Compressed sensing can be applied to a number of fields, so make sure you are reading material primarily concerned with your specific application, but the underlying concept remains the same. A perhaps more intuitive application CS has to start with however,  is CS applied to image reconstruction, where sparsity in natural images transformed in some sparse frequency domain , such as with a Discreet cosine transform(DCT) is exploited to allow the recovery of an image with far less samples than Nyquist.

This sparse domain has only a few non zero k coefficients. It is then possible to arrange such an implementation as an ill conditioned matrix inverse problem, which is solved by ''l1 minimisation'', (least squares is ''l2 minimisation'') and where l2 and l1 norms are Euclidean and 'taxicab' norms respectively.

The supplied link is the 'l1 magic' toolbox from Stanford which is a matlab based program which will preform compressed reconstruction of an ill-conditioned  2D matrix, where the 2D matrix should be in your sparse domain ( DCT space) and your 1D vector should be ideally less than number of rows in the 2D matrix, ( making it ill conditioned, but works if you make it a simple inverse problem too, which you may want to do to confirm the program gives the same answer when not ill conditioned. You should find that even at 50% of Nyquist, or 50% compression the image is almost fully recovered. 

Also see the compressive computational imaging paper by ori katz et al which is as applied to single pixel computational imaging, sometimes called ghost imaging.

http://statweb.stanford.edu/~candes/l1magic/

https://archive.org/details/arxiv-0905.0321

This is a good question.   In addition to the very good answer by @Stephen S. Welsh. there is a bit more to add.

A good overview of comprehensive sensing is given in

https://www.researchgate.net/publication/221285712_Interconnecting_Wireless_Sensor_and_Wireless_Mesh_Networks_Challenges_and_Strategies

See, also:

https://www.researchgate.net/publication/220144587_Mobile_Crowd_Sensing_Current_State_and_Future_Challenges

Conference Paper Interconnecting Wireless Sensor and Wireless Mesh Networks: ...

Article Mobile Crowd Sensing: Current State and Future Challenges

Dear Mitra

Actually, Stephen S. Welsh has mentioned a good point that the CS is applied to a number of fields. Basically, it's started with image processing (reconstruction) then it's employed to the signal processing and other fields.

Therefore, determine the area that you would like to work with, so we could send you a right material.