# Separating a signal into its constituent components

First off, I apologize for possibly incorrect use of jargon, as I am new in this area.

I have a type of signal that I am assuming can be described by an arbitrary number (let's say $n$) of basis vectors, which are linearly independent. I am measuring a signal through one sensor, which gives me a vector of amplitudes. I am able to measure an arbitrary number (let's say $m$) of mixtures of the basis signals, which gives me $m$ measured signals.

The length of my vectors is 150000, sampled at a frequency of 2.5 MHz.

So here is what I think my system of equations will look like.

$$Y = A X$$

where $Y$ is the matrix containing the mixtures (measurements) and has dimensions of $m \times 150000$, $X$ is the matrix containing the basis signals, and has dimensions of $n \times 150000$, and $A$ is a coefficient matrix that has dimensions of $m \times n$.

I need to solve for $A$. However, since $X$ is non-square ($n \ll 150000$), it does not have an inverse. I guess I could calculate the pseudo-inverse. Does this make sense?

I have also looked into Principal Component Analysis and how it is used in various types of spectroscopy. It seems to me that PCA assumes a set of basis signals, and I don't know whether that has a physical meaning in my case, since my signals are stochastic (but stationary). I used the PCA routine on matrix $Y$ in MATLAB, and it gave me some signals very similar to the input ones, but I did not know what to do with them.

Any ideas?

I think I do have a valid physical model for my signals. Should I perhaps use model-based blind source separation (BSS) ?

Thanks in advance for taking the time.

• Yes, if you have a rectangular system, you would want to use the pseudoinverse. One way to do that is using the singular value decomposition, which can also be used in PCA. – Jason R Dec 17 '13 at 20:07
• I have tried that. I calculate the coefficient matrix $A$, but the answer I get does not seem right. Maybe then my physical model is flawed? - that is, the assumed basis signals. – sigma Dec 17 '13 at 21:00
• Anyone? What about time-frequency analysis? – sigma Dec 18 '13 at 0:45
• Did you try Independent component analysis? The setup resembles it. – Deniz Dec 19 '13 at 23:23

You do a QR decomposition of $X^T$. Then the system reads as $$Y=AR^TQ^T.$$ Now multiply with $Q$ from the right, $$YQ=AR^T,$$ which already should give you hints on the reliability of the eventual solution, since ideally, the block of $YQ$ right of the leading $m\times n$ block should be zero. If that is true, then $$A=[YQ]_{[1:m,1:n]}[R^T]_{[1:n,1:n]}^{-1}$$.

Note that in the QR decomposition using Householder reflections, you can apply the reflections computed for $X$ simultaneously to $X$ and $Y$, if $$S=I-2 vv^T,$$ then $$X_{next}=XS=X-2(Xv)v^T\text{ and }Y_{next}=YS=Y-2(Yv)v^T$$ and $$X_{final}=R^T,\; Y_{final}=YQ$$.

• How can you QR decompose $X$ if it is unknown? – anderstood Feb 9 '18 at 19:49
• "$X$ is the matrix containing the basis signals", which is known as the input of the system. Or did you mean "when" instead of "if"? And in the case that $X$ really is unknown, what are the known inputs and the measured outputs in this hypothetical situation? – Lutz Lehmann Feb 9 '18 at 20:21
• Mmh, in my understanding only the measurement $Y$ was known (Blind Source Separation), the answer by Tarin Ziyaee suggests the same, but now I understand your answer, thank you. – anderstood Feb 9 '18 at 20:40

You can have a look to Non-negative Matrix Factorization, which try to find the factorization of a matrix (be careful, the matrix values have to be non negative) into a product of two matrix.

By adding L1 regularization to the cost function, you can ensure that each measure is encoded by a relatively small set of basis functions.

It has been used for example for music transcription, computer vision, text classification,....

• Thanks for your answer. My basis signals are uncorrelated in the time domain. In the frequency domain they all decay exponentially, and their correlation is non-zero. Does that make sense? – sigma Dec 19 '13 at 0:57
• @jason-r My measured signals are cyclostationary and as I mentioned previously, they decay exponentially in the frequency domain. I assume that each signal decays at the same rate, so that their linear mixture is what we measure. Do you think it is possible to separate the individual spectra if they all belong to the same family of curves? – sigma Dec 19 '13 at 1:26

What you are looking for is exactly Independent Component Analysis, (ICA). The setup for ICA is exactly, given a matrix $Y$, from $Y = AX$, find both $A$ and $X$.