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if $x(t)=b A e^{ j\omega t} + e(t)$ for $t= 1,2,...,N$ where $b$ is a parameter, $A$ is a vector $M \times 1$, $e(t)$ is a white Gaussian noise with covariance matrix of $Q$ theh what is log-likelihood function for that?

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  • $\begingroup$ why is it : $log|Q|+Trace[Q^{-1}*C]$ ; where $ C=1/N * summation((x-bAexp(jwt)(x-bA*exp(jwt))^{H}) $ $\endgroup$ – Reza Mahjoob Sep 30 '17 at 19:58
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    $\begingroup$ Hi Reza, instead of a comment, you can edit your question. $\endgroup$ – Atul Ingle Sep 30 '17 at 20:09
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I’m assuming that the trace is what is confusing, so let me give you some clues.

First an identity $$ Trace(\mathbf{ABC)})= Trace(\mathbf{BCA}) $$

and for scaler $x$, $$ Trace(x) =x$$

Let $\mathbf{a}$ and $\mathbf{b}$ be $N \times 1$ vectors and $\mathbf{C} $is a $N \times N$ matrix, so $$ \mathbf{ a^T C b} = Trace ( \mathbf {C \,b a^T}) $$

The rest you should be able to figure out using the definition of a Gaussian, independence, and properties of the log.

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  • $\begingroup$ Thanks a lot. What is the need for "abs" of "Q" in "log |Q| "? $\endgroup$ – Reza Mahjoob Oct 3 '17 at 18:22
  • $\begingroup$ not abs, determinant $\endgroup$ – Stanley Pawlukiewicz Oct 3 '17 at 18:24

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