# The optimal estimator using FIR Wiener filter

The received signal $$y(n)$$ is defined as

$$y(n) = h(n)*x(n) +v(n)$$ where $$h(n)$$ is unknown impulse response and $$v(n)$$ is a white Gaussian noise.

The value of $$x(n)$$ for $$n=1, ..., 32$$ are given. The task is to calculate an estimator of $$x(n)$$. I calculated the estimator using FIR Wiener filter which gives the optimal means error, when the impulse response $$h_{wie} = R_y^{-1} r_{xy}$$ where $$R_y=E[ YY^T]$$ and $$r_{xy}= E[xy]$$. My intuition is when we have more value of $$x(n)$$, we should get the better estimator but that not what i can observe from my computation i get optimal estimator when filter orders is around 24-30 by try and error.

Can someone explain a systematic way to find optimal taps/orders of FIR Wiener filter? And is there a way to improve the estimator i.e different types of filters?

• If $x(n)$ is given, then the estimator for $x(n)$ is the given $x(n)$. Done. And a Wiener filter requires that you know $h(n)$ -- so, can't be done. And -- for $h(n)$ known -- a Wiener filter is the optimal filter in the mean squared sense. If you want something better, you need to define "good" and "better". – TimWescott Jan 24 at 20:24