Closed-Form CRLBs for SNR Estimation From Turbo-Coded BPSK-, MSK-, and Square-QAM-Modulated Signals

In this contribution, we derive for the first time the closed-form expressions for the Cramér-Rao lower bounds (CRLBs) of the signal-to-noise ratio (SNR) estimates from BPSK-, MSK- and square-QAM modulated signals over turbo-coded transmissions. These CRLBs, relatively easy to derive from BPSK, MSK and QPSK transmissions, become extremely challenging with higher-order square-QAM-modulated signals. In the latter, by exploiting the structure of the Gray mapping, we are able to factorize the likelihood function thereby linearizing all the derivation steps for the Fisher information matrix (FIM) elements. We also propose another approach that allows the evaluation of the considered bounds using extensive Monte Carlo computer simulations. The analytical CRLBs coincide exactly with their empirical counterparts validating thereby our new analytical expressions. Numerical results suggest that the CLRBs for code-aided (CA) SNR estimates range between the CRLBs for non-data-aided (NDA) SNR estimates and those for data-aided (DA) ones, thereby highlighting the expected potential in SNR estimation improvement from the coding gain. Indeed, the CA CRLBs improve by decreasing the overall coding rate due to enhanced decoding capabilities. However they do increase with the modulation order for a given code rate. Finally, the derived bounds are also valid for LDPC coded systems and they can be evaluated when the latter are decoded using the turbo principal.