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Topic: Resonance transform for sound analysis (Read 7476 times) previous topic - next topic
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Resonance transform for sound analysis

Someone in Korea has found a new transform besides Fourier or wavelet transforms, and has released a basic introduction.

http://bbs.kldp.org/viewtopic.php?t=61263

http://user.chol.com/~kkb110/RT.pdf

Here's a rough translation of the abstract. Any errors are probably mine.

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In frequency analysis of signals Fourier or wavelet transforms have been widely used. While these are useful tools, it is a little different from the ear, the typical organ which acts to analyse frequencies. The ear measures frequencies through the resonance of cells, and this paper describes a method of frequency analysis which simulates these resonations.

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If anyone is interested, I might have a go at translating it all into English.

Resonance transform for sound analysis

Reply #1 –
It would be most interesting.
"You can fight without ever winning, but never win without a fight."  Neil Peart  'Resist'

Resonance transform for sound analysis

Reply #2 –
It looks like a simplified cochlea model, or not?

My Korean is not that good

Resonance transform for sound analysis

Reply #3 –
The author is a high-school student, who's figured out "real" (as opposed to "complex") FFT on his own before. This too might be something that's been found before.

I'm not familiar with the material, so my terms might be a bit off the mark. Because I don't understand what the author means, the sentences might not make sense.

I'll start from 3.1.

3.1 Frequency Analysis Method of Ear

Sound passes through various parts of the ear before reaching variously-sized hairs in the cochlea. Each hair cell has its resonant frequency and vibrates when exposed to sound. At the end of the hair cell there is a device into which calcium ions flow, depending on the amplitude of the cell's vibration, and through this mechanism the strength of the vibration according to frequency can be sensed.

3.2 Resonance Transform

As the ear senses frequencies through resonating hair cells, this method can be mimiced by simulating the resonations. When an external force F(t) applies to an object, its "decay vibration" can be shown as the differential equation below. Here, m is the mass, gamma the decay constant, k the elasticity constant, and function x is (INSERT SOMETHING ABOUT TIME HERE)

m * x''(t) + gamma * x'(t) + k * x(t) = F(t) (1)

Let Xm,gamma,k(t) be  the formula above as solved by t through function x, and the amplitude function Am,gamma,k(t) is as follows.

Am,gamma,k(t) = sqrt(  (Xm,gamma,k(t))^2 + (m/k) * (X'm,gamma,k(t))^2  ) (2)

The equation above is the vibration function of an object vibrating through external force, and is the formula for resonance transforms. What's needed further is to find values of m, gamma, and k that are appropriate for the object's resonant frequency and decay rate. In equation (1), when external force F(t) is 0 for all t,

x(t) = A0 * e^(-gamma*t/2m) * cos(sqrt(k/m - (gamma/2m)^2) * t + phi) (3)

When analysing various frequencies of sound spontaneously, it is better for the decay formula according to t, e^(-gamma*t/2m), to be constant regardless of the vibration rate f. This is when you ignore the vibration rate, and if you want to adjust the decay rate to be proportionate with the vibration rate, you can make (-gamma/2m) proportionate with f. If e^(-gamma*t/2m) is fixed, gamma will be proportionate with m, and f=sqrt(k/m - (gamma/2m)^2)/(2*pi). Using the vibration rate f and a, the time-frequency trade-off element, to make equations of m and gamma,

m = k/(4*pi^2*f^2) + (log(1-a))^2) (4)

gamma = -2m*log(1-a) (5)

When the above equations are maintained, k does not influence the vibration movement's shape, but the vibration amplitude is inversely proportional to k at all intervals. Hence k can be fixed to an arbitrary value.

3.3 Discrete Resonance Transform

Resonance transform of discrete signals ... at non-continous points the acceleration rate v will change due to external signals ... and the region where the signal is constant between the non-continous points can be analysed as developments of equation (3) of section 3.2. However, sound is normally sampled at a frequency of 44100Hz, so the development of equation (3) can be approximated to a high degree of accuracy through the use of resonant filters, at a low cost.[1] The following is pseudo-code, and S is the external signal.

Code: [Select]
function RTN(Sn)
    x<-0
    v<-0
    temp<-0
    for i = 0 to N − 1 do
        Fi   (Si − temp) * SamplesPerSecond
        temp = Si
        v<-v + Fi/m
        v<-v − (kx/m)/SamplesPerSecond
        v<-v * e^((−ir/2m)/SamplesPerSecond)
        x<-x + (v/SamplesPerSecond)
        Ai<-sqrt(x^2 + (m/k) * v^2)
    end for


Figure 1 is the result of analysing the first 6 seconds of Beethoven's Für Elise, and the "decay rate of amplitude" per second is 1-10^(-14).

4 Remarks

Resonance transforms offer a very precise resolution in frequency analysis of data which has a time axis. Further research of hair cells is neccesary for finding precise values or m, gamma, and k which match our ears, and for finding the reverse resonance transform and fast resonance transform algorithms.

References
[1] Robin Green: Faster Math Functions, http://www.research.scea.com/gdc2003/fast-...unctions_p2.pdf

Resonance transform for sound analysis

Reply #4 –
kjoonlee, I have to read it in detail later, but it sure looks like a simplification of the existing cochlea model a fellow student of mine was working on. That model was also based on the actual hair cell response. I could show this to him.

edit: Ok, I've read it.

Quote
function x is (INSERT SOMETHING ABOUT TIME HERE)

It probably says: "function x is the displacement as a function of time", because that's what it is. Could that be correct?

Quote
Resonance transforms offer a very precise resolution in frequency analysis of data which has a time axis. Further research of hair cells is neccesary for finding precise values or m, gamma, and k which match our ears, and for finding the reverse resonance transform and fast resonance transform algorithms.

As fas as I know, these values are already known. Furthermore, the hair cells in our ear can not be treated individually, they are coupled, something that doesn't show up in this model. It is highly simplified.

However, it is interesting.  His goal is probably not to make a good model, but to make a new approach of time-frequency analysis.

Resonance transform for sound analysis

Reply #5 –
This could be a very good way to design an equalizer, based on how the ear would actually hear the tones rather than straight up frequency analysis.

Resonance transform for sound analysis

Reply #6 –
Quote
This could be a very good way to design an equalizer, based on how the ear would actually hear the tones rather than straight up frequency analysis.
[a href="index.php?act=findpost&pid=319689"][{POST_SNAPBACK}][/a]

That is indeed one of the goals of cochlea research, to be able to make predictions on how an arbitrary audio signal will sound to our ear (mind, that the processing done in our brains is not included in those models). A lot of research is still performed in that field.

This model is too simple for that kind of analysis, I think. The human ear is a very complex device.

Resonance transform for sound analysis

Reply #7 –
It looks like a FIR in the time domain, a cosine transform followed by a FIR in the frequency domain.

Resonance transform for sound analysis

Reply #8 –
Quote
It looks like a FIR in the time domain, a cosine transform followed by a FIR in the frequency domain.
[a href="index.php?act=findpost&pid=319698"][{POST_SNAPBACK}][/a]


That's odd. It looks just like a bunch of state-variable IIR resonant filters to me.
I might have missed something, though.


Sebi

Resonance transform for sound analysis

Reply #9 –
Quote
Quote
It looks like a FIR in the time domain, a cosine transform followed by a FIR in the frequency domain.
[a href="index.php?act=findpost&pid=319698"][{POST_SNAPBACK}][/a]


That's odd. It looks just like a bunch of state-variable IIR resonant filters to me.
I might have missed something, though.


Sebi
[a href="index.php?act=findpost&pid=319702"][{POST_SNAPBACK}][/a]


You are right, jeez, I said 3 things and all of them were wrong. 

Thanks for the correction anyway