# Modified Combined-Step-Size Affine Projection Sign Algorithms for Robust Adaptive Filtering in Impulsive Interference Environments

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Affine Projection Sign Algorithm Based on Step-size Adjustment

#### 2.1. APSA

#### 2.2. VSS-APSA and CAPSA

## 3. Proposed Algorithms

#### 3.1. CSS-APSA

#### 3.2. Analysis of Variable Mixing Factors for CSS-APSA

- When the C value is small, such as $C=2$, we can get ${s}_{-}=-ln\left(\frac{2+1}{2-1}\right)=-1.0986$ and ${s}_{+}=1.0986$. When $\left|s\right(n\left)\right|=0.3365$, $\lambda \left(n\right)$ is approximately equal to 1 or 0. However, such two $\lambda \left(n\right)$ values will be slowly increased to 1 or decreased to 0 as $\left|s\right(n\left)\right|$ becomes larger. Therefore, the effective range for $s\left(n\right)$ can be treated as $[-0.3365,0.3365]$.
- From (23), as the C value becomes larger, $\left|s\right(n\left)\right|$ becomes closer to $0.3365$ because $ln\left(\frac{C+1}{C-1}\right)$ is a monotone decreasing function of C. Also, when the C value is relatively large, such as $C=6$, $\lambda \left(n\right)$ will be very close to 1 or 0. Therefore, for the large C value, we can constrain $s\left(n\right)\in [-0.3365,0.3365]$.

#### 3.3. Practical Considerations about Proposed Algorithms

#### 3.4. Computational Complexity

Algorithm 1: The proposed Algorithms |

Initialization:$0<{\mu}_{2}<{\mu}_{1}$, $0<{\xi}_{1}\le 1$, $s\left(0\right)={\xi}_{1}$, $1<C$, $1\le \beta $, and ${\mathbf{w}}_{1}\left(0\right)={\mathbf{w}}_{2}\left(0\right)=\mathbf{0}$. Computation:while ${\{\mathbf{u}\left(n\right),d\left(n\right)\}}_{n\ge 1}$ available do 1: ${\mathbf{w}}_{1}\left(n\right)={\mathbf{w}}_{1}\left(n\right)+{\mu}_{1}\frac{{\mathbf{U}}_{s}(n-1)}{\sqrt{{\xi}_{1}+{\parallel {\mathbf{U}}_{s}(n-1)\parallel}_{2}^{2}}},$ 2: ${\mathbf{w}}_{2}\left(n\right)={\mathbf{w}}_{2}\left(n\right)+{\mu}_{2}\frac{{\mathbf{U}}_{s}(n-1)}{\sqrt{{\xi}_{1}+{\parallel {\mathbf{U}}_{s}(n-1)\parallel}_{2}^{2}}},$ 3: $\mathbf{w}\left(n\right)=\lambda \left(n\right){\mathbf{w}}_{2}\left(n\right)+(1-\lambda \left(n\right)){\mathbf{w}}_{1}\left(n\right),$ 4: $e\left(n\right)=d\left(n\right)-{\mathbf{u}}^{T}\left(n\right)\mathbf{w}\left(n\right),$ 5: $s\left(n\right)=s(n-1)-{\mu}_{s}\frac{\partial \mid e\left(n\right)\mid}{\partial s(n-1)},$ for SCSS-APSA 6: ${\lambda}_{s}\left(n\right)=\frac{C}{2-s\left(n\right)}-\left(\frac{C}{2}-\frac{1}{2}\right),$ 7: if $s\left(n\right)<-ln\left(\frac{C+1}{C-1}\right)$: then $s\left(n\right)=-ln\left(\frac{C+1}{C-1}\right)$, 8: else if $s\left(n\right)>ln\left(\frac{C+1}{C-1}\right)$: then $s\left(n\right)=ln\left(\frac{C+1}{C-1}\right)$, 9: else $s\left(n\right)=s\left(n\right)$, end. for MCSS-APSA 6: ${\lambda}_{\beta}\left(n\right)=\frac{C}{1+exp(-\beta s(n\left)\right)}-\left(\frac{C}{2}-\frac{1}{2}\right),$ 7: if $s\left(n\right)<-ln\left(\frac{C+1}{C-1}\right)/\beta $: then $s\left(n\right)=-ln\left(\frac{C+1}{C-1}\right)/\beta $, 8: else if $s\left(n\right)>ln\left(\frac{C+1}{C-1}\right)/\beta $: then $s\left(n\right)=ln\left(\frac{C+1}{C-1}\right)/\beta $, 9: else $s\left(n\right)=s\left(n\right)$, end. for SMCSS-APSA 6: ${\lambda}_{s\beta}\left(n\right)=\frac{C}{2-\beta s\left(n\right)}-\left(\frac{C}{2}-\frac{1}{2}\right),$ 7: if $s\left(n\right)<\frac{-2}{(C-1)\beta}$: then $s\left(n\right)=\frac{-2}{(C-1)\beta}$, 8: else if $s\left(n\right)>\frac{2}{(C+1)\beta}$: then $s\left(n\right)=\frac{2}{(C+1)\beta}$, 9: else $s\left(n\right)=s\left(n\right)$, end. end while |

#### 3.5. Steady-State Analysis

## 4. Numerical Simulation Results

**Remark**

**1.**

- If the value of C is too large (such as $C=7$), the change rate of $\lambda \left(n\right)$ from 0 to 1 is increased in the process of iteration. In this case, $\lambda \left(n\right)$ tends to 1 too quickly, resulting in $\mathbf{w}\left(n\right)$ dominated by ${\mathbf{w}}_{2}\left(n\right)$, which is related to the lower steady-state error. Hence, the combined algorithm will still realize a slower convergence speed.

- CSS-APSA with $C=6$ realizes faster convergence than $C=5$ after an abrupt change;
- SCSS-APSA with $C=6$ outperforms CSS-APSA in terms of convergence rate in the whole filtering process.

- With $\beta =5$, MCSS-APSA realizes better filtering accuracy at a transient process than $\beta \in \{1,10,20\}$;
- With $\beta =1$, MCSS-APSA becomes to CSS-APSA, while MCSS-APSA with other $\beta $ values (such as $\beta =5$ or 10) outperforms CSS-APSA. This observation demonstrates the usefulness of modifications for CSS-APSA.
- Increasing the $\beta $ value from 1 to 20 cannot always improve the filtering performance of MCSS-APSA. Similarly to the effect of the C value, a $\beta $ value which is too large leads to $\mathbf{w}\left(n\right)$ dominated by ${\mathbf{w}}_{2}\left(n\right)$ in iterations, and slows down the convergence speed of MCSS-APSA.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The NMSD learning curves of CSS-APSA and SCSS-APSA with different C. (

**a**,

**c**): $C=5$; (

**b**,

**d**): $C=6$.

**Figure 7.**The NMSD learning curves of APSA, CAPSA, SCSS-APSA, and SMCSS-APSA with colored input data ${H}_{1}\left(z\right)$, (

**a**): ${\mu}_{1}=0.01$; (

**b**): ${\mu}_{2}=0.0001$; (

**c**): ${\mu}_{1}=0.01$, ${\mu}_{2}=0.0001$; (

**d**): $C=6$; (

**e**): $C=6$, $\beta =5$.

**Table 1.**Computational complexity of APSA, CAPSA, CSS-APSA, SCSS-APSA, MCSS-APSA, and SMCSS-APSA at each iteration.

Algorithms | Multiplications | Additions | Comparisons | Exponents |
---|---|---|---|---|

APSA | $2M(K+1)+1$ | $M(2K+1)-1$ | 0 | 0 |

CAPSA | $2M(2K+3)+12$ | $M(4K+4)+6$ | 2 | 0 |

CSS-APSA | $2M(K+1)+M+8$ | $M(2K+2)+4$ | 2 | 2 |

SCSS-APSA | $2M(K+1)+M+8$ | $M(2K+2)+4$ | 2 | 0 |

MCSS-APSA | $2M(K+1)+M+8$ | $M(2K+2)+4$ | 2 | 2 |

SMCSS-APSA | $2M(K+1)+M+8$ | $M(2K+2)+4$ | 2 | 0 |

Algorithms | Parameters | Execution Time (s) |
---|---|---|

CSS-APSA | $C=6$ | 1.7132 |

SCSS-APSA | $C=6$ | 1.6952 |

MCSS-APSA | $C=6,\beta =5$ | 1.7467 |

SMCSS-APSA | $C=6,\beta =5$ | 1.7108 |

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**MDPI and ACS Style**

Li, G.; Zhang, H.; Zhao, J.
Modified Combined-Step-Size Affine Projection Sign Algorithms for Robust Adaptive Filtering in Impulsive Interference Environments. *Symmetry* **2020**, *12*, 385.
https://doi.org/10.3390/sym12030385

**AMA Style**

Li G, Zhang H, Zhao J.
Modified Combined-Step-Size Affine Projection Sign Algorithms for Robust Adaptive Filtering in Impulsive Interference Environments. *Symmetry*. 2020; 12(3):385.
https://doi.org/10.3390/sym12030385

**Chicago/Turabian Style**

Li, Guoliang, Hongbin Zhang, and Ji Zhao.
2020. "Modified Combined-Step-Size Affine Projection Sign Algorithms for Robust Adaptive Filtering in Impulsive Interference Environments" *Symmetry* 12, no. 3: 385.
https://doi.org/10.3390/sym12030385