Introduction
My goal is to determine a variation on Wiener filter in time domain. The proof as presented on Wikipedia works when criterion is defined as mean square error of spectra, I intend to do the same proof with criterion in time domain.
Let $c$ be the input, $\hat{c}$ our estimate, $h$ system impulse response, $g$ a filter and $n$ additive centered white noise with known variance. Let $\epsilon$ be the mean square error and all defined as follows. $$\epsilon=\int_{\infty}^{\infty}\mathbf{E}[|c-\hat{c}|^2|c] dx$$ $$y=c \ast h+n$$ $$\hat{c}=g\ast y$$ The conditional probability means the filter is optimized for every realization of $c$. The criterion then becomes an expression with three expressions using the noise: $$c(1-g \ast h)\mathbf{E}[\overline{g \ast n}|c]$$ $$\overline{c(1-g \ast h)}\mathbf{E}[g \ast n|c]$$ $$\mathbf{E}[(g \ast n)(\overline{g \ast n})|c]$$ As the context is now established, let me ask the questions.
Claim 1
$$\mathbf{E}[g \ast n|c]=\mathbf{E}[\overline{g \ast n}|c]=0$$ I am quite positive about this one, because in frequency domain it becomes $$\mathbf{E}[GN|c]=\mathbf{E}[G|c]·\mathbf{E}[N|c]+cov(G,N)=G·\mathbf{E}[N]+0=G·0=0$$ As $G$ and $N$ are independent, the covariance is zero. There should exist only one $g$ for deterministic $c$ (further in the derivation I take the derivative of criterion with respect to $g$). Lastly I assume that $\mathbf{E}[n]=\mathbf{E}[N]=0$ which probably holds, but might need some further background to proof, probably via Karhunen-Loève theorem applied on Fourier series of Wiener process? Or simply state that $\mathbf{E}[n]=0 \rightarrow \mathbf{E}[N]=0$?
Claim 2
As $n$ is completely independent I have an intuitive feeling that $$\mathbf{E}[g \ast n ]=\mathbf{E}[ g·n]$$ which would imply $$\mathbf{E}[(g \ast n)(\overline{g \ast n})|c]=\mathbf{E}[g·n·\overline{g·n}|c]=g·\overline{g}\mathbf{E}[n·\overline{n}]=g·\overline{g}·var(n)$$ The main argument I have comes from the definition of white noise, $$\forall x: n(x)\sim \mathcal{N}(0,var(n)) $$ which could be used in the integral definition of convolution to obtain something useful, preferably a product. Any idea on this?
General question
Are there any general properties of operations {multiplication, convolution, expected value, Fourier transform} that could be exploited to obtain something uselful in this context?