# Damped Harmonic Oscillation as an LTI

The goal is to create an LTI filter which is exactly, or approximates, damping of harmonic modes.

The equation of course is:

$$\frac{d^2 x}{dt^2} + 2 \xi \omega \frac{dx}{dt}+\omega^2x=0$$

This can be done with very costly convolution, I suppose.

I was thinking that instead of treating the velocity term as a part of a time dependent potential (no longer time invariant), I can treat this as a two dimentional state and therefore eliminate second derivatives and have it a $z^{-1}$ feedback filter. It is working weird.

A continous alternative of something close is:

$$e^{-\xi t} \cos \omega t$$

How is this approximated efficiently?

• we call that a 2nd-order homogeneous differential equation. using Laplace Transform, that can be turned into a 2nd-order polynomial with $X(s)$ and powers of $s$. – robert bristow-johnson Jul 26 '18 at 23:29

• Can't see the earthquake/building analogy :). Given $y(n)=h(n) \ast x(n)$ and for LTI, after z-transfom $H(z) = \frac{Y(z)}{X(z)}$. FRF is defined as feeding an imaginary $z$ to the transfer function. $H$ is nice in that you can break down filters using multiplication in the $z$. One is sinosuidal, and the other is exponential in this case? Hann't read the entire thing yet. Thank you BTW. – user76568 Jul 26 '18 at 15:53