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< &frac14;.


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.

Update:  after giving it some more thought, this pdf is a standard normal
distribution:  pdf = exp[-x^2/2]/sqrt(2pi)

Now the concern is:  do I just put the modulated cosine value into the pdf
as the x value in order to proceed with setting up the algorithm?

As so:  pdf = exp[-(cos(2pi*fo*n)*w(n))^2/2]/sqrt(2pi)

??????

Any feedback on this is appreciated

>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< &frac14;.
>
>
>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.
>
>
>
>
>