PowerSpectralDensityDistance

Introduction

The PowerSpectralDensityDistance class calculates the distance between two signals based on their Power Spectral Density (PSD). PSD represents how the power of a signal is distributed over different frequency components, and comparing PSDs can reveal significant differences in frequency content between signals.

Sense of the Distance

The Power Spectral Density (PSD) distance measures the dissimilarity between the energy distributions of two signals across frequencies. It is particularly useful in comparing signals that differ in their spectral properties, and it captures variations in the frequency domain.

Formal Representation

The distance between two signals’ Power Spectral Densities is typically calculated using metrics such as: - Euclidean distance between the PSD vectors. - Other possible metrics include Kullback-Leibler divergence or spectral overlap.

Let ( P_x(f) ) and ( P_y(f) ) represent the Power Spectral Densities of two signals ( x(t) ) and ( y(t) ). The distance ( D ) can be formally expressed as: [ D(P_x, P_y) = sqrt{sum_f left( P_x(f) - P_y(f) right)^2} ] where ( f ) represents the frequency components over which the PSDs are computed.

from distancia import PowerSpectralDensityDistance

signal1 = [0.1 * math.sin(2 * math.pi * 440 * t / 16000) for t in range(16000)]
signal2 = [0.1 * math.sin(2 * math.pi * 880 * t / 16000) for t in range(16000)]

psd_calculator = PowerSpectralDensityDistance(sample_rate=16000)

psd_distance = psd_calculator.compute_psd_distance(signal1, signal2)

print("PSD Distance:", psd_distance)

>>>[(1.5+0j), (0.2381966011250105+0.1902113032590307j), (0.2999999999999998+0j), (0.36180339887498947-0.1902113032590307j), 0]

PSD Distance: 513514.8559149296 Academic Reference ——————

Stoica et al.[1]

[1]

Petre Stoica, Randolph L Moses, and others. Spectral analysis of signals. Volume 452. Pearson Prentice Hall Upper Saddle River, NJ, 2005.

Conclusion

The PowerSpectralDensityDistance class provides a robust mechanism for comparing the frequency content of two signals. By focusing on how power is distributed across frequencies, it allows for meaningful comparisons in applications such as signal classification, sound analysis, and noise detection.