Fourier Series

Fourier Series Representations

The following are the three different representations used on this page. Note that the index $ k $ is used with the exponential form and will take on values from $ -\infty $ to $ \infty $, while the index $ n $ is used with the other forms and will take on values from $ 1 $ to $ \infty $. In the translation table, only $ n $ is used.

• Exponential Form
  • Synthesis$ \begin{align*} x(t)&=\sum_{k=-\infty}^{\infty}\mathbb{X}[k]e^{jk\omega_0t} \end{align*} $
  • Analysis$ \begin{align*} \mathbb{X}[k]&=\frac{1}{T}\int_Tx(t)e^{-jk\omega_0t}\,dt \end{align*} $
• Trigonometric Form
  • Synthesis$ \begin{align*} x(t)&=a_0+\sum_{n=1}^{\infty}\left(a_n\cos(n\omega_0t)+b_n\sin(n\omega_0t)\right) \end{align*} $
  • Analysis $ \begin{align*} a_0&=\frac{1}{T}\int_Tx(t)\,dt\\ a_n&=\frac{2}{T}\int_Tx(t)\,\cos(n\omega_0t)\,dt\\ b_n&=\frac{2}{T}\int_Tx(t)\,\sin(n\omega_0t)\,dt \end{align*} $
• Cosine Form
  • Note: Different sources have slightly different versions of this representation. This page uses a phase angle of $ +\theta_n $, adapted from Lathi and Alexander & Sadiku. Other sources may define the phase angle as $ -\theta_n $, which changes the sign in the phase calculation.
  • Synthesis$ \begin{align*} x(t)&=c_0+\sum_{n=1}^{\infty}c_n\cos(n\omega_0t+\theta_n) \end{align*} $
  • Analysis $ \begin{align*} c_0&=\frac{1}{T}\int_Tx(t)\,dt\\ c_n&=\sqrt{a_n^2+b_n^2}\mbox{ with } a_n,b_n \mbox{ from Trigonometric Form equations above}\\ \theta_n&=\mbox{atan2}(-b_n,a_n)\mbox{ with } a_n,b_n \mbox{ from Trigonometric Form equations above} \end{align*} $
    • The $ \mbox{atan2}(y,x) $ function is defined as:$ \begin{align*} \mbox{atan2}(y,x)&=\left\{\begin{array}{lll} \tan^{-1}\left(\frac{y}{x}\right) & x>0 & \text{(Quadrants 1 and 4, and positive } x \text{ axis)}\\ \tan^{-1}\left(\frac{y}{x}\right)+\pi & x<0, y\ge 0 & \text{(Quadrant 2, and negative } x \text{ axis)}\\ \tan^{-1}\left(\frac{y}{x}\right)-\pi & x<0, y<0 & \text{(Quadrant 3)}\\ \frac{\pi}{2} & x=0, y>0 & \text{(Positive } y \text{ axis)}\\ -\frac{\pi}{2} & x=0, y<0 & \text{(Negative } y \text{ axis)}\\ \mbox{undefined} & x=0, y=0 & \text{(Origin)} \end{array}\right.\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 $ \mathbb{X}[n]=\mathbb{X}^*[-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) \end{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{X}[n]+\mathbb{X}[-n]=2\Re\{\mathbb{X}[n]\}\\ \hline b_n & b_n & -c_n\sin(\theta_n) & j\left(\mathbb{X}[n]-\mathbb{X}[-n]\right)=-2\Im\{\mathbb{X}[n]\}\\ \hline a_0=c_0 & a_0 & c_0 & \mathbb{X}[0] \\ \hline c_n & \sqrt{a_n^2+b_n^2} & c_n & |\mathbb{X}[n]|+|\mathbb{X}[-n]|=2|\mathbb{X}[n]|\\ \hline \theta_n & \mbox{atan2}(-b_n,a_n) & \theta_n & \angle \mathbb{X}[n]\\ \hline \mathbb{X}[0] & a_0 & c_0 & \mathbb{X}[0] \\ \hline \mathbb{X}[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{X}[n]\\ \hline \mathbb{X}[-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{X}[-n] \\ \hline \end{array} $

Analysis Equation Derivation

Assume you have some periodic signal $ x(t) $ with fundamental frequency $ \omega_0 $ that has a Fourier Series representation of $ \mathbb{X}[k] $. Using a summation variable of $ m $ instead of $ k $ to help avoid some confusion later, you can write $ x(t) $ as:

$ {\displaystyle \begin{align*} x(t)&=\sum_{m=-\infty}^{\infty}\mathbb{X}[m]e^{jm\omega_0t} \end{align*}} $

Assume you have another (decidedly complex-valued) signal $ y(t) $ that is simply a complex exponential oscillating at a negative integer multiple $ -k $ of $ \omega_0 $ (this is why we used $ m $ above!):

$ \begin{align*} y(t)&=e^{-jk\omega_0t} \end{align*} $

We will now look at the integral of the product of $ x(t)\,y(t) $ over one period. We will look at the integral two different ways - one with the Fourier Series representation substituted in and one without:

$ \begin{align*} \int_Tx(\tau)\,y(\tau)\,d\tau&= \int_Tx(\tau)\,y(\tau)\,d\tau & ~&\mbox{Initial tautology}\\ \int_T\sum_{m=-\infty}^{\infty}\mathbb{X}[m]e^{jm\omega_0\tau}\,e^{-jk\omega_0\tau}\,d\tau&= \int_Tx(\tau)\,e^{-jk\omega_0\tau}\,d\tau & ~&\mbox{Substitute for }y(t)\mbox{ both places and for }x(t)\mbox{ on left only}\\ \sum_{m=-\infty}^{\infty}\mathbb{X}[m]\int_Te^{j(m-k)\omega_0\tau}\,d\tau&= \int_Tx(\tau)\,e^{-jk\omega_0\tau}\,d\tau & ~&\mbox{Swap order of sum and integral, pull functions of }m\mbox{ only out of integral} \end{align*} $

Let us focus for a moment on the integral:

$ \begin{align*} \int_Te^{j(m-k)\omega_0\tau}\,d\tau\end{align*} $

Using Euler's relationship, we can see that:

$ \begin{align*} e^{j(m-k)\omega_0\tau}=\cos((m-k)\omega_0\tau)+j\sin((m-k)\omega_0\tau)\end{align*} $

If $ m\neq k $, this represents a sinusoidal signal oscillating at some integer multiple $ m-k $ of the fundamental frequency $ \omega_0 $. And if we are integrating that over some period $ T $, the integral will be zero - the integral of a sinusoid over an integer number of periods is 0!

If, however, $ m=k $, the signal becomes $ e^{j0\omega_0\tau}=1 $. So in the summation on the left above, the only term in the summation that is not zero will be when $ m=k $:

$ \begin{align*} \sum_{m=-\infty}^{\infty}\mathbb{X}[m]\int_Te^{j(m-k)\omega_0\tau}\,d\tau&= \int_Tx(\tau)\,e^{-jk\omega_0\tau}\,d\tau & ~&\mbox{Resume where we left things}\\ \mathbb{X}[k]\int_T\,d\tau&= \int_Tx(\tau)\,e^{-jk\omega_0\tau}\,d\tau & ~&\mbox{Pull out only }k^{th}\mbox{ summand}\\ \mathbb{X}[k]\,T&= \int_Tx(\tau)\,e^{-jk\omega_0\tau}\,d\tau & ~&\mbox{Integrate} \end{align*} $

meaning:

$ \begin{align*} \mathbb{X}[k]&=\frac{1}{T}\int_Tx(\tau)\,e^{-jk\omega_0\tau}\,d\tau=\frac{1}{T}\int_Tx(t)\,e^{-jk\omega_0t}\,dt\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 Haykin and Van Veen as well as Oppenheim and Willsky.

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:

$ \delta[k]= \begin{cases} 1, & k=0\\ 0, & k\neq 0 \end{cases} $ $ \begin{array}{l l l} \mbox{Name} & \mbox{Signal} & \mbox{Fourier Series} \\ \hline \mbox{Basic Signal} & x(t)\text{, Period } T & \mathbb{X}[k]\text{, } \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)=\begin{cases} 1, & |t|<T_1\\ 0, & T_1<|t|\leq\frac{T}{2} \end{cases}\\ \text{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} $

Common Exponential Fourier Series Properties

$ \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]=\mathbb{X}[k-k_0]\\ \hline \mbox{Conjugation} & z(t)=x^*(t) & \mathbb{Z}[k]=\mathbb{X}^*[-k]\\ \hline \mbox{Time Reversal} & z(t)=x(-t) & \mathbb{Z}[k]=\mathbb{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}\mathbb{X}[l]\mathbb{Y}[k-l]}\\ \hline \mbox{Differentiation} & z(t)=\frac{dx(t)}{dt} & \mathbb{Z}[k]=jk\omega_0\mathbb{X}[k]\\ \hline \mbox{Integration} & {\displaystyle z(t)=\int_{-\infty}^{t}x(\tau)~d\tau}, \mathbb{X}[0]=0& \mathbb{Z}[k]=\left(\frac{1}{jk\omega_0}\right)\mathbb{X}[k]\\ \hline \mbox{Properties of Real Signals} & z(t)\mbox{ real} & \left\{ \begin{array}{l} \mathbb{Z}[k]=\mathbb{Z}^*[-k]\\ \Re\{\mathbb{Z}[k]\}=\Re\{\mathbb{Z}[-k]\}\\ \Im\{\mathbb{Z}[k]\}=-\Im\{\mathbb{Z}[-k]\}\\ |\mathbb{Z}[k]|=|\mathbb{Z}[-k]|\\ \measuredangle \mathbb{Z}[k]=-\measuredangle \mathbb{Z}[-k] \end{array} \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} $