can you upload a sample sound?

why did you transform your original signal to matrix form? is it 2D matrix?

Thank you for replying. I attach a short sound (a single vowel).

I transformed the sound to a matrix because this is what I was told that is necessary to calculate the derivative of a signal. 

I also tested the method with a sound I prepared from a formula (simple sine wave), for which I could analytically calculate the second derivative, as well as the energy integrals. The results of the analytic calculations and those I described above yielded similar results (it makes sense that the integral of the second derivative produces much higher values compared to the integral of the original signal, because the second derivative of a sine function is the original function multiplied by a positive number). So I am actually concerned that I didn't understand something fundamental about how BRI is defined.

Positive number? what positive number?

what is dx in your calculation?

if the signal is sin(2*pi*f*t), where f is the frequency in Hz, then it's second derivative is (2*pi*f)2*sin(2*pi*f*t) (well, there should be a minus sign there, but since the energy is calculated for the squared signal, the sign doesn't matter).

I used the default dx: 2.2675736961451248e-005

This formula is true for continues signal.

here we have a discrete sound signal and derivative value of a discrete signal is x(n)-x(n-1). i.e, dx=1.

I yet do not know why you used matrix format?!!! can you upload the original paper.

Well, increasing dx, definitely changes the result. But I'm not sure about dx=1. It seems to me that the matrix index is not an integer. The basic unit is total time divided by the number of columns in the matrix (which equals to the sampling frequency). Thus, if the sound duration is x=1 and sampling frequency is 10kHz, then the matrix has 10000 columns and dx = 10-4.

You might be right that changing dx will solve my problem. I am less certain, though, that dx=1 is the best option.

As to why I use matrices, here is what Paul Boersma wrote when asked about derivations in Praat:

"But if you want to have the derivative as a time function, you convert the Pitch to a Matrix (unvoiced regions will be set to zero) and apply the following formula: (self [col+1] - self [col]) / dx"

And that's what I did (but with sound object rather than pitch).

Can you suggest a different approach?

Thanks!

On second thought, I expect the discrete calculation to give similar results to the continuous calculation (with some error, of course). The results I received are many orders of magnitude far from what I have seen in the Fukazawa paper (see link). So either, I misread their results (I'm referring to the first paragraph of their results), or didn't get the procedure right.

http://scitation.aip.org/content/asa/journal/jasa/83/3/10.1121/1.396012

Yes, mathematically "dx" is the movement in time. In this case, the time between one sample and the next (the reciprocal of sampling frequency) as you (Chen Gafni) say.

The value you report suggests to me that maybe you have a logarithmic step missing. Are you calculating based on the raw sound? This would be (analagous to) pressure. If the reference results are reported in dB, then an appropriate log-transform would probably get your answer back into a reasonable range.

(I cannot access the Fukazawa paper content at this moment, so I can't say exactly how that would need to be done.)

Thank you for your reply. The calculation is performed on a raw signal, and the results are reported in Pa2*sec. I should say again that when I perform an analytic calculation I get similar "crazy" numbers: let's say your sound wave is 0.5sin(2pi*440t) lasting for one second, and to get its energy you calculate the integral of the squared signal. Then you perform a similar calculation for the second derivative of the function. To get BRI you calculate the ratio of the results which is 0.25*(2pi*440)4  ~1013. So, unless I did some very fundamental error, this should be a typical value for BRI, that is, of course, assuming that I got the definition correctly. I can try sending you the paper in a private message.

Okay, an initial scan of the method doesn't suggest that they did log scaling, and they do not report their results in dB, as would be expected if they had. I'm afraid I don't have time to dig deeper here. I agree that your values suggest an important difference from the study in the paper, either in how you are making the measurement or in how you are calculating the result. Although it is an old paper, I would recommend trying to contact the authors, or seeing if anyone has used the measure more recently (check citations in Google Scholar) and asking them.

My familiarity with voice quality literature hasn't brought up this measure - which seems to be based on a clinical application (voice pathology). However, many other measures are used which you might look at (if you haven't already). For example, the Praat software offers a "Harmonicity" measure, which is a ratio (measured in dB) between the harmonic and the noise energy - sometimes referred to in the literature as "harmonic-to-noise ratio" or "HNR".

Helen Hanson has some publications from the late 1990s and early 2000s looking at voice quality generally. She and her colleagues tend to compare the amplitudes of certain harmonic and formant peaks - a more accessible measure. They have reference values and ranges for adule male and female speakers, so you can check your results to see if they are in a sensible range.

The "spectral emphasis" measure (Traunmüller & Eriksson 2000) might also be worth checking. They used it in the context vocal effort (speaking vs shouting), but the comparison of low-frequency energy (where harmonics dominate) to high-frequency energy (where noise would dominate more in breathy speech) might be useful. And the measure is a very easy one to implement.

Thanks for the effort to try to make sense of things. I didn't know the measure of spectral emphasis before. It looks similar to other measures I have used (energy difference between frequency bands, Hammarberg index, H/L). But it seems that, like the other measures, it is based on some arbitrary choice of divisions to frequency bands, whereas the BRI is claimed to provide a global measure.

Might have something to do with the way "intensity" is estimated in Praat. Last time I looked it was, I recall, F0-dependent. It might be worth checking this and if it is the case use a more appropriate estimate of intensity, perhaps via MATLAB's signal processing capabilities..

Ok, I will try that. Thanks!

The Traunmüller & Eriksson (2000) method is not arbitrary - it uses f0 to determine the frequency cutoff. The measure is definitely susceptible to confounding by vowel quality, but that can normally be controlled for. Anyway, good luck in your search for a useful measure!

As a postscript, I tend to use measures only when I have either a clear personal understanding of how the input data affects the measure, or a co-author / amiable colleague with such understanding who can vet my methodology and analysis before I put in too much work. It is perilous to take a published measure and use it without that understanding to hand.

Dear Chen Gafni,

I suggest you to analyze the meaning of the derivative operation in the spectral domain, and see if it can help explain the contradictory results:

The amplitude of the frequency components in the first derivative of the signal is multiplied by a constant, proportional to the frequency, and in the second derivative the amplitudes are amplified proportional to their squared frequencies (as you already pointed out, in one of previous comments).

The interesting point is to note that signals with similar spectral slopes, but with different sampling frequencies (something related to the dx term you also mentioned) would report quite dissimilar Br indexes.

The most simple and illustrative example I can imagine is using a flat spectrum signal (the denominator in the Br expression would not change with different Fs, if we analyse the "mean" energy in the spectral domain). But, what happens to the mean energy of the spectrum of the second derivative of flat-spectrum signals if sampled at Fs=8KHz (a common sampling frequency in the times of Fukazawa's paper) compared to the one sampled at 44.1 KHz (which seem to be the case according to your "dx") ?

On this case, both amplitude spectrums (the ones for second derivatives at different Fs) would have an "upward" parabolic shape, but they would reach very different amplitudes, being the one with the higher Fs a lot bigger. The effect is to be more noticeably when energies (actually mean energies) are to be calculated, requiring amplitudes to be squared. This means that energies in the second derivative would be proportional to the fourth power of the frequency, and the higher the Fs, the higher the frequencies considered.

I apologize for not giving you a rigorous mathematical derivation and only a descriptive idea, with the easiest signal (flat spectrum) to illustrate the idea. but I think it would suffice...

I guess Fukazawa's Br index ranges can only be obtained with his original sampling frequencies.

Nothing that I'm telling here contradicts what you already pointed out (the amplitude dependency with the sinusoidal frequency components, and the relevance of dx), I just hope it helps you figure out what is happening.

Dear Carlos Ariel Ferrer-Riesgo,

Thank you for the ideas.

I tried resampling the sound at 8 kHz and calculating  the integrals in the spectral domain. Though I received lower values than before, they are still orders of magnitude higher than those reported in the Fukazawa paper (they actually sampled at 20 kHz).

I still don't know what's going on, and, at this point, I don't think I'll use that measure in my study.

Timothy - I checked the spectral emphasis measure, and it seems to be very useful for my goal (apparently I was looking at the wrong paper earlier), so thank you very much!