ece205: add fourier transform

docs/2a/ece205.md

@@ -509,6 +509,40 @@ $$\frac{1}{2L}\int^L_{-L}\underbrace{[f(t)]^2}_\text{time domain}dt=\sum^\infty_
     \frac{\pi^2}{6}&=\sum^\infty_{n=1}\frac{1}{n^2}
     \end{align*}
 
+### Fourier transform
+
+To convert a function to a Fourier series:
+
+$$\mathcal F\{f(x)\}=\hat f(\omega)=\int^\infty_{-\infty}f(x)e^{-i\omega x}dx$$
+
+To convert a Fourier series back to the original function, the following conditions must hold:
+
+- there must not be any infinite discontinuities: $\int^\infty_{-\infty}|f(x)|dx<\infty$
+- in any finite interval, there must be a finite number of extrema and discontinuities
+
+Then, the **Fourier integral** / **inverse Fourier transform** converges to $f(x)$ wherever continuous and $\frac 1 2[f(x^+)+f(x^-)]$ at discontinuities.
+
+$$\mathcal F^{-1}\{\hat f(\omega)\}=f(x)=\frac{1}{2\pi}\int^\infty_{-\infty}\hat f(\omega)e^{i\omega x}d\omega$$
+
+!!! example
+    For $f(x)=\begin{cases} 1 & -1<x<1 \\ 0 & \text{else}\end{cases}$:
+    
+    \begin{align*}
+    \mathcal F\{f(x)\}&=\int^\infty_{-\infty}f(x)e^{-i\omega x}dx \\
+    &=\int^1_{-1}e^{-i\omega x}dx \\
+    &=\frac{i\omega}(e^{i\omega}-e^{-i\omega}) \\
+    &=\frac{2\sin\omega}{\omega}
+    \end{align*}
+
+Parseval's theorem can be generalised to non-periodic situations via Fourier transforms.
+
+$$\int^\infty_{-\infty}[f(t)]^2dt=\frac{1}{2\pi}\int^\infty_{-\infty}|\hat f(\omega)|^2d\omega$$
+
+#### Properties of the Fourier transform
+
+- FT/IFT are linear: $\mathcal F\{af+bg\}=a\mathcal F\{f\}+b\mathcal F\{g\}$
+- FT is scalable: $\mathcal F\{f(ax)\}=\frac 1 a\hat f\left(\frac{\omega}{a}\right)$
+
 ## Resources
 
 - [Laplace Table](/resources/ece/laplace.pdf)