The Cramer-Rao Bound for Training Sequence Design for Burst-Mode CPM

In this paper, we study the Cramer-Rao bound (CRB) for continuous phase modulation (CPM) signals where frequency offset, carrier phase, and symbol timing are jointly estimated when transmitted over an additive white Gaussian noise (AWGN) channel. We consider a data-aided (DA) estimation scenario in which the estimator takes advantage of a known training sequence at the start of each burst. Thus, we first derive the joint CRBs as functions of a known training sequence and CPM parameters. By analyzing the CRB expressions, we propose the optimum training sequence for which the CRB is minimized. We show that the same training sequence is optimum for all three estimation parameters. Additionally, we compare the performance of the optimum training sequence with a random one by providing a closed-form expression for the unconditional CRB (UCRB) for symbol timing estimation of CPM signals. Comparing the UCRB and the CRB for the optimum training sequence reveals that a DA estimator with the optimum training sequence leads to significant gains in terms of the mean-square error of the estimation parameter when the underlying CPM scheme is non-binary and/or partial response.

I. Introduction

In burst-mode transmissions, one usually embeds a fixed data sequence to each burst for the purpose of fast and accurate receiver synchronization. This data sequence is known a priori to the receiver and is referred to as a training sequence or preamble. It is called a training sequence because data-aided (DA) synchronization algorithms take advantage of it in order to estimate synchronization parameters such as frequency offset, carrier phase, and symbol timing [1]. The performance of an arbitrary DA estimator may depend on the training sequence being used. Hence, a design challenge in dealing with DA estimators is choosing the best training sequence for a given length. Another question that arises in the design of the preamble is the performance gain that is achieved in comparison with a preamble-less design in which non-data-aided (NDA) estimators are employed. The latter technique does not have a priori knowledge of the transmitted symbols and is also known as blind estimation.