Performance analysis of ℓ1-synthesis with coherent frames

Signals with sparse representations in frames comprise a much more realistic model of nature, it is therefore highly desirable to extend the compressed sensing methodology to redundant dictionaries (or frames) as opposed to orthonormal bases only. In the generalized setting, the standard approach to recover the signal is known as ℓ₁-synthesis (or Basis Pursuit). In this paper, we present the performance analysis of this approach in which the dictionary may be highly - and even perfectly - correlated. Our results do not depend on an accurate recovery of the coefficients. We demonstrate the validity of the results via several experiments.