# Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems

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## Abstract

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## 1. Introduction

- In this paper, with a unified massive MIMO framework, we compare the NMSE performance among three categories of algorithms, which is affected by the received SNR, the number of resolvable paths and pilot symbols, angular quantization error, hardware impairments and computational complexity. Through comprehensive comparison, the characteristics and application conditions of each algorithm are revealed and the factors that affect the estimated error and computational complexity are also presented.
- Through theoretical analysis and simulation results, we show that convex relation algorithms achieve the best estimation accuracy at the high SNR range and it is mainly affected by the received SNR and transmitter’s hardware impairments. At the low SNR range, greedy iteration algorithms outperform others and the estimated accuracy is then limited by the angle quantization error. Furthermore, a tradeoff between the estimated error and complexity is achieved in Bayesian inference algorithms, although its estimated error is sensitive to the number of available pilot symbols.
- We also analyze the overall computational complexities of three categories of algorithms and visually represent them by the running time. Through illustrating the runtime of different algorithms versus the sparseness, we show that the computational complexity in the convex relaxation algorithm is the highest, and it even squarely increases with the sparseness in the gradient descent-based convex algorithm, while that in greedy iteration algorithms is minimum and grows linearly with the sparseness. In contrast to them, the computational complexities of Bayesian inference algorithm decreases as the sparseness increases.

## 2. System Model

#### 2.1. System Model

#### 2.2. Channel Model

## 3. Formulation of the Channel Estimation Problem via Compressed Sensing

## 4. A Comparison of Sparse Signal Recovery Algorithms

#### 4.1. Convex Relaxation Algorithms

#### 4.2. Greedy Iterative Algorithms

#### 4.3. Bayesian Inference Algorithms

**Remark 1:**The overall computational complexities of three categories of algorithms are compared in Table 1. For convex relaxation such as IR-based algorithm, its computational complexity will grow squarely with the sparseness increase, since there are gradients of ${\mathit{\theta}}_{T}$ and ${\mathit{\theta}}_{R}$ that need to be calculated in each iteration [9]. For greedy iteration algorithms, its computational complexity will grow linearly with the sparseness due to the number of iterations depends on the sparseness [25]. With respect to Bayesian inference algorithms, its computational complexity will mainly depend on the amount of available pilot symbols at receiver where more available pilot symbols will let the algorithm break out early [30]. In Table 1, we can see that the computational complexity of all algorithms is quite high when deployed on massive MIMO antennas. The CS-based algorithms proposed in the previous works mainly aim at improving the estimation accuracy rather than reducing the computational complexity. However, reducing the computational complexity without significantly decreasing the accuracy is a direction that is worth studying.

## 5. Simulation Results

#### 5.1. Comprehensive Comparison of Estimation Quality

#### 5.2. Computation Complexity versus Sparseness

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Illustration of a hybrid MIMO architecture with the fully connected phase shifting network.

**Figure 2.**NMSE of the estimated CSI versus received SNR with sparseness $L=5$, pilot symbol frames $M=16$, angle quantization $G=32$ and without hardware impairment.

**Figure 3.**NMSE of the estimated CSI versus sparseness with received $SNR$ = 10 dB, pilot symbol frames $M=16$, angle quantization $G=32$ and without hardware impairment.

**Figure 4.**NMSE of the estimated CSI versus received pilot symbols with sparseness $L=5$, received $SNR$ = 10 dB, angle quantization $G=32$ and without hardware impairment.

**Figure 5.**NMSE of the estimated CSI versus received angle quantization (G) with received $SNR$ = 10 dB, sparseness $L=5$, pilot symbol frames $M=40$ and without hardware impairment.

**Figure 6.**NMSE of the estimated CSI versus received SNR with hardware impairment = 13 dB, sparseness $L=5$, angle quantization G = 32 and pilot symbol frames $M=40$.

**Figure 7.**Success probability of channel estimation versus the number of antennas (${N}_{T}$) with received $SNR=10$ dB, ${N}_{R}=4,{N}_{RF}^{R}=4,{\gamma}_{th}=-10$ dB, sparseness $L=5$ and pilot symbol frames $M=16$ and without hardware impairment.

**Figure 8.**Runtime of the estimation algorithms versus the sparseness with received $SNR=10$ dB, angle quantization $G=32$, pilot symbol frames $M=16$ and without hardware impairment.

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**MDPI and ACS Style**

Lu, X.; Yang, W.; Cai, Y.; Guan, X.
Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems. *Appl. Sci.* **2019**, *9*, 4346.
https://doi.org/10.3390/app9204346

**AMA Style**

Lu X, Yang W, Cai Y, Guan X.
Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems. *Applied Sciences*. 2019; 9(20):4346.
https://doi.org/10.3390/app9204346

**Chicago/Turabian Style**

Lu, Xingbo, Weiwei Yang, Yueming Cai, and Xinrong Guan.
2019. "Comparison of CS-Based Channel Estimation for Millimeter Wave Massive MIMO Systems" *Applied Sciences* 9, no. 20: 4346.
https://doi.org/10.3390/app9204346