MATLAB: Extract signal components from mixed signal

@image analyst

Hi,
I have signal that is consists of three individual signals. How can I separate these individual signals from the mixed signal. I did the FFT and then IFFT to get back to the original signal. The same way I wanna get the three individual signals (I_first, I_second, I_third) by performing FFT and IFFT.
The code I used:
clc
clear all;
close all;
format long
m=1000; I1=0.5; I2=0.3; I3=0.2; L1=100*m; L2=1000*m; n1=1; n2=1.446;
lam1=1530; lam2=1565;
inc=(lam2-lam1)/(2^10-1);
lam=lam1:inc:lam2;
Q12=(4*pi*n1*L1)./lam;
Q23=(4*pi*n2*L2)./lam;
Q13=Q12+Q23;
I_first=I1+I2+2*sqrt(I1*I2).*cos(Q12);  % first signal
I_second=I2+I3+2*sqrt(I2*I3).*cos(Q23);  % second signal
I_third=I1+I3+2*sqrt(I1*I3).*cos(Q13);  % third signal
I=I1+I2+I3+2*sqrt(I1*I2).*cos(Q12)+2*sqrt(I2*I3).*cos(Q23)+2*sqrt(I1*I3).*cos(Q13); % Mixed signal
N=length(lam);
fs=1/inc;
dt=1/fs;
df=1/(N*dt);
f=(-N/2:N/2-1)*df;
xxx=5;
subplot(xxx,1,1)
plot(lam,I)
title('Mixed signal')
y=fft(I);
y1=fftshift(y);
y2=abs(y1);
subplot(xxx,1,2)
plot(f,abs(y))
title('FFT')
subplot(xxx,1,3)
plot(f,y2)
title('FFTshift')
y3=ifft(ifftshift(y1));
subplot(xxx,1,4)
plot(lam,abs(y3))
title('IFFTshift')

Best Answer

Hi Sohel, May I first suggest that you clean this code up a bit. It isn't easy to follow exactly what kind of signals you are creating. For example, instead of 
lam1=1530; lam2=1565;
inc=(lam2-lam1)/(2^10-1);
Why not just ?
t = linspace(1530,1565,1024);
It appears from your code above you are using "lam" as your time vector. Then, you seemingly create a sine wave wtih expressions like in the case of I_first where I have used t for your "lam". What exactly are you trying to do here? A little explanation about what signal you are trying to create would help us determine if perhaps you have an inadvertent error in the signal model.
Note the equivalence of the following:
t = linspace(1530,1565,1024);
angle = 4*pi*1e5./t;
plot([angle' Q12'])
% Then you basically do 
plot(cos(angle))
Were you trying to create a frequency-modulated signal with the above?
At any rate, I don't see how you can cleanly separate I_second from I_third. The frequencies of these two components are so close, whether intentional or not, that you have essentially just created one amplitude modulated signal. Note
plot([abs(fft(I_second))' abs(fft(I_third))'])
Now if I plot the sum of those two in time, you see the amplitude modulated signal
plot(I_second+I_third)
With respect to bandpass filtering, you are not going to be able to separate these components in a way that when you sum them back you get the original signal. You can however accomplish that with multiresolution techniques. In this case I would recommend a wavelet packet technique, modwptdetails. Please see Practical MRA for a introduction.
So here:
mra = modwptdetails(I,5,'fk18');
% first component
plot(mra(1,:))
% second and third component together
plot(mra(3,:))
Again, you cannot expect to separate I_second and I_third with any technique I know of. Now note that if I sum all the mra components back together
ts = sum(mra);
max(abs(ts-I))
I get back the original signal perfectly. Now if you compare the extracted MRA components, you see that except for the expected DC shifts (shifts in the mean), they quite accurately reproduce, I_first and I_second+I_third
subplot(2,1,1)
plot([mra(1,:)' I_first']), title('First');
axis tight
subplot(2,1,2)
plot([mra(3,:)' (I_second+I_third)']), title('Second+Third')
axis tight
To see that more clearly, let's add the DC shift in and compare the AM component extracted by the wavelet packet MRA with the original.
mu = mean(I_second+I_third);
figure
plot([mra(3,:)'+mu (I_second+I_third)']);
title('Comparison of wavelet packet with original')
axis tight
Hope that helps,
Wayne
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