Signal Decomposition for a mixed signal

Hi friends,

Suppose we have a mixed signal X composed of three component signals x1, x2, and x3:

t=0:0.00001:0.3;
x1=(exp(-3*t)).*(0.2*sin(2*pi*400*t));
x2=1.2+(exp(-1.5*t)).*(1.1*sin(2*pi*40*t+pi/6));
x3=(exp(-5*t)).*(0.8*sin(2*pi*75*t+pi/3));
X=x1+x2+x3;
subplot(4,1,1);
plot(t,x1);title('Component signal: x1');
subplot(4,1,2);
plot(t,x2);title('Component signal: x2');
subplot(4,1,3);
plot(t,x3);title('Component signal: x3');
subplot(4,1,4);
plot(t,X);title('Mixed signal: X=x1+x2+x3');

Now, inversely, how can we obtain the samples of the three component signals x1, x2, and x3 without any additional information except the samples of the mixed signal X?

I would be very grateful if anyone could provide a code or efficient technique for this challenging example.

Thanks in advance for your help.

Note: Unfortunately, the ICA package and also the function emd() did not lead to a desired result. Is there any other practical solution for this example?

[Allen]

What it appears you are asking is not mathmatically possible. Since a composite signal can be created from an infinite number of possible s1+s2+s3 sample sets, there is no way to extract the original x1, x2, x3 signals without additional information.

This is similar to solving any linear solution, such as a+b+c = D, where a, b, and c are all real numbers.

[Coo Boo]

Although I don't know the answer, I think it's possible. We should not look at each sample individually. Rather, we should look at the mixed signal in general and identify the component signals through specific techniques like signal multiresolution analyses. Although the output of emd(.) function seems relatively acceptable, I am looking for a better solution and technique.

Try this

emd(X)

emd-->>> Empirical mode decomposition

[Ridwan Alam]

Since the components each has periodic parts with different frequencies, you can try to use band-pass filtering to separate out these components.

[Coo Boo]

The idea came to my mind, but the filtering would also have some losses and frequency overlap, and I think the result might not be very good. Can you provide the code for investigating the result of this idea for this example?

[Ridwan Alam]

Posted below as an answer. Hope this helps!

Answer 1

[Ridwan Alam]

t=0:0.00001:0.3;
x1=(exp(-3*t)).*(0.2*sin(2*pi*400*t));
x2=1.2+(exp(-1.5*t)).*(1.1*sin(2*pi*40*t+pi/6));
x3=(exp(-5*t)).*(0.8*sin(2*pi*75*t+pi/3));
X=x1+x2+x3;

fs = 1/.00001;
z1 = highpass(X,350,fs);
z3 = bandpass(X,[60 100],fs);
z2 = X - (z1 + z3);
Z = z1 + z2 + z3;

[Coo Boo]

Thanks a lot,

Although the decomposition is not ideal, this is the best answer I've received.

[Ridwan Alam]

Indeed. This answer assumes the pass bands are known beforehand. Moreover, the exponentially decaying function is not really decomposed well, as their frequency bands are different than the sin components. It just answers toy examples for learning filters, do not use in real general purpose applications.

[Srivatsa Dasa]

how to decompose a random .wav signal if its pass bands are unknown. Is it possible to decompose the signal