# Estimate $s$ from $y=s+n$

Now I receive $y=s+n$, $s$ is the signal and $n$ is the white guassion noise. Everytime $s$ will be change irregularly, have some methods to get a estimation of $s$ namely $\hat s$?

Namely $s$ is random. And I don't know $s$ how to change everytime,so the traditional kalman filter algorithm may can't be used in this circumstance.

• How is your noise $n$ defined ? – Gilles Feb 29 '16 at 12:44
• Just Gaussian or White Gaussian Noise ? And is your signal deterministic or random ? Please add these details to your question, it will help in answering. – Gilles Mar 1 '16 at 10:43
• Just White Gaussian Noise.And the signal is random.So maybe the exist kalman filter algorithm can't be used in this circumstance.Because I don't know how the signal changed.Thanks. – w xd Mar 1 '16 at 11:40
• Edit your question to include these details on the nature of $s$ and $n$ in your question please. – Gilles Mar 1 '16 at 11:52
• Without some structure to the way $s$ changes, it will be difficult to recommend any technique, I think. If it's random, how is its distribution different from that of $n$? If there is no difference, then I fear you may not be able to do much at all. The Kalman filter will only work if you know something about how $s$ changes. – Peter K. Mar 1 '16 at 14:43

I think you are seeking for an algorithm but this may help. If you don't know how $s$ is distributed but you know its moments of first and second order, you can try with the linear estimator of the linear Minimum-mean Square Error (lmmse), with the following form, $$\hat{s}(y) = ay + b$$ with $$a = \frac{ \mathrm{C}_{xy} }{ \sigma_{y}^{2} } \\ b = m_{s} - am_{y}$$ where $m$ denotes mean, $\sigma$ variance and $C$ covariance. It would help to know the moments of the noise, I mean, for example if is zero-mean and normalized to unity.