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Approximate Expressions for Cramer-Rao Bounds of Code Aided QAM Dynamical Phase Estimation

Abstract : —In this paper, we study Bayesian and hybrid Cramer-Rao bounds (BCRB and HCRB) for the code-aided (CA) dynamical phase estimation of QAM modulated signals. In order to avoid the calculus of the inverse of the Bayesian information matrix and of the hybrid information matrix, we present some analytical expressions for the various CRBs, which greatly reduce the computation complexity. I. 0BINTRODUCTION lassicaly there are three ways of performing estimation in a telecommunication system: data aided (DA), code aided (CA) and non data aided (NDA) estimations X[1]X. Earlier attempts of signal synchronization in the low-SNR regime focused either on the DA or NDA synchronization mode X[2],[3]X. On one hand, DA estimation techniques achieve the better performance but may lead to unacceptable losses in power and spectral efficiency. On the other hand, NDA estimation algorithms drop some information about the transmitted data and may lead to poor results at the benefit of transmission efficiency. However, with the developments of channel coding techniques X[4]XX,[5] X, more and more attention has focused on CA synchronization [6]X X,[7]X which uses the decoding gain to improve the estimation performance. A natural question which arises when designing estimators is the ultimate accuracy that one can achieve in the estimation operation. The lower bounds answer this question by providing a minimum mean square error (MMSE). Although there exists many lower bounds, the Cramer-Rao bound (CRB) family is the most commonly used and the easiest to determine [8]-[11]X. There are several works concerning the CA CRBs for carrier phase and frequency estimation. In [11]X , the CRB for the CA carrier phase estimation has been expressed in terms of the marginal a posteriori probabilities (APPs) of the coded symbols, allowing the numerical evaluation of the CA bounds. This method has been applied to the evaluation of the CRB for the Turbo code aided [12]X and the convolution code aided [13]X scenarios; the CA CRBs in X[11] X-X[13]X were derived from the first derivative of the joint probability function between the observations and parameters. All these papers refer to an idealized situation in which the phase offset is constant. However, in modern communications, it is common to take into account a time-varying phase noise mostly due to oscillator 1 This work was partially funded by the ANR LURGA program. J. Yang is with SATIE, ENS Cachan) A. Wei is with LATTIS, Université Toulouse II (e-mail:
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Jianxiao Yang, Benoit Geller, Anne Wei. Approximate Expressions for Cramer-Rao Bounds of Code Aided QAM Dynamical Phase Estimation. ICC, Jun 2009, Dresde, Germany. ⟨10.1109/ICC.2009.5198756⟩. ⟨hal-01224238⟩



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