Re: two cycles

["Pennington, Mark" <mwpennin@xxxxxxxxxxx>]

Dear List,

Rabiner and Schafer (1978: 145, Digital Processing of Speech Signals) 
state that "to get any indication of periodicity in the autocorrelation 
function, the window must have a duration of at least two periods of 
the waveform."

Using an autocorrelation (ACC) function with a 39.37 ms rectangular 
window, I tested this proposition with three 2.5 s sawtooth waves at 
27.5 Hz (pitch period = 36.36 ms), 38.9 Hz (25.71 ms), and 55 Hz (18.18 
ms).

R&S's statement holds if the ACC function is scanned for its maximum 
value. Only the F0 of the 55 Hz sawtooth is accurately determined (mean 
calculated over the 2.5 s waveform = 54.74 Hz). The other F0s are 
severely overestimated, the mean estimated F0s of the 27.5 and 38.9 Hz 
sawtooths being respectively 281.19 and 179.54 Hz. Because two periods 
of the 55 Hz sawtooth are equal to 36.36 ms, R&S are clearly correct if 
the ACC maximum value detection method is employed, given the 39.37 ms 
rectangular window.

On the other hand, using the ACC peak-peaking method developed in 
Section 4.2.2 of my thesis (p. 96-98), I found that the respective mean 
F0s of the 27.5, 38.9, and 55 Hz sawtooths to be 31.52, 41.30, and 
54.69 Hz (Table 5.1, p. 115). These results indicate that it is 
possible to obtain reasonable F0 estimates even when the pitch period 
is nearly of the same duration as the analysis window, on the condition 
that the appropriate ACC peak-peaking procedure is used.

Mark

Thesis URL:
http://mypage.iu.edu/~mwpennin/GlottalMannerDiss-Pennington.pdf

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The "pitch in two cycles" phenomenon is exactly what would be predicted by an
autocorrelation model of pitch.  With two copies of a pulse, you can 
measure a consistent interval as a peak in the ACF.

Dick

Hi list,

In many of the papers published by Georg von Békésy he makes the 
statement that the
fundamental frequency was determined by the auditory system "even when 
the stimulus was
only two cycles" in length. In at least one of his publications {The 
Missing Fundamental
and Periodicity Detection in Hearing  JASA 1972 512) 631-637) he 
attributes this to his
own experiments and to a paper by Savart.[Annalen der Physik und Chemie 
1840 53 ( )
555-561 in german]. It is quite true that Savart found that the 
fundamental was
determined in two cycles but it was published in an earlier paper by 
Savart (Ueber die
Empfindlichkeit des Gehörorgans Felix Savart Annalen der Physik und 
Chemie 1830 20( )
290-304 in german)and no mention of the two cycles is mentioned in the 
citation by
Békésy. Actually from a historical point of view the original paper by 
Felix Savart was
published in French in: Annales de chimie et de Physique 1830 44 ( )  
337-352 in French.

Can anyone point me to literature which shows how the auditory system 
performs the "two
cycle" feat, or to papers which show how such a "two cycle" feat might 
be acomplished
mathematically or to any papers of a more recent origin which discusses 
this ability.

Thanks,

Fred