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In this paper, we present a closed-form expression of a Bayesian Cramér-Rao lower bound for the estimation of a dynamical phase offset in a non-data-aided BPSK transmitting context. This kind of bound is derived considering two different scenarios: a first expression is obtained in an off-line context and then, a second expression in an on-line context logically follows. The SNR-asymptotic expressions of this bound drive us to introduce a new asymptotic bound, namely the Asymptotic Bayesian Cramér-Rao Bound. This bound is close to the classical Bayesian bound but is easier to evaluate.

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