TimeLagDistance

Introduction

The TimeLagDistance class calculates the optimal time lag between two signals to determine the best alignment. Time lag distance is often used in signal processing to find the temporal offset between two time-series signals, which maximizes their similarity or correlation.

Sense of the Distance

Time lag distance measures the temporal shift between two signals that produces the highest correlation or the least amount of difference. It is useful in applications where signals may be time-shifted versions of each other, and identifying the correct lag is necessary for proper comparison or synchronization.

Formal Representation

The time lag distance ( tau_{opt} ) is defined as the value of ( tau ) that maximizes the cross-correlation between two signals ( x(t) ) and ( y(t) ): [ tau_{opt} = arg max_{tau} int_{-infty}^{infty} x(t) y(t + tau) , dt ] In a discrete form, this can be written as: [ tau_{opt} = arg max_{k} sum_{n} x[n] y[n + k] ] where ( tau_{opt} ) represents the time shift that yields the highest similarity between the signals.

from distancia import TimeLagDistance

signal1: List[float] = [0.1 * math.sin(2 * math.pi * 440 * t / 16000) for t in range(16000)]
signal2: List[float] = [0.1 * math.sin(2 * math.pi * 440 * (t - 100) / 16000) for t in range(16000)]  # signal2 is shifted

time_lag_calculator = TimeLagDistance(sample_rate=16000)

best_lag: int = time_lag_calculator.compute_time_lag_distance(signal1, signal2, max_lag=500)

print("Optimal time lag:", best_lag)

>>>Optimal time lag: 9

Academic Reference

Rabiner and Gold[1]

[1]

Lawrence R Rabiner and Bernard Gold. Theory and Application of Digital Signal Processing:(by) Lawrence R. Rabiner (and) Bernard Gold. Prentice-Hall, 1975.

Conclusion

The TimeLagDistance class provides a useful tool for analyzing and aligning time-shifted signals. It is particularly helpful in fields like audio processing, speech recognition, and communications, where understanding the time delay between signals is crucial for proper analysis and synchronization.