Fourier Series

The orthogonal basis set for cartesian coordinates

$f(x)=\frac{a_0}{2}+\sum _{n=1}^{\infty }\left[a_n\cos \left(\frac{n\pi }{L}x\right)+b_n\sin \left(\frac{n\pi }{L}x\right)\right]$

• States that any continuous, periodic function on the interval $[-L,L]$ can be expanded as the above

$a_0=\frac{1}{L}{\int }_{x_0}^{2L+x_0}f(x)dx$

$a_n=\frac{1}{L}{\int }_{x_0}^{2L+x_0}f(x)\cos \left(\frac{n\pi }{L}x\right)dx=\frac{2}{L}{\int }_{x_0}^{L+x_0}f(x)\cos \left(\frac{n\pi }{L}x\right)\mathrm{d}x$

• Called Fourier’s trick
• Obtained by multiplying $f(x)=\frac{a_0}{2}+\sum _{n=1}^{\infty }\left[a_n\cos \left(\frac{n\pi }{L}x\right)+b_n\sin \left(\frac{n\pi }{L}\right)\right]$ on both sides by $\cos \left(\frac{n^{\prime }\pi }{L}x\right)$ and integrating from $x_0$ to $2L+x_0$ and noticing that the only contributing terms on the RHS are of the form ${\int }_{x_0}^{2L+x_0}{a}_n\cos \left(\frac{n\pi }{L}x\right)\cos \left(\frac{n^{\prime }\pi }{L}x\right)\mathrm{d}x$ for when $n=n^{\prime }$, which evaluates to $\frac{2L}{2}{a}_n$

$b_n=\frac{1}{L}{\int }_{x_0}^{2L+x_0}f(x)\sin (\frac{n\pi }{L}x)dx=\frac{2}{L}{\int }_{x_0}^{L+x_0}f(x)\sin \left(\frac{n\pi }{L}x\right)\mathrm{d}x$

Signal Processing

Topics

• Fourier Series Coefficients
• Parsavel’s Theorem

$x(t)=\sum _{k=-\infty }^{\infty }{c}_k{e}^{jk{\omega }_{0}t}\text{ if }x(t)\text{ is periodic}$

• A signal can be decomposed into a series of complex exponentials