# Exact Analytical H-BER for Ad Hoc XOR H-Map Detector for Two Differentially Modulated BPSK Sources in H-MAC Channel

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Materials and Methods

#### H-BER Formula Derivation

_{e}= H-BER is derived.

## 4. Results

#### 4.1. Analytical H-BER Formula for Ad Hoc (AH) Differential Detector

#### 4.2. Analytical H-BER Formula for Coherent Detector (Coh)

#### 4.3. Analytical H-BER Formula for Approximate Coherent (ApC) Detector

_{e}are

#### 4.4. Comparison of AH Differential Detector with Other Two Differential Detectors

#### 4.5. Other Numerical Results

- 1.
- Decision regions for coherent and differential detectors.
- 2.
- Comparison of H-BER performance of coherent vs. approximate coherent detector and coherent vs. ad hoc detector.
- 3.
- H-BER performance of the ad hoc differential detector vs. the approximate coherent detector for the Rayleigh channel.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

TWRC | Two-Way Relay Channel |

BPSK | Binary Phase Shift Keying |

HDF | Hierarchical Decode and Forward |

AF | Amplify and Forward |

PLNC/WPNC | (Wireless) Physical Layer Network Coding |

H-MAC | Hierarchical Multiple Access Channel |

H-BER | Hierarchical Bit Error Rate |

CSI | Channel State Information |

DBPSK | Differential BPSK |

SNR | Signal to Noise Ratio |

Probability Density Function | |

cdf | Cumulative Density Function |

JML | Joint Maximum Likelihood detector, maximum marginalized, from [13] |

JML2 | Joint Maximum Likelihood detector, marginalized by integration |

JCN | Joint Complex-Normal differential detector from [9] |

AH/AdH | Ad Hoc Differential Detector |

Coh | Coherent Detector |

ApC | Approximate Coherent Detector |

RV | Random Variable |

## Appendix A. Complex Normal and Normal Distributions

## Appendix B. Derivation of pdf of Detector Metrics

#### Appendix B.1. AH Differential Detector

#### Appendix B.2. Coherent Detector

_{Coh}follows. (${d}^{\left(1\right)},{d}^{\left(2\right)}$ are BPSK symbols, so ${{d}^{\left(i\right)}}^{*}={d}^{\left(i\right)}$).

#### Appendix B.3. Approximate Coherent Detector

## Appendix C. Operations with Random Variables

#### Appendix C.1. Joint Distribution of cosh(U), cosh(V) and Their Ratio Distribution

#### Appendix C.2. Joint Distribution of Absolute Values of U, V

#### Appendix C.3. Difference of Two Nonnegative RVs

#### Appendix C.4. Minimum of Two Independent RVs

## References

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**Figure 3.**H-BER of ad hoc (AH) vs. JCN detector for different ranges of SNR. (

**a**) Ad hoc (AH) vs. JCN detector for SNR from 3 dB to 30 dB. (

**b**) Ad hoc (AH) vs. JCN detector for SNR from 3 dB to 15 dB.

**Figure 4.**H-BER of ad hoc (AH) vs. JML detector for different ranges of SNR. (

**a**) Ad hoc (AH) vs. JML detector for SNR from 3 dB to 24 dB. (

**b**) Ad hoc (AH) vs. JML detector for SNR from 3 dB to 12 dB.

**Figure 5.**Decision regions of the coherent (

**a**) and differential detectors (

**b**). (

**a**) Decision regions for coherent and approximate coherent detector. For SNR = 0 dB. $\eta =0.9$, ${\phi}_{1}=0.1$ rad, ${\phi}_{2}=0.5$ rad. (

**b**) Decision regions for the differential ad hoc detector, JML detector (for two SNRs) and JML2 detector. Decision regions on ${y}_{1}$ for given ${y}_{0}$. For $\eta =0.9$.

**Figure 6.**H-BER of the coherent (coh) vs. the approximate coherent (apCoh) detector (

**a**) and the coherent vs. ad hoc (adH) detector (

**b**). (

**a**) H-BER of coherent vs. the approximate coherent detector. (

**b**) H-BER of coherent vs. ad hoc differential detector.

**Table 1.**Values of ${y}_{0}+{y}_{1}$ and ${y}_{0}-{y}_{1}$ for all tuples of $[{d}^{\left(1\right)},{d}^{\left(2\right)}]$.

Event Name | ${d}^{\left(1\right)}$ | ${d}^{\left(2\right)}$ | ${\mathit{y}}_{0}+{\mathit{y}}_{1}$ | ${\mathit{y}}_{0}-{\mathit{y}}_{1}$ | b |
---|---|---|---|---|---|

${A}_{1}$ | 1 | 1 | $2({u}_{A}+{u}_{B})$ | 0 | 0 |

${A}_{2}$ | 1 | −1 | $2{u}_{A}$ | $2{u}_{B}$ | 1 |

${A}_{3}$ | −1 | 1 | $2{u}_{B}$ | $2{u}_{A}$ | 1 |

${A}_{4}$ | −1 | −1 | 0 | $2({u}_{A}+{u}_{B})$ | 0 |

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**MDPI and ACS Style**

Lukac, J.; Sykora, J.
Exact Analytical H-BER for Ad Hoc XOR H-Map Detector for Two Differentially Modulated BPSK Sources in H-MAC Channel. *Mathematics* **2022**, *10*, 903.
https://doi.org/10.3390/math10060903

**AMA Style**

Lukac J, Sykora J.
Exact Analytical H-BER for Ad Hoc XOR H-Map Detector for Two Differentially Modulated BPSK Sources in H-MAC Channel. *Mathematics*. 2022; 10(6):903.
https://doi.org/10.3390/math10060903

**Chicago/Turabian Style**

Lukac, Jozef, and Jan Sykora.
2022. "Exact Analytical H-BER for Ad Hoc XOR H-Map Detector for Two Differentially Modulated BPSK Sources in H-MAC Channel" *Mathematics* 10, no. 6: 903.
https://doi.org/10.3390/math10060903