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()
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]:
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)
Plot the autocorrelation:
In [5]:
plt.figure(figsize=(14, 5))
plt.plot(r[:10000])
plt.xlabel('Lag (samples)')
plt.xlim(0, 10000)
Out[5]:
The second method is librosa.autocorrelate:
In [6]:
r = librosa.autocorrelate(x, max_size=10000)
print(r.shape)
In [7]:
plt.figure(figsize=(14, 5))
plt.plot(r)
plt.xlabel('Lag (samples)')
plt.xlim(0, 10000)
Out[7]:
librosa.autocorrelate conveniently only keeps one half of the autocorrelation function, since the autocorrelation is symmetric. Also, the max_size parameter prevents unnecessary calculations.
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]:
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)
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]:
Find the location of the maximum:
In [14]:
t_max = r.argmax()
print(t_max)
Finally, estimate the pitch in Hertz:
In [15]:
float(sr)/t_max
Out[15]:
Indeed, that is very close to the true frequency of C6:
In [16]:
librosa.midi_to_hz(84)
Out[16]:
When perfomed upon a novelty function, an autocorrelation can provide some notion of tempo.
For more, see the notebook Tempo Estimation.