改进Sigmoid函数的变步长LMS算法：提升收敛性能

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本文主要探讨了传统固定步长最小均方（Least Mean Square, LMS）算法存在的不足，并提出了一个基于改进Sigmoid函数的变步长LMS自适应滤波算法。LMS算法是一种广泛应用于信号处理领域的优化技术，主要用于在线估计系统中的未知参数，以最小化预测误差。然而，固定步长在处理不同信号特性时可能存在收敛速度慢或不稳定的问题。作者通过对现有算法的深入分析，特别是对Sigmoid函数的特性进行重新审视，引入了翻译变换，旨在增强Sigmoid函数与误差信号之间的非线性关系。这种改进使得算法能够动态调整步长，以适应信号变化，从而提高算法的收敛性和整体性能。研究还涉及参数的选择及其对算法效果的影响，理论分析和仿真结果显示，与传统固定步长LMS算法相比，提出的变步长LMS算法具有更快的收敛速度和更稳定的性能。关键词包括：最小均方算法、自适应滤波算法、翻译变换、变步长策略。这项工作不仅提升了LMS算法的适应性，而且可能为信号处理领域的其他应用提供新的思路和技术支持。在实际应用中，特别是在通信、信号处理和神经网络等领域，变步长LMS算法由于其高效性和鲁棒性，有望展现出更大的优势。

Variable Step Size LMS Algorithm Based on

Modified Sigmoid Function

Yong Chen, JinpengTian, Yanping Liu

Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Shanghai University

Shanghai 200072, China

cyongmail@163.com

Abstract—By studyingthe shortage of the traditional fixed

step size least mean square (LMS) algorithm. This paper builds a

nonlinear function relationship between ȝ and the error signal by

reviewing the existing algorithm and presents a novel variable

step size LMS adaptive filtering algorithm by improving Sigmoid

function based on translation transformation. The selective of

parameters and the performance of convergence are discussed.

Theoretical analysis and simulation results show that the

proposed variable step size LMS algorithm has better

performance. Comparing with some existing algorithms, the

algorithm improves their convergence performance.

Keywords—Leastmean square algorithm; adaptive filtering

algorithm; translation transformation; variable step

I. INTRODUCTION

The least square algorithm is a gradient estimation

algorithm and first proposed by Widrow and Hoff. Because of

its simplicity and robustness, it is used widely in many fields

such as Adaptive Control, Radar, System Identification and

Signal Processing. But the convergence rate of the traditional

fixed step size is very slow. What is more, if we increase the

step size to improve the convergence rate, it is bound to

enlarge the steady state error of the algorithm. What is worse,

it may cause divergence of algorithm. So the convergence

speed and steady-state error is contradictory for fixed step

LMS algorithm.

In order to overcome the contradictory, many researchers

have proposed some variable step least mean square algorithms

which build a nonlinear function relationship between ȝ and

the error signal. In [1] the authors propose an algorithm based

on Sigmoid function which can overcome the contradictory.

But the step size of the method changes very large in the steady

state. It will affect the steady state error. Reference [2]

improves [1] based on its shortage. But when [2] reduces the

error, it is lacking in convergence speed and anti-interference.

Reference [3] makes the step size change slowly by adding

index, when the error close to zero. However, the index will

reduce the performance of the anti-interference. In [4], authors

put forward an algorithm of the tongue-like curve which

reduces the calculation. But the improvement of the

convergence speed is very little. The basic idea of these

algorithm is that: the step size should be as large as possible at

the initial stage of algorithm, which can make the algorithm

has a quick convergence speed. And the step size should be as

small as possible to decrease the steady state error when the

error is small at the stage of the convergence steady-

state.References [5-16] also make the improvement based on

the theory.

Through the above idea, this paper puts forward a new

variable step size adaptive algorithm. We build a nonlinear

relationship based on the translation transformation of the

Sigmoid function. The method can reduce the effect of the

step size at the stage of convergence steady-state. What is

more, it also has a better performance to anti-interference.

Theoretical analysis and simulation results show the

advantages of this algorithm.

II. I

MPROVED LMS ALGORITHM

The basic LMS algorithm˖



ሺ



ሻ

ൌ

ሺ



ሻ

െܺ

்

ሺሻɘ

ሺ

ሻ

(1)

ሺ

൅ͳ

ሻ

ൌɘ

ሺ



ሻ

൅ ʹρሺሻሺሻ (2)

Where ɘሺሻ is the vector of the adaptive filter

weights,d(n) is the desired signalˈX(n) is the adaptive filter

input signal vector, and ρ is step size and it is a constant. The

condition of the algorithm achieving convergence isͳ൏ߤ൏

ͳȀߣ

௠௔௫

, where ߣ

௠௔௫

is the maximal eigenvalue of the input

signal autocorrelation matrix.

Due to the problem of the traditional LMS algorithm, [1]

proposes a variable step size algorithm based on Sigmoid

called SVSLMS algorithm. The step size of it is variable.

Different error will get different step size.



ሺ



ሻ

ൌ

ଵ

ଵାୣ୶୮ሺି௫ሻ

(3)

The step-size ρ is the nonlinear function of the error e(n)˖

ሺ



ሻ

ൌȾቀ

ଵ

ଵାୣ୶୮

ሺ

ି஑

௘

ሺ

௡

ሻ

െͲǤͷቁ (4)

This algorithm can meet the requirement of the step size.

However, the step size has great changes at the stage of steady

state[2].

Equation (3) is the Sigmoid function and (4) is the function

of [1]. By researching this function in [1], the reason why the

step size changes largely at the stage of the convergence is that

the function varies quickly in thesymmetry point. When we

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