On the Achievability of Cramér–Rao Bound in Noisy Compressed Sensing

Recently, it has been proved in Babadi [B. Babadi, N. Kalouptsidis, and V. Tarokh, “Asymptotic achievability of the Cramér-Rao bound for noisy compressive sampling,” IEEE Trans. Signal Process., vol. 57, no. 3, pp. 1233-1236, 2009] that in noisy compressed sensing, a joint typical estimator can asymptotically achieve the Cramér-Rao lower bound of the problem. To prove this result, Babadi used a lemma, which is provided in Akçakaya and Tarokh [M. Akçakaya and V. Trarokh, “Shannon theoretic limits on noisy compressive sampling,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 492-504, 2010] 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 Babadi 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 Babadi 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 Babadi et al.