Source Separation with Sparsity¶

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This numerical tour explore local Fourier analysis of sounds, and its application to source separation from stereo measurements.

In [1]:

from __future__ import division

import numpy as np
import scipy as scp
import pylab as pyl
import matplotlib.pyplot as plt

from nt_toolbox.general import *
from nt_toolbox.signal import *

import warnings
warnings.filterwarnings('ignore')

%matplotlib inline
%load_ext autoreload
%autoreload 2

Sound Mixing¶

We load 3 sounds and simulate a stero recording by performing a linear blending of the sounds.

Sound loading.

In [2]:

from nt_toolbox.load_sound import *

n = 1024*16
s = 3 #number of sounds
p = 2 #number of micros

x = np.zeros([n,3])
x[:,0] = load_sound("nt_toolbox/data/bird.wav",n)
x[:,1] = load_sound("nt_toolbox/data/female.wav",n)
x[:,2] = load_sound("nt_toolbox/data/male.wav",n)

Normalize the energy of the signals.

In [3]:

x = x/np.tile(np.std(x,0),(n,1))

We mix the sound using a $2\mathrm{x}3$ transformation matrix. Here the direction are well-spaced, but you can try with more complicated mixing matrices.

Compute the mixing matrix

In [4]:

theta = np.linspace(0, np.pi, s + 1)[:-1]
theta[0] = .2
M = np.vstack((np.cos(theta), np.sin(theta)))

Compute the mixed sources.

In [5]:

y = np.dot(x,np.transpose(M))

Display of the sounds and their mix.

In [6]:

plt.figure(figsize = (10,10))

for i in range(s):
    plt.subplot(s, 1, i+1)
    plt.plot(x[:, i])
    plt.xlim(0,n)
    plt.title("Source #%i" %(i+1))

Display of the micro output.

In [7]:

plt.figure(figsize = (10,7))
    
for i in range(p):
    plt.subplot(p, 1, i+1)
    plt.plot(y[:, i])
    plt.xlim(0,n)
    plt.title("Micro #%i" %(i+1))

Local Fourier analysis of sound.¶

In order to perform the separation, one performs a local Fourier analysis of the sound. The hope is that the sources will be well-separated over the Fourier domain because the sources are sparse after a STFT.

First set up parameters for the STFT.

In [8]:

w = 128   #size of the window
q = w//4  #overlap of the window

Compute the STFT of the sources.

In [9]:

from nt_toolbox.perform_stft import *
from nt_toolbox.plot_spectrogram import *

X = np.zeros([w,4*w+1,s])
Y = np.zeros([w,4*w+1,p])

for i in range(s):
    X[:,:,i] = perform_stft(x[:,i],w,q,n)
    plt.figure(figsize = (15,10))
    plot_spectrogram(X[:,:,i],"Source #%i" %(i+1))