Trivial echo separation Say we have a signal S, and we composite it with a 0.1-volume 30-sample-delayed version of itself, so:

T(k) = S(k) + 0.1*S(k-30)


How would one rearrange this equation to make S the subject?

i.e given the resultant signal T, and the fact that we know the parameters of the echo (0.1 and 30), how to reproduce the original S?
The way the signal $T(k)$ is generated is by applying an FIR (finite impulse response) filter to the signal $S(k)$. The transfer function of this FIR filter is

$$H(z)=1+0.1z^{-30}$$

If you want to compensate for such a filter, you need a filter with a transfer function which is the inverse of $H(z)$:

$$G(z)=1/H(z)=\frac{1}{1+0.1z^{-30}}$$

This is a recursive filter with an infinitely long impulse response (IIR). If you filter the signal $T(k)$ with this IIR filter, the result will be $S(k)$.