minimum cycles for fft, and limits of filling short sample out with zeros

p.kootsookos@remove.ieee.org (Peter K.) writes:

> Randy Yates <yates@ieee.org> writes:
>
>> Thank you for challenging these sorts of assertions from another
>> POV. My usual answer is similar - if you have a Hilbert transform of a
>> noiseless sine wave, then you can get its frequency using only two
>> [complex] samples: F = (p2 - p1)/T, where pi = arctan(yi/xi). 
>
> Agreed, with the caveat that you can never form the ideal Hilbert
> transform, 

Good point, but there are cases where you can perform quadrature
measurements directly without having to use the transform. For
example, a quadrature downconverter followed by dual A/Ds.

> and that you can only have a completely noiseless signal
> represented in a digital computer in pathological cases
> (e.g. 1,-1,1,-1 or -1, 0, 1, 0, -1, etc.).

But that is not the basis of your assertion. 
-- 
%  Randy Yates                  % "She's sweet on Wagner-I think she'd die for Beethoven.
%% Fuquay-Varina, NC            %  She love the way Puccini lays down a tune, and
%%% 919-577-9882                %  Verdi's always creepin' from her room." 
%%%% <yates@ieee.org>           % "Rockaria", *A New World Record*, ELO   
http://home.earthlink.net/~yatescr

Randy Yates <yates@ieee.org> writes:

> p.kootsookos@remove.ieee.org (Peter K.) writes:
> 
> > and that you can only have a completely noiseless signal
> > represented in a digital computer in pathological cases
> > (e.g. 1,-1,1,-1 or -1, 0, 1, 0, -1, etc.).
> 
> But that is not the basis of your assertion. 

Sorry, Randy, I'm not sure what you mean?  Can you elaborate?

Ciao,

Peter K.

p.kootsookos@remove.ieee.org (Peter K.) writes:

> Randy Yates <yates@ieee.org> writes:
>
>> p.kootsookos@remove.ieee.org (Peter K.) writes:
>> 
>> > and that you can only have a completely noiseless signal
>> > represented in a digital computer in pathological cases
>> > (e.g. 1,-1,1,-1 or -1, 0, 1, 0, -1, etc.).
>> 
>> But that is not the basis of your assertion. 
>
> Sorry, Randy, I'm not sure what you mean?  Can you elaborate?

Simply that your statements regarding limitations in measuring
frequency accurately had nothing to do with the noise but rather
with the time window of the measurement.
-- 
%  Randy Yates                  % "So now it's getting late,
%% Fuquay-Varina, NC            %    and those who hesitate
%%% 919-577-9882                %    got no one..."
%%%% <yates@ieee.org>           % 'Waterfall', *Face The Music*, ELO
http://home.earthlink.net/~yatescr

rhnlogic@yahoo.com wrote:
> Peter K. wrote:
> 
>>kroger@princeton.edu writes:
>>
>>
>>>My intent is to simple see the power at various frequencies. I'd like a
>>>short window and high frequency resolution, to the degree possible.
>>
>>You start running up against Heisenberg's uncertainty principle (or a
>>variant).  High frequency resolution implies longer time
>>durations... short time durations imply lower frequency resolution.
>>Whether you can trade off an acceptable amount of one for enough of
>>the other will depend on your application.
> 
> 
> The missing factor is noise.  In quantum uncertainty, Planck's
> constant is the limiting factor.  In measuring frequency, the
> signal-to-noise ratio is the limiter.  For instance, if the noise
> level is zero, you might be able to measure a single sine waves
> frequency perfectly with only three infinitely accurate samples.
> Increase the noise floor, and the frequency accuracy goes down,
> or the number of samples needed for a given frequency resolution
> goes up, or both.
> 
> With low enough noise and a large enough separation of
> frequencies in the signal, zero-padding might be one
> computational method of getting a bit more accuracy in the
> measurement of frequency peaks.  But it really depends a
> lot on the characteristics of the noise and some a priori
> knowledge about what's really in the signal.

Well, if you know in advance that there is only a single frequency in 
the signal, what about the spectrum do you want to measure? The OP is 
interested in a spectrum measurement, not a frequency identification.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Randy Yates <yates@ieee.org> writes:

> Simply that your statements regarding limitations in measuring
> frequency accurately had nothing to do with the noise but rather
> with the time window of the measurement.

OK.

I'm wondering where the discrepancy is... your example says that two
(complex) samples can estimate any frequency.  I'm saying that, in
order to localize a given frequency, you need a longer duration.

I suspect that part of the difference is that you're making the
assumption of one single, pure sinusoid.  That's a pretty big
assumption.

I'll ponder some more and let you know.

Thanks,

Peter K.

p.kootsookos@remove.ieee.org (Peter K.) writes:

> 
> I'll ponder some more and let you know.
> 

Not sure it helps, but a colleague of mine at ANU has written this note:

http://axiom.anu.edu.au/~williams/uncertainty.pdf

Ciao,

Peter K.

p.kootsookos@remove.ieee.org (Peter K.) writes:

> p.kootsookos@remove.ieee.org (Peter K.) writes:
>
>> 
>> I'll ponder some more and let you know.
>> 
>
> Not sure it helps, but a colleague of mine at ANU has written this note:
>
> http://axiom.anu.edu.au/~williams/uncertainty.pdf

Looks intereseting, Peter. I would like to take some time to read it
with understanding. 

I also have a book on the subject by Cohen - hasn't helped much. I guess
I need to read it, too.
-- 
%  Randy Yates                  % "She's sweet on Wagner-I think she'd die for Beethoven.
%% Fuquay-Varina, NC            %  She love the way Puccini lays down a tune, and
%%% 919-577-9882                %  Verdi's always creepin' from her room." 
%%%% <yates@ieee.org>           % "Rockaria", *A New World Record*, ELO   
http://home.earthlink.net/~yatescr

Peter K. wrote:
> p.kootsookos@remove.ieee.org (Peter K.) writes:
> 
> 
>>I'll ponder some more and let you know.
>>
> 
> 
> Not sure it helps, but a colleague of mine at ANU has written this note:
> 
> http://axiom.anu.edu.au/~williams/uncertainty.pdf
> 
> Ciao,
> 
> Peter K.
> 
> 
The problem with this, is that in quantum mechanics the location of the 
particle is governed by a probability density which is related to the 
observation.   In signal processing, the transform broadens but the 
location of peak governs the frequency measurement. The frequency 
content of a truncated signal increases but its not a probability 
density, i.e. the true frequency isn't a random parameter governed by 
chance, like a quantum particle.

Now, I know for a fact Peter that you know about parametric estimation 
and the CRLB, so what's up with the Heisenberg witchcraft.

"Peter K." <p.kootsookos@remove.ieee.org> wrote in message 
news:uoe75bsf1.fsf@remove.ieee.org...
> p.kootsookos@remove.ieee.org (Peter K.) writes:
>
>>
>> I'll ponder some more and let you know.
>>
>
> Not sure it helps, but a colleague of mine at ANU has written this note:
>
> http://axiom.anu.edu.au/~williams/uncertainty.pdf
>
> Ciao,
>
> Peter K.
>
>

Hello Peter,

This approach assumes one is determining freq., by a method one doesn't 
usually use (I.e., looking at 2nd moments about the mean of the magnitude 
functions). For example, if you know there is only one sinusoid, the noise 
is the limiting factor. Notice in the CS method, there is no allowance for 
noise. Yet, three real valued samples can determine a sinusoid (amplitude, 
phase, and frequency) if there is no noise.

Also unlike the QM approach, you can actually make multiple processing 
passes on the data. In QM, each observation pass changes the data (unless 
the measurements commute, but then they yield nothing new)), so this won't 
work in the QM world to yield new information.

I think the Fourier uncertainty principle has an application in signal 
processing, but I also think one has to be cognizant of the underlying 
assumptions. When a theory says something can't be done, it only means that 
it can't be done given the assumptions are true.

We certainly know that if you want to know when a pulse arrives, we don't 
simply look for the centroid of the energy, but if we know something more 
like something about the pulse, then we can use a matched filter. The fact 
that it is a pulse tells us some information. If all we know is there is a 
pulse and nothing more, then there is less that we can do.

Clay

Stan Pawlukiewicz wrote:

(snip)

> The problem with this, is that in quantum mechanics the location of the 
> particle is governed by a probability density which is related to the 
> observation.   In signal processing, the transform broadens but the 
> location of peak governs the frequency measurement. The frequency 
> content of a truncated signal increases but its not a probability 
> density, i.e. the true frequency isn't a random parameter governed by 
> chance, like a quantum particle.

Say I have a signal that in frequency space has a broad peak,
maybe not so peak shaped at all.  What do I call the frequency
of that signal?   Somewhere within the peak, but where?

-- glen