Bayesian signal detection with compressed measurements

• A general expression of the probability of error by using the compressed measurements of sparse signal is obtained.
• Upper and lower bounds for the probability of error are derived using the RIP constant and the mutual coherence.
• An approximate but simpler expression of the probability of error with compressed measurements is obtained.
• A tighter bound of the probability of error than the existing one is derived in terms of using a piecewise function.

This paper proposes the Bayesian approach to signal detection in compressed sensing (CS) using compressed measurements directly. A general expression of the probability of error is obtained where the prior probabilities of hypotheses could be equal or unequal and the additive noise is assumed to be uncorrelated Gaussian noise with possibly unequal variances. Upper and lower bounds of the probability of error are also derived using the restricted isometry property (RIP) constant and then the more computationally feasible mutual coherence of a given sampling matrix in CS. When the difference between the prior probabilities is sufficiently small and the signal to noise ratio is relatively large, an approximate but simpler expression of the probability of error is obtained. Furthermore, a new bound of the probability of error is derived in terms of a piecewise function. Numerical simulations are also provided to illustrate the new theoretical results.