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Partial power

Partial power quantifies the proportion of energy within a specified frequency band $[f_l,f_u)$ relative to the total energy. It is also known as the band energy ratio. The frequency band is defined by its lower ($f_l$) and upper ($f_u$) frequency bounds.

$$\text{PartialPower}=\frac{\sum _{m=m_l}^{m_u-1}{X}_p[m]}{\sum _{m=0}^{M-1}{X}_p[m]},$$

where:

• $X_p=|X{|}^2$ is the power spectrum,
• $M$ is the total number of bins,
• $m_l=\lfloor M\frac{f_l}{2f_s}\rfloor$ is the lower bin index corresponding to $f_l$,
• $m_u=\lfloor M\frac{f_u}{2f_s}\rfloor$ is the upper bin index corresponding to $f_u$,
• $m\in [0,M)$ represents the spectral bin indices.

Parameters

Name	Description	Unit	Limits
lower	Lower frequency bound	Hz	[0, $\frac{f_s}{2}$]
upper	Upper frequency bound	Hz	[0, $\frac{f_s}{2}$]

References

• Eyben, F. (2016). Real-time Speech and Music Classification by Large Audio Feature Space Extraction. https://doi.org/10.1007/978-3-319-27299-3
• Sause, M. G. R. (2016). In Situ Monitoring of Fiber-Reinforced Composites: Theory, Basic Concepts, Methods, and Applications (Vol. 242). Springer. https://doi.org/10.1007/978-3-319-30954-5

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 partial_power(
    spectrum: np.ndarray, samplerate: float, f_lower: float, f_upper: float
) -> float:
    ps = np.abs(spectrum) ** 2
    n = len(ps)
    n_lower = int(2 * n * f_lower / samplerate)
    n_upper = int(2 * n * f_upper / samplerate)
    return np.sum(ps[n_lower:n_upper]) / np.sum(ps)

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