Fourier Series


The Fourier series of a periodic function, $f(\theta )$ with period $2\pi$ is given by: $$f(\theta )=\frac{a_0}{2}+\sum_{n=1}^\infty \left(a_n\cos n\theta +b_n \sin n\theta \right)$$ where $$\begin{align*}a_0&=\frac1\pi\int_{-\pi}^{+\pi}{f(\theta})\,d\theta\\ a_n&=\frac1\pi\int_{-\pi}^{+\pi}{f(\theta})\cos n\theta\,d\theta,\quad n=1,2,\dots\\ b_n&=\frac1\pi\int_{-\pi}^{+\pi}{f(\theta})\sin n\theta\,d\theta,\quad n=1,2,\dots\end{align*}$$

The coefficients $a_0$ and $a_n$ re called the Fourier cosine coefficients and the $b_n$ are the Fourier sine coefficients.

If the function $f(\theta )$ has period $2L$ where $L$ is a finite real number, the limits in the integrations above are replaced by $\pm L$.

Every periodic function can be decomposed into a sum of one or more cosine and/or sine terms of selected frequencies determined solely by that of original function. Conversely, by superimposing cosines and/or sines of a certain selected set of frequencies we can reconstruct any periodic function.

Phase angles $\alpha$ and $\beta$

The cosine and sine terms in the Fourier series may be combined in a single cosine or sine series with phase angles $\alpha$ and $\beta$ respectively as $$f(\theta )=\frac{A_0}{2}+\sum_{n=1}^\infty A_n\cos \left(n\theta +\alpha_n\right)$$ where $$\begin{align*} A_n&=\sqrt{a_n^2+b_n^2}\\ \alpha_n&=\tan^{-1}\left(-\frac{b_n}{a_n}\right)\end{align*}$$ or $$f(\theta )=\frac{B_0}{2}+\sum_{n=1}^\infty B_n\sin \left(n\theta +\beta_n\right)$$ where $$\begin{align*} B_n&=\sqrt{a_n^2+b_n^2}\\ \beta_n&=\tan^{-1}\left(\frac{a_n}{b_n}\right)\end{align*}$$ with the $a_n$ and $b_n$ terms as defined in the previous section.

Fourier Convergence Theorem

If $f(x)$ is a periodic function with period $2\pi$ and $f(x)$ and $f'(x)$ are peicewise continuous on the interval $[-\pi, \pi ]$, then the Fourier series is convergent. The sum of the Fourier series is equal to $f(x)$ at all values of $x$ where $f(x)$ is continuous. At the values $x$ where $f(x)$ is discontinuous, the sum of the Fourier series is the average of the right hand and left hand limits of the function, that is $$\frac{f(x^+)+f(x^-)}{2}.$$

Fourier Series Types

Trigonometric Fourier Series (TFS)

Any function $x(t)$ in the interval $\left(t_0, t_0+\frac{2\pi}{\omega}\right)=(t_0,t_0+T)$ can be represented as $$x(t )=\frac{a_0}{2}+\sum_{n=1}^\infty \left(a_n\cos n\omega t +b_n \sin n\omega t \right), \quad (t_0<t<t_0+T)$$ where $$\begin{align*}a_0&=\frac1T\int_{t_0}^{t_0+T}{x(t)\,dt}\\ a_n&=\frac2T\int_{t_0}^{t_0+T}{x(t})\cos n\omega t\,dt,\quad n=1,2,\dots\\ b_n&=\frac2T\int_{t_0}^{t_0+T}{x(t)\sin n\omega t\,dt},\quad n=1,2,\dots\end{align*}$$

Exponential Fourier Series (EFS)

We consider the set of complex exponential functions $ \left\{e^{in\omega t}\right\}$ $n=0, \pm 1, \pm 2, \dots )$ which is orthogonal over the interval $(t_0, t_0+T)$, where $T=\frac{2\pi}{\omega}$. The function $f(t)$ can be represented as $$f(t)=\sum_{n=-\infty}^{\infty}F_ne^{in\omega t}\quad (t_0<t<t_0+T)$$ where $$F_n=\frac1T\int_{t_0}^{t_0+T}{f(t)e^{-in\omega t}\,dt}.$$

Relation between TFS and EFS

$$\begin{align*} a_0&-F_0\\ a_n&=F_n+F_{-n}\\ b_n&=i(F_n-F_{-n})\end{align*}$$

Similarly, $$\begin{align*} F_n&=\frac{a_n-ib_n}{2}\\ F_{-n}&=\frac{a_n+ib_n}{2}\end{align*}$$

Additional Resources