Closed-form expressions of the true Cramer-Rao bound for parameter estimation of BPSK, MSK or QPSK waveforms

This paper addresses the stochastic Cramer-Rao bound (CRB) pertaining to the joint estimation of the carrier frequency offset, the carrier phase and the noise and data powers of binary phase-shift keying (BPSK), minimum shift keying (MSK) and quaternary phase-shift keying (QPSK) modulated signals corrupted by additive white circular Gaussian noise. Because the associated models are governed by simple Gaussian mixture distributions, an explicit expression of the Fisher Information matrix is given and an explicit expression for the stochastic CRB of these four parameters are deduced. Refined expressions for low and high-SNR are presented as well. Finally, our proposed analytical expressions are numerically compared with the approximate expressions previously given in the literature.