Show me the numbers

On 6/2/15 10:26 PM, Eric Jacobsen wrote:
> On Tue, 02 Jun 2015 20:05:13 -0400, robert bristow-johnson
> <rbj@audioimagination.com>  wrote:
>
>> On 6/2/15 5:25 PM, Eric Jacobsen wrote:
>>> On Tue, 2 Jun 2015 20:33:09 +0000 (UTC), spope33@speedymail.org (Steve
>>> Pope) wrote:
>>>
>>>> Eric Jacobsen<eric.jacobsen@ieee.org>   wrote:
>>>>
>> ...
>>>>
>>>>> Many of the two-term estimators have
>>>>> large variance when the frequency is near the bin center as one of the
>>>>> terms approaches zero.
>>>>
>>>> Seems this might also apply to three-term estimators...  that is,
>>>> if the frequency is near the center of the middle bin out of three
>>>> consecutive bins, the other two terms approach zero.
>>>
>>> Obviously it's algorithm dependent, but most of the three-term
>>> estimators don't wind up with the denominator approaching zero as the
>>> adjacent samples approach the nulls, so the effect is mitigated much
>>> better.
>>
>> seems to me that, with a gaussian window (which implies a gaussian
>> function in the frequency domain) that in the log (dB) domain, the
>> quadratic three-point solution should be a perfect fit.
>
> That's correct, and that's covered in a few places (books, papers,
> etc.).   I think the performance isn't all that great under certain
> conditions, though.

well, for three points, there's no room for interference from noise or 
other components.

in a 2001 paper (the only one i ever did for an IEEE thingie) i derived 
an estimator for (middle) frequency, sweep rate, and ramp rate of a 
gaussian-windowed chirp (assuming linearly swept frequency and 
exponentially ramped amplitude within the narrow gaussian window) with a 
collection (more than three) of adjacent points.  it's based on the fact 
that such a chirp is fourier transformed into the same species of animal 
in the frequency domain and then, in the dB scale, doing a simple 
discrete differentiator, the quadratic becomes a straight line (with 
complex coefficients) and you can fit to that with a simple linear fit 
line and get parameters from the apparent coeffcients).

i've mentioned the paper before.  if anyone wants a little pdf from me, 
lemme know and i'll send it.  or get it from researchgate.


-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Wednesday, June 3, 2015 at 3:20:00 AM UTC-7, tsd82 wrote:

> ...
> By the way, I was wondering, does anyone knows about applications
> requiring to work at these very high SNR (60 - 100 dB) ? I have no idea
> for myself. Maybe very precise frequency measurements for atomic-based
> clock?
> 
> Kind Regards,
> 
> Julien

There is a considerable literature on frequency peak location and measurement for the application of ADC and DAC testing. Think 24 bit convertors.

There is a much smaller literature providing window designs for interference suppression for testing done by the people who design 24 bit convertors to evaluate their device internal implementations. Those are the minimum sidelobe windows in "Windows Connections" (pages 6 and 12) at:
http://www.abvolt.com/compdsp/presentations/Dalrymple/dbd.pdf

Dale B. Dalrymple

On Wed, 3 Jun 2015 12:14:42 -0700 (PDT), dbd
<d.dalrymple@sbcglobal.net> wrote:

>On Wednesday, June 3, 2015 at 3:20:00 AM UTC-7, tsd82 wrote:
>
>> ...
>> By the way, I was wondering, does anyone knows about applications
>> requiring to work at these very high SNR (60 - 100 dB) ? I have no idea
>> for myself. Maybe very precise frequency measurements for atomic-based
>> clock?
>>=20
>> Kind Regards,
>>=20
>> Julien
>
>There is a considerable literature on frequency peak location and measureme=
>nt for the application of ADC and DAC testing. Think 24 bit convertors.

That would do it.

>There is a much smaller literature providing window designs for interferenc=
>e suppression for testing done by the people who design 24 bit convertors t=
>o evaluate their device internal implementations. Those are the minimum sid=
>elobe windows in "Windows Connections" (pages 6 and 12) at:
>http://www.abvolt.com/compdsp/presentations/Dalrymple/dbd.pdf
>
>Dale B. Dalrymple

Ah, nostalgia.   I remember when you presented that.   Very cool.


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

On Wednesday, June 3, 2015 at 9:54:31 AM UTC-7, robert bristow-johnson wrote:
> ...
> in a 2001 paper (the only one i ever did for an IEEE thingie) i derived 
> an estimator for (middle) frequency, sweep rate, and ramp rate of a 
> gaussian-windowed chirp (assuming linearly swept frequency and 
> exponentially ramped amplitude within the narrow gaussian window) with a 
> collection (more than three) of adjacent points.
> ...
> r b-j                  rbj@audioimagination.com

There is a more general development of the process that has appeared under the term "frequency reassessment" at:

http://perso.ens-lyon.fr/patrick.flandrin/0065-Ch05.pdf
Time-frequency reassignment: from principles to algorithms
Flandrin, Auger, Chassandre-Mottin

and

http://arxiv.org/abs/0903.3080
A Unified Theory of Time-Frequency Reassignment
Kelly R. Fritz, Sean A. Fulop

Dale B. Dalrymple

Eric Jacobsen <eric.jacobsen@ieee.org> wrote:

>On Wed, 3 Jun 2015 12:14:42 -0700 (PDT), dbd

>>There is a much smaller literature providing window designs for interferenc=
>>e suppression for testing done by the people who design 24 bit convertors t=
>>o evaluate their device internal implementations. Those are the minimum sid=
>>elobe windows in "Windows Connections" (pages 6 and 12) at:
>>http://www.abvolt.com/compdsp/presentations/Dalrymple/dbd.pdf

>Ah, nostalgia.   I remember when you presented that.   Very cool.

Was that in Kansas City?  I had formed a plan to attend but in the
end did not.

I have, however, used dbd's windows subsequently.  They are very
good for measuring tiny spurs in the stopband of a much stronger
comm signal, therefore, useful for evaluating regulatory compliance.

Steve

On Wed, 3 Jun 2015 21:51:29 +0000 (UTC), spope33@speedymail.org (Steve
Pope) wrote:

>Eric Jacobsen <eric.jacobsen@ieee.org> wrote:
>
>>On Wed, 3 Jun 2015 12:14:42 -0700 (PDT), dbd
>
>>>There is a much smaller literature providing window designs for interferenc=
>>>e suppression for testing done by the people who design 24 bit convertors t=
>>>o evaluate their device internal implementations. Those are the minimum sid=
>>>elobe windows in "Windows Connections" (pages 6 and 12) at:
>>>http://www.abvolt.com/compdsp/presentations/Dalrymple/dbd.pdf
>
>>Ah, nostalgia.   I remember when you presented that.   Very cool.
>
>Was that in Kansas City?  I had formed a plan to attend but in the
>end did not.
>
>I have, however, used dbd's windows subsequently.  They are very
>good for measuring tiny spurs in the stopband of a much stronger
>comm signal, therefore, useful for evaluating regulatory compliance.
>
>Steve

Hi Steve,
  Yep, that was the paper Dale presented in Kansa City in 
2010.  That conference was both educational and a good time.
Vladimir Vassilevsky, and his lovely wife, put in much 
effort to make the conference a success.

[-Rick-]

On Monday, June 1, 2015 at 12:53:34 PM UTC-7, Cedron wrote:

...
> 
> Here is a candidate window function:
> 
> w_n = 2 [ cos( Pi/N ) - cos( (n+1/2) 2Pi/N ) ]
> 
>     = 4 sin( (n+1) Pi/N ) sin( n Pi/N )
> 
> Ced

Since we keep suggesting the use of windows with DFTs, we ought to help you with your understanding of the consequences of the 'digital' in DFT. Your window lacks the proper symmetry for a digital window.

Hint: the digital samples should be symmetric about the N/2 sample for a N point DFT, that is, if you have a continuous symmetric window of length N, zero only at the ends, only the first sample in the digital version of the window will be zero.

search comp.dsp for "DFT symmetric" 
or google: harris proceedings windows

Dale B. Dalrymple

>On Monday, June 1, 2015 at 12:53:34 PM UTC-7, Cedron wrote:
>
>...
>> 
>> Here is a candidate window function:
>> 
>> w_n = 2 [ cos( Pi/N ) - cos( (n+1/2) 2Pi/N ) ]
>> 
>>     = 4 sin( (n+1) Pi/N ) sin( n Pi/N )
>> 
>> Ced
>
>Since we keep suggesting the use of windows with DFTs, we ought to help
you
>with your understanding of the consequences of the 'digital' in DFT.
Your
>window lacks the proper symmetry for a digital window.

Here is the vector from my formula definition:

r = e^( -i 2Pi/N )

W = ( -1, 1 + r, -r )

Since the numerator and denominator are linear in W, W can be arbitrarily
rescaled and the formula will still work.

Let s = e^( i Pi/N )

Ws = ( -s, -s + 1/s, -1/s )

Ws ~=~ ( -1, 2 -1 )

The window function I provided in the time domain corresponds to Ws
applied in the frequency domain.  You keep assuming I don't understand
windows and what they do.  

So if you want to carry a window function through to the bin value
formulas and then solve a frequency equation from there, this window
function has the possibility of working well.

I'll repeat myself, for frequency determination of a signal that is steady
across the frame, I have not found any advantage in using a window
function.  Either in the real signal case or the complex signal case. 
Since my real signal frequency formula is clearly the best in the real
signal case according to Julien's Figure 4, and it is as good as any of
the DFT methods for the complex signal case in Figure 3, I would think the
burden would be on you to provide a window function and a corresponding
estimator to beat my unwindowed one.

>
>Hint: the digital samples should be symmetric about the N/2 sample for a
N
>point DFT, that is, if you have a continuous symmetric window of length
N,
>zero only at the ends, only the first sample in the digital version of
the
>window will be zero.
>
>search comp.dsp for "DFT symmetric" 
>or google: harris proceedings windows
>
>Dale B. Dalrymple

Everybody wants to send me to references to read, hunting in haystacks, so
to speak.  In your own words, please tell me why windows should be
symmetric.  At least you said "harris" and not "Harris".

Ced
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On Saturday, June 6, 2015 at 7:59:35 PM UTC-7, Cedron wrote:
...

> Everybody wants to send me to references to read, hunting in haystacks, so
> to speak.  In your own words, please tell me why windows should be
> symmetric.  At least you said "harris" and not "Harris".
> 
> Ced

You came to comp.dsp asking for references to read. People must have assumed that that implied that you were capable of benefiting from references. Maybe that's not true.

Window DFT-even symmetry is desirable for phase linearity and the independence of real and imaginary components of each bin.

Dale B. Dalrymple

[...snip...]
>
>You came to comp.dsp asking for references to read. People must have
assumed
>that that implied that you were capable of benefiting from references.
Maybe
>that's not true.
>
Close, but not correct.  I came to share my discovery of a better formula.
 After searching the literature and not finding it, after emailing many
professors and being sent on many haystack searches, I am asking if anyone
is aware of my formulas existing anywhere in the literature, not
references on the general topic.  How can I validate that it is actually a
new discovery not a rediscovery?

>Window DFT-even symmetry is desirable for phase linearity and the
>independence of real and imaginary components of each bin.
>

Yes, misspoke a little.  The window I presented is symmetric around
(N-1)/2.  Coincidentally (or not) that is the center point that Martin
Vicanek's DFT is rotated about to gain his solutions for his two bin
formula (see the "Show me some more numbers" thread for a link to his
paper).  Since there is a slight "twist" to the DFT of a real signal as in
reaches its flattened ends at the DC and Nyquist bins, the linearity and
separability might not actually be desirable goals.

>Dale B. Dalrymple

Thanks for your reply.

If you haven't read my blog article on the formula's derivation, there is
a section titled "Implicit Windowing" where I explain that the formula
effectively has a Von Hann like window factored in and how that dampens
the effects of other tones.

Ced
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