Spectral Convergence

Introduction

The Spectral Convergence is a distance measure used to assess the similarity between the frequency spectra of two signals. It is particularly useful in audio processing, where the frequency content of signals plays a key role in distinguishing between different sounds.

Sense of the Distance

Spectral Convergence measures the difference between the magnitudes of the frequency spectra of two signals. The closer the two spectra are, the smaller the spectral convergence, indicating that the signals are more similar in their frequency content.

Formal Definition

Given two signals, signal1 and signal2, their Spectral Convergence is computed as follows:

\[\text{Spectral Convergence} = \frac{\sum_{i} |M_1(i) - M_2(i)|}{\sum_{i} M_1(i)}\]

Where: - \(M_1(i)\) and \(M_2(i)\) are the magnitudes of the frequency components of the FFT of signal1 and signal2, respectively. - The numerator computes the absolute differences between the magnitudes of the frequency components. - The denominator normalizes the distance by the sum of the magnitudes of signal1.

from distancia import SpectralConvergence

signal1 = [0.5, 0.1, 0.2, 0.4, 0.3, 0.2, 0.1, 0.0]
signal2 = [0.4, 0.2, 0.2, 0.5, 0.3, 0.1, 0.2, 0.0]

convergence = SpectralConvergence().compute(signal1, signal2)
print(f"Spectral Convergence: {convergence}")

>>>Spectral Convergence: 0.14673244459294343

Academic Reference

Allen and Rabiner[1]

[1]

J.B. Allen and L.R. Rabiner. A unified approach to short-time fourier analysis and synthesis. Proceedings of the IEEE, 65(11):1558–1564, 1977. doi:10.1109/PROC.1977.10770.

Conclusion

The Spectral Convergence provides a practical way of comparing the frequency content of two signals, making it useful in audio analysis, speech processing, and signal comparison tasks.