This paper focusses on the sparse estimation in the situation where both the
the sensing matrix and the measurement vector are corrupted by additive
Gaussian noises. The performance bound of sparse estimation is analyzed and
discussed in depth. Two types of lower bounds, the constrained Cram\'{e}r-Rao
bound (CCRB) and the Hammersley-Chapman-Robbins bound (HCRB), are discussed. It
is shown that the situation with sensing matrix perturbation is more complex
than the one with only measurement noise. For the CCRB, its closed-form
expression is deduced. It demonstrates a gap between the maximal and nonmaximal
support cases. It is also revealed that a gap lies between the CCRB and the MSE
of the oracle pseudoinverse estimator, but it approaches zero asymptotically
when the problem dimensions tend to infinity. For a tighter bound, the HCRB,
despite of the difficulty in obtaining a simple expression for general sensing
matrix, a closed-form expression in the unit sensing matrix case is derived for
a qualitative study of the performance bound. It is shown that the gap between
the maximal and nonmaximal cases is eliminated for the HCRB. Numerical
simulations are performed to verify the theoretical results in this paper.Comment: 32 pages, 8 Figures, 1 Tabl
