Stephan M. Bernsee wrote:

>>>it looks
>>>like you're correlating (exponentially?) decaying cosines with your
>>>signal, probably with a higher "forgetting factor" for the upper
>>>frequencies, at a stride of one sample (the equivalent to a so-called
>>>"sliding transform").
>>
>>You are entirely wrong with that.
> 
> How so?

I will describe the FCT below.


>>I am actually using a less well known definition of Fourier cosine 
>>transform (FCT) as an integral over time from zero to plus infinity. In 
>>order to avoid mistakes I would like to stress that I do so within R^+, 
>>not in R. Consequently, my FCT is not a special case of complex Fourier 
>>transform.
> 
> 
> What exactly is *your* definition of the FCT? How does that
> integral/sum look like?

Apart from above discribed restriction to R^+, I am using in principle 
the same definition of FCT as does e.g. Wolfram. In contrast to 
complex-valued Fourier transform (FT), the kernel is not a complex 
exponential function but simply a cosine transform instead.
Perhaps, I should remind you of a consequence: While the inverse FT 
requires a kernel with opposite sign of the imaginary part, the inverse 
of FCT uses simply the same cosine kernel again. That's why literature 
prefers to split the factor 1/2pi and ascribe sqrt(2/pi) to each of both 
FCT and inverse FCT (iFCT). There is only one definition for both FCT 
and iFCT.


>>I am aware of these tons for many years with steadily decreasing 
>>interest, and I am very happy to be absolutely independent thereof.
> 
> 
> Well, I hope you don't mind my being sceptical here.

I am not just sceptical. I am convinced that abundant theory indicates 
a non-appropriate theoretical basis. There are obviously rather too much 
transforms: Fourier, Gabor, Hartley, Laplace, Mellin, Wigner,... and 
also too much wavelets instead of just one ubiquitously adequate one.



>>The natural spectrogram 
>>has quite a different basis: elapsed time with a zero that is sliding 
>>with reference to ordinary time.
> 
> 
> That's the definition of a "sliding transform" then, isn't it?

I expect all those who are trained in traditional theory to deliberately 
close they eyes to radical alternatives. Do not overlook what R^+ means: 
There is no negative value at all of a variable like elapsed time or 
radius or absolute temperature ( I know, some physicists even tried to 
fabricate arguments against the absolute zero of temperature).

While unbiased people are in position to understand my arguments, those 
who grew up with traditional theory behave like missionaries who feel a 
call to explain me the correct believe.


>>>For the purpose of approximating the auditory perception you should
>>>probably think about a "transform" that not only changes its time and
>>>frequency localization with frequency, but with time and amplitude as
>>>well...
>>
>>Having resolved the problem I am not interested in suggestions.
> 
> 
> Quite a bold claim, I must say.

Feel challenged. So far, you might approach the brink of new insight.



>>The FCT in R^+ is not the daughter but the adequate sister of complex FT 
>>in R. You can not just put the positive reals into the reals but also 
>>vice versa. Elapsed time is always positive as is radius.
> 
> 
> What does this FCT show you for a real function sin(k*w*t) with k
> being an integer? Moreover, you claimed several posts ago that it is
> invertible - what becomes of iFCT(FCT(f(t))) if f(t) = -f(-t)?

Within R^+, iFCT(FCT(f(t)))= f(t).
Stay with what the world was told so far, and you will never grasp why I 
am telling a different story, - or - be ready to not ignore what I 
consider essential: Well, engineers like you and me are not immediately 
familiar with R and R^+. Therefore, I will once again tell you that 
future time does not really have any influence on a physical process.
Only elapsed time matters. Nobody can analyze data that are not yet 
available. We have two possiblities to cope with that:
i) Traditional theory uses illusory zero continuation. Heaviside's trick 
tacitly fills the empty future with an even real mirror and an odd 
imaginary mirror of past as to introduce redundancy required for complex 
calculus.
ii) I am suggesting to accept that causality-bound reality demands that 
t is never negative at all.
Instead of calculating within R^+, you might prefer even continuation 
within R where negative values of t exist. This means e.g.
sin(kt) for kt>0 but -sin(kt) for kt<0.
In reality, there are no odd functions of elapsed time.
Time shift is not admissible in reality.


> I'm not accusing or blaming you of anything, I simply do not have
> enough information to do anything other than guess. I'm simply
> curious, but I have a hunch that there's a misconception on your part
> with regard to the Heisenberg uncertainty and the properties of
> whatever it is you're doing.

I faced a lot of distrust so far. It took me 18 month of fierce 
discussion until Hendrik van Hees, a quantum physicist apologised for 
wrongly blaming me wrong. Your wrong "hunch" is understandable to me.


> I am also sceptical about your claim that you have solved all the
> intricacies of the human auditory perception with your natural
> spectrogram. 

After I started dealing with the enigma of hearing more than two decades 
ago, I focused on main contradictions. Steven Greenberg provided to me 
the opportunity to learn from famous specialist in Il Ciocco in 1998. 

I am actually claiming that 'the' natural spectrogram has a sound basis 
while the traditional spectrogram is not just much less adequate but it 
  also lacks mathematical rigor.
Of course, we have to refine upon application of the natural spectrogram 
to cochlea in order to better consider some physiological details, in 
particular resulting in critical bandwidth and in Nelson's notch.
Based on the natural spectrogram I am suggesting a cepstrum-like 
autocorrelation mechanism of pitch. So far both spectral and temporal 
mechanisms have neither proven entirely correct nor entirely wrong.


> Unlike you I don't claim to have "solved the problem",
> but I can extrapolate from my research that the human auditory
> perception (not just the cochlea!) is doing things that are far beyond
> a simple trigonometric transform of any kind.

During DAGA 2000 in Oldenburg, I made a similar sceptical statement 
within one out of 106 theses which were put at the door of auditory.
I was perhaps largely correct in my criticism against overestimation of 
mechanical frequency analysis but underestimation of the role of brain. 
It was Christian K&#4294967295;rnbach who uttered to me that most likely, cochlea 
will soon be largely understood while function of less caudal nuclei 
will remain more or less obscule for quite a while. I agree with him 
with one caveat: I found out that most likely cochlea and cochlear 
nucleus must not merely be considered separately. Cepstrum-like 
autocorrelation jointly includes cochlea and midbrain.

Eckard Blumschein

Eckard Blumschein wrote:
> Hi Stephan, you wrote:
> 
> > If I recount what you've just said and add it all together, it looks
> > like you're correlating (exponentially?) decaying cosines with your
> > signal, probably with a higher "forgetting factor" for the upper
> > frequencies, at a stride of one sample (the equivalent to a so-called
> > "sliding transform").
> 
> You are entirely wrong with that.

How so?

> > This is nothing new, and calling this a Fourier Cosine Transform isn't
> > quite correct either. 
> 
> 
> I am actually using a less well known definition of Fourier cosine 
> transform (FCT) as an integral over time from zero to plus infinity. In 
> order to avoid mistakes I would like to stress that I do so within R^+, 
> not in R. Consequently, my FCT is not a special case of complex Fourier 
> transform.

What exactly is *your* definition of the FCT? How does that
integral/sum look like?

> I am aware of these tons for many years with steadily decreasing 
> interest, and I am very happy to be absolutely independent thereof.

Well, I hope you don't mind my being sceptical here.

> The natural spectrogram 
> has quite a different basis: elapsed time with a zero that is sliding 
> with reference to ordinary time.

That's the definition of a "sliding transform" then, isn't it?

> > For the purpose of approximating the auditory perception you should
> > probably think about a "transform" that not only changes its time and
> > frequency localization with frequency, but with time and amplitude as
> > well...
> 
> Having resolved the problem I am not interested in suggestions.

Quite a bold claim, I must say.

> The FCT in R^+ is not the daughter but the adequate sister of complex FT 
> in R. You can not just put the positive reals into the reals but also 
> vice versa. Elapsed time is always positive as is radius.

What does this FCT show you for a real function sin(k*w*t) with k
being an integer? Moreover, you claimed several posts ago that it is
invertible - what becomes of iFCT(FCT(f(t))) if f(t) = -f(-t)?

I'm not accusing or blaming you of anything, I simply do not have
enough information to do anything other than guess. I'm simply
curious, but I have a hunch that there's a misconception on your part
with regard to the Heisenberg uncertainty and the properties of
whatever it is you're doing.

I am also sceptical about your claim that you have solved all the
intricacies of the human auditory perception with your natural
spectrogram. Unlike you I don't claim to have "solved the problem",
but I can extrapolate from my research that the human auditory
perception (not just the cochlea!) is doing things that are far beyond
a simple trigonometric transform of any kind.

--smb

Hi Stephan, you wrote:

> If I recount what you've just said and add it all together, it looks
> like you're correlating (exponentially?) decaying cosines with your
> signal, probably with a higher "forgetting factor" for the upper
> frequencies, at a stride of one sample (the equivalent to a so-called
> "sliding transform").

You are entirely wrong with that.


> This is nothing new, and calling this a Fourier Cosine Transform isn't
> quite correct either. 


I am actually using a less well known definition of Fourier cosine 
transform (FCT) as an integral over time from zero to plus infinity. In 
order to avoid mistakes I would like to stress that I do so within R^+, 
not in R. Consequently, my FCT is not a special case of complex Fourier 
transform.


 > You could implement something like this easily
> with a set if IIR filters with poles spaced closely to the unit
> circle, or by modifying a Goertzel or DFCT. Come to think of it, it
> can even be viewed as a multiresolution filterbank.

In contrast to the traditional spectrogram, my result actually resembles 
that of a multiresolution filterbank pretty well. My solution is, 
however, much more straightforward.


> There are actually tons of papers (and code) on this subject, look for
> multiresolution analysis and/or the names Malvar (lapped transforms),
> Wickerhauser (best basis wavelets), Mandelshtam (eigenmodes), Larson
> (sliding FFT), Levine (multiresolution sinusoidal modeling) etc. Many
> of them cover this topic, at least to some extent. If you're really
> interested in that sort of thing you might also want to consider
> subscribing to the Wavelet Digest (http://www.wavelet.org) which is
> highly relevant and quite interesting reading on that kind of topic.

I am aware of these tons for many years with steadily decreasing 
interest, and I am very happy to be absolutely independent thereof.
As I already claimed: There are many wavelets but only one natural 
spectrogram.

> Someone here (Steven G. Johnson - the "FFTW guy" - I believe) recently
> posted code for a program named "harminv" IIRC it does something
> closely related, although it also finds the decay rate (your
> "forgetting factor") from the actual signal.

Maybe, I gave rise to a mistake. In hearing, the decay factor does not 
depend on the signal, at least not directly.


> So, the bottom line is that your "transform" isn't really evading the
> Heisenberg uncertainty, it's simply varying the time and frequency
> localization tradeoff with frequency. 

The bottom line of what? You are wrong in that. The natural spectrogram 
  shows frequency chirping up towards the uncertainty limit according to 
delta f * delta t.

> It's the same thing some wavelets do, 

IIRC, wavelets were invented for that purpose. The natural spectrogram 
has quite a different basis: elapsed time with a zero that is sliding 
with reference to ordinary time.

> and although it may be nice for some purposes it is
> nothing magical, 

No. It is not magical but natural.

> and you're subject to the exact same restrictions as
> is the rest of the discrete universe.

Of course. In particular I will have to pass away, and I cannot 
completely anticipate the future. The latter is the basis for my reasoning.


> For the purpose of approximating the auditory perception you should
> probably think about a "transform" that not only changes its time and
> frequency localization with frequency, but with time and amplitude as
> well...

Having resolved the problem I am not interested in suggestions.


> PS: I still don't see why you would use the FCT (Fourier Cosine
> Transform) for this and not the Hartley transform, because the FCT
> provides meaningful output only for even and real (or odd and
> imaginary) signals. 

I will try and patiently reitereate:
If there are no negative values of (elapsed) time then there is neiter 
an even nor an odd function of time. I know the Hartley transform for 
long. It is not appropriate for what I was after.

> You seem to think that the FCT is a real valued Fourier transform. That is not correct!.

The FCT in R^+ is not the daughter but the adequate sister of complex FT 
in R. You can not just put the positive reals into the reals but also 
vice versa. Elapsed time is always positive as is radius.

It took me 18 months of fierce discussion until Hendrik van Hees in 
de.sci.physik apologized for wrongly blaming me wrong.

Eckard

Hi Eckard,

I'll skip the other comments for now because I don't have much time to
argue right now - here's what I think based on what I know so far,
reduced to the mere facts.

If I recount what you've just said and add it all together, it looks
like you're correlating (exponentially?) decaying cosines with your
signal, probably with a higher "forgetting factor" for the upper
frequencies, at a stride of one sample (the equivalent to a so-called
"sliding transform").

This is nothing new, and calling this a Fourier Cosine Transform isn't
quite correct either. You could implement something like this easily
with a set if IIR filters with poles spaced closely to the unit
circle, or by modifying a Goertzel or DFCT. Come to think of it, it
can even be viewed as a multiresolution filterbank.

There are actually tons of papers (and code) on this subject, look for
multiresolution analysis and/or the names Malvar (lapped transforms),
Wickerhauser (best basis wavelets), Mandelshtam (eigenmodes), Larson
(sliding FFT), Levine (multiresolution sinusoidal modeling) etc. Many
of them cover this topic, at least to some extent. If you're really
interested in that sort of thing you might also want to consider
subscribing to the Wavelet Digest (http://www.wavelet.org) which is
highly relevant and quite interesting reading on that kind of topic.

Someone here (Steven G. Johnson - the "FFTW guy" - I believe) recently
posted code for a program named "harminv" IIRC it does something
closely related, although it also finds the decay rate (your
"forgetting factor") from the actual signal.

So, the bottom line is that your "transform" isn't really evading the
Heisenberg uncertainty, it's simply varying the time and frequency
localization tradeoff with frequency. It's the same thing some
wavelets do, and although it may be nice for some purposes it is
nothing magical, and you're subject to the exact same restrictions as
is the rest of the discrete universe.

For the purpose of approximating the auditory perception you should
probably think about a "transform" that not only changes its time and
frequency localization with frequency, but with time and amplitude as
well...

--smb

PS: I still don't see why you would use the FCT (Fourier Cosine
Transform) for this and not the Hartley transform, because the FCT
provides meaningful output only for even and real (or odd and
imaginary) signals. You seem to think that the FCT is a real valued
Fourier transform. That is not correct!.

Hi Richard,

the sample rate is 44.1kHz.

I'm so used to working at 44.1 I often forget to mention it. Sorry for that.

--smb

Wnat's the sample rate?

For float samples, +-1.0 is OK, so long as it is converted (as IMO it 
always should be) +-32767 if written to shorts. But I know the arguments 
about this can run and run...

I have made a soundfile at 22050, and needless to say, all one hears is 
two broadband clicks in rapid succession, just close enough in time to 
hint at a pitch (two cycles of some ultra-low fundamental - this will of 
course depend on the intended srate). The waveform has hard transitions 
from -1 to +1 etc, so is inevitably not band-limited, and for a 
quasi-square wave signal is also strictly-speaking over-range with those 
amplitudes - will clip/distort at the dac.  No reason that I can see not 
to scale everything by 0.5. At what SPL are we supposed to listen to it? 
The ear's trick of making a ~loud~ tone sound flatter is well-known 
(easily experienced using headphones!); is that what this is all about?



Richard Dobson



Stephan M. Bernsee wrote:
> Richard - the file doesn't help much, he sent it to me via email this
> afternoon. It contains the following data - transcribed to C,
> actually, the +1. should be something like 0.9999... to avoid
> wrapping, I cut it short to keep the post (relatively) small. ;-)
> 
> --smb
> 
> const float data[2640]={1.,1.,1.,1.,1.,1.,1.,1.,1.,
...

Stephan M. Bernsee wrote:

> Too bad the .wav file doesn't work. It's a dead link - we already told
> you that.

I just sent it as an attachment.


>>I will wonder if you can tell me something new.

I meant concerning Ohm.


>>I did not directly refer to Ohm but to Ohm's law and the fact that 
>>hearing is actually highly insensitive against phase, not entirely 
>>though. You may understand it from natural spectrogram.
>  
> How?

As already did Nelson Kiang from his records. While magnitude in the 
traditional spectrogram is always positive, BM actually oscillates up 
and down. It depends on polarity of the click or onset how the rectified 
pattern looks like.


> Our hearing is *not* insensitive to phase. It is insensitive to
> *absolute* phase, but actually quite capable of perceiving relative
> phase. For example, the phase in dichotic stimuli is important for the
> perception of localization cues.

I referred to what you called absolute phase. The binaural phenomena are 
better understood in terms of interaural time disparity. This does not 
mean I dislike the results by Benedikt Grothe. I just prefer a more 
accurate terminology.


>>>Phase is discarded simply because <<people are>>
>>>interested in the magnitude of the partial frequencies - that's
>>>practically the definition of a spectrogram. 
>>
>>When the spectrogaph was invented in the late fourties, there was no 
>>alternative as to restrict to magnitude.
> 
> 
> I doubt it. I know for a fact that Henrik Bode was known to plot
> magnitude and phase in his frequency response plots of complex
> functions since 1938 and the sound spectrograph was invented by Potter
> a little later, so I highly doubt that they were unable to plot phase
> in a spectrogram if they had wanted to.
> 
> Maybe Jerry has some additional information, IIRC he provided valuable
> information on Bode plots some time ago.

Bode relations relate magnitude and phase to each other. If Bode and 
Potter actually worked together, you might be right in that. Nonetheless 
the first spectrograph was a rather primitive mechanical solution.


>>>Displaying the phase, too, would only clutter the readout. 
>>
>>It was not imaginable at all.
> 
> 
> I'd rather think it was not necessary! Why would someone plot phase in
> a spectrogram? How would you be able to say anything meaningful about
> the magnitude at a certain point if your display keeps oscillating
> wildly?

I am still not sure how do you imagine a plot of magnitude and phase at 
a time.


>>>Bear in mind that the spectrogram does not make the claim to be 
>> > in any way related to something our auditory system does,
>>
>>It has been in use as to mimic the cochlea or even the auditory system.
> 
> 
> It has been used to study harmonic signals, most importantly speech.

If speech was actually a harmonic signal, then it would be boring.

> Since it has been observed that the auditory system localizes stimuli
> in time and frequency it seemed obvious (but not equivalent!) to use
> the spectrogram as a means for comparison and study. In the beginning
> this choice was easy, because the spectrogram was more or less all
> there was for such a purpose

Yes, I own an old book by Ladefoged.

> But the analogy is crude, even though it helps understanding some of
> the principles wrt. frequency. 

That's why I am after a better solution for at lest a decade.

> Going so far as to claim that the
> spectrogram was used as a means to "mimic the cochlea or even the
> auditory system" is nonsense 

I should separate two reproaches:
You are impolite.
You did not understand what I meant.
IIRC it was David Poeppel who recently started his explanation of 
cortical auditory function at the spectrogram. The traditional 
spectrogram is highly suggestive. Many attemts were made to read it out.

- it has been (and still is) a means for
> plotting frequency dependent magnitude of periodic signals over time,
> nothing more.

I also recall spectrograms of non-periodic signals.


>>The spectrogram shows rather a misleading pattern.
> 
> No. The spectrogram shows exactly the pattern it was designed to
> display: the frequency and time dependend magnitude of partial waves
> in a signal. This is not misleading, it is, in fact, highly relevant
> to what it was designed to do. The misleading part comes from you
> believing it should do otherwise.

Well, I agree to some extent. The spectrogram presents Ohm's wrong point 
of view in his dispute with Seebeck.


>>Neither the natural spectrogram nor cochlea itself are designed just for 
>>the first stage.

With first stage I did nor mean the most caudal element of a chain but 
the very beginning of onset.

> The cochlea *is* the first stage of our auditory perception (if we
> leave aside peripheral factors like the pinnae that contribute to the
> perception, but are not actually involved in the "active process").
> The cochlea is the place where sound waves are converted into
> information in the form of nerve pulses.

Yes.


>>On what input are automatic speech-recognizers based if not a current 
>>spectral analysis?
> 
> I never questioned whether ASR software is using "spectral analysis"
> (btw. are we talking just about a spectrogram or the whole "spectral
> analysis" ballgame here with all its possible flavors?).

As far as I know, the spectrogram performs exactly the same sort of 
spectral analysis as does ASR.

> I said I doubt that someone claims that the *human hearing* uses a
> spectrogram like the one in CoolEdit, 

Schoolgirls will perhaps not claim that but they do not have alternative 
models yet as to imagine hearing.

> therefore it is my belief that
> you're barking up the wrong tree with your rant about the (natural or
> not) spectrogram.

The word rant is impolite and perhaps not justified.


>>>>_The_ natural spectrogram does not omit phase. It conveys all 
>>>>information and would correspondingly be reversible.
>>>
>>>Yes and no. If you're using the FCT (Fourier Cosine Transform), this
>>>is only true for even signals.
>>
>>I do not know you, and I have to be polite in any case.
> 
> 
> Well, you don't have to, but it would be nice. I also see no reason
> why you shouldn't. I just said that the Fourier Cosine Transform is
> the even part of the complete Fourier Transform. That is how it is
> defined, and I see nothing wrong with that definition (neither do I
> find that statement in any way objectionable).

You are quite right that this is a pretty common tenet.

> According to its definition, the Fourier Cosine Transform is only
> meaningful for even signals. Does that fact disturb you? And if so,
> why?

FCT has also been defined with restriction of argument to merely 
positive values where there is neither even nor odd symmetry.


>>However, I feel 
>>deeply disappointed because you did not even understand the simplest 
>>basics of what I found out.
> 
> 
> I'm sorry if I disappoint you, but you must admit you don't give us
> much to understand this from. Maybe you can tell me what you've found
> out *exactly*, so I can provide a less disappointing response?
> 
> 
>>Use of even and odd functions of time is 
>>only necessary iff you decide to tacitly accept Heaviside's trick and 
>>perform a complex-valued Fourier Transform.
> 
> 
> You mean I should paint my fingernails in pink like he did? :-)

:-))) No no. Wasn't it black? Anyway, I suggest the opposite. Do _not_ 
always perform your FT like a complex one but as a real-valued one with 
just a cosine kernel. Fourier himself already thought of the possibility 
to integrate from zero to infinity instead against all common sense from 
minus infinity to plus infinity except for his ringshaped subject.


> The Fourier Transform is inherently complex, because that's how it is
> defined. 

The actual reason dated back at least to Ren&#4294967295; Descartes who introduced 
bilateral coordinates.

> Perhaps it is my limited knowledge but I have not seen any
> other definition called the "Fourier Transform" that omits the "i".

Euler's equation can be used back and forth. Accordingly, real-valued 
(FCT) and complex-valued (FT) representations are equivalent, in 
principle. So you may choose FCT a special case of FT or vice versa. The 
same is already true for all reals in relation to all positive reals. 
Yes, paradoxically one may put all reals into all positive reals.


>>In this case one has to have 
>>a time function that extends from minus infinite to plus infinite 
>>including past as well as future time. Windowing does not matter in that 
>>respect.
> 
> 
> It does, on the discrete side of things, if you want to get any
> meaningful representation of your signal. There are no true infinities
> in a discrete world.

I understand what you object. I meant, the key question is not infinity 
but the sign of time. Future time is of no influence on the past. This 
fact is independent of more or less windowing. Do not cheat yourself by 
shifting the window into future.


> The basis functions of the DFT are complex exponentials. 

It might be a hurdal to you, but you have to realize that there is 
nothing complex with only positive values of time, radius, frequency, 
and wavenumber. Also get aware that this kind of consideration is not 
just absolutely correct as long as you restrict to a description of the 
real world because of causality. It is also distinguished by simplicity 
while complex calculus demands Hermitean redundancy. If necessary, one 
can perform complex calculus independently after a real-valued analysis.


> As I said in
> my last post, they have perfect frequency localization but do not have
> time localization. 

My approach is quite different. I understand the theory you are 
referring to at least in principle. However, I also understand that it 
is doomed to fail if it comes to the discrepancy between the fixed 
ordinary scale of time and the permanently moving border between past 
and future.

> This boils down to the fact that for each of them,
> time has no global significance. You cannot tell if you're *right
> here*, or one period away. This is no different for the Fourier Cosine
> Transform, except for the added restriction that it assumes even
> symmetry of your sequence.

Given the theory you learned was the ultima ratio, then you would be 
perfectly correct, too. Do not confuse the FCT as a special case of FT 
with a FCT which is equivalent to the whole FT because it only exists 
within what looks from the traditional perspective just like a halfplane.


> And you don't have to do any interpretation of negative time and
> negative frequency - these are just "labels" for the axes that are
> somewhat arbitrary and, as you said, redundant in all cases but for
> complex signals. If that redundancy really bothers you, why not use
> the Hartley transform instead - it is purely real and contains the
> same information. Now that we got rid of negative time and frequency,
> what have we gained over the Fourier Transform?

Do not reckon me among those stupid people who just feel bothered by 
negative frequencies.


> Nothing really. For both the DFT and the Fourier Cosine Transform, the
> basis functions are periodic, which means they extend indefinitely
> into the past and the future. If you limit the transform to a finite
> window, they will be spaced in a way that their periods are in integer
> relationship to the transform size, which defines the resolution of
> the analysis. 

Yes. However, R^+ is also not closed. That's why frequency resolution of 
the natural spectrogram is not at all limited.

> But they will always have constant amplitude and
> frequency (ie. no time-dependent change, that is the definition of
> "lack of time localization"), so the time localization is actually a
> function of your transform size. 

Yes, see above.

 > And without some sort of window,
> there is no way the transform will produce any meaningful result in
> practice.

Your reasoning is understandable but not complete. The natural 
spectrogram behaves in that respect even a little bit better than does 
our cochlea due to physiological restrictions. It is gradually limited 
to the side of large time by attenuation acting like 'forgetting'.
The other side of its natural time window is set be the border between 
known and unknown input. So there is no arbitrary window.


> So, with that in mind, what is it now your FCT-based "natural
> spectrogram" does differently that would it make it "superior" to a
> "traditional spectrogram"?

a)In contrast to the traditional spectrogram, is is based on the same 
sliding timescale as is natural perception.
b)It does not use an arbitrarily chosen window. One needs no 
arbitrariness at all, except for attenuation settings.
c)There is no need to decide how often to relocate the window.
d)It is not subject to the notorious trade-off between spectral and 
temporal resolution. One has no longer to either favor narrow-band or 
wideband-settings.
e)It does not need an arbitrary starting point of analysis which is also 
missing in natural hearing.
f)It does not discard phase information because it is not based on a 
complex-valued analysis that would provides results in terms of 
magnitude and phase but on a real-valued analysis that yields positive 
and negative amplitudes as does basilar membrane.
g)Consequently, there would not be an obstacle against inverse transform 
whithout reconstruction of phase by means of Bode relations.
h)The result can immediately be subject to one-way rectification as it 
happens within cochlea.
i) It is causal while the traditional one, in particular with narrowband 
settings, strikingly shows absurd output before any input
j) It shows a more realistic pattern including glides.
k) It pretty clearly shows something like the first wave-front of the 
epiphenomenal traveling wave.
l) It additionally shows an even earlier reaction that parallels an 
obsterved early response at high SPL. This reaction is not at odds with 
the uncertainty principle, because frequency gradually chirps up to the 
correct value.
m)...


> [Btw. in your last post you wrote: "Actually the natural spectrogram
> shows output before one could expect it..." Since I don't have code to
> reproduce what you see, I can only infer from this statement and the
> fact you are using the Fourier Cosine Transform that your "natural
> spectrogram" is subject to the same time localization restrictions as
> all other STFT-based spectrograms. 

What you are calling time localization restrictions is in the case of an 
optimally designed wavelet perhaps about the same as the first wave 
front of the epiphenomenal traveling wave. You might judge the 
difference between wavelets and STFT-based spectrograms yourself.

> One more piece of evidence that it is not *superior*, just different!]

No, just a wrong suspition.


>>Future sound is not available to the ear and also not to the 
>>spectrograph. Only the latter is using a zero-padded half-axis of time 
>>resulting in an even real and an odd imaginary function of frequency. 
>>With real-valued analysis - as it is performed in cochlea as well as 
>>with the natural spectrogram - there is neither negative time nor 
>>negative frequency at all. I do not use FCT as a special case of FT in 
>>IR but my ear and me restrict to reality, which can sufficiently be 
>>reflected in IR^+. Notice, redundancy only belongs to complex FT.
> 
> 
> I agree that for purely real signals the Fourier Transform has
> redundancy. 

All signals of the world do not yet exist in future. Predicted 'causal' 
signals did not yet exist before switching on.

> I assert that this redundancy is not influencing the
> outcome 

It is, e.g. in case of so called optimal filters.

> and can be avoided from the result 

Well, one could economize a lot of paper. Some 'experts', however, 
consider negative frequency and not yet elapsed time physical realities.


> I believe you assign a meaning to the Fourier Cosine Transform that it
> does not have. 

Why do you think so?

> No matter how you think about it, the Fourier Cosine
> Transform, as per its definition, is the even half of the Fourier
> Transform. Either you're talking about something other than the
> Fourier Cosine Transform (please define then!), or you are simply
> mistaken.

Of course I know this definition, and I do not use it in IR. The formula 
is the same except for the lower limit. IR^+ makes the difference.

Yes, I am actually often mistaken because people like Laurent Schwartz 
spent much effort as to make complex Fourier transform a universal and 
rigorous tool. They succeeded almost completely. Earlier doubts were 
suppressed. Nonetheless, the notorious flaws of the traditional 
spectrogram cannot be overcome this way. Complex analysis has proven an 
obstacle at least for theory of function of cochlea. I found a different 
error by Eisenmanger in PRL. Nimtz relied on complex analysis when he 
came to the obviously wrong conclusion that he measured superluminal 
propagation of signals. It is just a vague guess of mine that the 
standard model of quantum physics suffers from similar basic problems.


>>I should tell you that elapsed time does not share its zero with 
>>ordinary time.
> 
> 
> I'm not sure what you mean by "elapsed time" (elapsed since when? The
> start of our measurement? The time I was born?) and "ordinary time" (=
> time since the Big Bang?). Please explain.

We are used to count time from an arbitrarily chosen point zero. We can 
only measure time intervals. Why not fix the opposite endpoint? This is 
actually the only natural border of the whole world to rely on: the 
actual border between past and future. The scale of elapsed time is 
continuously sliding against the ordinary time.


>>>You may also realize that you have just indirectly admitted that your
>>>resolution is actually not superior to the traditional (STFT-based)
>>>spectrogram. 

By no means.

>>>If you're using the Fourier Cosine Transform (which is
>>>what you said) and create a spectrogram from it that you admit being
>>>subject to the Heisenberg uncertainty principle we must conclude that
>>>it cannot have superior resolution. 

The traditional spectrogram is far from exhausting this limit. Wavelets 
were invented as to reach a better adaptation. However they are still 
suboptimal altogether. There are many wavelets. Not a single one 
deserves to be called the best one. However, there is only one natural 
spectrogram.

>>>It may present your data
>>>differently, but that doesn't change the fact that it cannot be beyond
>>>anything we have already seen.
>>
>>Your thinking is not precise in that case.
> 
> 
> At what point?

You did not even consider the possibility that I construct the natural 
spectrogram as a sum of elementary singularity functions. You did not 
even formalize the uncertainty relation, etc.


>>I appreciate that you are a little bit familiar with Laurent Carney,
>>http://www.ima.umn.edu/biology/wkshp_abstracts/carney1.html
>>hopefully with Mario Ruggero, Roy Patterson and others, too.
>>I do not correlate in the usual manner cosine basis functions as you 
>>imagine.
> 
> 
> In what manner do you correlate them instead?

I do not use cosine functions which span from minus infinite to plus 
infinite. My approach is different.


> And yes, I am quite familiar with the work you mentioned. I am partly
> involved in research in that direction, although on a commercial basis
> so I can't publish.

Maybe you can tell me something new from the literature.


>>>Yes it is. But that doesn't have to mean that their perceptual
>>>processes evade the Heisenberg restriction. 
>>
>>Do not conclude on a wrong basis.
> 
> 
> Do you think that's what I do? Why?

I am not sure, and I would recommend to you: Be cautious too.
Overall, hearing has definitely a smaller delta t * delta f (down to 
less than 10 microseconds and to about 1 Hz) than it should have. 
Nonetheless, after a closer look, it does not much violate the limit.
So the matter depends much on details and definitions.

>>I gave the code (about 100 lines) to a few 
>>people, and nobody found an error so far.
> 
> 
> Well, try me for a change :-) I'm pretty sure I can sort out the
> discrepancies if I have a clue of what's actually going on. But I must
> assume from previous correspondence that you do not want to share
> solid evidence with us but instead have us believe you from the little
> information you provide. I agree with you that this is disappointing.
> 
> Maybe it's time to inquire about a separate comp.dsp.religion NG? :-)

Those who are seriously interested might contact me privately.
I am even ready to cope with arrogance, distrust, curiosity and insult, 
provided I will met someone who is finally in position to contribute.

Eckard Blumschein

Richard - the file doesn't help much, he sent it to me via email this
afternoon. It contains the following data - transcribed to C,
actually, the +1. should be something like 0.9999... to avoid
wrapping, I cut it short to keep the post (relatively) small. ;-)

--smb

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Hi Eckard,

Eckard Blumschein wrote:
> Once again, this link will guide you to the mentioned example of a 
> natural spectrogram:
> http://iesk.et.uni-magdeburg.de/~blumsche/M275.html
> My sketch and or linked wav file show the input.

Too bad the .wav file doesn't work. It's a dead link - we already told
you that.

> I will wonder if you can tell me something new.

I was rather hoping you could tell me something new!

> I did not directly refer to Ohm but to Ohm's law and the fact that 
> hearing is actually highly insensitive against phase, not entirely 
> though. You may understand it from natural spectrogram.

How?

Our hearing is *not* insensitive to phase. It is insensitive to
*absolute* phase, but actually quite capable of perceiving relative
phase. For example, the phase in dichotic stimuli is important for the
perception of localization cues.

> > Phase is discarded simply because <<people are>>
> > interested in the magnitude of the partial frequencies - that's
> > practically the definition of a spectrogram. 
> 
> When the spectrogaph was invented in the late fourties, there was no 
> alternative as to restrict to magnitude.

I doubt it. I know for a fact that Henrik Bode was known to plot
magnitude and phase in his frequency response plots of complex
functions since 1938 and the sound spectrograph was invented by Potter
a little later, so I highly doubt that they were unable to plot phase
in a spectrogram if they had wanted to.

Maybe Jerry has some additional information, IIRC he provided valuable
information on Bode plots some time ago.

> > Displaying the phase, too, would only clutter the readout. 
> 
> It was not imaginable at all.

I'd rather think it was not necessary! Why would someone plot phase in
a spectrogram? How would you be able to say anything meaningful about
the magnitude at a certain point if your display keeps oscillating
wildly?

> > Bear in mind that the spectrogram does not make the claim to be 
>  > in any way related to something our auditory system does,
> 
> It has been in use as to mimic the cochlea or even the auditory system.

It has been used to study harmonic signals, most importantly speech.
Since it has been observed that the auditory system localizes stimuli
in time and frequency it seemed obvious (but not equivalent!) to use
the spectrogram as a means for comparison and study. In the beginning
this choice was easy, because the spectrogram was more or less all
there was for such a purpose

But the analogy is crude, even though it helps understanding some of
the principles wrt. frequency. Going so far as to claim that the
spectrogram was used as a means to "mimic the cochlea or even the
auditory system" is nonsense - it has been (and still is) a means for
plotting frequency dependent magnitude of periodic signals over time,
nothing more.

> The spectrogram shows rather a misleading pattern.

No. The spectrogram shows exactly the pattern it was designed to
display: the frequency and time dependend magnitude of partial waves
in a signal. This is not misleading, it is, in fact, highly relevant
to what it was designed to do. The misleading part comes from you
believing it should do otherwise.

> Neither the natural spectrogram nor cochlea itself are designed just for 
> the first stage.

The cochlea *is* the first stage of our auditory perception (if we
leave aside peripheral factors like the pinnae that contribute to the
perception, but are not actually involved in the "active process").
The cochlea is the place where sound waves are converted into
information in the form of nerve pulses.

> > I think no one claims that human hearing uses a
> > spectrogram like the one you have in Cooledit!
> 
> On what input are automatic speech-recognizers based if not a current 
> spectral analysis?

I never questioned whether ASR software is using "spectral analysis"
(btw. are we talking just about a spectrogram or the whole "spectral
analysis" ballgame here with all its possible flavors?).

I said I doubt that someone claims that the *human hearing* uses a
spectrogram like the one in CoolEdit, therefore it is my belief that
you're barking up the wrong tree with your rant about the (natural or
not) spectrogram.

> >>_The_ natural spectrogram does not omit phase. It conveys all 
> >>information and would correspondingly be reversible.
> > 
> > Yes and no. If you're using the FCT (Fourier Cosine Transform), this
> > is only true for even signals.
> 
> I do not know you, and I have to be polite in any case.

Well, you don't have to, but it would be nice. I also see no reason
why you shouldn't. I just said that the Fourier Cosine Transform is
the even part of the complete Fourier Transform. That is how it is
defined, and I see nothing wrong with that definition (neither do I
find that statement in any way objectionable).

According to its definition, the Fourier Cosine Transform is only
meaningful for even signals. Does that fact disturb you? And if so,
why?

> However, I feel 
> deeply disappointed because you did not even understand the simplest 
> basics of what I found out.

I'm sorry if I disappoint you, but you must admit you don't give us
much to understand this from. Maybe you can tell me what you've found
out *exactly*, so I can provide a less disappointing response?

> Use of even and odd functions of time is 
> only necessary iff you decide to tacitly accept Heaviside's trick and 
> perform a complex-valued Fourier Transform.

You mean I should paint my fingernails in pink like he did? :-)

The Fourier Transform is inherently complex, because that's how it is
defined. Perhaps it is my limited knowledge but I have not seen any
other definition called the "Fourier Transform" that omits the "i".

> In this case one has to have 
> a time function that extends from minus infinite to plus infinite 
> including past as well as future time. Windowing does not matter in that 
> respect.

It does, on the discrete side of things, if you want to get any
meaningful representation of your signal. There are no true infinities
in a discrete world.

The basis functions of the DFT are complex exponentials. As I said in
my last post, they have perfect frequency localization but do not have
time localization. This boils down to the fact that for each of them,
time has no global significance. You cannot tell if you're *right
here*, or one period away. This is no different for the Fourier Cosine
Transform, except for the added restriction that it assumes even
symmetry of your sequence.

And you don't have to do any interpretation of negative time and
negative frequency - these are just "labels" for the axes that are
somewhat arbitrary and, as you said, redundant in all cases but for
complex signals. If that redundancy really bothers you, why not use
the Hartley transform instead - it is purely real and contains the
same information. Now that we got rid of negative time and frequency,
what have we gained over the Fourier Transform?

Nothing really. For both the DFT and the Fourier Cosine Transform, the
basis functions are periodic, which means they extend indefinitely
into the past and the future. If you limit the transform to a finite
window, they will be spaced in a way that their periods are in integer
relationship to the transform size, which defines the resolution of
the analysis. But they will always have constant amplitude and
frequency (ie. no time-dependent change, that is the definition of
"lack of time localization"), so the time localization is actually a
function of your transform size. And without some sort of window,
there is no way the transform will produce any meaningful result in
practice.

So, with that in mind, what is it now your FCT-based "natural
spectrogram" does differently that would it make it "superior" to a
"traditional spectrogram"?

[Btw. in your last post you wrote: "Actually the natural spectrogram
shows output before one could expect it..." Since I don't have code to
reproduce what you see, I can only infer from this statement and the
fact you are using the Fourier Cosine Transform that your "natural
spectrogram" is subject to the same time localization restrictions as
all other STFT-based spectrograms. One more piece of evidence that it
is not *superior*, just different!]

> Future sound is not available to the ear and also not to the 
> spectrograph. Only the latter is using a zero-padded half-axis of time 
> resulting in an even real and an odd imaginary function of frequency. 
> With real-valued analysis - as it is performed in cochlea as well as 
> with the natural spectrogram - there is neither negative time nor 
> negative frequency at all. I do not use FCT as a special case of FT in 
> IR but my ear and me restrict to reality, which can sufficiently be 
> reflected in IR^+. Notice, redundancy only belongs to complex FT.

I agree that for purely real signals the Fourier Transform has
redundancy. I assert that this redundancy is not influencing the
outcome and can be avoided from the result (sortof a truism, because
that's how redundancy is defined). So why would you have a problem
with that?

I believe you assign a meaning to the Fourier Cosine Transform that it
does not have. No matter how you think about it, the Fourier Cosine
Transform, as per its definition, is the even half of the Fourier
Transform. Either you're talking about something other than the
Fourier Cosine Transform (please define then!), or you are simply
mistaken.

> > Yes of course it does. If you're using the FCT (Fourier Cosine
> > Transform), it is still very much subject to the Heisenberg
> > restriction. As I said above, the basis functions of the FCT are
> > cosines, which have perfect frequency localization but no time
> > localization. This fact does not change depending on whether or not
> > you include phase in your plot.
> 
> I should tell you that elapsed time does not share its zero with 
> ordinary time.

I'm not sure what you mean by "elapsed time" (elapsed since when? The
start of our measurement? The time I was born?) and "ordinary time" (=
time since the Big Bang?). Please explain.

> > You may also realize that you have just indirectly admitted that your
> > resolution is actually not superior to the traditional (STFT-based)
> > spectrogram. If you're using the Fourier Cosine Transform (which is
> > what you said) and create a spectrogram from it that you admit being
> > subject to the Heisenberg uncertainty principle we must conclude that
> > it cannot have superior resolution. It may present your data
> > differently, but that doesn't change the fact that it cannot be beyond
> > anything we have already seen.
> 
> Your thinking is not precise in that case.

At what point?

> I appreciate that you are a little bit familiar with Laurent Carney,
> http://www.ima.umn.edu/biology/wkshp_abstracts/carney1.html
> hopefully with Mario Ruggero, Roy Patterson and others, too.
> I do not correlate in the usual manner cosine basis functions as you 
> imagine.

In what manner do you correlate them instead?

And yes, I am quite familiar with the work you mentioned. I am partly
involved in research in that direction, although on a commercial basis
so I can't publish.

> > Yes it is. But that doesn't have to mean that their perceptual
> > processes evade the Heisenberg restriction. 
> 
> Do not conclude on a wrong basis.

Do you think that's what I do?
Why?

> I gave the code (about 100 lines) to a few 
> people, and nobody found an error so far.

Well, try me for a change :-) I'm pretty sure I can sort out the
discrepancies if I have a clue of what's actually going on. But I must
assume from previous correspondence that you do not want to share
solid evidence with us but instead have us believe you from the little
information you provide. I agree with you that this is disappointing.

Maybe it's time to inquire about a separate comp.dsp.religion NG? :-)

> Sorry I have even to stop reading now.

That's too bad! I thought we were finally getting somewhere...

--smb

The problem is that the link to "stimulus.wav" doesn't work. It is 
actually there, or are permissions set incorrectly?


Richard Dobson


Eckard Blumschein wrote:

> 
> Once again, this link will guide you to the mentioned example of a 
> natural spectrogram:
> http://iesk.et.uni-magdeburg.de/~blumsche/M275.html
 > My sketch and or linked wav file show the input.