On the Achievability of Cramér-Rao Bound In Noisy Compressed Sensing

Recently, it has been proved in [1] that in noisy compressed sensing, a joint typical estimator can asymptotically achieve the CramérRao lower bound of the problem. To prove this result, [1] used a lemma, which is provided in [2], that comprises the main building block of the proof. This lemma is based on the assumption of Gaussianity of the measurement matrix and its randomness in the domain of noise. In this correspondence, we generalize the results obtained in [1] by dropping the Gaussianity assumption on the measurement matrix. In fact, by considering the measurement matrix as a deterministic matrix in our analysis, we find a theorem similar to the main theorem of [1] for a family of randomly generated (but deterministic in the noise domain) measurement matrices that satisfy a generalized condition known as The Concentration of Measures Inequality. By this, we finally show that under our generalized assumptions, the Cramér-Rao bound of the estimation is achievable by using the typical estimator introduced in [1].