Correlation

Measure similarity between signals for pattern matching and analysis.

Cross-Correlation

$$ (x \star y)[n] = \sum_{k=-\infty}^{\infty} x[k] y[n + k] $$

Auto-Correlation

$$ R_{xx}[n] = (x \star x)[n] = \sum_{k=-\infty}^{\infty} x[k] x[n + k] $$

Implementation

 1import numpy as np
 2
 3# Cross-correlation
 4corr = np.correlate(x, y, mode='full')
 5
 6# Auto-correlation
 7autocorr = np.correlate(x, x, mode='full')
 8
 9# Normalized cross-correlation
10def normalized_xcorr(x, y):
11    corr = np.correlate(x, y, mode='full')
12    norm = np.sqrt(np.sum(x**2) * np.sum(y**2))
13    return corr / norm

Applications

• Pattern matching: Find template in signal
• Time delay estimation: Find lag between signals
• Pitch detection: Find fundamental frequency

Further Reading

• Cross-correlation - Wikipedia

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