3.3 Adaptive interference (noise) cancellation

The object is to subtract out as much of the noise as possible.

Analysis of the interference cancellor

$$({}_k^2)=((s_k+n_k-y_k)^2)=(s_k^2)+2(s_k(n_k-y_k))+((n_k-y_k)^2)$$ We assume $s_k$ , $n_k$ , and $n_k^{\text{'}}$ are zero-mean signals, and that $s_k$ is independent of $n_k$ and $n_k^{\text{'}}$ . Then $$(s_k(n_k-y_k))=(s_k)(n_k-y_k)=0$$ $$({}_k^2)=(s_k^2)+((n_k-y_k)^2)$$ Since the input signal has no information about $s_k$ in it, minimizing $({}_k^2)$ can only affect the second term, which is the standard Wiener filtering problem, with solution $$W=R_{n^{\text{'}}{n}^{\text{'}}}^{(-1)}{P}_{n{n}^{\text{'}}}$$