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):
\(
\begin{align}
\mbox{Trigonometric Series}&~ & f(t)&=a_0+
\sum_{n=1}^{\infty}\left(a_n~\cos(n\omega_0 t) +
b_n~\sin(n\omega_0 t)\right)\\
\mbox{Cosine Series} &~ & f(t)&=
c_0 + \sum_{n=1}^{\infty}c_n~\cos(n\omega_0 t+\theta_n)\\
\mbox{Exponential Series} &~ & f(t)&=
\sum_{k=-\infty}^{\infty}\mathbb{C}_n~e^{jn\omega_0 t}
\end{align}
\)
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:
\(
\begin{align}
e^{j\theta}&=\cos(\theta)+j\sin(\theta)\\
\cos(\theta+\phi)&=\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)\\
\mbox{atan2}(b_n,a_n)&=
\begin{cases}
\tan^{-1}\left(\frac{b_n}{a_n}\right) & a_n>0\\
\tan^{-1}-180^{\circ}\left(\frac{b_n}{a_n}\right) & a_n<0
\end{cases}\\
\end{align}
\)
\(
\begin{align}
\begin{array}{|c|c|c|c|} \hline
\mbox{Find:} & \mbox{From trig} & \mbox{From cosine} & \mbox{From exponential} \\
\hline
a_n & a_n & c_n\cos(\theta_n) & \mathbb{C}_n+\mathbb{C}_{-n}=2\Re\{\mathbb{C}_n\}\\ \hline
b_n & b_n & -c_n\sin(\theta_n) & j\left(\mathbb{C}_n-\mathbb{C}_{-n}\right)=-2\Im\{\mathbb{C}_n\}\\ \hline
a_0=c_0 & a_0 & c_0 & \mathbb{C}_0 \\ \hline
c_n & \sqrt{a_n^2+b_n^2} & c_n & |\mathbb{C}_n|+|\mathbb{C}_{-n}|=2|\mathbb{C}_n|\\ \hline
\theta_n & -\mbox{atan2}(b_n,a_n) & \theta_n & \angle \mathbb{C}_n\\ \hline
\mathbb{C}_0 & a_0 & c_0 & \mathbb{C}_0 \\ \hline
\mathbb{C}_n & \frac{a_n}{2}+\frac{b_n}{2j}=
\frac{a_n}{2}-j\frac{b_n}{2}
& \frac{c_n}{2}\angle \theta_n &
\mathbb{C}_n\\ \hline
\mathbb{C}_{-n} & \frac{a_n}{2}-\frac{b_n}{2j}=
\frac{a_n}{2}+j\frac{b_n}{2}
& \frac{c_n}{2}\angle -\theta_n &\mathbb{C}_{-n}
\\ \hline
\end{array}
\end{align}
\)
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