# Re: First moment of a spectrum ("James W. Beauchamp" )

```Subject: Re: First moment of a spectrum
From:    "James W. Beauchamp"  <j-beauch(at)UX1.CSO.UIUC.EDU>
Date:    Mon, 12 Jun 2000 10:41:32 -0500

Al, Eric, and all,

A number of researchers have been using a definition for the "spectral
centroid" as a physical measure closely correlating with perceptual
"brightness". I think centroid is the same as the first moment.

One definition that we use is

SUM(k=1,N) A[k]*f[k]
fc  =  --------------------
SUM(k=1,N) A[k]

where k is the spectral component, f[k] is its frequency, and A[k] is
its amplitude. N is the number of components. If f[k] is a harmonic
series, i.e., f[k] = k*f0, then

SUM(k=1,N) k*A[k]
fc  =  ----------------- * fo
SUM(k=1,N) A[k]

fc can also be thought of as the relative-amplitude-weighted average
frequency of the spectrum. A normalized version of fc is NC = fc/fo.

The minimum value of fc happens when all A[k] = 0 except A[1], and then
fc = fo. On the other hand if all A[] = 0 except A[j], where 1 <= j <= N,
then fc = j*fo. For these case, NC = 1 and j, respectively.

In order to cause the centroid to go to zero when only A[1] is nonzero,
we sometimes use the definition

fc' = fc - fo = fo*(NC - 1) .

I have found the centroid to be a very useful parameter for sound
synthesis, particularly for brass sounds. It has also been used a lot as
a primary physical correlate in MDS (multi-dimensional-scaling) studies.
McAdams, Menneguzzi, and I [1] used it in a study to determine which
parameters are most significant as judged by the ability of subjects to
detect their absence. What we did was eliminate centroid changes over the
durations of several musical instrument sounds. Our conclusion was that
this is a highly detectable modification for most instruments.

Lately it appears that some researchers are using higher-ordered moments
for automatic instrument classification. Fraser and Fujinaga [2] have
used spectral moments 0 through 3 for timbre recognition and Brown [3]
used spectral centroid and time domain moments for automatic identification
of instruments.

It appears that the centroid, and perhaps higher-ordered moments, are
pretty useful for instrument analysis, synthesis, perception, and
identification.

Jim Beauchamp
j-beauch(at)uiuc.edu

[1] McAdams, S., Beauchamp, J, and Menneguzzi, S., 1999. "Discrimination
of musical instrument sounds resynthesized with simplified spectrotemporal
parameters", J. Acoust. Soc. Am., Vol. 105, No. 2, pp. 882-897.

[2] Fraser, A. and Fujinaga, I., 1999. "Toward real-time recognition of
acoustic musical instruments", Proc. 1999 Int. Computer Music Conf.,
pp. 175-177.

[3] Brown, J. C., 2000. "Automatic identification of musical woodwind
instruments using pattern recognition" (A), J. Acoust. Soc. Am., Vol. 107,
No. 5, pt. 2, p. 2818.
```

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Electrical Engineering Dept., Columbia University