  The sparse signal processing literature often uses random sensing matrices to
obtain performance guarantees. Unfortunately, in the real world, sensing
matrices do not always come from random processes. It is therefore desirable to
evaluate whether an arbitrary matrix, or frame, is suitable for sensing sparse
signals. To this end, the present paper investigates two parameters that
measure the coherence of a frame: worst-case and average coherence. We first
provide several examples of frames that have small spectral norm, worst-case
coherence, and average coherence. Next, we present a new lower bound on
worst-case coherence and compare it to the Welch bound. Later, we propose an
algorithm that decreases the average coherence of a frame without changing its
spectral norm or worst-case coherence. Finally, we use worst-case and average
coherence, as opposed to the Restricted Isometry Property, to garner
near-optimal probabilistic guarantees on both sparse signal detection and
reconstruction in the presence of noise. This contrasts with recent results
that only guarantee noiseless signal recovery from arbitrary frames, and which
further assume independence across the nonzero entries of the signal---in a
sense, requiring small average coherence replaces the need for such an
assumption.
