Fourier series - by Ray Betz

Fourier Series

If

1. $x(t)=x(t+T)$
2. Dirichlet conditions are satisfied

then we can write

$\mathbf{x}(t)=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k{e}^{\frac{j2\pi kt}{T}}$

The above equation is called the complex fourier series. Given $x(t)$, we may determine ${\alpha }_k$ by taking the inner product of ${\alpha }_k$ with $x(t)$. Let us assume a solution for ${\alpha }_k$ of the form $e^{\frac{j2\pi nt}{T}}$. Now we take the inner product of ${\alpha }_k$ with $x(t)$. $<{\alpha }_k|x(t)>=<e^{\frac{j2\pi nt}{T}}|\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k{e}^{\frac{j2\pi kt}{T}}>$ $=\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}x(t)e^{\frac{-j2\pi nt}{T}}dt$ $=\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k{e}^{\frac{j2\pi kt}{T}}{e}^{\frac{-j2\pi nt}{T}}dt$ $=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{\frac{j2\pi (k-n)t}{T}}dt$

If $k=n$ then,

$\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{\frac{j2\pi (k-n)t}{T}}dt=\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}1dt=T$

If $k\ne ;n$ then,

$\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{\frac{j2\pi (k-n)t}{T}}dt=0$

We can simplify the above two conclusion into one equation.

$\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\alpha }_k\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}{e}^{\frac{j2\pi (k-n)t}{T}}dt=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}T{\delta }_{k,n}{\alpha }_k=T{\alpha }_n$

So, we may conclude ${\alpha }_n=\frac{1}{T}\underset{-\frac{T}{2}}{\overset{\frac{T}{2}}{\int }}x(t)e^{\frac{-j2\pi nt}{T}}dt$

Orthogonal Functions

The function $y_n(t)$ and $y_m(t)$ are orthogonal on $(a,b)$ if and only if $<y_n(t)|y_m(t)>=\underset{a}{\overset{b}{\int }}{y}_n^{*}(t)y_m(t)dt=0$.

The set of functions are orthonormal if and only if $<y_n(t)|y_m(t)>=\underset{a}{\overset{b}{\int }}{y}_n^{*}(t)y_m(t)dt={\delta }_{m,n}$.