In the article Compressive sensing by Richard Baraniuk ($2007$), measurements are formulated as: $$\mathbf{y} = \Phi \mathbf{x} = \Phi \Psi \mathbf{s} $$ Where $\mathbf{y} \in \mathbb{R}^m$ are measurements of a signal, $\mathbf{x} \in \mathbb{R}^n$ is the signal sampled in time domain, $\mathbf{s}\in \mathbf{R}^n$ is the coefficients of the signal represented in some basis $\Psi \in \mathbb{R}^{n \times n}$ and $\Phi \in \mathbb{R}^{m\times n}$ is a random Gaussian measurement matrix mapping measurements to the signal.
My questions are:
- Assuming we are sampling a signal $\mathbf{x}$ with one sensor. How is it possible that the mapping $\Phi$ between the measurements $\mathbf{y}$ and the signal sampled $\mathbf{x}$ is nothing more than a binary matrix with maximum one non-zero value for each row of $\Phi$ so that each element of $\mathbf{y}$ correspond to one element of $\mathbf{x}$, so that the measurement is an actual value of the signal $\mathbf{x}$? In other words, given a random Gaussian measurement matrix $\Phi$, how can one sample of the signal actually be a random linear combination of several values of the signal, and how do we know when the measurement is sampled form the signal $\mathbf{x}$
R. G. Baraniuk, "Compressive Sensing [Lecture Notes]," in IEEE Signal Processing Magazine, vol. 24, no. 4, pp. 118-121, July 2007, doi: 10.1109/MSP.2007.4286571.
I was taught that we should think of the sensing matrix as an experimental tool that a practitioner can choose. Each row of the matrix corresponds to a single experiment, or measurement. Each of these experiments can be designed by the practitioner. What if, instead of using a binary matrix -- which is just observing the superposition of a few entries of $x$ either "on" or "off" -- we just take a random linear combination of them? It turns out that this experimental setup (using a random matrix) actually has some really exceptional properties; they are almost always better than a deterministically-chosen binary matrix. Since the practitioner can choose what type of experiments they want to perform, it makes sense to pick a matrix from this class.