Fourier Series

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.

It is currently a work in progress and has some mixed notation between using subscripts and parameters (i.e. $$a_n$$ versus $$X[n]$$) - cleaning that up in Fall of 2019 / Spring of 2020.

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):

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

In the series above, \(a_0\), \(a_n\), \(b_n\), \(c_0\), \(c_n\), and \(\theta_n\) are real numbers while \(\mathbb{C}_n\) 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:

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

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: $$\begin{align*} \delta[k]&= \left\{ \begin{array}{cl} k=0 & 1\\ k\neq 0 & 0 \end{array} \right. \end{align*}$$

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

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*} $$

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