Analyzing Cross Spectrum for Pitch.

Eric Jacobsen wrote:
> On Fri, 20 May 2016 00:00:57 +0300, Evgeny Filatov
> <e.v.filatov@ieee.org> wrote:
>
>> On 19.05.2016 23:38, dbd wrote:
>>> On Thursday, May 19, 2016 at 5:57:32 AM UTC-7, Evgeny Filatov wrote:
>>>> On 18.05.2016 15:24, Cedron wrote:
>>>>> [...snip...]
>>>>>> Also, one can find the true frequency value by fitting three FFT bins
>>>>>> near the maximum value with a parabola. That option is discussed e.g. in
>>>
>>>>>> "Understanding DSP" by Lyons.
>>>
>>>>>> Gene
>>>
>>>>> Sorry, this is another estimator, not a "true" value.
>>>
>>>> I stand corrected.
>>>> Gene
>>>
>>> Actually, any calculation performed on real world sampled data is an estimator except to the intellectually dishonest, or impaired, who expect that -you- live in -their- imaginary world that they don't reveal (or perhaps understand) the details of. Estimators are often chosen in the real world because they are more accurate than proposed "true" value formulas that make unadmitted assumptions that are not met.
>>>
>>> Dale B. Dalrymple
>>>
>>
>> Indeed, there's no way to estimate true frequency of a noisy sinusoid.
>> What you care about is whether your estimator attains the CRB (among
>> other questions).
>>
>> I agreed with him because Cedron's claim may be correct (I don't know
>> anything about his estimator) in the case of infinite SNR, which
>> corresponds to CRB = 0. Which is, of course, utterly unrealistic, unless
>> you are dealing with a mathematical model of something.
>
> And if it's a numeric model with quantized samples there will be, at a
> minimum, quantization noise.
>
> Plus the whole finite-time observation thing.
>
> So, yeah, they're all estimators.
>
>


For something like  guitar tuner, it's got "better" accuracy than
the device being measured ( a guitar string ). +/- one cent is
typical for a string.

-- 
Les Cargill

[...snip...]

>>>
>>> Actually, any calculation performed on real world sampled data is an
>estimator except to the intellectually dishonest, or impaired, who
expect
>that -you- live in -their- imaginary world that they don't reveal (or
>perhaps understand) the details of. Estimators are often chosen in the
real
>world because they are more accurate than proposed "true" value formulas
>that make unadmitted assumptions that are not met.
>>>
>>> Dale B. Dalrymple
>>>

There is no intellectually dishonesty here, at least on my part.  I
assumed by "true", the poster meant mathematically exact in a theoretical
sense.  The assumptions behind the formula are pretty simple and they are
rigid.  It applies to a signal that is a single pure sinusoidal of
constant amplitude throughout the measuring frame.

On the other hand, even at the theoretical level, an "estimator" will have
some sort of discarded higher order terms and thus become somewhat
inaccurate even under the most ideal conditions.  Whether that is
significant needs to be situationally determined.

In my testing, none of the estimators have performed consistently better
than my formula.  The less noise there is, the better mine does in
comparison.  This is clearly shown in Julien's paper as well.

As pointed out by Martin Vicanek, my equation is actually overdetermined
so it is not unique.  He has derived a two-bin formula which can be found
here:

http://www.vicanek.de/articles/FreqFromTwoBins.pdf

I have a different two bin formula which I have not published.  Mainly
because his seems to outperform mine in many cases.  However, both are
"true" meaning in the absence of noise or other tones both give an answer
that is good to the limit of the precision used.

>>
>>Indeed, there's no way to estimate true frequency of a noisy sinusoid. 
>>What you care about is whether your estimator attains the CRB (among 
>>other questions).
>>
>>I agreed with him because Cedron's claim may be correct (I don't know 
>>anything about his estimator) in the case of infinite SNR, which 
>>corresponds to CRB = 0. Which is, of course, utterly unrealistic, unless

>>you are dealing with a mathematical model of something.

The CRB is a bound, not a limit.  

If you are interested in verifying my claim for yourself, the math is not
that difficult and is fully presented in my blog articles.

>
>And if it's a numeric model with quantized samples there will be, at a
>minimum, quantization noise.
>

Quite true.


>Plus the whole finite-time observation thing.
>

Which comes from the continuous case with the integration definition and
is totally irrelevant in the discrete case.  It is quite possible to get
an exact answer on a finite interval if the tone is a pure sinusoidal of
lower frequency than the Nyquist frequency.  Indeed, it is possible in the
mathematically pure case to get an exact answer on an interval that is
much shorter than the actual wavelength.

>So, yeah, they're all estimators.

In real world cases, this is correct.  The difference is whether you have
an understanding of the math you are working with.

Ced
---------------------------------------
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On 20.05.2016 23:20, Cedron wrote:
(snip)
> The CRB is a bound, not a limit.

Do you use the word "limit" in a mathematical sense?

> If you are interested in verifying my claim for yourself, the math is not
> that difficult and is fully presented in my blog articles.
>

Thanks, not now.

Gene

On Friday, May 20, 2016 at 1:20:16 PM UTC-7, Cedron wrote:
> [...snip...]
> 
> There is no intellectually dishonesty here, at least on my part.  I
> assumed by "true", the poster meant mathematically exact in a theoretical
> sense.  

There is no 'mathematically exact' in any non-trivial dsp. Non-trivial includes operations including any of: sampling, quantization, representation in fixed or floating point, external noise and the presence of multiple components as a start. The OP indicates analysis of sample data with results that indicate one or more of the non-trivialities are likely. The OP sought a solution that indicated the first harmonic instead of the second harmonic of a component. He didn't complain about the error due to rounding to the nearest bin. That is not a 'mathematically exact' situation. This is comp.dsp. It has been explained to you before that the context here is not 'mathematically exact' unless explicitly stated.

> The assumptions behind the formula are pretty simple and they are
> rigid.  It applies to a signal that is a single pure sinusoidal of
> constant amplitude throughout the measuring frame.
> 

But you did not post these limitations on your claim of 'true' until challenged. That's simply dishonest and your usual mode of operation here.

...
> 
> >So, yeah, they're all estimators.
> 
> In real world cases, this is correct.  The difference is whether you have
> an understanding of the math you are working with.
> 

A better understanding comes from posters who begin with an honest presentation of the characteristics of the methods they present.

Dale B. Dalrymple

>On Friday, May 20, 2016 at 1:20:16 PM UTC-7, Cedron wrote:
>> [...snip...]
>> 
>> There is no intellectually dishonesty here, at least on my part.  I
>> assumed by "true", the poster meant mathematically exact in a
theoretical
>> sense.  
>
>There is no 'mathematically exact' in any non-trivial dsp. Non-trivial
>includes operations including any of: sampling, quantization,
representation
>in fixed or floating point, external noise and the presence of multiple
>components as a start. The OP indicates analysis of sample data with
results
>that indicate one or more of the non-trivialities are likely. The OP
sought a
>solution that indicated the first harmonic instead of the second
harmonic
>of a component. He didn't complain about the error due to rounding to
the
>nearest bin. That is not a 'mathematically exact' situation. This is
>comp.dsp. It has been explained to you before that the context here is
not
>'mathematically exact' unless explicitly stated.
>

To some people, understanding the theoretical foundation of practical
techniques is important.  To others it is not.  You seem to be in the
latter category, but I don't think you should presume to speak for
everyone.

I was not responding to the OP, I was responding to the assertion that a
parabolic equation was a "true" formula.  It is not, it is an
approximation.

>> The assumptions behind the formula are pretty simple and they are
>> rigid.  It applies to a signal that is a single pure sinusoidal of
>> constant amplitude throughout the measuring frame.
>> 
>
>But you did not post these limitations on your claim of 'true' until
>challenged. That's simply dishonest and your usual mode of operation
here.

Not explicitly, but I did say "Whether the precision of an exact formula
is worth the extra computations
compared to a good estimator depends on the cost of computation, the
accuracy of your data, and the level of noise in the signal."  I think
anybody understanding the general use of mathematics within engineering
should understand that that means about the same thing.

I also referenced a paper which evaluates the formula in the presence of
noise.  I would think any reasonable reader would also understand from
that, and an understanding of math in engineering, that the formula won't
be exact with noise present.

>
>...
>> 
>> >So, yeah, they're all estimators.
>> 
>> In real world cases, this is correct.  The difference is whether you
have
>> an understanding of the math you are working with.
>> 
>
>A better understanding comes from posters who begin with an honest
>presentation of the characteristics of the methods they present.
>
>Dale B. Dalrymple

The characteristics of my method are fully explained in my referenced blog
article.  I think your usual mode of operation here, in particular with
regards to me, is to be nitpicky and argumentative.  I would go as far as
to call you a bully.  Others may see it differently, but to call me
dishonest is just disingenuous.

Ced
---------------------------------------
Posted through http://www.DSPRelated.com

On Fri, 20 May 2016 20:11:38 -0500, "Cedron" <103185@DSPRelated>
wrote:

>>On Friday, May 20, 2016 at 1:20:16 PM UTC-7, Cedron wrote:
>>> [...snip...]
>>> 
>>> There is no intellectually dishonesty here, at least on my part.  I
>>> assumed by "true", the poster meant mathematically exact in a
>theoretical
>>> sense.  
>>
>>There is no 'mathematically exact' in any non-trivial dsp. Non-trivial
>>includes operations including any of: sampling, quantization,
>representation
>>in fixed or floating point, external noise and the presence of multiple
>>components as a start. The OP indicates analysis of sample data with
>results
>>that indicate one or more of the non-trivialities are likely. The OP
>sought a
>>solution that indicated the first harmonic instead of the second
>harmonic
>>of a component. He didn't complain about the error due to rounding to
>the
>>nearest bin. That is not a 'mathematically exact' situation. This is
>>comp.dsp. It has been explained to you before that the context here is
>not
>>'mathematically exact' unless explicitly stated.
>>
>
>To some people, understanding the theoretical foundation of practical
>techniques is important.  To others it is not.  You seem to be in the
>latter category, but I don't think you should presume to speak for
>everyone.
>
>I was not responding to the OP, I was responding to the assertion that a
>parabolic equation was a "true" formula.  It is not, it is an
>approximation.
>
>>> The assumptions behind the formula are pretty simple and they are
>>> rigid.  It applies to a signal that is a single pure sinusoidal of
>>> constant amplitude throughout the measuring frame.
>>> 
>>
>>But you did not post these limitations on your claim of 'true' until
>>challenged. That's simply dishonest and your usual mode of operation
>here.
>
>Not explicitly, but I did say "Whether the precision of an exact formula
>is worth the extra computations
>compared to a good estimator depends on the cost of computation, the
>accuracy of your data, and the level of noise in the signal."  I think
>anybody understanding the general use of mathematics within engineering
>should understand that that means about the same thing.
>
>I also referenced a paper which evaluates the formula in the presence of
>noise.  I would think any reasonable reader would also understand from
>that, and an understanding of math in engineering, that the formula won't
>be exact with noise present.
>
>>
>>...
>>> 
>>> >So, yeah, they're all estimators.
>>> 
>>> In real world cases, this is correct.  The difference is whether you
>have
>>> an understanding of the math you are working with.
>>> 
>>
>>A better understanding comes from posters who begin with an honest
>>presentation of the characteristics of the methods they present.
>>
>>Dale B. Dalrymple
>
>The characteristics of my method are fully explained in my referenced blog
>article.  I think your usual mode of operation here, in particular with
>regards to me, is to be nitpicky and argumentative.  I would go as far as
>to call you a bully.  Others may see it differently, but to call me
>dishonest is just disingenuous.
>
>Ced

I think one thing that is very telling is that the majority of people
here post under their real names.   You don't.

That actually says quite a bit.

>
>I think one thing that is very telling is that the majority of people
>here post under their real names.   You don't.
>
>That actually says quite a bit.

If you can't attack the message, attack the messenger.  This is known as
an "ad hominem" fallacy, generally considered to be in poor form which
says a bit about you, no?

I have my reasons, and they are none of your business.  I think your
intended aspersions are baseless.  I am not dishonest about it either, if
you check my blogger profile you will see it starts with "Cedron Dawg is
the pen name..."

If you want to fault what I have said you can point out that I never
backed up my claim that a parabolic fit was an approximation, not a true
equation.  My response would be, especially to a theory adverse crowd,
that a numerical example should be sufficient and you can do that
yourself.

Getting back to the OP's original question.  As far as I know, there is no
true equation using a DFT for the situation he describes.  However, any
signal composed of pure tones, whether harmonic or not, can be decomposed
to any desired degree of precision (within noise and numerical
representation constraints) using an iterative method.

If a signal is composed of only harmonic tones, as the OP's seems to be,
then the signal will be periodic.  For signals of this type, the
autocorrelation and angrydude's methods will work.  A less computationally
intense approach would be to analyse the location of zero crossings.

The autocorrelation and zero crossing techniques will get you an answer
precise to the level of the number of samples.  The precision can be
improved by doing a parabolic fit at the autocorrelation peak for the
former and linear best fits at the zero crossings for the latter. 
angrydude can chime in about his technique if he wishes.

To put things in perspective, Les Cargill pointed out that one percent was
adequate for a guitar string.  An 'A' tone at 440Hz in a CD quality
44,100Hz recording is roughly 100 samples per wavelength.  A one sample
error on a one wavelength samping will then be about one percent.

In comparison, every semitone on the standard chromatic music scale is
about a 6% increase in frequency.  ( 2^(1/12) = 1.059463094 )  I've met
musicians who can tune their instruments by ear that can easily discern
less than a 1% change in pitch.

The percentage frequency change between bins in a DFT depend on the bin
number and can be simply calculated using the ratio of the bin numbers.

Ced
---------------------------------------
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On 21.05.2016 6:09, Eric Jacobsen wrote:

(snip)

> I think one thing that is very telling is that the majority of people
> here post under their real names.

And often not just names, but short forms of real names; like it helps 
to have an informal conversation?

Gene

On Sat, 21 May 2016 08:03:20 -0500, Cedron wrote:


[snip]

> To put things in perspective, Les Cargill pointed out that one percent
> was adequate for a guitar string.  An 'A' tone at 440Hz in a CD quality
> 44,100Hz recording is roughly 100 samples per wavelength.  A one sample
> error on a one wavelength samping will then be about one percent.

You're misquoting Les Cargill there.  He said "1 cent" rather than "1 
percent".

1 cent is 1 percent of 1 semitone.  That's about 578ppm, and is not 
really perceptible to humans (unless it's in a chord, etc.).

Regards,
Allan

On Friday, May 20, 2016 at 6:11:40 PM UTC-7, Cedron wrote:
> [...snip...]

> 
> The characteristics of my method are fully explained in my referenced blog
> article.  I think your usual mode of operation here, in particular with
> regards to me, is to be nitpicky and argumentative.  I would go as far as
> to call you a bully.  Others may see it differently, but to call me
> dishonest is just disingenuous.
> 
> Ced

We have discussed before, in detail, why statements that might be understandable in the context of a comp.soft-sys.symbolic-manipulation would be considered misleading in the context of comp.dsp.  That's treating you as an adult. You have been told that further misleading behavior would bring further discussion. Now you have repeated the behavior you know is considered misleading, apparently to market your blog, and here we are in the promised discussion. Your behavior hasn't gotten you bullied. It's gotten you recognized as behaving as a troll, whatever your reasons may be.

Dale B. Dalrymple