Filtering signals

After trying out the SciPy signal module and IIR/FIR filter design and all that jazz (and finding it mind-numbing and mostly useless), I've regressed back to just multiplying signals by various frequency domain transfer functions. Here is a set of useful transfer functions for filtering signals and looking at power spectra. The examples are for filtered white gaussian noise.  

Bandpass

 Lowpass

Cameras and Python

Fun with OpenCV, to acquire video feeds and process them.

Even zetas are sometimes rational

This calculation uses the amazing abilities of SymPy, to evaluate the even-valued zeta functions using Parseval's theorem applied to the functions $t^{s}$.

The last two lines evaluate to
$$2 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{3},$$
$$\pi^2 \sum_{n=1}^\infty \frac{1}{n^2} + \sum_{n=1}^\infty \frac{1}{n^4} = \frac{8 \pi^4}{45}.$$

Incidentally, this recursive definition provides an easy proof of why all the even $\zeta$s are rational multiples of $\pi^{2s}$.

 I wonder if the polynomials in $t$ that evaluate to $\zeta(s)$ are special somehow ...