Re: ACF of a rectangle

[Eckard Blumschein <Eckard.Blumschein@xxxxxxxxxxxxxxxxxxxxxxxxxx>]

Dear Cristian,

I appreciate your criticism. You are formally correct. However, I referred
to practical measurement rather than theory. Admittedly, a rectangular
waveform with small pulse ratio is only a poor approximation of some vowels
as well as sound pulses from arc welding I studied.
On one hand, I considered spectra, not energy spectra of them. Due to
physical reasons and A-weighting, the spectral comb tends to be not short
at all. On the contrary, effective amplitudes of some resolved cochlear
harmonics might even exceed the fundamental one. Why do I hear largely just
one pitch? On this list, Chen-gia reported some time ago that the pitch of
a rather complicated flute tone well coincided with the smallest
autocorrelation lag constituting the largest peak in the ACF. He provided
the pertaining file including the ACF.
So, at the other hand, I got almost the same "ACF" by means of using FCT,
rectification and FCT again. Such mechanism was physiologically and
evolutionary plausible to me. Of course, I had to understand why do we not
perceive the multiples of this lag. Let's assume f0 = 100 Hz. Then the
first autocorrelation lag equals ten milliseconds. Our neural system does
perhaps not look much at multiples of it. Well I should correct myself:
In case of repetetive pulses we perhaps tend to perceive just the first and
most prominent positive autocorrelation lag as if there was only a single one.

Being short of time I did not carefully read Peter Cariani's response. So I
didn't get aware him commenting  on my attempt to reconsile his point of
view with yours. Why not considering the possibility that you and he too
are correct? Isn't your evidence against the possibility of autocorrelation
within the brain still valid? Doesn't he correctly supports the idea of
autocorrelation as also do others, while there is no convincing evidence
for a physiological correlate inside brain?

I fully agree with Israel Nelken's comment on Gaussian vs. uniform noise.
The relationship between magnitude and phase may be expressed as
Kramers-Kronig relations. I do not much like such sophistication because I
consider complex FT an academic alternative rather than a natural language
which will be the best basis for bridging the still wide gap between
physiology and psychoacoustics. Those who are interested in this topic
might ask me for a copy of
"The natural spectrogram: no arbitrary window, no trade-off".
I ( blumschein@et.uni-magdeburg.de ) already offered it elsewhere and
responded to more than 60 requests, so far. Because I am still collecting
requests, you might have to wait one or two days. Here are just a few of my
conclusions:
Complex-valued magnitude-phase representation differs from real-valued
time-frequency representation merely by a single value to be chosen
arbitrarily. Audition lacks this arbitrary choice. So it definitely
performs a real-valued Fourier cosine transform, not the complex-valued
Fourier transform. Arbitrary choices start with the usual notion of time
and continue with the neglect of either exp(i omega t) or exp(-i omega t),
and they further imply an arbitrary choice of a time window, non-causality,
and other flaws.

I sadly realized distracting discussions on questionable analogies. Do not
be worried by the term spatial frequency. The more correct term is
wavenumber.


Best wishes,
Eckard


At 08:54 20.01.2004 +0100, Christian Kaernbach wrote:
>Eckard Blumschein wrote:
>> The spectrum of the pure tone is a single frequency. Its ACF resembles a
>> comb. It sounds harmonical.
>>
>> The spectrum of the rectangle is comb-like. Its ACF is a single
>> autocorrelation lag. It sounds sharp.
>
>The energy spectrum of a periodic rectangle waveform is "comb-like" with
>peaks decreasing as 1/n^2 (odd peaks only). The second peak is 1/9 of
>the first. A short comb, indeed...
>
>The ACF of a periodic rectangle waveform is a periodic triangluar waveform.
>
>Best,
>Christian Kaernbach
>