# log-likelihhood function for N sample of data

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?

• why is it : $log|Q|+Trace[Q^{-1}*C]$ ; where $C=1/N * summation((x-bAexp(jwt)(x-bA*exp(jwt))^{H})$ – Reza Mahjoob Sep 30 '17 at 19:58
• Hi Reza, instead of a comment, you can edit your question. – Atul Ingle Sep 30 '17 at 20:09

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 $$\mathbf{ a^T C b} = Trace ( \mathbf {C \,b a^T})$$