaystyle Z[k]=\sum_{l=-\infty}^{\infty}X[l]Y[k-l]}\\ \hline | + | {\displaystyle \mathbb{Z}[k]=\sum_{l=-\infty}^{\infty}X[l]Y[k-l]}\\ \hline |
% | % | ||
\mbox{Differentiation} & | \mbox{Differentiation} & | ||
z(t)=\frac{dx(t)}{dt} & | z(t)=\frac{dx(t)}{dt} & | ||
− | Z[k]=jk\ | + | \mathbb{Z}[k]=jk\omega_x\mathbb{X}[k]\\ \hline |
% | % | ||
\mbox{Integration} & | \mbox{Integration} & | ||
{\displaystyle z(t)=\int_{-\infty}^{t}x(\tau)~d\tau}, X[0]=0& | {\displaystyle z(t)=\int_{-\infty}^{t}x(\tau)~d\tau}, X[0]=0& | ||
− | Z[k]=\left(\frac{1}{jk\omega_x}\right)X[k]\\ \hline | + | \mathbb{Z}[k]=\left(\frac{1}{jk\omega_x}\right)\mathbb{X}[k]\\ \hline |
% | % | ||
\mbox{Properties of Real Signals} & | \mbox{Properties of Real Signals} & | ||
Line 217: | Line 217: | ||
\renewcommand{\arraystretch}{1.0} | \renewcommand{\arraystretch}{1.0} | ||
\begin{array}{l} | \begin{array}{l} | ||
− | Z[k]=Z^*[-k]\\ | + | \mathbb{Z}[k]=Z^*[-k]\\ |
− | \Re\{Z[k]\}=\Re\{Z[-k]\}\\ | + | \Re\{\mathbb{Z}[k]\}=\Re\{Z[-k]\}\\ |
− | \Im\{Z[k]\}=-\Im\{Z[-k]\}\\ | + | \Im\{\mathbb{Z}[k]\}=-\Im\{Z[-k]\}\\ |
− | |Z[k]|=|Z[-k]|\\ | + | |\mathbb{Z}[k]|=|Z[-k]|\\ |
− | \measuredangle Z[k]=-\measuredangle Z[-k] | + | \measuredangle \mathbb{Z}[k]=-\measuredangle Z[-k] |
\end{array} | \end{array} | ||
\renewcommand{\arraystretch}{2.0} | \renewcommand{\arraystretch}{2.0} | ||
Line 227: | Line 227: | ||
% | % | ||
\mbox{Properties of Real, Even Signals} & | \mbox{Properties of Real, Even Signals} & | ||
− | z(t)\mbox{ real and even}&Z[k]\mbox{ real and even}\\ \hline | + | z(t)\mbox{ real and even}&\mathbb{Z}[k]\mbox{ real and even}\\ \hline |
% | % | ||
\mbox{Properties of Real, Odd Signals} & | \mbox{Properties of Real, Odd Signals} & | ||
− | z(t)\mbox{ real and odd}&Z[k]\mbox{ imaginary and odd}\\ \hline | + | z(t)\mbox{ real and odd}&\mathbb{Z}[k]\mbox{ imaginary and odd}\\ \hline |
% | % | ||
\mbox{Isolation of Even Part} & | \mbox{Isolation of Even Part} & | ||
z(t)=x_e(t)\mbox{ with x(t) real}& | z(t)=x_e(t)\mbox{ with x(t) real}& | ||
− | Z[k]=\Re\{X[k]\} | + | \mathbb{Z}[k]=\Re\{\mathbb{X}[k]\} |
\\ \hline | \\ \hline | ||
% | % | ||
\mbox{Isolation of Odd Part} & | \mbox{Isolation of Odd Part} & | ||
z(t)=x_o(t)\mbox{ with x(t) real}& | z(t)=x_o(t)\mbox{ with x(t) real}& | ||
− | Z[k]=j\Im\{X[k]\} | + | \mathbb{Z}[k]=j\Im\{\mathbb{X}[k]\} |
\\ \hline | \\ \hline | ||
% | % | ||
\mbox{Parseval's Relation (Power)} & | \mbox{Parseval's Relation (Power)} & | ||
{\displaystyle P_{ave}=\frac{1}{T}\int_{T}|z(t)|^2~dt}& | {\displaystyle P_{ave}=\frac{1}{T}\int_{T}|z(t)|^2~dt}& | ||
− | {\displaystyle P_{ave}=\sum_{k=-\infty}^{\infty}|Z[k]|^2} | + | {\displaystyle P_{ave}=\sum_{k=-\infty}^{\infty}|\mathbb{Z}[k]|^2} |
\end{array} | \end{array} | ||
\end{align*} | \end{align*} | ||
$$ | $$ | ||
+ | |||
==Examples== | ==Examples== | ||
Revision as of 14:17, 28 September 2021
Contents
Introduction
This document takes a look at different ways of representing real periodic signals using the Fourier series. It will provide translation tables among the different representations as well as (eventually) example problems using Fourier series to solve a mechanical system and an electrical system, respectively.
Notation
The notation for this page has been updated to match the zyBook for Fall of 2021 with two exceptions - the phase angle for the cosine series is represented by $$\theta_n$$ versus $$\phi_n$$ and the exponential Fourier Series coefficients is represented by $$\mathbb{X}[n]$$ instead of $$\bf{x}_n$$
Synthesis Equations
There are three primary Fourier series representations of a periodic signal \(f(t)\) with period \(T\) and fundamental frequency \(\omega_0=\frac{2\pi}{T}\); note that different sources may use different symbols for the series coefficients!
In the series above, \(a_0\), \(a_n\), \(b_n\), \(c_0\), \(c_n\), and \(\theta_n\) are real numbers while \(\mathbb{X}[n]\) may be complex.
Analysis Equations
The formulas for obtaining the Fourier series coefficients are:
Translation Table
The table below summarizes how to get one set of Fourier Series coefficients from any other representation. Note that it is assumed the function being represented is real - meaning \(a_n=a_{-n}^*\). Also, \(n>0\) in the table. The core equations at use in the translation table are:
Common Fourier Series Pairs and Properties
The next two subsections present tables of common Fourier series pairs and Fourier series properties. The information in these tables has been adapted from:
- Signals and Systems, 2nd ed. Simon Haykin and Barry Van Veen. John Wiley & Sons, Hoboken, NJ, 2005. pp. 774, 777.
- Signals and Systems, 2nd ed. Alan V. Oppenheim and Alan S. Willsky with S. Hamid Nawab. Prentice Hall, Upper Saddle River, NJ, 1997. p. 206.
Common Exponential Fourier Series Pairs
Note in the table below, the discrete form of the Dirac delta function $$\delta[k]$$ is used. The definition of this function is: $$\begin{align*} \delta[k]&= \left\{ \begin{array}{cl} k=0 & 1\\ k\neq 0 & 0 \end{array} \right. \end{align*}$$
$$ \renewcommand{\arraystretch}{2.1} \begin{align*} \begin{array}{l l l} \mbox{Name} & \mbox{Signal} & \mbox{Fourier Series} \renewcommand{\arraystretch}{2.1} \\ \hline % \mbox{Basic Signal} & x(t)\mbox{, Period $T$} & \mathbb{X}[k]\mbox{, $\omega_0=\frac{2\pi}{T}$}\\ \hline % \mbox{Complex Exponential}& {\displaystyle x(t)=e^{jp\omega_0t}}& \mathbb{X}[k]=\delta[k-p]\\ \hline % \mbox{Cosine}& {\displaystyle x(t)=\cos(p\omega_0t)}& {\displaystyle \mathbb{X}[k]=\frac{1}{2}\left(\delta[k-p]+\delta[k+p]\right) }\\ \hline % \mbox{Sine}& {\displaystyle x(t)=\sin(p\omega_0t)}& {\displaystyle \mathbb{X}[k]=\frac{1}{j2}\left(\delta[k-p]-\delta[k+p]\right) }\\ \hline % \mbox{Constant}& {\displaystyle x(t)=c}& \mathbb{X}[k]=c\delta[k]\\ \hline % \mbox{Periodic Square Wave}& {\displaystyle \begin{array}{l} x(t)=\left\{ \renewcommand{\arraystretch}{1.2} \begin{array}{ll} 1, & |t|<T_1\\ 0, & T_1<|t|\leq\frac{T}{2} \end{array}\right.\\ \mbox{and }x(t+T)=x(t) \end{array}}& {\displaystyle \mathbb{X}[k]=\frac{\sin(k\omega_0T_1)}{k\pi}}\\ \hline % \mbox{Impulse Train}& {\displaystyle x(t)=\sum_{n=-\infty}^{\infty}\delta(t-nT)}& {\displaystyle \mathbb{X}[k]=\frac{1}{T}}\\ \hline \end{array} \end{align*} $$
Common Exponential Fourier Series Properties
$$ \renewcommand{\arraystretch}{2.0} \newcommand{\cc}{\circlearrowleft\!\!\!\!\!\!\!\!\!\!\;*~} \begin{align*} \begin{array}{l l l} \mbox{Property} & \mbox{Periodic Signal} & \mbox{Fourier Series}\\ \hline % \mbox{Basic Signals} & x(t), y(t), z(t);~T_x=T_y=T & \mathbb{X}[k], \mathbb{Y}[k], \mathbb{Z}[k];~\omega_0=\frac{2\pi}{T}\\ \hline % \mbox{Linearity} & z(t)=Ax(t)+By(t) & \mathbb{Z}[k]=A\mathbb{X}[k]+B\mathbb{Y}[k]\\ \hline % \mbox{Time Shifting} & z(t)=x\left(t-t_0\right) & \mathbb{Z}[k]=\mathbb{X}[k]e^{-jk\omega_0t_0}\\ \hline % \mbox{Frequency Shifting} & z(t)=e^{jk_0\omega_0t}x(t) & \mathbb{Z}[k]=X[k-k_0]\\ \hline % \mbox{Conjugation} & z(t)=x^*(t) & \mathbb{Z}[k]=X^*[-k]\\ \hline % \mbox{Time Reversal} & z(t)=x(-t) & \mathbb{Z}[k]=X[-k]\\ \hline % \mbox{Time Scaling} & z(t)=x(\alpha t), \alpha>0 & \mathbb{Z}[k]=\mathbb{X}[k], T_z=\frac{T_x}{\alpha}\\ \hline % \mbox{Periodic Convolution} & z(t)={\displaystyle \int_{T}x(\tau)y(t-\tau)d\tau} & \mathbb{Z}[k]=T\mathbb{X}[k]\mathbb{Y}[k]\\ \hline % \mbox{Multiplication} & z(t)=x(t)y(t) & {\displaystyle \mathbb{Z}[k]=\sum_{l=-\infty}^{\infty}X[l]Y[k-l]}\\ \hline % \mbox{Differentiation} & z(t)=\frac{dx(t)}{dt} & \mathbb{Z}[k]=jk\omega_x\mathbb{X}[k]\\ \hline % \mbox{Integration} & {\displaystyle z(t)=\int_{-\infty}^{t}x(\tau)~d\tau}, X[0]=0& \mathbb{Z}[k]=\left(\frac{1}{jk\omega_x}\right)\mathbb{X}[k]\\ \hline % \mbox{Properties of Real Signals} & z(t)\mbox{ real} & \left\{ \renewcommand{\arraystretch}{1.0} \begin{array}{l} \mathbb{Z}[k]=Z^*[-k]\\ \Re\{\mathbb{Z}[k]\}=\Re\{Z[-k]\}\\ \Im\{\mathbb{Z}[k]\}=-\Im\{Z[-k]\}\\ |\mathbb{Z}[k]|=|Z[-k]|\\ \measuredangle \mathbb{Z}[k]=-\measuredangle Z[-k] \end{array} \renewcommand{\arraystretch}{2.0} \right.\\ \hline % \mbox{Properties of Real, Even Signals} & z(t)\mbox{ real and even}&\mathbb{Z}[k]\mbox{ real and even}\\ \hline % \mbox{Properties of Real, Odd Signals} & z(t)\mbox{ real and odd}&\mathbb{Z}[k]\mbox{ imaginary and odd}\\ \hline % \mbox{Isolation of Even Part} & z(t)=x_e(t)\mbox{ with x(t) real}& \mathbb{Z}[k]=\Re\{\mathbb{X}[k]\} \\ \hline % \mbox{Isolation of Odd Part} & z(t)=x_o(t)\mbox{ with x(t) real}& \mathbb{Z}[k]=j\Im\{\mathbb{X}[k]\} \\ \hline % \mbox{Parseval's Relation (Power)} & {\displaystyle P_{ave}=\frac{1}{T}\int_{T}|z(t)|^2~dt}& {\displaystyle P_{ave}=\sum_{k=-\infty}^{\infty}|\mathbb{Z}[k]|^2} \end{array} \end{align*} $$
Examples
External Links
- Fourier Series Animation using Circles - YouTube user meyavuz
- Fourier Transform, Fourier Series, and frequency spectrum - YouTube user Eugene Khutoryansky
- Fourier series pen - André Michelle
- Uneven Pouring Voyage - Tick Slayer