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< 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.
>
>
>
>
>
Reply by booboojerkers●April 29, 20082008-04-29
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.