On the Use of Extrinsic Probabilities in the Computation of Non-Bayesian Cramér–Rao Bounds for Coded Linearly Modulated Signals
This contribution considers the non-Bayesian Cramér–Rao bound (CRB) related to parameter estimation from a linearly modulated signal observed in additive white Gaussian noise. We compare the exact CRB expression for coded modulation with two ad hoc CRB (ACRB) expressions; the first ACRB is obtained by substituting in the exact CRB expression for uncoded modulation the a priori symbol probabilities by the extrinsic symbol probabilities; the second ACRB, which has received some attention in recent scientific publications, additionally assumes that the real and imaginary parts of the symbols are independent. Our exposition focuses on the particular case of phase shift estimation. By means of examples we show that although for some coded modulation schemes the exact and ACRBs yield virtually the same numerical result, for other coded modulation schemes the ACRBs differ considerably among themselves and from the exact CRB. We provide some explanations for this behavior. We also argue that both ACRBs are expected to virtually coincide with the exact CRB in the case of bit-interleaved coded modulation combined with a rectangular constellations with independent in-phase and quadrature mapping and a binary code for which the factor graph does not contain short cycles.