Compressed Sensing Mathematical Concept in Signal Processing I am new in the field of compressive sensing, I've read many papers explaining that compressive sensing is used widely in sparse signal reconstruction. I've tried to understand how compressive sensing is used in signal processing, but I couldn't understand the concept. 

According to the paper On some common compressive sensing recovery algorithms and applications - Review paper in page 2, THE MATHEMATICAL BACKGROUND OF THE COMPRESSIVE SENSING CONCEPT, I got the best description for compressing sensing mathematical concept. Assuming we are dealing with signal $x$ with length $N$, that is sparse in time domain. So according to equation 5, we can write the system equation as $Y = AX$, where $A$ is the CS matrix. My questions, 

1- What is the signal $x$ with length $N$ which we want to reconstruct ? Is it $X$ or $Y$ ? And what does $A$ represent? 

2- Is there any tutorial or file which explain that point with details and in easy way?
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])