# Re: A new paradigm?(On pitch and periodicity (was "correction to post"))

```Hi again, Randy,

A brief postscript:

Another problem which I forgot to mention, is that the displacement of the basilar membrane and/or cilia is rarely a simple damped sinusoid, especially at the apical end. Even if the speech signal is "clean", you can have two or more harmonics at similar amplitudes at the same point on the membrane, the simple x(t).dx(t)/dt expression you mentioned can give rapidly varying (even negative) values for instantaneous "energy". You'll need to do something to convert these values to a more "robust" form - in my RAR analysis I dealt with it by applying a signal-dependent weighting function to the instantaneous values and integrating over a short period of time. This is simple and can be justified mathematically, but you might be able to come up with something better!

Cheers.

Steve Beet

On Thu, 8 Sep 2011 17:17:58 +0100
Steve Beet <steve.beet@xxxxxxxx> wrote:

> Dear Randy,
>
> I think we might be talking at cross purposes here. The "filter bank" I referred to could equally well be called a "linear model of wave propagation within the cochlea". The filter-bank I used was simply an empirical linear approximation to a real (non-linear, and quite possibly non-deterministic) transfer function between the incident pressure wave and the displacement of the cilia on the basilar membrane.
>
> I should mention perhaps that there is no way I would claim my "reduced auditory representation" is either complete or precise in it's characterisation of human perception - that was not my intention.
>
> The point I was trying to make in my email was that your characterisation of "signal energy" as "x(t).dx(t)/dt" is essentially the same as the Teager energy operator which has been widely investigated, both in the context of auditory-inspired, and conventional linear (Fourier-based) signal analysis.
>
> I also wanted to point out that this direct characterisation of signal energy as the sum of kinetic and potential energy in the system, calculated as "x(t).dx(t)/dt", is fine if you're analysing a simple resonant system, but in the real world, the acoustic environment is complicated (even if it is only a single voice) and the numerical results usually become swamped by any higher frequencies in the signal being analysed. I was merely trying to suggest that you need to consider the practicalities of how you measure "energy" before you commit yourself to a particular set of equations.
>
> It's all well and good to say "start with the highest of the series" but with the definition of energy which you propose, the energy at higher frequencies will be over-emphasised and I suspect you will have trouble differentiating between background and/or quantisation noise and the highest harmonics of the signal. However, I might be wrong - and I'd love to hear about the results of any experiments you do.
>
> Again, good luck with all this!
>
> Steve Beet
>
>
>
> On Thu, 08 Sep 2011 11:44:12 -0400
> Ranjit Randhawa <rsran@xxxxxxxxxxx> wrote:
>
> > Dear Steve,
> > The model I am proposing depends on analyzing frequency at each point
> > along the BM (no filter banks), which then means that magnitude of that
> > frequency can be given in terms of the magnitude of its harmonics, based
> > on using the rate of change of energy directly by summation. What this
> > then means is that the harmonic series is limited by the upper range of
> > the cochlea, 20 khz, and the number of terms of the harmonic series will
> > decrease as higher level frequencies are considered. Since the number of
> > terms for the higher frequencies is limited, it was conjectured by me
> > that it was the reason why phase locking tends to decrease above about 4
> > khz., and the quality of the sound decreases as compared with a tone at
> > much lower frequencies which will have many more terms in the harmonic
> > series.
> > The only way to proceed with the analyses, at least as discovered by me
> > so far, requires that the analyses start with the highest component of
> > the series, meaning that the highest associated frequency is first
> > evaluated and therefore subtracted before the next lower harmonic is
> > evaluated. Meaning that by the time the lower numbered harmonics are
> > evaluated, the ones that tend to define pitch, the signal is fairly
> > clean. Hence, noise enhancement due to the dx(t)/dt part of the rate of
> > change of energy (x(t)*dx(t)/dt) is removed automatically.
> > Since magnitude is available directly from the summation of the rate of
> > change of energy, phase for each of the harmonics can be determined by
> > using the criteria of choosing the maximum magnitude from the results
> > derived by rotating the input vector, sized to be equal to the
> > wavelength of the frequency being analyzed. The amount of rotation is
> > limited as it depends on the harmonic being analyzed, and the point at
> > which the maximum is found, also defines the phase of the harmonic. The
> > use of energy allows for such a criteria. For a periodic signal, there
> > will be one frequency at which the maximum sums of the magnitudes of the
> > harmonic series components will equal the total evaluated by summing the
> > absolute value of the rate of change of energy, providing a means of
> > choosing the fundamental. This is more complicated than using a modified
> > form of auto-correlation but I felt required to allow explanation of the
> > "party" effect.
> > I did want to clarify that one is not using a filter bank at all, since
> > I don't believe that such a thing actually exists in wetware. Hence, it
> > was necessary that the method include a method by which the higher
> > frequency components can be removed and its impact to the overall signal
> > noted. I have tried to understand your reference to the Teager energy
> > operator, and have to admit that my mathematical skills were not up to
> > it. I have tried to approach the problem at a more fundamental level and
> > hope that this clarification provides additional details of this.
> > Regards,
> > Randy Randhawa
> >
> >
> > On 9/7/2011 7:05 AM, Steve Beet wrote:
> > > Hi Ranjit,
> > >
> > > In respect of the paragraph below, what you're suggesting is essentially the same as the Teager energy operator. I applied a "stabilised" form of this idea to the output of an auditory filter-bank, loosely based on a very early version of Dick Lyon's auditory model, in the late 1980s. I extended it to include estimates of the signal energy, the phase velocity of the travelling wave within the cochlea (analogous to Yegnanarayana's "modified group delay"), and the dominant frequency at each point along the basilar membrane. There are some examples of these parameters in this paper:
> > >
> > > http://stevebeet.supanet.com/assets/archives/IOA92.zip
> > >
> > > and a more detailed description of the analysis method is in this one (I don't have an electronic copy for this I'm afraid):
> > >
> > > "Automatic speech recognition using a reduced auditory representation and position-tolerant discrimination. S. W. Beet. Computer Speech and Language, Vol. 4, pp 17-33. January 1990."
> > >
> > > It might be worth taking a look at these before trying your ideas out - the presence of the dx(t)/dt term in your equation will make any results very susceptible to background noise and distortion unless you take some measures akin to those described in the Computer Speech and Language paper.
> > >
> > > Good luck with your ideas!
> > >
> > > Steve Beet
> > >
> > >
> > >
> > > On Tue, 6 Sep 2011 12:53:12 -0400
> > > Ranjit Randhawa<rsran@xxxxxxxxxxx>  wrote:
> > >
> > >> If one were to consider a pure sinusoid in the phase domain (one where
> > >> the axis are x(t) and dx(t)/dt), the locus would be a circle. The area
> > >> of this circle would give us the magnitude, though how to determine this
> > >> requires a different approach as the integration over 2pi would be zero.
> > >> If we consider the product x(t)*dx(t)/dt as the rate of change of energy
> > >> it would have a sign associated with it, then it is possible to
> > >> determine this area, though the resulting algorithm would be too simple
> > >> and fall apart for more complex signals since we don't know the period.
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