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 [2]:

using PyPlot
using NtToolBox
using WAV
# using Autoreload
# reload("NtToolBox")

Sound Mixing¶

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

Sound loading.

In [3]:

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

x = zeros(n,3)
x[:,1] = load_sound("NtToolbox/src/data/bird.wav",n)
x[:,2] = load_sound("NtToolbox/src/data/female.wav",n)
x[:,3] = load_sound("NtToolbox/src/data/male.wav",n);

Normalize the energy of the signals.

In [4]:

x = x./repeat(std(x,1), outer=(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 [5]:

theta = Array(linspace(0, pi, s + 1)); theta = theta[1:3]
theta[1] = 0.2
M = vcat(cos(theta)', sin(theta)');

Compute the mixed sources.

In [6]:

y = x*M';

Display of the sounds and their mix.

In [7]:

figure(figsize = (10,10))

for i in 1:s
    subplot(s, 1, i)
    plot(x[:, i])
    xlim(0,n)
    title("Source #$i")
end

Display of the micro output.

In [8]:

figure(figsize = (10,7))
    
for i in 1:p
    subplot(p, 1, i)
    plot(y[:, i])
    xlim(0,n)
    title("Micro #$i")
end

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 [9]:

w = 128   #size of the window
q = Base.div(w,4);  #overlap of the window

Compute the STFT of the sources.

In [10]:

X = complex(zeros(w,4*w+1,s))
Y = complex(zeros(w,4*w+1,p))

for i in 1:s
    X[:,:,i] = perform_stft(x[:,i],w,q,n)
    figure(figsize = (15,10))
    plot_spectrogram(X[:,:,i],"Source #$i")
end