# The Precompression Processing of LMS Algorithm in Noise Elimination

Authors: Pengfei Lin; Chunsheng Lin; Ning Zhang; Xingya wu
Journal: International Journal of Antennas and Propagation
 (2019)

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## Abstract

In this study, the authors propose a novel precompression processing (PCP) of the least mean squares (LMS) algorithm based on a regulator factor. The novelty of the PCP algorithm is that the compressed input signals vary from each other on different components at each iteration. The input signal of the improved LMS algorithm is precompressed based on the regulator factor. The precompressed input signal is not only related to the regulator factorα and the current value of the input signal at each iteration but also related to the amplitude of the input signal before this iteration. The improved algorithm can eliminate the influence of input signal mutation on the filter performance. In the numerical simulations, we compare the improved LMS algorithm and NLMS algorithm in the cases of normal input signal and input signal with mutation and the influence of different regulator factors on the noise elimination. Results show that the PCP algorithm has good noise elimination effect when the input signal changes abruptly and the regulator factor α = 0.01 can meet the requirements.


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## Body

## 1. Introduction

Adaptive filters are widely used in system identification, adaptive line spectrum enhancement, echo cancellation, and other fields [1–3]. In signal processing, researchers have been trying to pursue adaptive algorithms with fast convergence speed, good stability, and lower computational complexity. Because of its robustness, low computational complexity, and good convergence to steady-state signals, the least mean squares (LMS) algorithm has widely been used in adaptive filters.Since the least mean squares (LMS) algorithm was proposed by Widrow and Hoff in 1960, there have been too many researches on variable step size and variable order of the LMS algorithm. In 1997, Gan proposed a new approach in adjusting the step size of the least mean squares (LMS) using the fuzzy logic technique, and the earlier work was extended by giving a complete design methodology and guidelines for developing a reliable and robust fuzzy step size LMS (FSS-LMS) algorithm [4]. In 1999, So proved that bias removal can be achieved by proper scaling of the optimal filter coefficients, and a modified least mean squares (LMS) algorithm is then developed for accurate system identification in noise [5]. And Keratiotis and Lind proposed an optimum time-varying step-size sequence for adaptive filters employing the least mean squares algorithm [6]. In 2013, Kang et al. proposed a new bias-compensated normalized least mean squares (NLMS) algorithm for parameter estimation with a noisy input. The algorithm is obtained from an approximated cost function based on the statistical properties of the input noise, and a condition checking constraint is involved to decide whether the weight coefficient vector must be updated [7]. In 2016, Huang et al. proposed a novel component-wise variable step-size (CVSS) diffusion distributed algorithm for estimating a specific parameter over sensor networks [8]. In 2018, Wu et al. proposed a multistage least mean squares (MLMS) algorithm based on polynomial fitting in time-varying systems [9].The LMS algorithm has greatly been improved according to different applications. In noise elimination, the input signal sequence may mutate, the conventional LMS algorithm will be greatly affected in this case, and the impact of mutation signal on the filter cannot be eliminated, thus affecting the filtering effect. And the NLMS algorithm has good adaptability to local signal fluctuation, but it cannot judge the big mutation of an individual signal very well. The precompression processing (PCP) of the least mean squares (LMS) algorithm based on a regulator factor can reduce the effect of signal mutation on the noise filtering well. The simulation results show that regardless of the degree of signal mutation, the improved LMS algorithm can eliminate the impact of mutation, while other algorithms will be seriously affected by the mutation signal. And the improved LMS algorithm has good noise filtering effect compared with other algorithms. The advantages of the improved LMS algorithm eliminating the mutational signal are obvious.


## 2. The LMS Algorithm

In 1960, Widrow and Hoff proposed the least mean squares (LMS) algorithm, which uses instantaneous values to estimate gradient vectors [11, 11]. The obvious advantage of the LMS is that the algorithm is simple, and it does not need to calculate the correlation matrix and inverse matrix. At the same time, it easily achieves stability and robustness because of which it is widely used [12, 13].


### 2.1. The Main Rationale of the LMS Algorithm

The structure of the LMS adaptive filter is shown in Figure1 [14].Figure 1
The structure of the LMS adaptive filter.In Figure1, xn=xnxn−1…xn−mT is the input vector; wn=w1nw2n…wMnT is the weight vector; and en, dn, and yn denote the error signal, desired signal, and output of the adaptive filter, respectively.(1)yn=xn⋅w1n+xn−1⋅w2n+⋯+xn−M+1⋅wMn=∑i=1Mxn−i+1⋅win=wTn⋅xn,where the notation “T” denotes the transposition of matrix;(2)en=dn−yn.Estimating the gradient vector with instantaneous values, the following can be obtained:(3)∇^n=∂e2n∂wn=−2enxn.According to the relationship between the gradient vector and the weight vectors of the adaptive filter, the following can be obtained:(4)w^n+1=w^n+12⋅μ−∇^n=w^n+μ⋅en⋅xn,where μ denotes the step size.Substituting formulas (1) and (2) into formula (5), the weight vectors can be rewritten as(5)w^n+1=w^n+μ⋅xn⋅dn−w^Hn⋅xn=I−μ⋅xn⋅xHn⋅w^n+μ⋅xn⋅dn.It can be seen from formula (5) that the adaptive LMS algorithm is a model with feedback form. The signal flow diagram is shown in Figure 2.Figure 2
The signal flow diagram of the LMS algorithm.


### 2.2. The Normalized LMS (NLMS) Algorithm

The normalized LMS (NLMS) algorithm is that the product vectoren⋅xn is normalized with respect to the square Euclidean norm of the input vector, and NLMS can be regarded as LMS with time-varying step parameters; the weight vectors of the adaptive filter can be obtained:(6)w^n+1=w^n+μ˜α+xn2⋅xn⋅en,where α is a preset correction, and 0<μ˜<2 guarantees the convergence of the NLMS algorithm.The NLMS algorithm has good stability and is widely used in engineering applications.


### 2.3. The Improved LMS Algorithm

#### 2.3.1. The Main Rationale of the Improved LMS Algorithm

In noise elimination, the input signal sequence may mutate, so the NLMS algorithm will be greatly affected in this case, thus affecting the filtering effect. In order to eliminate the impact of the mutation signal on the filter, this study proposes the precompression processing (PCP) of the least mean squares (LMS) algorithm based on the regulator factor. The improved algorithm reduces the effect of signal mutation on noise filtering and has good noise filtering effect even when the input signal changes abruptly.Set(7)R0=∑l=1Mxl⋅xl,where xl denotes the input signals.(8)Rn=α⋅x2n+1−α⋅Rn−1,where α is the regulator factor.Set(9)An=2⋅Rn.Then, the input signal is converted to formula (9):(10)xn⟶x′n=xnAn.So, the output of the adaptive filter is as follows:(11)yn=wTn⋅x′n=wTn⋅xn2⋅α⋅x2n+1−α⋅Rn−1.The error signal is obtained as follows:(12)en=dn−yn=dn−x′nT⋅w^n=dn−xn2⋅α⋅x2n+1−α⋅Rn−1T⋅wn.And the weight vector of the adaptive filter is(13)w^n+1=w^n+μ⋅en⋅x′n=w^n+μ⋅en⋅xn2⋅α⋅x2n+1−α⋅Rn−1.It can be seen from formulas (7) and (8) that An is not only related to the regulator factor α and the amplitude of the nth value of the input sequence, that is, the amplitude of xn but also related to Rn−1, that is, related to the amplitude of the input sequence before xn. By controlling the value of α, we can adjust the influence of the current input signal xn on the value of An. For example, when the input signal xn increases abruptly, the value of An increases a little. xn adjusts the value of An slightly. Through formula (9), we can see that the input signal is converted to xn/An, and the value of An is much bigger than the amplitude of xn, so the influence of the mutational signal xn will be controlled and the improved LMS algorithm can get good adaptability to the input signal.


#### 2.3.2. Steps of Algorithm

The steps of calculating the improved LMS algorithm are as follows:(1)
The input signalxn of the improved LMS algorithm is precompressed based on the regulator factor α. It is x′n=xn/An.(2)
Whenn = 0, the starting value of the filter weight vector is w0, arbitrarily. And the value of the step size μ is set.(3)
The error signalen is obtained through the filter weight vector wn, the input signal x′n, and the desired signal dn:(14)en=dn−x′nH⋅w^n.(4)
The updated value of the filter weight vector is calculated using the recursive method:(15)w^n+1=w^n+μ⋅en⋅x′n.(5)
Whenn increases by 1, the updated value of the filter weight vector wn+1 is substituted into Step 3, and then the following steps are carried out successively until the objective function of the adaptive filter is minimized, that is, until the steady state is achieved.From the steps of the improved LMS algorithm and formulas (7)–(10), six computations are added in each cycle compared with the traditional LMS algorithm, that is, four multiplications, one addition, and one extraction of a root; and only one computation is added compared with the NLMS algorithm.


## 2.1. The Main Rationale of the LMS Algorithm

The structure of the LMS adaptive filter is shown in Figure1 [14].Figure 1
The structure of the LMS adaptive filter.In Figure1, xn=xnxn−1…xn−mT is the input vector; wn=w1nw2n…wMnT is the weight vector; and en, dn, and yn denote the error signal, desired signal, and output of the adaptive filter, respectively.(1)yn=xn⋅w1n+xn−1⋅w2n+⋯+xn−M+1⋅wMn=∑i=1Mxn−i+1⋅win=wTn⋅xn,where the notation “T” denotes the transposition of matrix;(2)en=dn−yn.Estimating the gradient vector with instantaneous values, the following can be obtained:(3)∇^n=∂e2n∂wn=−2enxn.According to the relationship between the gradient vector and the weight vectors of the adaptive filter, the following can be obtained:(4)w^n+1=w^n+12⋅μ−∇^n=w^n+μ⋅en⋅xn,where μ denotes the step size.Substituting formulas (1) and (2) into formula (5), the weight vectors can be rewritten as(5)w^n+1=w^n+μ⋅xn⋅dn−w^Hn⋅xn=I−μ⋅xn⋅xHn⋅w^n+μ⋅xn⋅dn.It can be seen from formula (5) that the adaptive LMS algorithm is a model with feedback form. The signal flow diagram is shown in Figure 2.Figure 2
The signal flow diagram of the LMS algorithm.


## 2.2. The Normalized LMS (NLMS) Algorithm

The normalized LMS (NLMS) algorithm is that the product vectoren⋅xn is normalized with respect to the square Euclidean norm of the input vector, and NLMS can be regarded as LMS with time-varying step parameters; the weight vectors of the adaptive filter can be obtained:(6)w^n+1=w^n+μ˜α+xn2⋅xn⋅en,where α is a preset correction, and 0<μ˜<2 guarantees the convergence of the NLMS algorithm.The NLMS algorithm has good stability and is widely used in engineering applications.


## 2.3. The Improved LMS Algorithm

### 2.3.1. The Main Rationale of the Improved LMS Algorithm

In noise elimination, the input signal sequence may mutate, so the NLMS algorithm will be greatly affected in this case, thus affecting the filtering effect. In order to eliminate the impact of the mutation signal on the filter, this study proposes the precompression processing (PCP) of the least mean squares (LMS) algorithm based on the regulator factor. The improved algorithm reduces the effect of signal mutation on noise filtering and has good noise filtering effect even when the input signal changes abruptly.Set(7)R0=∑l=1Mxl⋅xl,where xl denotes the input signals.(8)Rn=α⋅x2n+1−α⋅Rn−1,where α is the regulator factor.Set(9)An=2⋅Rn.Then, the input signal is converted to formula (9):(10)xn⟶x′n=xnAn.So, the output of the adaptive filter is as follows:(11)yn=wTn⋅x′n=wTn⋅xn2⋅α⋅x2n+1−α⋅Rn−1.The error signal is obtained as follows:(12)en=dn−yn=dn−x′nT⋅w^n=dn−xn2⋅α⋅x2n+1−α⋅Rn−1T⋅wn.And the weight vector of the adaptive filter is(13)w^n+1=w^n+μ⋅en⋅x′n=w^n+μ⋅en⋅xn2⋅α⋅x2n+1−α⋅Rn−1.It can be seen from formulas (7) and (8) that An is not only related to the regulator factor α and the amplitude of the nth value of the input sequence, that is, the amplitude of xn but also related to Rn−1, that is, related to the amplitude of the input sequence before xn. By controlling the value of α, we can adjust the influence of the current input signal xn on the value of An. For example, when the input signal xn increases abruptly, the value of An increases a little. xn adjusts the value of An slightly. Through formula (9), we can see that the input signal is converted to xn/An, and the value of An is much bigger than the amplitude of xn, so the influence of the mutational signal xn will be controlled and the improved LMS algorithm can get good adaptability to the input signal.


### 2.3.2. Steps of Algorithm

The steps of calculating the improved LMS algorithm are as follows:(1)
The input signalxn of the improved LMS algorithm is precompressed based on the regulator factor α. It is x′n=xn/An.(2)
Whenn = 0, the starting value of the filter weight vector is w0, arbitrarily. And the value of the step size μ is set.(3)
The error signalen is obtained through the filter weight vector wn, the input signal x′n, and the desired signal dn:(14)en=dn−x′nH⋅w^n.(4)
The updated value of the filter weight vector is calculated using the recursive method:(15)w^n+1=w^n+μ⋅en⋅x′n.(5)
Whenn increases by 1, the updated value of the filter weight vector wn+1 is substituted into Step 3, and then the following steps are carried out successively until the objective function of the adaptive filter is minimized, that is, until the steady state is achieved.From the steps of the improved LMS algorithm and formulas (7)–(10), six computations are added in each cycle compared with the traditional LMS algorithm, that is, four multiplications, one addition, and one extraction of a root; and only one computation is added compared with the NLMS algorithm.


## 2.3.1. The Main Rationale of the Improved LMS Algorithm

In noise elimination, the input signal sequence may mutate, so the NLMS algorithm will be greatly affected in this case, thus affecting the filtering effect. In order to eliminate the impact of the mutation signal on the filter, this study proposes the precompression processing (PCP) of the least mean squares (LMS) algorithm based on the regulator factor. The improved algorithm reduces the effect of signal mutation on noise filtering and has good noise filtering effect even when the input signal changes abruptly.Set(7)R0=∑l=1Mxl⋅xl,where xl denotes the input signals.(8)Rn=α⋅x2n+1−α⋅Rn−1,where α is the regulator factor.Set(9)An=2⋅Rn.Then, the input signal is converted to formula (9):(10)xn⟶x′n=xnAn.So, the output of the adaptive filter is as follows:(11)yn=wTn⋅x′n=wTn⋅xn2⋅α⋅x2n+1−α⋅Rn−1.The error signal is obtained as follows:(12)en=dn−yn=dn−x′nT⋅w^n=dn−xn2⋅α⋅x2n+1−α⋅Rn−1T⋅wn.And the weight vector of the adaptive filter is(13)w^n+1=w^n+μ⋅en⋅x′n=w^n+μ⋅en⋅xn2⋅α⋅x2n+1−α⋅Rn−1.It can be seen from formulas (7) and (8) that An is not only related to the regulator factor α and the amplitude of the nth value of the input sequence, that is, the amplitude of xn but also related to Rn−1, that is, related to the amplitude of the input sequence before xn. By controlling the value of α, we can adjust the influence of the current input signal xn on the value of An. For example, when the input signal xn increases abruptly, the value of An increases a little. xn adjusts the value of An slightly. Through formula (9), we can see that the input signal is converted to xn/An, and the value of An is much bigger than the amplitude of xn, so the influence of the mutational signal xn will be controlled and the improved LMS algorithm can get good adaptability to the input signal.


## 2.3.2. Steps of Algorithm

The steps of calculating the improved LMS algorithm are as follows:(1)
The input signalxn of the improved LMS algorithm is precompressed based on the regulator factor α. It is x′n=xn/An.(2)
Whenn = 0, the starting value of the filter weight vector is w0, arbitrarily. And the value of the step size μ is set.(3)
The error signalen is obtained through the filter weight vector wn, the input signal x′n, and the desired signal dn:(14)en=dn−x′nH⋅w^n.(4)
The updated value of the filter weight vector is calculated using the recursive method:(15)w^n+1=w^n+μ⋅en⋅x′n.(5)
Whenn increases by 1, the updated value of the filter weight vector wn+1 is substituted into Step 3, and then the following steps are carried out successively until the objective function of the adaptive filter is minimized, that is, until the steady state is achieved.From the steps of the improved LMS algorithm and formulas (7)–(10), six computations are added in each cycle compared with the traditional LMS algorithm, that is, four multiplications, one addition, and one extraction of a root; and only one computation is added compared with the NLMS algorithm.


## 3. Simulation Results

Assume that the desired signaldn is composed of multiple sinusoidal signals and Gaussian white noise with zero mean and unit variance, and the input signal xn is a Gaussian white noise signal with zero mean and unit variance, and the output error signal en is the signal that the noise is eliminated. In the following simulation, the paper simulates the effect of noise removal of the normalized least mean squares (NLMS) algorithm and the improved LMS algorithm.


### 3.1. Filtering Simulation of the Input Signal without Mutation

When no signal mutation occurs in the input signal, the regulator factorα = 0.01.It can be seen from Figure3 that NLMS and the improved LMS algorithms have little difference in the effect of noise removal. When no signal mutation occurs in the input signal, the two algorithms can all extract the primitive signal from the mixed signal, and it is concluded that the improved LMS algorithm is smoother than the NLMS algorithm from Figure 4. The filter performance of the improved LMS algorithm is better than that of the NLMS algorithm.Figure 3
The error signal of the NLMS and the improved LMS algorithms. (a) Signal. (b)en of the NLMS algorithm. (c) en of the improved LMS algorithm.
(a)
(b)
(c)Figure 4
Local comparison of the NLMS algorithm and the improved LMS algorithm.It can be seen from Figures5 and 6 that the residual noise of the improved LMS algorithm is slightly smaller than that of the NLMS algorithm, and the residual noise of the NLMS algorithm is slightly smaller than that of the conventional LMS algorithm, but the advantages are not obvious.Figure 5
The residual noise of the NLMS algorithm. (a) Signal anden. (b) Residual noise.
(a)
(b)Figure 6
The residual noise of the improved LMS algorithm. (a) Signal anden. (b) Residual noise.
(a)
(b)In conclusion, when no signal mutation occurs in the input signal, the improved LMS algorithm and the NLMS algorithm can eliminate the noise and obtain the primitive signal. The filtering effect of the improved LMS algorithm is slightly better than that of the NLMS and conventional LMS algorithms, but its advantages are not obvious.


### 3.2. Filtering Simulation of the Input Signal with Mutation

When mutation occurs in the input signal, assume the input signalxn is as shown in formula (13).(16)x10000:10100=2⋅x10000:10100,x30000:30020=4⋅x30000:30020,x50000:50040=8⋅x50000:50040.It can be seen from Figures7 and 8 that the improved LMS algorithm has obvious advantages when the input signal changes abruptly. The improved LMS algorithm can recognize the mutation signal and eliminate noise well. However, the NLMS algorithm cannot recognize the mutation signal, so the error signal of the NLMS algorithm is changed abruptly at the 50000th point of the input signal, and the error signal of the NLMS algorithm at this point changes up to 5. Although the NLMS algorithm has good adaptability to local signal fluctuation, it cannot judge the big mutation of the individual signal very well.Figure 7
The error signal of the NLMS and the improved LMS algorithms with signal mutation. (a) Signal. (b)en of the NLMS algorithm. (c) en of the improved LMS algorithm.
(a)
(b)
(c)Figure 8
Local comparison of the NLMS algorithm and improved LMS algorithm.It can be seen from Figures9–12 that the residual noise of the improved LMS algorithm is less than that of the NLMS algorithm when mutation occurs in the input signal, and the weight vectors wn of the improved LMS algorithm are more stable than those of the NLMS algorithm. The advantages are obvious at the points of signal mutation.Figure 9
The residual noise of the NLMS algorithm with signal mutation. (a) Signal anden. (b) Residual noise.
(a)
(b)Figure 10
The residual noise of the improved LMS algorithm with signal mutation. (a) Signal anden. (b) Residual noise.
(a)
(b)Figure 11
The weight vectorswn of the NLMS algorithm with signal mutation.Figure 12
The weight vectorswn of the improved LMS algorithm with signal mutation.The simulation results show that regardless of the degree of signal mutation, the improved LMS algorithm can eliminate the impact of mutation, while other algorithms will be seriously affected by the mutation signal, so we get the conclusion that the precompression processing of the input signal can effectively eliminate the impact of signal mutation.


### 3.3. Effect of Regulator Factor on the Filtering Effect

(1)
Whenα=0.001,0.005,and0.01, respectively, the simulation of the filtering effect is as shown in Figures 13 and 14.Figure 13
The residual noise of the improved LMS algorithm with signal mutation when (a)α = 0.001, (b) α = 0.005, and (c) α = 0.01 residual noise.
(a)
(b)
(c)Figure 14
The error signalen of the improved LMS algorithm with signal mutation when (a) signal, (b) α = 0.001, (c) α = 0.005, and (c) α = 0.01.
(a)
(b)
(c)
(d)It can be seen from Figures13 and 14 that when α = 0.001, 0.005, and 0.01, respectively, the original signal can be obtained well and the differences are tiny.(2)
Whenα = 0.01, 0.05, and 0.5, respectively, the simulation of the filtering effect is as shown in Figures 15 and 16.Figure 15
The residual noise of the improved LMS algorithm with signal mutation when (a)α = 0.01, (b) α = 0.05, and (c) α = 0.5 residual noise.
(a)
(b)
(c)Figure 16
The error signalen of the improved LMS algorithm with (a) signal mutation when (b) α = 0.01, (c) α = 0.05, and (d) α = 0.5.It can be seen from Figures15 and 16, when α = 0.5, the filtering effect of the filter becomes worse, and the curve of the error signal en is less smooth than that of the error signal en when a = 0.1 and 0.01. Estimating the range of mutation signal and setting the value of α reasonably can make the filter to achieve better filtering performance. In general, a = 0.01 is enough.


## 3.1. Filtering Simulation of the Input Signal without Mutation

When no signal mutation occurs in the input signal, the regulator factorα = 0.01.It can be seen from Figure3 that NLMS and the improved LMS algorithms have little difference in the effect of noise removal. When no signal mutation occurs in the input signal, the two algorithms can all extract the primitive signal from the mixed signal, and it is concluded that the improved LMS algorithm is smoother than the NLMS algorithm from Figure 4. The filter performance of the improved LMS algorithm is better than that of the NLMS algorithm.Figure 3
The error signal of the NLMS and the improved LMS algorithms. (a) Signal. (b)en of the NLMS algorithm. (c) en of the improved LMS algorithm.
(a)
(b)
(c)Figure 4
Local comparison of the NLMS algorithm and the improved LMS algorithm.It can be seen from Figures5 and 6 that the residual noise of the improved LMS algorithm is slightly smaller than that of the NLMS algorithm, and the residual noise of the NLMS algorithm is slightly smaller than that of the conventional LMS algorithm, but the advantages are not obvious.Figure 5
The residual noise of the NLMS algorithm. (a) Signal anden. (b) Residual noise.
(a)
(b)Figure 6
The residual noise of the improved LMS algorithm. (a) Signal anden. (b) Residual noise.
(a)
(b)In conclusion, when no signal mutation occurs in the input signal, the improved LMS algorithm and the NLMS algorithm can eliminate the noise and obtain the primitive signal. The filtering effect of the improved LMS algorithm is slightly better than that of the NLMS and conventional LMS algorithms, but its advantages are not obvious.


## 3.2. Filtering Simulation of the Input Signal with Mutation

When mutation occurs in the input signal, assume the input signalxn is as shown in formula (13).(16)x10000:10100=2⋅x10000:10100,x30000:30020=4⋅x30000:30020,x50000:50040=8⋅x50000:50040.It can be seen from Figures7 and 8 that the improved LMS algorithm has obvious advantages when the input signal changes abruptly. The improved LMS algorithm can recognize the mutation signal and eliminate noise well. However, the NLMS algorithm cannot recognize the mutation signal, so the error signal of the NLMS algorithm is changed abruptly at the 50000th point of the input signal, and the error signal of the NLMS algorithm at this point changes up to 5. Although the NLMS algorithm has good adaptability to local signal fluctuation, it cannot judge the big mutation of the individual signal very well.Figure 7
The error signal of the NLMS and the improved LMS algorithms with signal mutation. (a) Signal. (b)en of the NLMS algorithm. (c) en of the improved LMS algorithm.
(a)
(b)
(c)Figure 8
Local comparison of the NLMS algorithm and improved LMS algorithm.It can be seen from Figures9–12 that the residual noise of the improved LMS algorithm is less than that of the NLMS algorithm when mutation occurs in the input signal, and the weight vectors wn of the improved LMS algorithm are more stable than those of the NLMS algorithm. The advantages are obvious at the points of signal mutation.Figure 9
The residual noise of the NLMS algorithm with signal mutation. (a) Signal anden. (b) Residual noise.
(a)
(b)Figure 10
The residual noise of the improved LMS algorithm with signal mutation. (a) Signal anden. (b) Residual noise.
(a)
(b)Figure 11
The weight vectorswn of the NLMS algorithm with signal mutation.Figure 12
The weight vectorswn of the improved LMS algorithm with signal mutation.The simulation results show that regardless of the degree of signal mutation, the improved LMS algorithm can eliminate the impact of mutation, while other algorithms will be seriously affected by the mutation signal, so we get the conclusion that the precompression processing of the input signal can effectively eliminate the impact of signal mutation.


## 3.3. Effect of Regulator Factor on the Filtering Effect

(1)
Whenα=0.001,0.005,and0.01, respectively, the simulation of the filtering effect is as shown in Figures 13 and 14.Figure 13
The residual noise of the improved LMS algorithm with signal mutation when (a)α = 0.001, (b) α = 0.005, and (c) α = 0.01 residual noise.
(a)
(b)
(c)Figure 14
The error signalen of the improved LMS algorithm with signal mutation when (a) signal, (b) α = 0.001, (c) α = 0.005, and (c) α = 0.01.
(a)
(b)
(c)
(d)It can be seen from Figures13 and 14 that when α = 0.001, 0.005, and 0.01, respectively, the original signal can be obtained well and the differences are tiny.(2)
Whenα = 0.01, 0.05, and 0.5, respectively, the simulation of the filtering effect is as shown in Figures 15 and 16.Figure 15
The residual noise of the improved LMS algorithm with signal mutation when (a)α = 0.01, (b) α = 0.05, and (c) α = 0.5 residual noise.
(a)
(b)
(c)Figure 16
The error signalen of the improved LMS algorithm with (a) signal mutation when (b) α = 0.01, (c) α = 0.05, and (d) α = 0.5.It can be seen from Figures15 and 16, when α = 0.5, the filtering effect of the filter becomes worse, and the curve of the error signal en is less smooth than that of the error signal en when a = 0.1 and 0.01. Estimating the range of mutation signal and setting the value of α reasonably can make the filter to achieve better filtering performance. In general, a = 0.01 is enough.


## 4. Conclusion

The input signal of the improved LMS algorithm is precompressed based on the regulator factor in this study. The improved algorithm can eliminate the influence of input signal mutation on the filter performance. In the numerical simulations, we compare the improved LMS algorithm and the NLMS algorithm in the cases of the normal input signal and input signal with mutation and the influence of different regulator factors on the filtering effect. Results show that regardless of the degree of signal mutation, the improved LMS algorithm can eliminate the impact of mutation, while the NLMS algorithm will be seriously affected by the mutation signal, and the regulator factorα = 0.01 can meet the requirements.



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