Signal sparsity: with noise or without noise?
Actually, the beauty of compressive-sensing lies in fact that it works for signals which are approximately sparse. Recovery algorithms in CS are based on L1 norm minimization and have moderately tolerate noise in measurements. However noise surely reduces probability of exact recovery and degrades quality of recovered signal. If there is noise in a sparse signal you might need to take more measurements for exact recovery.
Take a look at this for further information:
http://statweb.stanford.edu/~candes/papers/spm-robustcs-v05.pdf