Newton-Rapshon method over a multiplicative or modulated noise process

I am trying to implement a newton-raphson iteration to estimate sinusoidal
frequency mixed with noise. Here's the problem statement:

A white Gaussian noise process w(n) with variance sigma^2 = 1 is

modulated by a sinusoid cos(2pif0n)*w(n) to yield the data set

x(n) = cos(2pif0n)*w(n), n = 0, 1, ... , N-1,

where N = 1000. The frequency f0 of the sinusoid is known to be in

the range 0 < f0< ¼.


Here's how the problem is seen, please correct me if I'm wrong:

The best way it seems to solve this problem is to write down the pdf, ie,
likelihood function, and then search it over the interval 0<fo<1/4.
(this will be done in matlab).

In this problem there is actually no signal but a noise process that is
modulated. The nth sample is Gaussian with mean zero and variance sigma^2
= 1, which is time varying.

All the noise samples are independent.

The way to solve is to just write down the pdf and maximize over f_0. A
maximum should be seen at the correct f_0.

My problem is when initially writing down the PDF.

With additive noise, I have typically done this as w(n) = x(n) - cos(*),
here though (for multiplicative noise) if solving for w(n) we have w(n) =
x(n)/cos(*)????

With additive noise the pdf =>
p(x;f)=[(2*pi*sigma^2)^-N/2]*exp[(-1/2sigma^2)*Esum(x(n)-cos2pifn)^2] -->as
Gaussian form

But my question is, how do I write out the pdf with multiplicate (rather
than additive) noise???


This is where I am stuck.




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