Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

There's so much going on here I don't know where to start:

$$ d_{t} = d_{t - 1} + \left(\frac{1}{r}\right)v_{t - 1} $$

$$ v_{t} = x_t + v_{t - 1} - \left(\frac{f^2}{r}\right) (4\pi^2 d_{t - 1} + D v_{t - 1}) $$

$r$, $f$, and $D$ are constants. How can I solve this for $d_t$?

share|cite|improve this question
Where does $x_t$ come from? – erlking Dec 16 '12 at 1:45
erlking: The context is audio signal processing, and $x_t$ is the input sample at time $t$. – Andreas Jansson Dec 16 '12 at 1:47

First, rename the constants appropriately so that we get $$d_t = d_{t-1} + A v_{t-1} \\ v_t = x_t + B v_{t-1} + Cd_{t-1}$$

Then use the second equation to substitute $d_t$ and $d_{t-1}$ in the first one: $$ (v_{t+1}-x_{t+1}-Bv_t)/C = (v_t - x_t -Bv_{t-1})/C+Av_{t-1} \\ \Leftrightarrow v_{t+1} -(B+1)v_t +(-AC+B)v_{t-1}=(x_{t+1}-x_t) $$

This is an inhomogeneous second order difference equation. There are standard methods to solve this. You first find the solutions for the homogeneous case (i.e. $x_{t+1}-x_t=0$) and then you find one particular solution to the inhomogeneous case. This will obviously depend on the nature of the $x_t$, and there may not be a closed form. Then you can substitute back to get $d_t$. So if your input is real data you may have to put up with numerical iterative solutions for $d_t$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.