ece205: add more complex fouriers

docs/2a/ece205.md

@@ -490,6 +490,25 @@ To convert it to a real Fouier series:
     \therefore f(x)&=\sum^\infty_{\substack{n=-\infty \\ n\neq0}}\frac{(-1)^ni}{n\pi}e^{in\pi x}
     \end{align*}
 
+The Fourier coefficients $c_n$ map to the amplitude spectrum $|c_n|$. **Parseval's theorem** maps the frequency domain ($\{c_n\}$) to and from the time domain ($f(t)$):
+
+If a 2L-periodic function $f(t)$ has a complex Fourier series $f(t)=\sum^\infty_{n=-\infty}c_ne^{\frac{in\pi x}{L}}$:
+
+$$\frac{1}{2L}\int^L_{-L}\underbrace{[f(t)]^2}_\text{time domain}dt=\sum^\infty_{n=-\infty}\underbrace{|c_n|^2}_\text{time domain}$$
+
+!!! example
+    For the Sawtooth function, $f(t)=t, -1 < t < 1, f(t+2)=f(t)$:
+    
+    \begin{align*}
+    f(x)&=\sum^\infty_{\substack{n=-\infty \\ n\neq 0}}\frac{ni}{n\pi}e^{in\pi t}+0 \\
+    \frac 1 2\int^1_{-1}t^2dt&=\sum^\infty_{\substack{n=-\infty \\ n\neq 0}}\left|\frac{(-1)^ni}{n\pi}\right|^2+|0|^2 \\
+    \tag{$\left|\frac{(-1)^ni}{n\pi}\right|=\frac{1}{n\pi}$}\frac 1 3 &=\sum^\infty_{\substack{n=-\infty \\ n\neq 0}}\left(\frac{1}{n\pi}\right)^2 \\
+    &=\sum^{-1}_{n=-\infty}\left(\frac{1}{n\pi}\right)^2+\sum^\infty_{n=1}\left(\frac{1}{n\pi}\right)^2 \\
+    \tag{$\frac 1 n^2$ sign doesn't matter}&=2\sum^\infty_{n=1}\frac{1}{n^2\pi^2} \\
+    \frac 1 3 &=\frac{2}{\pi^2}\sum^\infty_{n=1}\frac{1}{n^2} \\
+    \frac{\pi^2}{6}&=\sum^\infty_{n=1}\frac{1}{n^2}
+    \end{align*}
+
 ## Resources
 
 - [Laplace Table](/resources/ece/laplace.pdf)