# Low-Complexity Channel Estimation in 5G Massive MIMO-OFDM Systems

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## Abstract

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## 1. Introduction

- (a)
- It improves the existing CoSaMP [14] algorithm from the dynamic sparsity adaptive and structural aspects.
- (b)
- The proposed algorithm has a feature to estimate the channel without unknown sparsity. This is realized by the threshold parameter $\beta $ which provides information on when to stop the iterations to get the required sparsity of channel.
- (c)
- Based on the adaptive sparsity, the massive MIMO channel space–time common sparse characteristics are used for structural processing. This not only makes the acquisition of sparsity faster but also improves estimation performance and accuracy.
- (d)

## 2. Space–Time Common Sparsity Modeling

#### 2.1. The Spatial Common Sparsity

#### 2.2. Time Correlation

## 3. System Model

## 4. Proposed Algorithm

#### 4.1. Sparse Degree Adaptive Principle

- (a)
- Sparseness adaptation: In the case of unknown sparsity, the number of iterations can still be controlled by Equation (12), and finally the iteration is stopped, and the sparsity is obtained, thereby improving the ability of the algorithm in practical engineering applications.
- (b)
- Stop the dynamic setting of the iteration parameter $\beta $: The influence of noise on the observation value will be different under different channel states, so the stop iteration parameter cannot be set to a fixed value but should be in different channel states. It is different, and the channel state is characterized by SNR, that is, the stop iteration parameter $\beta $ is different under different SNRs.

#### 4.2. Proposed SSA-CoSaMP Algorithm

Algorithm 1. SSA-CoSaMP |

Input: Signal observation value $Y$, observation matrix $\mathsf{\Psi}$, maximum channel length $L$, number of antenas configured on the base station $M$, number of consecutive OFDM symbols $R$, algorithm stop parameter $\beta $. |

Output: Estimated CIR matrix $\widehat{H}$. |

1: Initialization: Initial residual ${V}_{0}=Y$; sparsity $S=1$; number of cycles $k=1$; support set ${\Omega}_{0}=\varphi $ (Equations (1) and (2)) |

2: $Z={\mathsf{\Psi}}^{H}{V}_{k}$ (using the method of Reference [14]) |

3: Combine multiple vectors of the remainder into a vector using massive MIMO common space–time sparsity, i.e., $c\left(\tau \right)={\displaystyle {{\displaystyle \sum}}_{r=1,i=0}^{R,M-1}}{\left|{Z}^{\left(\tau +iL,r\right)}\right|}^{2}$, $0\le \tau \le L-1$ where $c\left(\tau \right)={\left[c\left(0\right),\dots ,c\left(L-1\right)\right]}^{T}$, ${Z}^{\left(m,n\right)}$ is the $m\mathrm{th}$ row and n$\mathrm{th}$ column of $Z$ (using Equation (6) and Reference [9]) |

4: $\Omega =\Omega {\displaystyle \cup}\mathrm{supp}\left\{c{\rangle}_{2K}\right\}$ merged set (from Equation (2) and References [9,14,15]) |

5: $\Gamma =\Omega {\displaystyle \cup}\left[\Omega +L\right]{\displaystyle \cup},\dots ,{\displaystyle \cup}\left[\Omega +L\left(M-1\right)\right]$ merged set |

6: ${\widehat{H}}_{r}={\widehat{S}}_{r}$ |

7: ${V}_{k}=y-{\mathsf{\Psi}}^{H}\mathsf{\Delta}\widehat{h}$ updates the margin (using the method of References [8,9,10,11,12,28]) |

8: Stop iteration if $\Vert {V}_{k}\Vert {}_{2}\le \beta \times \Vert {Y}_{2}\Vert $ is satisfied (using Equation (12)); otherwise let, $k=k+1$, goto Step 2. |

- (a)
- The algorithm has reconstructed the result, but the algorithm itself needs to continue the loop iteration in order to satisfy the given sparsity, resulting in a waste of time and energy.
- (b)
- The result of the algorithm reconstruction is not accurate enough, but the given sparsity has already met the number of loop iterations, and the iteration stops, resulting in an insufficient accuracy of the estimation.

#### 4.3. Computational Complexity

## 5. Simulation Results

- (a)
- Estimation accuracy comparison experiment: The estimated multipath component is compared with the actual modeling, and the estimation performance is characterized by an intuitive method.
- (b)
- Performance simulation experiment with the change of pilot frequency: The decrease of the number of pilots will inevitably affect the estimation performance, and the number of pilot changes to characterize the estimation performance can better reflect whether the estimation algorithm can reduce the pilot overhead, without affecting the estimation performance.
- (c)
- Performance simulation experiment with SNR variation: SNR can reflect the channel state well, and SNR variation to characterize the estimation performance can well reflect the estimation of estimation algorithm under different channel conditions.

#### 5.1. Stop Iteration Parameter $\beta $

#### 5.2. Accuracy Comparison

#### 5.3. Channel Estimation Comparison with Number of Pilots

#### 5.4. Channel Estimation Comparison with SNR Variation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Normalized mean square error (NMSE) performance comparison of algorithms under a different number of pilots.

Parameter | Value |
---|---|

Number of base station antennas (M) | 128 |

Number of users (UEs) | 8 |

Number of subcarriers | 1024 |

Number of multipath channels | 6 |

Number of consecutive OFDM symbols (R) | 5 |

Channel characteristics | Independently distributed Rayleigh fading |

Pilot allocation scheme | Non-orthogonal |

Channel model | Rayleigh fading |

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**MDPI and ACS Style**

Saraereh, O.A.; Khan, I.; Alsafasfeh, Q.; Alemaishat, S.; Kim, S.
Low-Complexity Channel Estimation in 5G Massive MIMO-OFDM Systems. *Symmetry* **2019**, *11*, 713.
https://doi.org/10.3390/sym11050713

**AMA Style**

Saraereh OA, Khan I, Alsafasfeh Q, Alemaishat S, Kim S.
Low-Complexity Channel Estimation in 5G Massive MIMO-OFDM Systems. *Symmetry*. 2019; 11(5):713.
https://doi.org/10.3390/sym11050713

**Chicago/Turabian Style**

Saraereh, Omar A., Imran Khan, Qais Alsafasfeh, Salem Alemaishat, and Sunghwan Kim.
2019. "Low-Complexity Channel Estimation in 5G Massive MIMO-OFDM Systems" *Symmetry* 11, no. 5: 713.
https://doi.org/10.3390/sym11050713