Cramér-Rao Lower Bounds for Frequency andPhase NDA Estimation From Arbitrary Square QAM-Modulated Signals

In this paper, we derive for the first time the analytical expressions of the exact Cramér-Rao lower bounds (CRLBs) of the carrier frequency and the carrier phase from square quadrature amplitude modulated (QAM) signals, assuming the noise power and the signal amplitude to be completely unknown. The signal is assumed to be corrupted by additive white Gaussian noise (AWGN). The main contribution of this paper consists in deriving the analytical expressions for the non-data-aided (NDA) Fisher information matrix (FIM) for higher-order square QAM-modulated signals. We prove that the problem of estimating the synchronization parameters is separable from the one of estimating the signal and the noise powers by showing that the FIM is block diagonal. Besides, we show analytically that the phase CRLB is higher than the frequency CRLB, implying that it is much easier to estimate the frequency than the distortion phase. It will be seen that the CRLBs differ widely from one modulation order to another in the medium SNR range. The newly derived expressions corroborate previous attempts to numerically or empirically compute the considered CRLBs as well as their asymptotical expressions derived only in special SNR regions.