Difference between revisions of "Fourier Series"

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There are three primary Fourier series representations of a periodic
 
There are three primary Fourier series representations of a periodic
 
signal <math>f(t)</math>  
 
signal <math>f(t)</math>  
−
with period <math>T</math> and fundamental frequency <math>\omega_0=\frac{2\pi}{T}</math>  
+
with period <math>T</math> and fundamental frequency <math>\omega_0=\frac{2\pi}{T}</math>; note that different sources may use different symbols for the series coefficients!
−
(using the notation in Svoboda & Dorf, Introduction to Electric Circuits, 9th Edition - please note that Oppenheim & Willsky, Signals & Systems, 2nd edition uses <math>a_k</math> instead of <math>\mathbb{C}_k</math> for the exponential Fourier series coefficients; Alexander and Sadiku, Fundamentals of Electric Circuits, 7th Edition uses $$A_n$$ and $$\phi_n$$ for the cosine series and $$c_n$$ for the exponential series):
 
 
<center><math>
 
<center><math>
 
\begin{align}
 
\begin{align}
 
\mbox{Trigonometric Series}&~ & f(t)&=a_0+
 
\mbox{Trigonometric Series}&~ & f(t)&=a_0+
−
\sum_{n=1}^{\infty}\left(a_n~\cos(n\omega_0 t) +
+
\sum_{n=1}^{\infty}\left(a_n\,\cos(n\omega_0 t) +
−
b_n~\sin(n\omega_0 t)\right)\\
+
b_n\,\sin(n\omega_0 t)\right)\\
 
\mbox{Cosine Series} &~ & f(t)&=
 
\mbox{Cosine Series} &~ & f(t)&=
−
c_0 + \sum_{n=1}^{\infty}c_n~\cos(n\omega_0 t+\theta_n)\\
+
c_0 + \sum_{n=1}^{\infty}c_n\,\cos(n\omega_0 t+\theta_n)\\
 
\mbox{Exponential Series} &~ & f(t)&=
 
\mbox{Exponential Series} &~ & f(t)&=
−
\sum_{n=-\infty}^{\infty}\mathbb{C}_n~e^{jn\omega_0 t}
+
\sum_{n=-\infty}^{\infty}\mathbb{X}[n]\,e^{jn\omega_0 t}
 
\end{align}
 
\end{align}
 
</math></center>
 
</math></center>
 
In the series above, <math>a_0</math>, <math>a_n</math>, <math>b_n</math>, <math>c_0</math>, <math>c_n</math>,
 
In the series above, <math>a_0</math>, <math>a_n</math>, <math>b_n</math>, <math>c_0</math>, <math>c_n</math>,
 
and <math>\theta_n</math> are real
 
and <math>\theta_n</math> are real
−
numbers while <math>\mathbb{C}_n</math>  may be complex.
+
numbers while <math>\mathbb{X}[n]</math>  may be complex.
−
<!--
 
−
Also note that the index <math>n</math> is used for summations between 1 and <math>\infty</math>
 
−
while <math>k</math>  is used for the summation between <math>-\infty</math>  to <math>\infty</math>
 
−
primarily to demonstrate the different ranges of the summations.
 
−
-->
 
  
 
==Analysis Equations==
 
==Analysis Equations==

Revision as of 14:10, 28 September 2021

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!

\( \begin{align} \mbox{Trigonometric Series}&~ & f(t)&=a_0+ \sum_{n=1}^{\infty}\left(a_n\,\cos(n\omega_0 t) + b_n\,\sin(n\omega_0 t)\right)\\ \mbox{Cosine Series} &~ & f(t)&= c_0 + \sum_{n=1}^{\infty}c_n\,\cos(n\omega_0 t+\theta_n)\\ \mbox{Exponential Series} &~ & f(t)&= \sum_{n=-\infty}^{\infty}\mathbb{X}[n]\,e^{jn\omega_0 t} \end{align} \)

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:

\( \begin{align} a_n&=\frac{2}{T}\int_{T}f(t)~\cos(n\omega_0t)~dt & b_n&=\frac{2}{T}\int_{T}f(t)~\sin(n\omega_0t)~dt \\ a_0=c_0&=\frac{1}{T}\int_{T}f(t)~dt & c_n&= \sqrt{a_n^2+b_n^2} \\ \theta_n&= \begin{cases} -\tan^{-1}\left(\frac{b_n}{a_n}\right) & a_n>0\\ 180^{\circ}-\tan^{-1}\left(\frac{b_n}{a_n}\right) & a_n<0 \end{cases}\\ \mathbb{C}_n&=\frac{1}{T}\int_Tf(t)~e^{-jn\omega_0t}~dt & \end{align} \)

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:

\( \begin{align} e^{j\theta}&=\cos(\theta)+j\sin(\theta)\\ \cos(\theta+\phi)&=\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)\\ \mbox{atan2}(b_n,a_n)&= \begin{cases} \tan^{-1}\left(\frac{b_n}{a_n}\right) & a_n>0\\ \tan^{-1}\left(\frac{b_n}{a_n}\right)-180^{\circ} & a_n<0 \end{cases}\\ \end{align} \)
\( \begin{align} \begin{array}{|c|c|c|c|} \hline \mbox{Find:} & \mbox{From trig} & \mbox{From cosine} & \mbox{From exponential} \\ \hline a_n & a_n & c_n\cos(\theta_n) & \mathbb{C}_n+\mathbb{C}_{-n}=2\Re\{\mathbb{C}_n\}\\ \hline b_n & b_n & -c_n\sin(\theta_n) & j\left(\mathbb{C}_n-\mathbb{C}_{-n}\right)=-2\Im\{\mathbb{C}_n\}\\ \hline a_0=c_0 & a_0 & c_0 & \mathbb{C}_0 \\ \hline c_n & \sqrt{a_n^2+b_n^2} & c_n & |\mathbb{C}_n|+|\mathbb{C}_{-n}|=2|\mathbb{C}_n|\\ \hline \theta_n & -\mbox{atan2}(b_n,a_n) & \theta_n & \angle \mathbb{C}_n\\ \hline \mathbb{C}_0 & a_0 & c_0 & \mathbb{C}_0 \\ \hline \mathbb{C}_n & \frac{a_n}{2}+\frac{b_n}{2j}= \frac{a_n}{2}-j\frac{b_n}{2} & \frac{c_n}{2}\angle \theta_n & \mathbb{C}_n\\ \hline \mathbb{C}_{-n} & \frac{a_n}{2}-\frac{b_n}{2j}= \frac{a_n}{2}+j\frac{b_n}{2} & \frac{c_n}{2}\angle -\theta_n &\mathbb{C}_{-n} \\ \hline \end{array} \end{align} \)

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: $$\begin{align*} \delta[k]&= \left\{ \begin{array}{cl} k=0 & 1\\ k\neq 0 & 0 \end{array} \right. \end{align*}$$ Also, the table uses $$X[k]$$ instead of $$\Bbb{C}_k$$ for the Fourier Series coefficients.

$$ \renewcommand{\arraystretch}{2.1} \begin{align*} \begin{array}{l l l} \mbox{Name} & \mbox{Signal} & \mbox{Fourier Series} \renewcommand{\arraystretch}{2.1} \\ \hline % \mbox{Basic Signal} & x(t)\mbox{, Period $T$} & X[k]\mbox{, $\omega_0=\frac{2\pi}{T}$}\\ \hline % \mbox{Complex Exponential}& {\displaystyle x(t)=e^{jp\omega_0t}}& X[k]=\delta[k-p]\\ \hline % \mbox{Cosine}& {\displaystyle x(t)=\cos(p\omega_0t)}& {\displaystyle X[k]=\frac{1}{2}\left(\delta[k-p]+\delta[k+p]\right) }\\ \hline % \mbox{Sine}& {\displaystyle x(t)=\sin(p\omega_0t)}& {\displaystyle X[k]=\frac{1}{j2}\left(\delta[k-p]-\delta[k+p]\right) }\\ \hline % \mbox{Constant}& {\displaystyle x(t)=c}& X[k]=c\delta[k]\\ \hline % \mbox{Periodic Square Wave}& {\displaystyle \begin{array}{l} x(t)=\left\{ \renewcommand{\arraystretch}{1.2} \begin{array}{ll} 1, & |t|<T_1\\ 0, & T_1<|t|\leq\frac{T}{2} \end{array}\right.\\ \mbox{and }x(t+T)=x(t) \end{array}}& {\displaystyle X[k]=\frac{\sin(k\omega_0T_1)}{k\pi}}\\ \hline % \mbox{Impulse Train}& {\displaystyle x(t)=\sum_{n=-\infty}^{\infty}\delta(t-nT)}& {\displaystyle X[k]=\frac{1}{T}}\\ \hline \end{array} \end{align*} $$

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 & X[k], Y[k], Z[k];~\omega_0=\frac{2\pi}{T}\\ \hline % \mbox{Linearity} & z(t)=Ax(t)+By(t) & Z[k]=AX[k]+BY[k]\\ \hline % \mbox{Time Shifting} & z(t)=x\left(t-t_0\right) & Z[k]=X[k]e^{-jk\omega_0t_0}\\ \hline % \mbox{Frequency Shifting} & z(t)=e^{jk_0\omega_0t}x(t) & Z[k]=X[k-k_0]\\ \hline % \mbox{Conjugation} & z(t)=x^*(t) & Z[k]=X^*[-k]\\ \hline % \mbox{Time Reversal} & z(t)=x(-t) & Z[k]=X[-k]\\ \hline % \mbox{Time Scaling} & z(t)=x(\alpha t), \alpha>0 & Z[k]=X[k], T_z=\frac{T_x}{\alpha}\\ \hline % \mbox{Periodic Convolution} & z(t)={\displaystyle \int_{T}x(\tau)y(t-\tau)d\tau} & Z[k]=TX[k]Y[k]\\ \hline % \mbox{Multiplication} & z(t)=x(t)y(t) & {\displaystyle Z[k]=\sum_{l=-\infty}^{\infty}X[l]Y[k-l]}\\ \hline % \mbox{Differentiation} & z(t)=\frac{dx(t)}{dt} & Z[k]=jk\omega_xX[k]\\ \hline % \mbox{Integration} & {\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 % \mbox{Properties of Real Signals} & z(t)\mbox{ real} & \left\{ \renewcommand{\arraystretch}{1.0} \begin{array}{l} Z[k]=Z^*[-k]\\ \Re\{Z[k]\}=\Re\{Z[-k]\}\\ \Im\{Z[k]\}=-\Im\{Z[-k]\}\\ |Z[k]|=|Z[-k]|\\ \measuredangle 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}&Z[k]\mbox{ real and even}\\ \hline % \mbox{Properties of Real, Odd Signals} & z(t)\mbox{ real and odd}&Z[k]\mbox{ imaginary and odd}\\ \hline % \mbox{Isolation of Even Part} & z(t)=x_e(t)\mbox{ with x(t) real}& Z[k]=\Re\{X[k]\} \\ \hline % \mbox{Isolation of Odd Part} & z(t)=x_o(t)\mbox{ with x(t) real}& Z[k]=j\Im\{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}|Z[k]|^2} \end{array} \end{align*} $$

Examples

External Links

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