最小符号误差率的归一化自适应均衡器设计

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"该文研究了基于最小符号错误率的归一化自适应信道均衡器，通过约束优化方法改进了最小符号错误率均衡器的实现。" 在无线通信和数字信号处理领域，信道均衡是解决信道衰落和多径传播导致的信号失真问题的关键技术。传统的自适应信道均衡器主要采用最小均方误差（Mean Squared Error, MSE）准则进行优化，但这种准则并不一定能最小化实际的符号错误率。"基于最小符号错误率的归一化自适应信道均衡器"这篇论文聚焦于提高信道均衡的性能，特别是针对符号错误率的降低。描述中的"Minimum-Symbol-Error-Rate (MSER) equalizers"是指旨在最小化传输符号错误率的均衡器，这种均衡器的目标函数复杂，使得其数学推导并不直接。论文提出了一种新的方法来求解这个问题，将问题重新表述为在保持符号错误率最小化的约束下，寻找使两个连续参数向量之间范数最小化的解决方案。 "Constrained optimization problem" 是指带约束条件的优化问题，文中使用拉格朗日乘数法（Lagrangian multiplier method）解决了这一问题。这种方法允许在满足特定约束（如符号错误率最小化）的同时，寻找优化目标（范数最小化）的解。通过这种方式，他们设计了一种带有正常化的自适应算法，以改善均衡器的收敛速度和稳态性能。 "Adaptive channel equalizer" 是一种能够根据接收信号的变化动态调整其参数的均衡器，它可以根据信道条件的变化实时更新其滤波器系数，以提供最佳的信号恢复。 "Normalized adaptive algorithm" 指的是经过归一化的自适应算法，通过归一化可以确保算法的稳定性，并控制收敛速度，防止滤波器系数的过大波动。模拟结果证明，所提出的算法在收敛速度和稳态性能上优于现有的自适应最小符号错误率均衡器。这表明，采用这种新方法的信道均衡器在实际应用中能更好地减少误码率，提高通信系统的整体性能。关键词包括"Adaptive channel equalizer"（自适应信道均衡器）、"Minimum Symbol-Error-Rate"（最小符号错误率）和"constrained optimization"（约束优化），这些是理解本文核心内容的关键术语。

1374 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 4, APRIL 2013

Normalized Adaptive Channel Equalizer Based on

Minimal Symbol-Error-Rate

Meiyan Gong, Fangjiong Chen, Member, IEEE,HuaYu,Member, IEEE, Zhaohua Lu, and Liujun Hu

Abstract—Existing minimum-symbol-error-rate equalizers

were derived based on the symbol-error-rate objective function.

Due to the complexity of the objective function the derivation is

not straightforward. In this paper we present a new approach

to derive the minimum-symbol-error-rate adaptive equalizers.

The problem is formulated as minimizing the norm between two

subsequent parameter vectors under the constraint of symbol-

error-rate minimization. The constrained optimization problem

then is solved with the Lagrange multiplier method, which

results in an adaptive algorithm with normalization. Simulation

results show that the proposed algorithm outperforms the

existing adaptive minimum-symbol-error-rate equalizer in

convergence speed and steady-state performance.

Index Terms—Adaptive channel equalizer; Minimum Symbol-

Error-Rate; constrained optimization.

I. INTRODUCTION

ONVENTIONAL criteria for receive ﬁlter design, such

as the minimum-mean-squared-error (MMSE) criterion

and the least-square (LS) criterion, adopt the squared error

as the performance measure. These criteria intend to min-

imize the mean or cumulant of the squared error between

the ﬁlter output and the target signal. However, in practice

it is the system’s symbol error rate (SER), not the mean

squared error (MSE), that really matters. It has been illustrated

by various simulations that minimizing the MSE does not

necessarily produce the minimum SER (MSER) performance

[1]. Moreover, the theoretical analysis on MMSE receiver

also indicates that, when channel coding is not applied, the

achievable SER of MMSE receiver is not optimal [2]

.The

MSER criterion, which is directly based on minimizing the

SER, has attracted increasing attention in the past two decades.

The MSER criterion can be traced back to the error rate

analysis in multipath channel [4]. Recently the MSER criterion

has shown promising performance in various research ﬁelds,

Manuscript recei ved September 15, 2012; revised January 16 and January

30, 2013. The associate editor coordinating the review of this letter and

approving it for publication was G. Colavolpe.

M. Gong, F. Chen, and H. Yu (corresponding author) are with the

School of Electronic and Information Engineering, South China University

of Technology, Guangzhou, China (e-mail: myan.gong@mail.scut.edu.cn,

{eefjchen, yuhua}@scut.edu.cn). This work was supported in part

by the National Natural Science Foundation of China (No.61171083,

No.61071212, No.61271209), the Key Grant Project of Chinese Min-

istry of Education (No.313021) and Guangdong Provincial research project

(No.S2011010001241, 2012B091100138).

Z. Lu and L. Hu are with the Wireless Pre-Research Department, ZTE Coor-

poration, Shenzhen, China (e-mail: {lu.zhaohua, hu.liujun}@zte.com.cn).

Their work is supported by the Chinese Next-generation Broadband Wireless

Mobile Communication Network Major Science and Technology Issues

(No.2010ZX03003-002-02).

Digital Object Identiﬁer 10.1109/TCOMM.2013.13.120698

including chann el equalization [5]–[9], multiuser detection

[10]–[12], beamforming [13], [14], power control/allocation

[15], [16], carrier phase recovery [17], timing recovery [18]

and precoding [19]. A thorough review on the application of

the MSER criterion can be found in [1].

In this paper we focus on MSER-based channel equal-

ization. In existing research, the MSER equalizer is derived

based on the bit error rate (BER) or SER objective function,

which are for BPSK sources and QAM sources, respectively.

The BER/SER objective functions are non-convex and conse-

quently it is difﬁcult to obtain a closed-form solution. I terative

approaches usually are applied. In [5], [7]–[9] the gradient-

descent method is applied to ﬁnd a local minimum of the

BER/SER objective functions. By setting the gradient to zero,

ref. [6] obtains a nonlinear function for the MSER equalizer

and solves the nonlinear function by an iterative approach.

The iterative approach and the gradient-descent method are

not practical sample-by-sample adaptive equalizers since they

need a batch of samples to build the objective f unctions. To

derive a sample-by-sample adaptive algorithm, approximations

have been applied to simplify the objective functions [5]–[8].

Due to the complicated structure of the BER/SER objective

functions, the derivation o f the adaptive MBER/MSER (AM-

BER/AMSER) equalizers is not straightforward. Moreover, the

resulting adaptive equalizers in existing work usually have

complicated structures. For in stance, the adaptive equalizer

in [6] needs the channel parameters to calculate the symbol

detection indicator (see (31) in [6]). However, the channel

parameters may not be available to the equalizer in practice.

The adaptive equalizer in [8] needs the estimate of the noise

variance to calculate the instantaneous stochastic gradient (see

(39) in [8]).

In this paper we present a n ew approach to derive the

AMBER/AMSER equalizers. Consider a sample-by-sample

adaptive equalizer, where the equalizer parameters are updated

when a new sample is received. The problem is formulated

as minimizing the norm between two subsequent parameter

vectors under the constraint of minimal BER/SER. Compared

with the existing work that derives the equalizers based on the

BER/SER objective functions, our proposed problem model

has the following properties: (1) The MSER criterion is

reformulated as the constraint of correct symbol detection. It

does not need to calculate the BER/SER expressions and h ence

has a simpler derivation. (2) For different source modulation

The recent work of Rusek et al [3] indicates that in the presence of a

good coding and decoding scheme, the MMSE receiver is almost optimal. We

remark that the theoretical analysis of [2], [3] both assume that the channel

input is zero-mean Gaussian process. The conclusion may not be applicable

to other types of channel input.

0090-6778/13$31.00

 2013 IEEE

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