Compressed Sensing Mathematical Concept in Signal Processing
If the signal $x$ is already sparse in the original/time domain, then it does not need to be transformed, in other words the transform $\mathcal{J}$ can be taken to be the identity. So $x$ and $X=\mathcal{J}x$ are equal. What we want is to recover, from observed $y$, the  unknown $x$, via the recovery of $X$, since $x=\mathcal{J}^{-1}X$. In that case, $A$ is just the "random" sensing matrix.

The above paper does not seem too complicated, and I am not sure there are many "much simpler" explanation. You can try:


a YouTube primer: Introduction to Compressed Sensing
some links: Understanding Compressed Sensing
and perhaps from Emmanuel Candès: slides (Compressive Sensing - A 25 Minute Tour) and a tutorial (An Introduction To Compressive Sampling [A sensing/sampling paradigm that goes against the common knowledge in data acquisition])