Spectral Centroid
The spectral centroid (Beauchamp 1982) is commonly associated with the measure of the brightness of a sound (Grey and Gordon 1978). This measure is obtained by evaluating the "center of gravity" using the Fourier transform's frequency and magnitude information (Equation 2.7). Generally speaking, it has been found that increased loudness also increases the amount of high spectrum content of a signal thus making a sound brighter.
(2.7)

X[k] is the magnitude corresponding to bin k and N is the length of the DFT. This measure has also been used in MDS (multidimensional scaling) based systems.  Figure 2.10 shows examples of the spectral centroid for the french horn and the electric bass guitar.


Figure 2.10 Spectral centroid of french horn and electric bass at 44.1 kHz

        Spectral Smoothness
The spectral smoothness (McAdams 1999) measures the smoothness of the frame to frame spectral envelope obtained via the short time Fourier transform. The algorithm basically takes the average of adjacent amplitudes of the spectral bins and compares them to the current amplitude at bin k as shown in equation 2.8.
(2.8)

Figure 2.11 shows examples of the spectral smoothness for white noise, sine wave and electric bass sampled at 44,100 Hz.


Figure 2.11 White noise, sine wave and electric bass spectral smoothness

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