Spectral centroid
Encyclopedia
The spectral centroid is a measure used in digital signal processing
to characterise a spectrum
. It indicates where the "center of mass" of the spectrum is. Perceptually, it has a robust connection with the impression of "brightness" of a sound.
It is calculated as the weighted mean
of the frequencies present in the signal, determined using a Fourier transform
, with their magnitudes as the weights:
where x(n) represents the weighted frequency value, or magnitude, of bin
number n, and f(n) represents the center frequency of that bin.
of the spectrum. This is a different statistic, the difference being essentially the same as the difference between the unweighted median and mean
statistics. Since both are measures of central tendency
, in some situations they will exhibit some similarity of behaviour. But since typical audio spectra are not normally distributed, the two measures will often give strongly different values. Grey and Gordon in 1978 found the mean a better fit than the median.
.
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
to characterise a spectrum
Spectrum
A spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
. It indicates where the "center of mass" of the spectrum is. Perceptually, it has a robust connection with the impression of "brightness" of a sound.
It is calculated as the weighted mean
Weighted mean
The weighted mean is similar to an arithmetic mean , where instead of each of the data points contributing equally to the final average, some data points contribute more than others...
of the frequencies present in the signal, determined using a Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, with their magnitudes as the weights:
where x(n) represents the weighted frequency value, or magnitude, of bin
Histogram
In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson...
number n, and f(n) represents the center frequency of that bin.
Alternative usage
Some people use "spectral centroid" to refer to the medianMedian
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
of the spectrum. This is a different statistic, the difference being essentially the same as the difference between the unweighted median and mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
statistics. Since both are measures of central tendency
Average
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
, in some situations they will exhibit some similarity of behaviour. But since typical audio spectra are not normally distributed, the two measures will often give strongly different values. Grey and Gordon in 1978 found the mean a better fit than the median.
Applications
Because the spectral centroid is a good predictor of the "brightness" of a sound, it is widely used in digital audio and music processing as an automatic measure of musical timbreTimbre
In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...
.