Maximum Likelihood Estimation of Phase

Question

I'm trying to solve an exercise where I need to derive the MLE of the phase parameter.

I'm given the below signal with white noise $w[n] \sim \mathcal{N}(0, \sigma^2) $, amplitude $A$, angular frequency $\omega \notin \{0, \pi\}$, $N$ measurements and unknown phase $\phi$.

$$ x[n]=A \cos (\omega n+\phi)+w[n], \; \; n= 1,...,N $$

Usually I'd get the log-likelihood function, compute the first derivative, set it to zero and solve for the unknown parameter which is $\phi$ in this case. However, I am unable to solve for $\phi$ in the 1st derivative as I am can't extract it from inside the sine and cosine functions.

$$ \begin{gathered} \frac{\partial}{\partial \phi} \ln p(x ; \phi)=-\frac{1}{\sigma^{2}} \sum_{1}^{N}(x[n]-A \cos (\omega n+\phi))(A \sin (\omega n+\phi)) \\ \frac{\partial}{\partial \phi} \ln p(x ; \phi)=0 \end{gathered} $$

Can someone please help me solve this?

Answer 1

I'm not sure that there's a closed-form solution for the ML estimate of phase, but if you recast the definition of $x[n]$ to $$x[n] = a \cos (\omega n) + b \sin (\omega n) + w[n], \; \; n= 1,...,N $$ then you can easily find the ML estimates of $a$ and $b$, and take $$\hat \phi = \tan^{-1}\frac{\hat b}{\hat a}.$$

This PDF seems to back up the assumption, though it calls the resulting estimator only an approximate maximum likelihood estimator.

Answer 2

Maybe you cannot get an analytical solution. Or maybe you can assume the phase is very small, and use some approximation like $\cos \left( \phi \right) = 1$ and $\sin \left( \phi \right) = \phi$.