On the True Cramer-Rao Lower Bound for the DA Joint Estimation of Carrier Phase and Timing Offsets

Y. Jiang, F.W. Sun and John S. Baras

Number: CSHCN TR 2000-14, Year: 2000, Advisor: John S. Baras

Abstract

The Cramer-Rao lower bound (CRB) plays a pivotal role in parameter estimation theory, such as timing, frequency and phase synchronization. Therefore, it receives considerable attention in the literature. This paper concerns the CRB for data-aided (DA) timing and/or phase recovery, i.e. the parameter synchronization is aided by a training sequence known to the receiver.

For DA parameter synchronization, the CRB typically varies with the training sequence. This indicates that different training sequences offer fundamental different performance. Therefore, it is very important to be able to compute the CRB for any particular training sequence to understand the fundamental limit that a particular training sequence has. However, in the literature, the closed-form CRB for an arbitrary training sequence is not available. In principle, it is possible to use brute-force numerical approach to compute CRB for any given training sequence. Such brute-force computation involves evaluation of derivatives numerically and matrix inversion. Besides the computational complexity, brute-force approach does not provide any insight on the interaction between training sequence and the resultant CRB.

In the literature, the widely cited close-form data-aided CRB for timing and phase recovering was derived under the assumption that the training sequence is independently identical distributed (i.i.d.) and the length of the training sequence is sufficiently long. We found that the CRB for a particular training sequence can be significantly lower than that with the long i.i.d. assumption. Therefore, the widely cited data-aided CRB actually does not give the fundamental limit for a particular training sequence.

In this manuscript, we derive a closed-form formula for data-aided CRB fortiming and phase synchronization with respect to arbitrary training sequence. The bound illustrates the close relation between the training sequence and the fundamental limit on timing and phase synchronization. This bound provides additional insights on the sequence design.