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.

  • $\begingroup$ 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
  • 1
    $\begingroup$ I had made a terrible blunder with the notations, thank you for telling me. I have corrected them. $\endgroup$
    – Ria George
    Jul 1, 2015 at 16:08

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.