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I am looking for methods which are able to simultaneously estimate the
frequency (f_i) and phase (p_i) of the
mixture (superposition) the simple "sinusoidal" signals with additive noise:

s_i(t) = A_i*cos(2*pi*f_i*t+p_i)+n_i(t) ,

where i=1,2, ..., N and n_i is additive gaussian noise.


Michal

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Michal Kvasnicka wrote:
> I am looking for methods which are able to simultaneously estimate the
> frequency (f_i) and phase (p_i) of the
> mixture (superposition) the simple "sinusoidal" signals with additive noise:
> 
> s_i(t) = A_i*cos(2*pi*f_i*t+p_i)+n_i(t) ,
> 
> where i=1,2, ..., N and n_i is additive gaussian noise.
> 
> 
> Michal
> 
> 
> 
> 

Hello Michal,

change your estimation model into:

  s_i(t) = A_i*cos(2*pi*f_i*t)+ B_i*sin(2*pi*f_i*t)+n_i(t)

  linearize ...

  solve the system for f_i, B_i (other than p_i)

then find:
p_i = angle(complex(A_i./B_i)

note: if you estimate A_i, too, your system becomes very unlinear and 
can be solved only with good initial values.

Christian

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Michal Kvasnicka wrote:
> I am looking for methods which are able to simultaneously estimate the
> frequency (f_i) and phase (p_i) of the
> mixture (superposition) the simple "sinusoidal" signals with additive noise:
> 
> s_i(t) = A_i*cos(2*pi*f_i*t+p_i)+n_i(t) ,
> 
> where i=1,2, ..., N and n_i is additive gaussian noise.
> 
> 
> Michal
> 

What about the Fourier transform?

Paul

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Hi Paul,

I am looking for adaptive high-resolution methods. FFT is standard but not
always suitable for this type of problems.

Michal (ERA)
"Paul Howland" <h...@wanadoo.nl> píse v diskusním príspevku
news:4086a8bf$0$54152$1...@news.wanadoo.nl...
> Michal Kvasnicka wrote:
> > I am looking for methods which are able to simultaneously estimate the
> > frequency (f_i) and phase (p_i) of the
> > mixture (superposition) the simple "sinusoidal" signals with additive
noise:
> >
> > s_i(t) = A_i*cos(2*pi*f_i*t+p_i)+n_i(t) ,
> >
> > where i=1,2, ..., N and n_i is additive gaussian noise.
> >
> >
> > Michal
> >
>
> What about the Fourier transform?
>
> Paul

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Hi Michal

Ah OK - I missed the significance of the index i on f_i and p_i.  So the 
frequency and phase and amplitude are all slowly varying within the 
observation period?  What kind of SNR do you expect? - I think many of 
the hi-res algorithms require high SNR to get good results.  Some also 
fail in the presence of correlated signals (eg. multipath).  I'm not an 
expert in these areas, but my guess is that a few more details like this 
may help those who are provide the best guidance to you.

Paul



Michal Kvasnicka wrote:

> Hi Paul,
> 
> I am looking for adaptive high-resolution methods. FFT is standard but not
> always suitable for this type of problems.
> 
> Michal (ERA)
 >
> "Paul Howland" <h...@wanadoo.nl> píse v diskusním príspevku
> news:4086a8bf$0$54152$1...@news.wanadoo.nl...
>

 >>
 >>What about the Fourier transform?
 >>
 >>Paul
 >>
>>Michal Kvasnicka wrote:
>>
>>>I am looking for methods which are able to simultaneously estimate the
>>>frequency (f_i) and phase (p_i) of the
>>>mixture (superposition) the simple "sinusoidal" signals with additive noise:
>>> 
>>>s_i(t) = A_i*cos(2*pi*f_i*t+p_i)+n_i(t) ,
>>>
>>>where i=1,2, ..., N and n_i is additive gaussian noise.
>>>
>>>
>>>Michal

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On Wed, 21 Apr 2004 05:46:40 -0700, Michal Kvasnicka wrote:
> I am looking for methods which are able to simultaneously estimate the
> frequency (f_i) and phase (p_i) of the mixture (superposition) the
> simple "sinusoidal" signals with additive noise

This is pretty standard stuff; I think you will find what you are
looking for in Chapter 8 of Steven Kay's _Fundamentals of Statistical
Signal Processing: Estimation Theory (Vol. 1)_ Prentice Hall, 1993.

-Leif

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Leif Harcke <l...@stanford.edu> writes:

> On Wed, 21 Apr 2004 05:46:40 -0700, Michal Kvasnicka wrote:
>> I am looking for methods which are able to simultaneously estimate the
>> frequency (f_i) and phase (p_i) of the mixture (superposition) the
>> simple "sinusoidal" signals with additive noise
>
> This is pretty standard stuff; I think you will find what you are
> looking for in Chapter 8 of Steven Kay's _Fundamentals of Statistical
> Signal Processing: Estimation Theory (Vol. 1)_ Prentice Hall, 1993.
>
> -Leif

Come on, Leif, fundamental? How often have we discussed the FFT, which
is much more so than this? I was wondering about the answer to this
question as well. A single sinusoid, well even I can estimate that.
But simultaneously estimating multiple sinusoids of unknown frequencies???

Why don't you give the rest of us a little insight on how such and
algorithm would operate.
-- 
%  Randy Yates                  % "So now it's getting late,
%% Fuquay-Varina, NC            %    and those who hesitate
%%% 919-577-9882                %    got no one..."
%%%% <y...@ieee.org>           % 'Waterfall', *Face The Music*, ELO
http://home.earthlink.net/~yatescr

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"Michal Kvasnicka" <m...@_NO_ZpaMM-.quick.cz> writes:

> I am looking for adaptive high-resolution methods. FFT is standard but not
> always suitable for this type of problems.

Hi Michal,

High resolution methods (e.g. MUSIC etc) generally don't help with
phase information... and possibly don't estimate as robustly as more
standard techniques.

Does the technique have to be on-line? Or is it batched?

Generally, if you have multiple sinusoids, you can just estimate the
parameters (amplitude, phase, frequency) of the largest, regress it
out of the data, estimate the second largest, regress it out, etc. for
N total steps.

Of course, if you want to do it on-line, then this is not really an
option.

Ciao,

Peter K.

-- 
Peter J. Kootsookos

"I will ignore all ideas for new works [..], the invention of which
 has reached its limits and for whose improvement I see no further
 hope."

- Julius Frontinus, c. AD 84

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Leif Harcke <l...@stanford.edu> wrote in message
news:<p...@stanford.edu>...
> On Wed, 21 Apr 2004 05:46:40 -0700, Michal Kvasnicka wrote:
> > I am looking for methods which are able to simultaneously estimate the
> > frequency (f_i) and phase (p_i) of the mixture (superposition) the
> > simple "sinusoidal" signals with additive noise
> 
> This is pretty standard stuff; I think you will find what you are
> looking for in Chapter 8 of Steven Kay's _Fundamentals of Statistical
> Signal Processing: Estimation Theory (Vol. 1)_ Prentice Hall, 1993.
> 
> -Leif

Well... I don't have Kay's book available, so I don't know what this
"standard stuff" is, but I assume you are talking about frequency 
estimators like MUSIC, ESPRIT and so on. That kind of material is 
found in lots of textbooks and there are tens of articles published 
per year on MUSIC, with variations, alone. 

In my experience, the people who write articles and books about these 
types of techniques, do not always understand the (im)practical aspects 
of any given technique. In most presentations of MUSIC, there are usually 
two significant practical aspects missing. Occationally, one sees one 
aspects commented upon very breafly, but never both at the same time. 
I am thinking of the reason why the MUSIC "pseudo spectrum" has no 
physical significance whatsoever, and the importance of calibrating 
the sensor system properly.

A more interesting observation is that papers that describe these types
of methods applied to real-world data are simply missing from the journals.
I have yet to see any article describing any frequency estimator being 
applied to real data in real environments in "production type" 
applications. Heck, I don't even know of any published real-world lab 
tests (as opposed to computer-generated synthetic data) with these types 
of techniques. 

I find this deafening silence with respect to practical applications 
and real-world results quite worrying. 

Rune

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Hi Rune.

Rune> A more interesting observation is that papers that describe
Rune> these types of methods applied to real-world data are simply
Rune> missing from the journals.  I have yet to see any article
Rune> describing any frequency estimator being applied to real data in
Rune> real environments in "production type" applications. Heck, I
Rune> don't even know of any published real-world lab tests (as
Rune> opposed to computer-generated synthetic data) with these types
Rune> of techniques.

Rune> I find this deafening silence with respect to practical
Rune> applications and real-world results quite worrying.

Well, ESPRIT has been used (and you could very well use MUSIC, I
believe) for sinusoidal modeling and coding of speech and audio, and
there are papers on this in IEEE Trans. on Speech and Audio
Processing. Since different applications have different journals, no
wonder that one does not encounter these in the more theoretical
journals such as IEEE Trans. on Signal Processing. They don't belong
there. From your post I take it, that these methods are not applied in
your field. That doesn't mean that there aren't applications, though.

-- 
/Mads (http://kom.aau.dk/~mgc)

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