On the sensing matrix performance for support recovery of noisy sparse signals

Recovery of sparse signals from few linear measurements is a central task of the recently emerged area of compressed sensing. Evidently, the design of the measurement plays a key role in the signal recoverability. In this contribution we analyze the explicit dependence between a deterministic sensing matrix and the support recovery performance. We do so by deriving the probability of wrong support recovery and output SNR in the presence of additive input noise. Due to tractability, a closed-form analytical expression can only be found for the 1-sparse case. However, we present numerical evidence that the expressions obtained for 1-sparse case qualitatively capture the trend for the more general iV-sparse case as well. Additionally, the investigations reveal that when designing a measurement, along with the low coherence one has to ensure a stable output SNR. We provide an example of a sensing matrix that, despite having slightly higher coherence, is superior compared to the conventional random matrix with i.i.d. Gaussian entries in terms of the support recovery performance due to providing a constant output SNR.