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%matplotlib inline
import seaborn
import numpy, scipy, matplotlib.pyplot as plt, pandas, librosa, IPython.display as ipd, urllib
Autocorrelation¶
The autocorrelation of a signal describes the similarity of a signal against a time-shifted version of itself. For a signal $x$, the autocorrelation $r$ is:
$$ r(k) = \sum_n x(n) x(n-k) $$In this equation, $k$ is often called the lag parameter. $r(k)$ is maximized at $k = 0$ and is symmetric about $k$.
The autocorrelation is useful for finding repeated patterns in a signal. For example, at short lags, the autocorrelation can tell us something about the signal's fundamental frequency. For longer lags, the autocorrelation may tell us something about the tempo of a musical signal.
Let's download and listen to a file:
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urllib.urlretrieve('http://audio.musicinformationretrieval.com/c_strum.wav')
x, fs = librosa.load('c_strum.wav', sr=44100)
ipd.Audio(x, rate=fs)
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librosa.display.waveplot(x, fs, alpha=0.5)
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numpy.correlate¶
Use numpy.correlate to compute the autocorrelation:
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# Because the autocorrelation produces a symmetric signal, we only care about the "right half".
r = numpy.correlate(x, x, mode='full')[len(x)-1:]
print x.shape, r.shape
Plot the autocorrelation:
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plt.plot(r[:10000])
plt.xlabel('Lag (samples)')
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librosa.autocorrelate¶
Or use librosa.autocorrelate:
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r = librosa.autocorrelate(x, max_size=10000)
plt.plot(r)
plt.xlabel('Lag (samples)')
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