In [1]:
%matplotlib inline
import numpy, scipy, matplotlib.pyplot as plt, IPython.display as ipd
import librosa, librosa.display
import stanford_mir; stanford_mir.init()

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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 load a file:

In [2]:
x, sr = librosa.load('audio/c_strum.wav')
ipd.Audio(x, rate=sr)
Out[2]:
In [3]:
plt.figure(figsize=(14, 5))
librosa.display.waveplot(x, sr)
Out[3]:
<matplotlib.collections.PolyCollection at 0x1172659e8>

numpy.correlate

There are two ways we can compute the autocorrelation in Python. The first method is numpy.correlate:

In [4]:
# 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)

(102400,) (102400,)

Plot the autocorrelation:

In [5]:
plt.figure(figsize=(14, 5))
plt.plot(r[:10000])
plt.xlabel('Lag (samples)')
plt.xlim(0, 10000)
Out[5]:
(0, 10000)

librosa.autocorrelate

The second method is librosa.autocorrelate:

In [6]:
r = librosa.autocorrelate(x, max_size=10000)
print(r.shape)

(10000,)
In [7]:
plt.figure(figsize=(14, 5))
plt.plot(r)
plt.xlabel('Lag (samples)')
plt.xlim(0, 10000)
Out[7]:
(0, 10000)

librosa.autocorrelate conveniently only keeps one half of the autocorrelation function, since the autocorrelation is symmetric. Also, the max_size parameter prevents unnecessary calculations.

Pitch Estimation

The autocorrelation is used to find repeated patterns within a signal. For musical signals, a repeated pattern can correspond to a pitch period. We can therefore use the autocorrelation function to estimate the pitch in a musical signal.

In [8]:
x, sr = librosa.load('audio/oboe_c6.wav')
ipd.Audio(x, rate=sr)
Out[8]:

Compute and plot the autocorrelation:

In [9]:
r = librosa.autocorrelate(x, max_size=5000)
plt.figure(figsize=(14, 5))
plt.plot(r[:200])
Out[9]:
[<matplotlib.lines.Line2D at 0x116f41240>]

The autocorrelation always has a maximum at zero, i.e. zero lag. We want to identify the maximum outside of the peak centered at zero. Therefore, we might choose only to search within a range of reasonable pitches:

In [10]:
midi_hi = 120.0
midi_lo = 12.0
f_hi = librosa.midi_to_hz(midi_hi)
f_lo = librosa.midi_to_hz(midi_lo)
t_lo = sr/f_hi
t_hi = sr/f_lo
In [11]:
print(f_lo, f_hi)
print(t_lo, t_hi)

16.351597831287414 8372.018089619156
2.633773573344376 1348.4920695523206

Set invalid pitch candidates to zero:

In [12]:
r[:int(t_lo)] = 0
r[int(t_hi):] = 0
In [13]:
plt.figure(figsize=(14, 5))
plt.plot(r[:1400])
Out[13]:
[<matplotlib.lines.Line2D at 0x118225208>]

Find the location of the maximum:

In [14]:
t_max = r.argmax()
print(t_max)

21

Finally, estimate the pitch in Hertz:

In [15]:
float(sr)/t_max
Out[15]:
1050.0

Indeed, that is very close to the true frequency of C6:

In [16]:
librosa.midi_to_hz(84)
Out[16]:
1046.5022612023945

Tempo Estimation

When perfomed upon a novelty function, an autocorrelation can provide some notion of tempo.

For more, see the notebook Tempo Estimation.

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