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Re: two cycles


I think what you're showing is that applying a rectangular window and then computing an ACF of the windowed signal is a poor way to use the ACF concept, since it can severely degrade the "repetition" property that you're looking for. Especially if the window is too short, as R&S show.

If instead you use the "running" method, of applying an LPF to the running product of a signal with a delayed version of itself, with no need for windowing, simple peak picking will work much more robustly. This is what Licklider proposed in 1952 in his "Duplex model of pitch perception".

R&S's window condition should not be taken the be the same as the psychophsical result that two cycles of a waveform are sufficient to hear a pitch.


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

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.


Thesis URL:


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.


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.