I am trying to do blind system identification of a univariate linear FIR model:
I am unsure if the approach is correct or not and any help to further proceed with the maximization will be great. Thank you.
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$\begingroup$ Your equations (1) and (2) do not make sense to me. How is $u(t)$ used? What is $w(t)$ in the paragraph below (2). It looks like (1) should read something like: $z(t) = h_1 z(t-1) + h_2 z(t-2) + w(t)$ for this to start to make sense... Or is it something else? $\endgroup$– Peter K. ♦Jul 1, 2015 at 11:56
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1$\begingroup$ I had made a terrible blunder with the notations, thank you for telling me. I have corrected them. $\endgroup$– Ria GeorgeJul 1, 2015 at 16:08
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1 Answer
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I think your E-step is correct(only one term missing in the last expression: $-N\ln\sigma_w$ ).
To obtain the M-step you have to differentiate with respect to all your parameters. You don't have to include $u(n)$ in $\theta$, since it's defined by $p$. So you compute $\frac{\partial Q}{\partial\theta}=0$. Solving for $\theta$, and this is your new $\theta$ i.e. $\theta_{m+1}$. E.g. for $h$ we gets $\frac{\partial Q}{\partial h}=-(-2\sum y_nz_n^s+2h\sum P_n^s)/(2\sigma_w^2)=0$.
Solving: $h=h_{m+1}=\sum y_nz_n^s(\sum P_n^s)^{-1}$.
-For $p$: $Q'(p)=\sum z_n^s/p -(N-\sum z_n^s)/(1-p)=0,p=\sum z_n^s/N$.
This is typical Bernoulli/binomial. So it seems right.
-For $\sigma_w$: $Q'(\sigma_w)=-(\sum y_n^2-2h^T\sum y_nz_n^s+h^Th\sum P_n^s)\cdot (-2/(2\sigma_w^3))-N/\sigma_w=0, \sigma_w^2=(\sum y_n^2-2h^T\sum y_nz_n^s+h^Th\sum P_n^s)/N$
and updating the parameters in a loop until convergence.