# Low-Computational Extended Orthogonal Matched Filter Structure for Multiuser Detection

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Principle of the EOMF

#### 2.1. Formulation of an EOMF

#### 2.2. Computational Complexity of an EOMF

## 3. Proposed EOMF Structure

#### 3.1. Approach to an EOMF with Multiuser Detection

#### 3.2. Structure of an EOMF with Multiuser Detection

#### 3.3. Computational Complexity of the Proposed EOMF Structure

## 4. Performance Evaluation and Discussion

#### 4.1. Purpose of Computer Simulation

#### 4.2. Results and Discussion

#### 4.2.1. MUDT

^{−2}. Furthermore, both squared error performances are confirmed to be similar. In other words, both users can be detected while suppressing the squared error.

#### 4.2.2. CQET

^{−1}, which is not significantly different from that of the proposed structure when the number of desired users increases. In other words, the adaptive array antenna cannot sufficiently suppress the interference signal by null steering, and the replica generation in an OMF does not function sufficiently in all three cases. On the other hand, the BER is below 10

^{−3}in all three cases under the condition that the DIR is −10 dB or higher. The reason is that the replica generation in an OMF works well even if the interference signal cannot be sufficiently suppressed by the adaptive array antenna since the interference signal energy is small originally. Then, Figure 8 shows the BER result for scenario 2; the number of total users is 30. The array antenna characteristics are greatly deteriorated due to the close arrival angles of the desired signal and the interference signal and the lack of antenna flexibility. Hence, both the conventional EOMF and the proposed structure cannot remove interferences that are up to DIR = −20 dB for the same reason as in the case that the number of total users is eight. Under the condition of DIR = −10 dB, good BER performance is shown in the order of the conventional EOMF, two desired users, and 15 desired users. However, the BER is less than 10

^{−4}in all three cases.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Basic structure of the extended orthogonal matched filter (EOMF). The adaptive array antenna is arranged in the front stage. The OMF is placed in the rear stage.

**Figure 2.**Proposed Structure of the EOMF. This schematic shows an example of two-user detection. Unlike the conventional EOMF, ${\mathrm{MF}}_{2}$ have the same sequences as the spreading sequences of desired 2 ${\mathit{c}}_{2}$.

**Figure 3.**Example of the computational complexity of the conventional EOMF and the proposed structure. ${M}_{d}=2\mathrm{to}8$; $K=8;$ $N=31$. The red and blue bars represent the result of the conventional EOMF and the proposed structure, respectively.

**Figure 4.**Squared error of desired users 1 and 2 as a function of the number of normalized least mean square (NLMS) algorithm updates. The blue and green lines show the results of desired users 1 and 2, respectively.

**Figure 5.**Situation of each scenario. (

**a**) Scenario 1: the signal arrival angles of the other users are evenly spaced at ±180°. (

**b**) Scenario 2: the other users are assumed to exist within ±180° of desired user 1.

**Figure 6.**BER result of scenario 1. The number of total users is 30. The blue, red, and green plots show the results when the number of desired users among total users is one (the conventional EOMF case), two and 15 (the proposed structure case), respectively.

**Figure 7.**BER result of scenario 2. The number of total users is eight. The blue, red, and green plots show the results when the number of desired users among total users is one (the conventional EOMF case), two and four (the proposed structure case), respectively.

**Figure 8.**BER result of scenario 2. The number of total users is 30. The blue, red, and green plots show the results when the number of desired users among total users is one (the conventional EOMF case), two and 15 (the proposed structure case), respectively.

Computation | Computational Complexity |
---|---|

(a) ${y}_{\mathrm{beam}}=Re\left\{{\mathit{U}}^{H}\mathit{X}\right\}$ | $K$ |

(b) ${y}_{\mathrm{null}}=Re\left\{{\mathit{V}}^{H}\mathit{X}\right\}$ | $K$ |

(c) ${y}_{{\mathrm{MF}}_{1}}{}^{\prime}=\frac{{\mathit{c}}_{1}{}^{T}{\mathit{y}}_{\mathrm{beam}}}{\alpha}$ | $N$ |

(d) ${y}_{{\mathrm{MF}}_{i}}{}^{\prime}=\frac{{\mathit{u}}_{i}{}^{T}{\mathit{y}}_{\mathrm{null}}}{\alpha}$ | $N$ |

(e) $y={y}_{{\mathrm{MF}}_{1}}{}^{\prime}+{\mathit{w}}^{T}{\mathit{y}}_{{\mathrm{MF}}_{i}}{}^{\prime}$ | $N-1$ |

Computation | Computational Complexity |
---|---|

(f) ${y}_{\mathrm{beam}}\left(t\right)=Re\left\{{\mathit{U}}^{H}\left(t\right)\mathit{X}\left(t\right)\right\}$ | $K$ |

(g) ${e}_{U}\left(t\right)={d}_{U}\left(t\right)-{y}_{\mathrm{beam}}\left(t\right)$ | 1 |

(h) $\mathit{U}\left(t+1\right)=\mathit{U}\left(t\right)+\frac{\mu {e}_{U}^{*}\left(t\right)\mathit{x}\left(t\right)}{\Vert \mathit{x}{\left(t\right)\Vert}^{2}}$ | $K$ |

Parameter | Detail |
---|---|

Channel Model | Additive white Gaussian noise channel |

Modulation | BPSK and DSSS |

Spreading sequence | Gold sequence |

Spreading factor | 31 |

Carrier frequency | 760 (MHz) |

Transmit power | 1 (desired users 1 and 2), DIR −30 (dB) equally divided (interference users) |

Energy per bit to noise power spectral density ratio, ${E}_{b}/{N}_{0}$ | 60 (dB) |

Number of interference users | 30 |

Number of antenna elements | 8 |

Direction of arrival | 0 (degrees) (desired user 1), 180 (degrees) (desired user 2), equidistant to 180 (degrees) (interference users) |

Information data length | 30,000 (bits) |

Step size of NLMS algorithm | 0.01 |

Array antenna weight vector | Winner solution |

Synchronization, normalization | Ideal |

Parameter | Detail |
---|---|

Channel Model | AWGN |

Modulation | BPSK and DSSS |

Spreading sequence | Gold sequence |

Spreading factor | 31 |

Carrier frequency | 760 (MHz) |

Transmit power | 1 (desired user 1), Randomly distributed according to DIR (other users) |

${E}_{b}/{N}_{0}$ | 15 (dB) |

Number of total users | 8, 30 |

Number of antenna elements | 8 |

Number of desired users | 1 (Conventional EOMF), 2, Half the total number of users |

Direction of arrival | 0 (degrees) (desired user 1), scenario 1, 2 (other users) |

Information data length | 10,000 (bits) |

Step size of NLMS algorithm | 0.01 |

Number of weight updates | 10,000 (array antenna, linear combiner) |

Synchronization, normalization | Ideal |

**Table 5.**Bit error ratio (BER) result of scenario 1 when the number of total users is eight and DIR = −50 dB.

Number of Desired Users | BER (10^{−3}) |
---|---|

1 (Conventional EOMF) | 1.16 |

2 (Proposed structure) | 2.59 |

4 (Proposed structure) | 3.41 |

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**MDPI and ACS Style**

Takabayashi, K.; Harada, S.; Kobayashi, T.; Sakakibara, K.; Kohno, R.
Low-Computational Extended Orthogonal Matched Filter Structure for Multiuser Detection. *Telecom* **2020**, *1*, 32-47.
https://doi.org/10.3390/telecom1010004

**AMA Style**

Takabayashi K, Harada S, Kobayashi T, Sakakibara K, Kohno R.
Low-Computational Extended Orthogonal Matched Filter Structure for Multiuser Detection. *Telecom*. 2020; 1(1):32-47.
https://doi.org/10.3390/telecom1010004

**Chicago/Turabian Style**

Takabayashi, Kento, Shuhei Harada, Takumi Kobayashi, Katsumi Sakakibara, and Ryuji Kohno.
2020. "Low-Computational Extended Orthogonal Matched Filter Structure for Multiuser Detection" *Telecom* 1, no. 1: 32-47.
https://doi.org/10.3390/telecom1010004