﻿ 基于MEEMD与FLP的光纤陀螺去噪
 光电工程  2020, Vol. 47 Issue (6): 190137      DOI: 10.12086/oee.2020.190137

1. 海军航空大学，山东 烟台 264000;
2. 海军92728部队，上海 200040;
3. 海军92214部队，浙江 宁波 315000

De-noising algorithm for FOG based on MEEMD and FLP algorithm
Dai Shaowu1, Chen Qiangqiang1,2, Liu Zhihao3, Dai Hongde1
1. Naval Aviation University, Yantai, Shandong 264000, China;
2. Naval 92728, Shanghai 200040, China;
3. Naval 92214, Ningbo, Zhejiang 315000, China
Abstract: In order to reduce the influence of noise on the output signal of FOG, a de-noising algorithm of FOG based on modified ensemble empirical mode decomposition (MEEMD) and forward linear prediction (FLP) is proposed. Firstly, the concept of permutation entropy is introduced, and the FOG signal is decomposed and reconstructed by using MEEMD. Secondly, the low-order IMF terms of the mixed noise after decomposition is filtered and de-noised by the FLP algorithm. Finally, the signal processed by the MEEMD-FLP is reconstructed to get the result. The static test of a FOG is carried out. The experimental results show that compared with the original FOG signal, the RMSE after de-noising is reduced by 76.77%, and the standard deviation is reduced by 76.76%. It can effectively reduce the influence of noise on the FOG output signal and has higher de-noising accuracy.
Keywords: fiber optic gyroscope    signal de-noising    empirical mode decomposition    FLP algorithm

1 引言

2 MEEMD-FLP去噪算法

2.1 信号分解

 $S(t) = \sum\limits_{i = 1}^n {{F_{{\rm{imf}} - i}}} (t) + {r_n}(t){\kern 1pt} {\kern 1pt} {\kern 1pt} ,$ (1)

1) 信号中零点数和极值点数相等或至多相差1个；

2) 极大值包络线和极小值包络线的均值相等且为0。

1) 模态混叠问题。即同一个IMF分量中出现了不同尺度和频率的信号，或同一尺度及频率的信号被分解到多个IMF分量中。多个模态混杂在若干个IMF分量中，影响了IMF分量的物理意义，不利于后续的去噪处理。

2) 噪声信号的确定。在分解产生的多个IMF分量中，如何选择需要滤除的IMF分量以及后续如何对所得到的多个IMF分量进行处理。

1) 针对原始信号S(t)，分别添加均值为0且互为相反数的白噪声信号序列ni(t)和-ni(t)，用表示噪声幅值，i表示添加的白噪声对数，即：

 $\left\{ {\begin{array}{*{20}{l}} {S_i^ + (t) = S(t) + {a_i}{n_i}(t)}\\ {S_i^ - (t) = S(t) - {a_i}{n_i}(t)} \end{array}} \right.。$ (2)

2) 对式(2)中的序列进行EMD分解，得到第一阶IMF分量为Ii1+(t)及Ii1-(t)。集成这一对IMF分量：

 ${I_1}(t) = \frac{1}{{2N}}\mathop \sum \limits_{i = 1}^{{N_e}} [I_{i1}^ + (t) + I_{i1}^ - (t)]。$ (3)

3) 引入排列熵(permutation entropy，PE)概念，对式(3)中所得的IMF分量进行PE判断。PE是一种检测时间序列随机性和动力学突变的方法，其概念简单，运行速度快，抗干扰能力强，适用于具有非线性的FOG输出数据[16]。如果PE值大于设定值，则认为序列为噪声序列，否则近似认为平稳序列。

4) 经过步骤3)的PE检测后，如果式(3)中所得的IMF分量是噪声序列，则继续执行步骤1)，直至该IMF分量平稳。

5) 将已分解得到的相对平稳的IMF分量从原始信号中分离出：

 $r(t) = S(t) - \sum\limits_{i = 1}^{p - 1} {{I_j}} (t)。$ (4)

6) 对式(4)中得到的剩余信号进行EMD分解，将所得到的所有IMF分量按照频率的高低进行排列。

2.2 信号去噪

FOG输出数据在时刻t的估计值为

 $\hat x(t) = \sum\limits_{p = 1}^K {{a_p}} x(t - p) = {\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{X}}(t - 1){\kern 1pt} {\kern 1pt} ,$ (5)

 $\mathit{\boldsymbol{X}}(t - 1) = {\{ x(t - 1), x(t - 2), \ldots , x(t - K)\} ^{\rm{T}}}。$ (6)

 $J(t) = E[{e^2}(t)],$ (7)

 $A(t - 1) = A(t) + \mu e(t)\mathit{\boldsymbol{X}}(t - 1)。$ (8)

2.3 算法实现

1) 在MEEMD分解过程中，通过设定PE阈值，将FOG输出数据分解为

 ${F_{{\rm{FOG}}}} = \sum\limits_{i = 1}^m {{F_{{\rm{imf}} - i}}} (t) + \sum\limits_{i = m + 1}^q {{F_{{\rm{imf}} - i}}} (t) + \sum\limits_{i = q + 1}^n {{F_{{\rm{imf}} - i}}} (t){\kern 1pt} {\kern 1pt} ,$ (9)

2) 针对MEEMD分解过程中得到的第m+1至第q个分量(混合IMF分量)进行分析，这些IMF分量中包含了噪声和输出信号，通过FLP滤波算法进行噪声滤除，从而实现对IMF分量中有效信号的提取。通过对混合IMF分量进行更进一步的FLP处理，可以有效提高对FOG输出数据的降噪精度。

3) 针对MEEMD分解过程中得到的第q+1至第n个分量(余项IMF分量)进行分析，余项IMF分量反映了FOG数据序列的趋势项，包含着FOG数据序列的信息。在实际处理过程中对余项IMF分量予以保留。

4) 经过对噪声IMF分量的滤除，以及步骤2)和步骤3)的处理，处理后的FOG输出数据可表示为

 $F_{{\rm{FOG}}}^\prime = \sum\limits_{l = m + 1}^q {{F_{{\rm{FLP}}}}} ({F_{{\rm{imf}} - l}}(t)) + \sum\limits_{l = q + 1}^n {{F_{{\rm{imf}} - l}}} (t),$ (10)

MEEMD-FLP算法框图如图 1所示。在分解过程中，通过设置排列熵阈值，对FOG输出信号中的较为强烈的噪声进行滤除，这一步骤提升了FOG输出信号噪声滤除过程中的效率，避免了对噪声信号的复杂处理；FLP滤波算法的引入，提升了FOG输出信号去噪的精确性；最后通过对MEEMD-FLP处理之后的信号进行重构，完成FOG输出数据的滤波。

 图 1 MEEMD-FLP算法流程图 Fig. 1 Flowchart of MEEMD-FLP
3 测量实验与结果

 图 2 光纤陀螺的原始含噪信号 Fig. 2 The orignanoiy signal of FOG

 图 3 EMD分解结果 Fig. 3 The results of EMD

 图 4 MEEMD分解结果 Fig. 4 The results of MEEMD

 图 5 序列的排列熵 Fig. 5 Permutation entropy of IMFs

 图 6 MEEMD-FLP处理结果 Fig. 6 Detailed processing by the MEEMD-FLP

 图 7 三种去噪方法的性能对比 Fig. 7 Performance comparison of three methods

 图 8 去噪后的均值及标准差 Fig. 8 Standard deviations and means of the FOG outputs after de-noising

 ${{E_{{\rm{RMS}}}} = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {{{({x_i} - \hat x)}^2}} } , }$ (11)
 ${{E_{{\rm{SSE}}}} = \sum\limits_{i = 1}^N {{{({x_i} - \hat x)}^2}} , }$ (12)

 指标 原数据 FLP EMD MEEMD-FLP RMSE 0.001421 0.00099 0.000738 0.00033 SSE 0.008078 0.003927 0.00218 0.000436 R 0.013235 0.0082 0.0058 0.0026

4 结论

1) 通过MEEMD分解算法，可以有效缓解模态混叠现象，提高分解能力。

2) 通过对IMF分量进行排列熵分析，可以有效判断信号的复杂程度，实现对噪声项、混合项及余项的判断。

3) MEEMD-FLP算法可以有效降低噪声对FOG输出信号的影响，具有更高的去噪精度。

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