I need to solve: $$\vec a_i\circ \vec b_i=\sum_{j}^n\alpha_{ij}\vec c_j\circ(\vec d_j-2\pi\tau_{ij})$$
For $\vec d_j$ and $\tau_{ij}$ when $\circ$ denotes the Hadamard product and $\vec a_i,\vec b_i,\vec c_j, \alpha_{ij}$ are known.

I ran across an equation of this form in a PhD thesis and the author recommended that it be solved numerically via an alternating least squares approach. However, the Hadamard product is really tripping me up.

Can I get rid of the summation and convert the expression to vector/matrix products? If so how?

Is an alternating least squares approach really appropriate?

For a little context I am attempting to do blind source separation with delayed signals. Normal separation techniques like PCA and ICA don't work with delayed signals. However, if you compute the Fourier transform of all your measurements it turns out that the amplitude spectra form a stationary mixture that can be separated using methods like PCA and ICA with non-negativity constraints. Once the separation is complete the trick is to recover the phase information and delays.

In this case $\vec a_i$ is the squared magnitude of the Fourier transform of the $i^{th}$ measurement, $\vec b_i$ is the first derivative of the argument of the Fourier transform of the $i^{th}$ measurement, $\alpha_{ij}$ is one element of the mixing matrix $\mathbf A$, $\vec c_j$ is the squared magnitude of the Fourier transform of the $j^{th}$ extracted source, $\vec d_j$ is the first derivative of the argument of the Fourier transform of the $j^{th}$ source, and $\tau_{ij}$ is one element of the delay matrix $\mathbf T$.

Once $\vec d_j$ has been found for all sources then the IDFT of each source can be calculated. $\mathbf T$ also needs to be estimated but I'm fairly certain I know how to do that so it could be taken as known.

I did try contacting the author but he appears to have stopped publishing about 5 years ago and I cannot find any current contact info. He provided a link to the code he used in the thesis but the code wasn't actually at the link. A copy of the thesis can be downloaded from here for those who are interested. The equation in question can be found in section 6.2 on the top of page 74.

I decided to post in Mathematics rather than Signal Processing because the question is about a specific mathematical step.

  • $\begingroup$ I don't see why you couldn't convert this to a matrix-vector product, but is this problem not inherently undetermined? If your vectors have length $M$ and you have $N$ terms in the summation, your linear system will be of the form $Ax=b$, where $x,b\in\mathbb{R}^{M}$ and $A\in\mathbb{R}^{M\times MN}$. $\endgroup$ – AnonSubmitter85 Oct 23 '16 at 20:46
  • $\begingroup$ So your comment just made me realize that what I have written makes no sense mathematically. I know that $\tau_{ij}$ is a scalar and I have it subtracted from a vector meaning I totally misinterpreted the original equation and $\vec a_i$, $\vec b_i$, $\vec c_j$ and $\vec d_j$ all have to be scalars. Which pretty much invalidates the entire question and puts me back at square one. Thanks for your suggestion. It was very helpful. Now I feel like I have a shot at finding a solution assuming I can ever properly understand the problem. $\endgroup$ – ShawnS Oct 24 '16 at 8:33

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