# Deep Learning for Joint Pilot Design and Channel Estimation in MIMO-OFDM Systems

^{*}

## Abstract

**:**

## 1. Introduction

- Compared with the deep learning-based channel estimation methods in the existing references [9,10,11,12,13,14], this paper applies a hybrid deep learning framework to integrate the two functions of pilot design and channel estimation. Pilot optimization and channel estimation are performed simultaneously during offline training.
- In order to obtain the optimal pilot design and higher channel estimation accuracy, this paper adopts the Concrete selector layers and the fully connected neural networks as the encoder and decoder respectively, the decoder also acts as the generator of the conditional generative adversarial network, and we use the combined loss function to constrain the optimization direction of the entire network.
- Finally, the simulation results show that the method proposed in this paper can significantly improve the channel estimation accuracy and save the pilot overhead in the MIMO-OFDM systems.

## 2. Network Architecture and Implementation Principles

## 3. System Model

#### 3.1. Receive Signal

_{t}transmit antennas and M

_{r}receive antennas. In the time-frequency resources of the NR air interface, it is assumed that one-time slot contains N

_{s}OFDM symbols, and one resource block group (RBG) contains N

_{f}subcarriers. Then the ith subcarrier signal of the jth OFDM symbol received by the m

_{r}th receiving antenna [20] can be expressed as

_{t}th transmit antenna, ${\mathit{x}}_{i,j}$ is the transmit signal, ${\mathit{z}}_{{i,j,m}_{r}}$ is additive white Gaussian noise (AWGN).

_{r}th receiving antenna can be expressed as

_{f}× N

_{s}dimensional matrices, and the elements at the (i, j) position are respectively ${\mathit{y}}_{{i,j,m}_{r}}$, ${\tilde{\mathit{h}}}_{i,j,{m}_{r}}$, ${\mathit{x}}_{i,j}$ and ${\mathit{z}}_{{i,j,m}_{r}}$, $\odot $ represents Hadamard product, which denotes the element-wise product.

#### 3.2. Pilot Design Based on Concrete AE Network

**H**

_{noisy}, the size of the grid is N

_{f}× N

_{s}. In the offline training phase,

**H**

_{noisy}can be obtained through the pilot frequency and the received signal, i.e.,

**H**

_{noisy}=

**Y**

_{mr}$\varnothing \mathit{X}$, $\varnothing $ means dividing the corresponding elements of the two matrices. Then flattening the matrix

**H**to obtain its vectorized representation

_{noisy}**h**

_{noisy}= [

**h**

_{1},

**h**

_{2}, …,

**h**

_{D}], where D = N

_{f}× N

_{s}is the vectorized time-frequency grid length.

**h**

_{noisy}as the input of concrete AE network, correspondingly, the output of the encoder (concrete selector layer) is

**h**

_{p},

_{noisy}= [

**h**

_{p}

_{,1}

**, h**

_{p}

_{,2},…,

**h**

_{p,L}], L < D,

**h**

_{p,noisy}is the most informative feature subset of

**h**

_{noisy}, its lth element

**h**

_{p}

_{,l}is the output of the lth node of the specific selector layer, which can be expressed as

_{j}is sampled from a Gumbel distribution.

#### 3.3. Channel Estimation Based on cGAN Network

**h**

_{ideal}(

**h**

_{idea}

_{l}is the matrix ${\mathit{H}}_{{m}_{r}}$ after leveling 1 × D dimensional vectorized representation). During the offline training phase, the generator is responsible for estimating the vectorized channel $\stackrel{\wedge}{\mathit{h}}=[{\stackrel{\wedge}{\mathit{h}}}_{1},{\stackrel{\wedge}{\mathit{h}}}_{2},\cdots ,{\stackrel{\wedge}{\mathit{h}}}_{D}]$ from the conditional input ${\mathit{h}}_{p,noisy}^{n}$, and the discriminator can recognize the given input as a true label “1” or a false label “0”. After the training is successful, the trained generator can be used to perform channel estimation on the new pilot input.

_{φ}represents the generator parameterized by φ.

_{φ}and maximize D

_{θ}in the loss function. Therefore, the cGAN objective function is

## 4. Simulation Results and Analysis

^{−3}to 10

^{−4}magnitude. It can be seen that the CAGAN scheme has better robustness to the number of pilots at a light higher signal-to-noise ratio, and can effectively improve the spectral efficiency of the system under the premise of ensuring the accuracy of channel estimation.

^{−1}and 10

^{−2}, its performance is still worse than CAGAN. When the number of pilots is 16, under the same SNR condition, CAGAN has obvious advantages over the other three algorithms, the estimation accuracy is improved to between 10

^{−3}and 10

^{−4}, and under high signal-to-noise ratio, CAGAN and ideal MMSE is much closer.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Performance comparison between uniform pilot design and concrete AE pilot optimization; (

**a**) number of pilots = 16; (

**b**) number of pilots = 8.

**Figure 3.**Comparison of channel estimation performance of different algorithms under different pilot numbers when SNR = 15 dB.

Methods | Number of Flops | Number of Parameters |
---|---|---|

CAGAN | 2557 K | 1281 K |

CAGAN (CNN replace DNN) | 548 K | 275 K |

ChannelNet | 1731 K | 865 K |

ChannelNet2 | 3903 K | 1954 K |

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**MDPI and ACS Style**

Kang, X.-F.; Liu, Z.-H.; Yao, M.
Deep Learning for Joint Pilot Design and Channel Estimation in MIMO-OFDM Systems. *Sensors* **2022**, *22*, 4188.
https://doi.org/10.3390/s22114188

**AMA Style**

Kang X-F, Liu Z-H, Yao M.
Deep Learning for Joint Pilot Design and Channel Estimation in MIMO-OFDM Systems. *Sensors*. 2022; 22(11):4188.
https://doi.org/10.3390/s22114188

**Chicago/Turabian Style**

Kang, Xiao-Fei, Zi-Hui Liu, and Meng Yao.
2022. "Deep Learning for Joint Pilot Design and Channel Estimation in MIMO-OFDM Systems" *Sensors* 22, no. 11: 4188.
https://doi.org/10.3390/s22114188