High-order approximation algorithm using Gram-Schmidt QR transformations coupled with an iterative correction process for tracking frequency drift in OFDM systems

Rainfield Y Yen,Hong-Yu Liu,Yung-Chang Yin

doi:10.1186/1687-1499-2014-16

Abstract

Abstract To jointly track frequency drift (or fine frequency offset) and channel state variation for orthogonal frequency division multiplexing (OFDM) communications over mobile wireless channels, the maximum-likelihood (ML) estimation techniques are commonly adopted. A major difficulty arises from the highly nonlinear nature of the log-likelihood function which renders local extrema or multiple solutions. The use of mathematical approximation coupled with an adaptive iteration algorithm is a viable approach to ease problem solving. The approximation methods used in existing works are usually of the first-order level. In this work, we devise a high-order approximation algorithm to improve the tracking performance. As shall be seen, the problem amounts to finding the roots of an approximate high-degree polynomial equation derived from the log-likelihood function. To this end, we triangularize the companion matrix constructed from the high-degree polynomial using iterative Gram-Schmidt QR transformations, an eigenvalue problem in matrix computation, coupled with another iterative correction process to produce a frequency offset estimate to good accuracy. It is found that a high-order approximation algorithm formulated can achieve much better tracking performance in terms of estimation accuracy, tracking range, and error rate results than various other tracking algorithms appearing in the literature.

Highlights

To operate an orthogonal frequency division multiplexing (OFDM) system in a mobile wireless environment, initial acquisition, sometimes referred to as the coarse synchronization, is required for both time and frequency synchronization [1,2,3,4,5,6]
Either perfect channel estimation is assumed in deriving frequency offset algorithms or vice versa, perfect frequency synchronization is assumed in deriving channel estimation algorithms [10]
For the gradient method to converge to the desired global solution, it is critical to properly choose the starting point as well as the adaptation step size

Summary

Introduction

To operate an orthogonal frequency division multiplexing (OFDM) system in a mobile wireless environment, initial acquisition, sometimes referred to as the coarse synchronization, is required for both time and frequency synchronization [1,2,3,4,5,6]. To overcome the local extrema difficulty, a viable approach is to use mathematical approximation as done in the expectation-maximization (EM)-based algorithm of [12,24,25] and the joint maximum-likelihood channel and frequency estimation (JML-CFE) algorithm of [12]. The problem amounts to finding the roots of a highdegree real coefficient polynomial equation resulting from the high-order Taylor series truncation (the derivative of the log-likelihood function is expanded into a Taylor series) To this end, we triangularize the companion matrix constructed from the high-degree polynomial using iterative QR transformations by way of the Gram-Schmidt orthogonalization process (to be called the Gram-Schmidt QR transformations) [26,27,28]. The reason explains why we make efforts on improving the fine synchronization with wide tracking range, high estimation accuracy, and low computational complexity

Signal and system model

N hH FHv XH FN DHδ Dδ FHN XFvh

Simulation results and discussions

Conclusions