Fourier Series
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 example problems using Fourier series to solve a mechanical system and an electrical system, respectively.
Synthesis Equations
There are three primary Fourier series representations of a periodic signal \(f(t)\) with period \(T\) and fundamental frequency \(\omega_0=\frac{2\pi}{T}\) (using the notation in Svoboda & Dorf, Introduction to Electric Circuits, 9th Edition - please note that Oppenheim & Willsky, Signals & Systems, 2nd edition uses \(a_k\) instead of \(\mathbb{C}_k\) for the exponential Fourier series coefficients):
In the series above, \(a_0\), \(a_n\), \(b_n\), \(c_0\), \(c_n\), and \(\theta_n\) are real numbers while \(\mathbb{C}_n\) 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: $$\begin{align*} \delta[k]&= \left\{ \begin{array}{cl} k=0 & 1\\ k\neq 0 & 0 \end{array} \right. \end{align*}$$
$$ \renewcommand{\arraystretch}{2.1} \begin{align*} \begin{array}{l l l} \mbox{Name} & \mbox{Signal} & \mbox{Fourier Series} \renewcommand{\arraystretch}{2.1} \\ \hline % \mbox{Basic Signal} & x(t)\mbox{, Period $T$} & X[k]\mbox{, $\omega_0=\frac{2\pi}{T}$}\\ \hline % \mbox{Complex Exponential}& {\displaystyle x(t)=e^{jp\omega_0t}}& X[k]=\delta[k-p]\\ \hline % \mbox{Cosine}& {\displaystyle x(t)=\cos(p\omega_0t)}& {\displaystyle X[k]=\frac{1}{2}\left(\delta[k-p]+\delta[k+p]\right) }\\ \hline % \mbox{Sine}& {\displaystyle x(t)=\sin(p\omega_0t)}& {\displaystyle X[k]=\frac{1}{j2}\left(\delta[k-p]-\delta[k+p]\right) }\\ \hline % \mbox{Constant}& {\displaystyle x(t)=c}& X[k]=c\delta[k]\\ \hline % \mbox{Periodic Square Wave}& {\displaystyle \begin{array}{l} x(t)=\left\{ \renewcommand{\arraystretch}{1.2} \begin{array}{ll} 1, & |t|<T_1\\ 0, & T_1<|t|\leq\frac{T}{2} \end{array}\right.\\ \mbox{and }x(t+T)=x(t) \end{array}}& {\displaystyle X[k]=\frac{\sin(k\omega_0T_1)}{k\pi}}\\ \hline % \mbox{Impulse Train}& {\displaystyle x(t)=\sum_{n=-\infty}^{\infty}\delta(t-nT)}& {\displaystyle X[k]=\frac{1}{T}}\\ \hline \end{array} \end{align*} $$
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
- Fourier Series Animation using Circles - YouTube user meyavuz
- Fourier Transform, Fourier Series, and frequency spectrum - YouTube user Eugene Khutoryansky
- Fourier series pen - André Michelle