The Cramer Rao bound of estimating a Sinusoid

The Cramér-Rao bound (CRB) for estimating a Sinusoid was only discussed numerically under some special cases in references. It will be investigated under a theoretical frame for a general case. It is shown that, firstly, the critical CRB depends on the cycles in the record and the sole sample at the recorded window center. Secondly, for a short record, the critical phase corresponding to the minimum CRB requires the sinusoid wave to be zero at the recorded window center. Thirdly, for a long record, CRB depends on the initial phase in an approximate sine way.