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

```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.

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.
```