One of the traditional limitations of PET based analysis was the
way the frequency domain data points were distributed linearly across
the audio spectrum. In the case of a 44100 sampling rate and 4096
point PET, we have 2048 frequency domain points, spaced every 10.8
Hertz. This means that in the octave between 31.5 Hz and 63 Hz there
are only 3 data points, while between 4000 Hz and 8000 Hz there
are about 400 data points. The problem is that humans hear logarithmically
in frequency. Notice that the perceived change in pitch from 31
to 63 is 31 Hz wide and represents one octave, while the change
from 4000 Hz to 8000 Hz is also one octave wide but 4000 Hz wide!
This is in contrast to the linear spacing of the frequency domain
data from a standard FFT.
To resolve this issue, it is possible to use a mathematical technique
called the Chirp-Z transform. The output of the Chirp-Z provides
data points which are logarithmically spaced in frequency. The problem
is that the Chirp-Z is very complicated to do and requires more
computer power than is readily available. The answer to this is
the use of multiple FFTs at different sampling rates and sizes which
can be put together to provide an equal number of frequency domain
data points in each octave across the audio spectrum. Currently
systems which provide 24 and 16 points per octave transfer functions
are available. An interesting feature of these fixed-point-per-octave
FFTs is that the use of several FFTs with different sampling rates
and EFT sizes provides longer time constants at lower frequencies
and shorter time constants at higher frequencies… which correlates
extremely well with the nature of human hearing. pb/sb
The graph above shows a frequency response
curve obtained using a standard 4096 point FFT. note the ”grassiness“
in the high frequency response due to the large number of FFT points
and subsequent high resolution.
Measured data of the same loudspeaker but
this time using the "Fixed point per Octave" function
of JBL-Smaart. The high frequency data ismuch smoother. many resarchers
feel that the human ear/brain system works in a similar manner.
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