This question is an extension of another question of mine asked earlier here Help in understanding if the maximum likelhood estimation is working properly

In that question the inputs were real valued. Now, in this question the inputs are real valued and hence the confusion.

I am having difficulties in the notation when applying to the complex domain. What notations and operators should I use when all the data are complex valued? Please help

  • $\begingroup$ A likelihood is a (possibly conditional) probability density function/mass function -- it has to be real valued. $\endgroup$ – Batman Nov 29 '16 at 1:26
  • $\begingroup$ To me it's not yet clear: What are deterministic/known inputs, what is random, what do you want to estimate? What is $P_n$? what is $x^s$? Also, your log-likelihood in the bottom does not match the L-expression in the top? $\endgroup$ – Maximilian Matthé Nov 29 '16 at 6:35
  • $\begingroup$ Please let me know if more information is required. Thank you once again $\endgroup$ – SKM Nov 29 '16 at 7:12
  • $\begingroup$ I dont want to go through a full kalman-filter derivation here. But if it is just about adapting the notation: you define your signal model as you want. When it's supposed to be convolution, keep it with transpose without conjugation. However, take care that your signal model is matching your numeric calculations. Whenever you have squared terms, e.g. $z_n(z_n-hx)$, you should conjugate the second term. This should yield the correct result. $\endgroup$ – Maximilian Matthé Nov 29 '16 at 7:26
  • 1
    $\begingroup$ No, its the power of 2 in Python Notation. Or abs(x-y).^2 in Matlab $\endgroup$ – Maximilian Matthé Nov 29 '16 at 21:53

Without seeing your overall signal model we cannot really help. I suppose $y$ is your estimate? And hence, $L=L(y)$. Where is $h$ in your Likelihood equation? However, some general observations:

  • As Batman pointed out: likelihood corresponds to probability, which is necessarily a real value. So, if your equations yield complex values, you did something wrong. (Note that, depending on numerical accuracies etc. it can happen, that there is a very small imaginary part ($\approx 10^{-16}$) in your numeric computations. This part can be ignored.)
  • Commonly (but not always!) when in the real-domain you have a transpose in your equation, it changes to conjugate transpose (i.e. Hermitian operation) in the complex domain.
  • Noise variances commonly remain real-valued. However, noise realizations will be complex-valued.
  • You should understand the expression in your exponential as the squared distance between the measurement and the estimate. In real domain, the squared distance of $x,y$ is equal to $(x-y)^2$. However, in the complex domain, it becomes $\|x-y\|^2=(x-y)(x-y)^*$.
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