# A Low Rank Channel Estimation Scheme in Massive Multiple-Input Multiple-Output

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Research Problem

## 4. System Model

#### 4.1. Channel Estimation Based on Antenna Grouping and Block Sparsity

#### 4.2. Low Complexity Channel Estimation Based on Block Matching Pursuit

Algorithm 1. BMP |

Input: Received signal ${\mathit{y}}_{G}$, perceptual matrix $\mathit{A}$, All zero $\left(M\times {N}_{c}\right)\times 1$ vector $\widehat{\mathit{h}}$Step 1: According to Equation (14), the received signal ${\mathit{y}}_{G}$ is correlated with each block of the matrix $\mathit{A}$ according to a previously known group situation: $r\left(m\right)={\Vert {\mathit{y}}_{G}^{H}{\mathit{A}}_{m}\Vert}_{2}$, $m=1,2,3,\dots ,M$Step 2: Sort the results of Step 1 in descending order: $r\left({i}_{1}\right)>r\left({i}_{2}\right)>\dots >r\left({i}_{M}\right)$Step 3: Perform a difference operation on the results obtained in Step 2: $\delta \left(k\right)=r\left({i}_{k}\right)-r\left({i}_{k+1}\right),k=1,2,\dots ,M-1$Step 4: Find the index ${k}_{S}$ corresponding to the maximum value in the result of Step 3, which is the estimation of the sparsity: $\widehat{S}={}_{k=1,2,\text{}\dots ,\text{}M-1}{}^{\mathrm{argmax}\text{}}\delta \left(k\right)$Step 5: Determine the first $\widehat{S}$ indexes in Step 2: ${i}_{1},{i}_{2},\dots ,{i}_{S}$, find the matrix ${\mathit{A}}_{\widehat{S}}$: ${\mathit{A}}_{\widehat{S}}=\left[{\mathit{A}}_{{i}_{1}}\text{}{\mathit{A}}_{{i}_{2}}\dots {\mathit{A}}_{{i}_{\widehat{S}}}\right]$Step 6: Calculate the LS estimation of the sparse channel: ${\widehat{\mathit{h}}}_{S\_LS}={\left({\mathit{A}}_{\widehat{S}}^{H}{\mathit{A}}_{\widehat{S}}\right)}^{-1}{\mathit{A}}_{\widehat{S}}^{H}{\mathit{y}}_{G}$Step 7: According to the index: ${i}_{1},{i}_{2},\dots ,{i}_{S}$ assigns each sub-block of ${\widehat{\mathit{h}}}_{S\_LS}$ obtained in Step 6 to $\widehat{\mathit{h}}$:${\widehat{\mathit{h}}}_{{i}_{1}}={\widehat{\mathit{h}}}_{S\_LS(1)},{\widehat{\mathit{h}}}_{{i}_{2}}={\widehat{\mathit{h}}}_{S\_LS(2)},\dots ,{\widehat{\mathit{h}}}_{{i}_{\widehat{S}}}={\widehat{\mathit{h}}}_{S\_LS(\widehat{S})}$. Where ${\widehat{\mathit{h}}}_{S\_LS(s)}$ means taking the $s$ sub-block of ${\widehat{\mathit{h}}}_{S\_LS}$ Step 8: The sub-blocks in $\widehat{\mathit{h}}$ that are not assigned continue to remain at 0Output: Final estimated channel $\widehat{\mathit{h}}$ |

## 5. Simulation Results and Analysis

_{b}/N

_{0}for the proposed BMP algorithm and the existing LS and OMP algorithm for different groupings. It can be seen from Figure 7 that if the grouping is not performed, that is, each antenna transmits a pilot at the wrong time, the performance of the proposed BMP algorithm is better than that of the LS algorithm. This is because, in the case where the antenna is not grouped, even if the E

_{b}/N

_{0}is low, the proposed algorithm can accurately estimate the position of the non-zero sub-block (as shown in Figure 4), so its performance is better than the performance of the LS algorithm. Moreover, the BER performance for G = 2 of the proposed BMP is better than LS at G = 2, but worse than LS at G = 1. Also, for G = 4, the BMP performance degrades and is worse than that of the OMP at G = 2. Therefore, such results indicate that proper grouping affects the system performance, which must be considered.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Proposed sparsity adaptive block matching pursuit (BMP) algorithm-based differential value distribution with different signal-to-noise ratios (SNRs).

**Figure 4.**Proposed sparsity adaptive BMP algorithm-based accuracy of non-zero position estimation with different SNRs.

**Figure 5.**Comparison of mean square error (MSE) of the proposed BMP algorithm and other algorithms with different SNRs.

**Figure 6.**Comparison of the runtime between the proposed BMP algorithm and the convex optimization algorithm with different SNRs.

**Figure 7.**Comparison of the proposed BMP algorithm with LS and orthogonal matching pursuit (OMP) algorithms under different E

_{b}/N

_{0}values.

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

Number of antennas ($M$) | 120 |

Number of subcarriers (${N}_{c}$) | 128 |

The sparsity of each sub-carrier ($S$) | 10 |

SNR | 6–20 dB |

Sub-block index | 60 |

User groups ($G$) | 1–6 |

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**MDPI and ACS Style**

Shahjehan, W.; Shah, S.W.; Lloret, J.; Leon, A.
A Low Rank Channel Estimation Scheme in Massive Multiple-Input Multiple-Output. *Symmetry* **2018**, *10*, 507.
https://doi.org/10.3390/sym10100507

**AMA Style**

Shahjehan W, Shah SW, Lloret J, Leon A.
A Low Rank Channel Estimation Scheme in Massive Multiple-Input Multiple-Output. *Symmetry*. 2018; 10(10):507.
https://doi.org/10.3390/sym10100507

**Chicago/Turabian Style**

Shahjehan, Waleed, Syed Waqar Shah, Jaime Lloret, and Antonio Leon.
2018. "A Low Rank Channel Estimation Scheme in Massive Multiple-Input Multiple-Output" *Symmetry* 10, no. 10: 507.
https://doi.org/10.3390/sym10100507