I am reading this page on Compressed Sensing. I quoted maybe more than I needed to for context. Knowledgeable people might want to skip to the end of the quote and read backwards.
You might be asking: what the hell is that $A$ matrix? Well, it’s the key to the whole party. Let me explain.
In order to perform the minimization, we must somehow finagle our problem into a linear system of equations: $$Ax=b$$ Specifically, we want to derive a matrix $A$ that can be multiplied with a solution candidate $x$ to yield $b$, a vector containing the data samples. In the context of our current problem, the candidate solution x exists in the frequency domain, while the known data b exists in the temporal domain. Clearly, the matrix performs both a sampling and a transformation from spectral to temporal domains.
Compressed sensing really comes down to being able to correctly derive the $A$ operator. Fortunately, there’s a methodology. Start off by letting $f$ be the target signal in vector form (if your signal is $2$-dimensional or higher, flatten it) and $\Phi$ be the sampling matrix. Then: $$b=\Phi f$$
I tried Google, but thanks to the stemming of words ("sample" vs "sampling") and different definitions of the word "matrix", I can't find a good link.
How do I construct a sampling matrix?