This paper introduces a simple and very general theory of compressive
sensing. In this theory, the sensing mechanism simply selects sensing vectors
independently at random from a probability distribution F; it includes all
models - e.g. Gaussian, frequency measurements - discussed in the literature,
but also provides a framework for new measurement strategies as well. We prove
that if the probability distribution F obeys a simple incoherence property and
an isotropy property, one can faithfully recover approximately sparse signals
from a minimal number of noisy measurements. The novelty is that our recovery
results do not require the restricted isometry property (RIP) - they make use
of a much weaker notion - or a random model for the signal. As an example, the
paper shows that a signal with s nonzero entries can be faithfully recovered
from about s log n Fourier coefficients that are contaminated with noise.