# An Enhanced Affine Projection Algorithm Based on the Adjustment of Input-Vector Number

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## Abstract

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## 1. Introduction

## 2. Conventional Affine Projection Algorithm

## 3. Enhanced Affine Projection Algorithm Based on the Adjustment of Input-Vector Number

Algorithm 1: Re-initialization of the input-vector number. |

${e}_{th}\triangleq \mu {\sigma}_{v}^{2}{M}_{max}/(2-\mu )$, flag $=0$, ${e}_{avg}={e}_{0}^{2}$, |

$\lambda ,{\alpha}_{1},{\alpha}_{2}$: user defined. |

for each i do |

if (${e}_{i}^{2}<{\alpha}_{1}\ast {e}_{th}$) |

flag $=1$ |

else if ( flag = 1 and ${\alpha}_{2}\ast {e}_{avg}<{e}_{i}^{2}$ ) |

flag $=0$, ${e}_{avg}={e}_{i}^{2}$, ${M}_{i}={M}_{max}$, ${P}_{i}={M}_{max}$ |

end if |

${e}_{avg}=\lambda {e}_{avg}+(1-\lambda ){e}_{i}^{2}$ |

end for |

## 4. Experimental Results

#### 4.1. System Identification Verification for Correlated Input

#### 4.2. Speech Input Verification Including a Double-Talk Situation

#### 4.3. Comparison for Computational Complexity

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**MSD learning curves of the conventional APA, DS-APA, SR-APA, E-APA, and proposed APA when the input signals are generated using ${G}_{1}\left(z\right)$, with n = 16 and SNR = 30 dB.

**Figure 3.**MSD learning curves of the conventional APA, DS-APA, SR-APA, E-APA, and proposed APA when the unknown system is abruptly changed from ${\mathbf{w}}_{o}$ to $-{\mathbf{w}}_{o}$ at iteration $5\times {10}^{3}$; the input signals are generated using ${G}_{1}\left(z\right)$, with n = 16 and SNR = 30 dB.

**Figure 4.**MSD learning curves of the conventional APA, DS-APA, SR-APA, E-APA, and proposed APA when the input signals are generated using ${G}_{1}\left(z\right)$, with n = 256 and SNR = 30 dB.

**Figure 5.**MSD learning curves of the conventional APA, DS-APA, SR-APA, E-APA, and proposed APA when the input signals are generated using ${G}_{2}\left(z\right)$, with n = 16 and SNR = 30 dB.

**Figure 6.**Comparison of the input-vector numbers over one trial for the E-APA and proposed APA when the input signals are generated using ${G}_{1}\left(z\right)$, with n = 16 and SNR = 30 dB.

**Figure 7.**MSD learning curves of the conventional APA, DS-APA, SR-APA, E-APA, and proposed APA when the unknown system is abruptly changed from ${\mathbf{w}}_{o}$ to $-{\mathbf{w}}_{o}$ at iteration $5\times {10}^{3}$; the input signals are generated using ${G}_{2}\left(z\right)$, with n = 16 and SNR = 30 dB.

**Figure 8.**MSD learning curves of the proposed APAs with several values of $\alpha $ and $\gamma $ when the input signals are generated using ${G}_{1}\left(z\right)$, with n = 16 and SNR = 30 dB.

**Figure 9.**MSD learning curves of the proposed APAs with several values of $\delta $ when the input signals are generated using ${G}_{1}\left(z\right)$, with n = 16 and SNR = 30 dB.

**Figure 13.**Accumulated sum of multiplications for the conventional APA, DS-APA, E-APA, and proposed algorithm.

APA | DS-APA | E-APA | Proposed APA | |
---|---|---|---|---|

Input-Vector | ||||

Number | M | ${M}_{j}$ | ${M}_{k}$ | ${M}_{i}$ |

$\#(\times /\xf7)$ | $({M}^{2}+2M)n$ | $({M}_{j}^{2}+{M}_{j}+M)n$ | $({M}_{k}^{2}+2{M}_{k})n+$ | $({M}_{i}^{2}+2{M}_{i})n+$ |

$+{M}^{3}+{M}^{2}$ | $+{M}_{j}^{3}+{M}_{j}^{2}$ | ${M}_{k}^{3}+{M}_{k}^{2}$ | ${M}_{i}^{3}+{M}_{i}^{2}$ | |

$+{M}^{3}+{M}^{2}$ | $+{M}_{j}^{3}+{M}_{j}^{2}$ | $+{M}_{k}+1$ | $+{M}_{i}+2$ | |

#(comparisons) | 0 | M | 2 | 2 |

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**MDPI and ACS Style**

Shin, J.; Kim, J.; Kim, T.-K.; Yoo, J.
An Enhanced Affine Projection Algorithm Based on the Adjustment of Input-Vector Number. *Entropy* **2022**, *24*, 431.
https://doi.org/10.3390/e24030431

**AMA Style**

Shin J, Kim J, Kim T-K, Yoo J.
An Enhanced Affine Projection Algorithm Based on the Adjustment of Input-Vector Number. *Entropy*. 2022; 24(3):431.
https://doi.org/10.3390/e24030431

**Chicago/Turabian Style**

Shin, Jaewook, Jeesu Kim, Tae-Kyoung Kim, and Jinwoo Yoo.
2022. "An Enhanced Affine Projection Algorithm Based on the Adjustment of Input-Vector Number" *Entropy* 24, no. 3: 431.
https://doi.org/10.3390/e24030431