﻿ 基于辅助数据的OFDM时间同步算法
 计算机系统应用  2020, Vol. 29 Issue (2): 140-144 PDF

1. 广东电网有限责任公司 电力调度控制中心, 广州 510600;
2. 广东省电信规划设计院有限公司, 广州 510630;
3. 华南理工大学, 广州 510641

OFDM Time Synchronization Algorithm Based on Auxiliary Data
WANG Ying1, ZHANG Pei-Ming1, SHI Zhan1, WANG Jin2, LI Jia-Liang2, WANG Chao-Xiong3
1. Dispatching and Control Center, Guangdong Electric Power Co. Ltd., Guangzhou 510600, China;
2. Guangdong Planning and Designing Institute of Telecommunications Co. Ltd., Guangzhou 510630, China;
3. South China University of Technology, Guangzhou 510641, China
Abstract: In order to realize high-precision synchronization of OFDM (Orthogonal Frequency Division Multiplexing) under arbitrary carrier frequency offset, this study proposes a system timing offset estimation method based on auxiliary data. Firstly, under the Gaussian white noise channel, based on the auxiliary data without special structure, the optimal synchronization algorithm is derived under the maximum likelihood criterion. Then, under the consideration that the maximum likelihood method is too complex, a sub-optimal method, which has reduced computational complexity, is proposed. Finally, the performance of the proposed timing method is evaluated by Monte Carlo simulations under the frequency selective Rayleigh fading channel. The simulation results show that the timing performance of the proposed method is significantly better than the traditional algorithms.
Key words: OFDM     inter-carrier interference     timing     synchronization     estimation     maximum likelihood

1 信号模型

 $x(n) = \frac{1}{{\sqrt N }}\sum\limits_{k = 0}^{{N_{\rm{use}}} - 1} {{X_k}{e^{j2\pi kn/N}},{0} \le n \le N{\rm{ - }}1}$ (1)

 $\tilde x(n) = \left\{ {\begin{array}{*{20}{c}} {x(n + N), \; - G \le n \le - 1} \\ {x(n),\quad \quad 0 \le n \le N - 1} \end{array}} \right.$ (2)

 $r[n] = x[n - \tau ]{e^{j(2\pi \varepsilon n/N + \theta)}} + \omega [n]$ (3)

2 同步算法 2.1 ML同步

 $\bar r = \{ r[n]|n = 0,1, \cdots ,\tau , \cdots ,MN - 1\}$ (4)

 f(r(n)|\tau ,\varepsilon ,\theta)\! =\! \left\{ {\begin{aligned} & {\frac{1}{{\pi \sigma _\omega ^2}}\exp \left( - \frac{{{{\left| {r(n) \!- \!{s_{n - \tau }}{e^{j(2\pi \varepsilon n/N + \theta)}}} \right|}^2}}}{{\sigma _\omega ^2}}\right),n \!\in\! {{{I}}_p}} \\ & {\frac{1}{{\pi \sigma _1^2}}\exp \left( - \frac{{{{\left| {r(n)} \right|}^2}}}{{\sigma _1^2}}\right),n \in {{{I}}_d}} \end{aligned}} \right. (5)

 $f(\bar r|\tau ,\varepsilon ,\theta) = \prod\limits_{n \in {{\rm{I}}_p} \cup {{\rm{I}}_d}} {f(r(n)|\tau ,\varepsilon ,\theta)}$ (6)

 $\Lambda (\tau ,\varepsilon ,\theta) =\log\left( {f(\bar r|\tau ,\varepsilon ,\theta)} \right)$ (7)

 ${\Lambda ^{'}}(\tau ,\varepsilon ,\theta) = - {\rho _0}\sum\limits_{n \in {{\rm{I}}_d}} {{{\left| {r(n)} \right|}^2}} - \sum\limits_{n \in {{\rm{I}}_p}} {{{\left| {r(n) - {s_{n - \tau }}{e^{j(2\pi \varepsilon n/N + \theta)}}} \right|}^2}}$ (8)

 \begin{aligned} \left( {{\tau _o},{\varepsilon _o},{\theta _o}} \right) & = \mathop {{\rm{argmax}}}\limits_{(\tau ,\varepsilon ,\theta)} - {\rho _0}\sum\limits_{n \in {{\rm{I}}_d}} {{{\left| {r(n)} \right|}^2}} - \sum\limits_{n \in {{\rm{I}}_p}} {{{\left| {r(n) - {s_{n - \tau }}{e^{j(2\pi \varepsilon n/N + \theta)}}} \right|}^2}} \\ & = \mathop {{\rm{argmax}}}\limits_{(\tau ,\varepsilon ,\theta)} \left\{ {2{\rm{Re}}\left\{ {{e^ - }^{j(2\pi \varepsilon \tau /N + \theta)}\sum\limits_{i = 0}^{N - 1} {r(\tau + i)s_i^{\rm{*}}{e^ - }^{j2\pi \varepsilon i/N}} } \right\} - {\rho _1}\sum\limits_{i = 0}^{N - 1} {{{\left| {r(\tau + i)} \right|}^2}} - {\rm{c}}} \right\} \end{aligned} (9)

 $c = {\rho _0}\sum\limits_{i = 0}^{MN - 1} {{{\left| {r(i)} \right|}^2}} + \sum\limits_{i = 0}^{N - 1} {{{\left| {{s_i}} \right|}^2}}$ (10)

 $2\left| {\sum\limits_{i = 0}^{N - 1} {r(\tau + i)s_i^{\rm{*}}{e^ - }^{j2\pi \varepsilon i/N}} } \right| - {\rho _1}\sum\limits_{i = 0}^{N - 1} {{{\left| {r(\tau + i)} \right|}^2}} - c$ (11)

 $\theta = - \frac{{2\pi \varepsilon \tau }}{N} + \angle \left\{ {\sum\limits_{i = 0}^{N - 1} {r(\tau + i)s_i^{\rm{*}}{e^ - }^{j2\pi \varepsilon i/N}} } \right\}$ (12)

 $\left( {{\tau _o},{\varepsilon _o}} \right)\! =\! \mathop {{\rm{argmax}}}\limits_{(\tau ,\varepsilon)} \left\{\!\! {2\left| {\sum\limits_{i = 0}^{N - 1} {r(\tau \!+\! i)s_i^{*}{e^ - }^{j2\pi \varepsilon i/N}} } \right| \!- \!{\rho _1}\sum\limits_{i = 0}^{N - 1} {{{\left| {r(\tau \!+\! i)} \right|}^2}} } \!\!\right\}$ (13)

 $\left( {{\tau _o},{\varepsilon _o}} \right)\mathop { = {\rm{argmax}}}\limits_{(\tau ,\varepsilon)} \left\{ {\left| {\sum\limits_{i = 0}^{N - 1} {r(\tau + i)s_i^{\rm{*}}{e^ - }^{j2\pi \varepsilon i/N}} } \right|} \right\}$ (14)

2.2 简化的同步算法

 $\hat \tau \mathop { = {\rm{argmax}}}\limits_d \left\{ {M(d)} \right\}$ (15)

 $M(d) = {\rm{max}}\left\{ {\left| {\sum\limits_{i = 0}^{N - 1} {r(d + i)s_i^{\rm{*}}{e^{ - j\frac{{2\pi }}{N}ki}}} } \right|,0 \le k < N} \right\}$ (16)

2.3 算法复杂度分析

 $M(d) = \max \{ |({R_d}\circ {S^*})W_k^{\rm{H}}|,0 \le k < {\rm{N}}\}$ (17)

 ${R_d} = [r(d),r(d + 1),\cdots,r(d + N - 1)]$ (18)
 ${W_k} = [1,{e^{j\tfrac{{2\pi }}{N}k}},{e^{j\frac{{2\pi }}{N}2k}},\cdots,{e^{j\tfrac{{2\pi }}{N}(N - 1)k}}]$ (19)

3 仿真及结果

 图 1 在SUI信道下不同定时算法的MSE对比

 图 2 在一个5抽头多径信道下不同算法的MSE

4 结语

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