# A Space-Time Correlation Model for MRC Receivers in Rayleigh Fading Channels

## Abstract

**:**

## 1. Introduction

## 2. Background and Previous Works

#### 2.1. MRC Receiver Formulation

#### 2.2. Previous Works

- Extension of MRC receivers (conventionally analyzed in the space domain) to the time domain. This means the use of either a retransmission diversity mechanism or a space-time coding repetition approach.
- The MRC performance model uses spatial and temporal correlation coefficients. To the best of our knowledge, this is not commonly used in the literature of MRC receivers, and
- A frequency-domain approach (via the characteristic function) is used to calculate the statistics of the space-time MRC receiver. The result is a polynomial function whose roots can be obtained via numerical methods. The inverse transform is then used to obtain the exact probability density function (PDF). This is opposed to conventional methodologies where all conditioning and integration work are conducted exclusively in the PDF domain, which in some cases can lead to complex infinite series expressions.

## 3. System Model

## 4. Statistics Derivation

#### 4.1. First Stage

#### 4.2. Second Stage

#### 4.3. Third Stage

#### 4.4. Final Statistics

#### 4.5. Adaptive Activation

## 5. Results

## 6. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. CF Calculation in (26)

**Theorem**

**A1.**

**Proof.**

## Appendix B. Derivation of the Conditioned CCDF in (30)

**Theorem**

**A2.**

**Proof.**

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**Figure 7.**CCDF of the instantaneous SNR $\mathrm{\Gamma}$ for different values of spatial and temporal correlation coefficients and number of antennas of the MRC receiver.

**Figure 8.**CCDF of the instantaneous SNR $\mathrm{\Gamma}$ conditional on ${\gamma}_{n-1}<\beta $ for different values of spatial and temporal correlation coefficients and number of antennas of the MRC receiver.

Variable | Meaning |
---|---|

${h}_{m,n}$ | Channel between the terminal and antenna m in time slot n |

M | Number of Rx antennas. |

N | Number of time slots. |

$\gamma $ | Channel variance |

${\rho}_{s}$ | Space correlation coefficient |

${\rho}_{t}$ | Temporal correlation coefficient |

$\mathbf{r}$ | Rx Signal |

$\mathbf{v}$ | Noise vector at the receiver |

Variable | Value |
---|---|

M | Variable from 2 to 10 |

$\gamma $ | 1. |

$\beta $ | $Mn$. |

${\rho}_{s}$ | Variable: 0.1, 0.3, 0.7 |

${\rho}_{t}$ | Variable: 0.1, 0.3, 0.7 |

N | set to 10 time slots |

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**MDPI and ACS Style**

Sámano-Robles, R.
A Space-Time Correlation Model for MRC Receivers in Rayleigh Fading Channels. *Technologies* **2020**, *8*, 41.
https://doi.org/10.3390/technologies8030041

**AMA Style**

Sámano-Robles R.
A Space-Time Correlation Model for MRC Receivers in Rayleigh Fading Channels. *Technologies*. 2020; 8(3):41.
https://doi.org/10.3390/technologies8030041

**Chicago/Turabian Style**

Sámano-Robles, Ramiro.
2020. "A Space-Time Correlation Model for MRC Receivers in Rayleigh Fading Channels" *Technologies* 8, no. 3: 41.
https://doi.org/10.3390/technologies8030041