Trivial echo separation
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 $S(k)$.