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Spectral skewness

Spectral skewness is a measure of the asymmetry of the power spectrum around its mean, the spectral centroid $f_{centroid}$. It indicates whether the bulk of the spectral power lies to the left or right of the centroid:

• Positive skewness: Indicates a spectrum with a tail extending towards higher frequencies.
• Negative skewness: Indicates a spectrum with a tail extending towards lower frequencies.
• Zero skewness: Indicates a symmetric spectrum around the centroid.

It is computed from the power spectrum $X_p=|X{|}^2\in {\mathbb{R}}^M$.

$\begin{array}{rl}SpectralSkewness& =\frac{m_3}{{\sqrt{m_2}}^3}\\ m_x& =\sum _{m=0}^{M-1}(f(m)-f_{centroid}{)}^x\cdot p(m)\\ f(i)& =\frac{i\cdot f_s}{2(M-1)}\\ p(i)& =\frac{X_p[i]}{\sum _{m=0}^{M-1}{X}_p[m]}\end{array}$

References

• https://www.mathworks.com/help/signal/ref/spectralskewness.html
• Peeters, G. (2004). A large set of audio features for sound description (similarity and classification) in the CUIDADO project. In CUIDADO IST Project Report (Vol. 54).

Code

INFO

The following snippet is written in a generic and unoptimized manner. The code aims to be comprehensible to programmers familiar with various programming languages and may not represent the most efficient or idiomatic Python practices. Please refer to implementations for optimized implementations in different programming languages.

code.py

import numpy as np


def _spectral_centroid(spectrum: np.ndarray, samplerate: float):
    ps = np.abs(spectrum) ** 2
    ps_sum = 0.0
    ps_sum_weighted = 0.0
    for i, magnitude in enumerate(ps):
        ps_sum += magnitude
        ps_sum_weighted += magnitude * i
    return 0.5 * samplerate / (len(ps) - 1) * (ps_sum_weighted / ps_sum)


def spectral_skewness(spectrum: np.ndarray, samplerate: float):
    f_centroid = _spectral_centroid(spectrum, samplerate)
    ps = np.abs(spectrum) ** 2
    ps_sum = 0.0
    ps_sum_weighted_2 = 0.0
    ps_sum_weighted_3 = 0.0
    for i, magnitude in enumerate(ps):
        f = 0.5 * samplerate / (len(ps) - 1) * i
        ps_sum += magnitude
        ps_sum_weighted_2 += magnitude * (f - f_centroid) ** 2
        ps_sum_weighted_3 += magnitude * (f - f_centroid) ** 3
    return (ps_sum_weighted_3 / ps_sum) / np.sqrt(ps_sum_weighted_2 / ps_sum) ** 3

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