Closed-form CRLBs for SNR estimation from turbo-coded 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 turbo-coded square-QAM transmissions. By exploiting the structure of the Gray mapping, we are able to factorize the likelihood function thereby linearizing all the derivation steps for the FIM elements. 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 effect of the coding gain. At sufficiently high SNR levels, the three CRLBs coincide. The derived bounds are also valid for LDPC-coded systems and they can be evaluated in the same way when the latter are decoded using the turbo principal.