LYU Siting ,LI Xiaohui,* ,FAN Tao ,LIU Jiawen ,and SHI Mingli

1.School of Telecommunication Engineering,Xidian University,Xi’an 710071,China;2.State Key Laboratory of Integrated Service Networks,Xidian University,Xi’an 710071,China

Abstract:Channel estimation has been considered as a key issue in the millimeter-wave (mmWave) massive multi-input multioutput (MIMO) communication systems,which becomes more challenging with a large number of antennas.In this paper,we propose a deep learning (DL)-based fast channel estimation method for mmWave massive MIMO systems.The proposed method can directly and effectively estimate channel state information (CSI) from received data without performing pilot signals estimate in advance,which simplifies the estimation process.Specifically,we develop a convolutional neural network (CNN)-based channel estimation network for the case of dimensional mismatch of input and output data,subsequently denoted as channel (H) neural network (HNN).It can quickly estimate the channel information by learning the inherent characteristics of the received data and the relationship between the received data and the channel,while the dimension of the received data is much smaller than the channel matrix.Simulation results show that the proposed HNN can gain better channel estimation accuracy compared with existing schemes.

Keywords:millimeter-wave (mmWave),channel estimation,deep learning (DL),dimensional mismatch,channel state information (CSI).

1.Introduction

Millimeter-wave (mmWave) massive multiple-input multiple-output (MIMO) wireless transmission is one of the most promising research areas for future wireless communication systems,which cannot only extend new spectrum resources but also improve wireless transmission rate effectively.The application of mmWave massive MIMO systems [1,2] requires the development of impactful techniques,the most important of which is the hybrid beamforming (HBF) technology [3,4].The design of the HBF generally relies on sufficient channel state information (CSI).However,it is intractable to estimate the perfect CSI for the significant feedback overhead and a low signal to noise ratio (SNR) caused by large-scale antennas and path loss in the mmWave channel.

Traditional channel estimation methods mainly include pilot-based least square (LS) [5],minimum mean square error (MMSE) [6],and compressed sensing (CS) [7,8]based on the mmWave channel sparsity.Because of the influence of noise,the error of the LS method is proportional to the square of the number of transmitting antennas.The MMSE method requires the information of the noise and the correlation of the channel to achieve more accurate channel estimation,which leads to high computational complexity.The CS method restores the channel matrix by compressing and reconstructing the signal.Nevertheless,the nonlinear optimization problem in CS is suject to longer processing time and higher computational complexity when solved by an iterative method.A direction of arrival (DOA)-aided channel estimation method was proposed in [9],whose performance is approaching the Cramer-Rao lower bounds (CRLBs) in the high SNR region.However,this method requires obtaining an initial channel estimate first,followed by DOA and channel gain estimates.The mmWave channel estimation was expressed as an atomic norm minimization problem in [10].The proposed estimator can effectively track the time-varying channel dynamics,but it does not perform well in the high SNR region.Besides,Liu et al.[11] proposed a channel estimation method based on the clustering block sparse Bayesian learning (CBSBL) algorithm,which does not require the prior information of the channel,but the complexity is high due to a lot of matrix inversion.

Recently,researches have focused on using deep learning (DL) methods to solve difficult problems in the communication field.Mei et al.[12] analyzed the mean square error (MSE) performance of DL-based channel estimation from the theoretical perspective and derived an upper bound on the MSE,which provides theoretical support for DL-based channel estimation.Chun et al.[13]developed a two-stage estimation method based on DL for the condition that the pilot length is less than the number of transmit antennas,including pilot-based channel estimation and data-based channel estimation.In [14],the channel response was modeled as a low-resolution image,and then the image super-resolution and image restoration method were applied for channel estimation.Balevi et al.[15] proposed a channel estimation method for massive MIMO systems with limited multicell interference,which employed a deep neural network(DNN) based on a deep image prior network to denoise the received data and then performed the LS method.Hirose et al.[16] assumed that the DL-assisted channel estimation was used to reduce pilot pollution in multi-cell layout.A DNN-based channel estimation method for multi-input single-output (MISO) systems was proposed in [17],and it also demonstrated that the DL-based estimation performance was sensitive to the quality of the training data.However,most of the references above adopted a simple fully connected structure with the same dimensions of input and output data.Fully connected networks do not exploit the location information between the channel matrix elements,i.e.,the spatial correlation of the channels.For channel matrices,especially mmWave channel matrices,the connection between each element and its surrounding elements is relatively tight.Moreover,the training parameters of the fully connected network are large and prone to overfitting.Li et al.[18] proposed a channel estimation scheme for high-movement mmWave systems.The CSI to be estimated was regarded as a twodimensional (2D) picture,and a generative adversarial network (GAN) was employed to learn the relationship between the received data and the channel.However,the application environment of this method is relatively special,and the estimation target is the covariance matrix of the channel.

In this paper,we propose a new channel estimation method based on DL and develop a convolutional neural network (CNN)-based channel estimation network combining convolution and deconvolution,which is referred to as “channel (H) neural network (HNN)” for convenience of representation.It can estimate channel information directly and effectively from the received data without the help of the pilot,which reduces the system overhead and does not consume spectrum resources.Compared to existing DL-based methods with the same dimension for both the input and output of networks,our work tackles the most challenging task of recovering highdimensional data from low-dimensional data.The proposed method acquires channel information by fully learning the inherent features of the received data itself and the correlation between the received data and the channel.The proposed HNN includes a feature learning module and a channel estimation module.In the feature learning module,we apply a convolutional layer and a deconvolutional layer,and the skip layer is combined to fully learn the received data to extract channel features.The learned channel characteristics are analyzed through two fully connected layers to gain estimated channel information in the channel estimation module.The simulation results show that the proposed HNN exhibits good estimation performance.

The rest of this paper is organized as follows: We introduce the channel model and the system model in Section 2.In Section 3,we introduce the main idea of the proposed method,and the architecture and parameters of HNN are described in detail.The simulation results are given in Section 4.Finally,the paper is summarized in Section 5.

2.Channel model and system model

2.1 Channel model

This paper considers the classic Saleh-Valenzuela (SV)channel model [2,4].Both the receiver and the transmitter adopt the uniform planar arrays (UPAs).Suppose that there areNtantennas at the transmitter andNrantennas at the receiver.It is assumed that there areNclscattering clusters,and each cluster containsNrayscattering paths.Then the normalized narrowband mmWave channel model can be written as

The antenna array response can be written as

whereP×Qis the total number of antennas in the UPAs with 0 ≤p＜Pand 0 ≤q＜Qantenna elements on they-axis andz-axis,respectively.λ is the wavelength,anddis the antenna spacing.

2.2 System model

In this paper,we consider an mmWave massive MIMO downlink communication system [20].There areNtantennas andMt≤Ntradio frequency (RF) chains at the transmitter andNrantennas andMr≤NrRF chains at the receiver.AndNs≤min{Mt,Mr} data streams can be transmitted.

Fig.1 Diagram of an mmWave MIMO system

From the above analysis,we can see that the received signalris a complex vector with dimensionsand the channel matrixHis a complex matrix with dimensions.We consider a fast channel estimation method based on DL to directly recover the channel matrix from the received data.

3.Proposed method

In this section,we introduce a DL method for fast channel estimation in mmWave massive MIMO systems.We explain the main idea of the proposed method in Subsection 3.1,and then describe the architecture of HNN in Subsection 3.2.

3.1 Main idea

This work aims to recover channel information from the received data via DL.Traditional pilot-based channel estimate requires a lot of pilot overhead.And most of the existing DL-based channel estimation methods have used channel matrices after pilot estimation as network inputs,as in [12,21].The difference is that we try to recover the channel matrix using only the received data,i.e.,perform blind channel estimation,which reduces the system overhead as no additional pilot information is required.Few DL methods use the received signal as network input.They are mainly for MISO systems,as shown in [17].All the above DL-based channel estimation methods keep the same matrix dimension for both input and output of network.However,it is clear from the analysis in Subsection 2.2 that the received data of the MIMO system considered in this study is a column vector withNrelements,and the channel matrix is a two-dimensional matrix withNr×Ntelements.It is tough to reconstruct a complete channel matrix from a received signal since the dimensions are small and the amount of information included is limited.The field of communication research based on DL is currently attracting a lot of attention.Inspired by this,we design a neural network that can be used for channel estimation named HNN.It should be noticed that we aim to recover the channel information directly from the received data,thus learning enough about the received data is critical.We consider developing a feature learning module that does not change the data size,but only learns the data features.Then we employ the linear fully connected layer to form the channel estimation module.The proposed DL-based channel estimation method is shown in Fig.2.The out_channel×L×Win Fig.2(b)shows the dimensional change of the data from input to output,where out_channel is the number of output channels of the convolutional layer,Lis the data height,andWis the data width.

Fig.2 Illustration of the channel estimation method based on DL

3.2 HNN architecture

The proposed HNN consists of two modules.One is the feature learning module that relies on convolution (blue part),deconvolution (orange part),and skip connections(gray connection lines).Convolution operation can quickly explore key features from input data and generalize results to unknown data of the same type,so it is widely used in the field of image recognition.Deconvolution is commonly used to restore visual data,deblurring,and other processing methods.In the stage of feature learning,we output the feature weights through the convolutional layer and reverse the feature information through the deconvolutional layer.This process allows the network to learn the characteristics of the input data completely.However,the output of the features by deconvolution is rough,so we apply a skip layer to fill in the missing detail data.The other is the channel estimation module (purple part) with the fully connected (FC)layers,which can effectively estimate the interesting data from the extracted feature output.Here,a Linear(·) with BN(·),ReLU(·),and Dropout(0.5) requires to be executed before outputting the regression layer Linear(·).In addition,we adopt the Xavier_normal initialization [22]for the convolutional and the Kaiming_normal initialization [23] for the Linear(·).

The HNN architecture is shown in Fig.2(a).The feature learning module includes three convolutional (Conv)layers and three deconvolutional (Deconv) layers.It should be noted that a Conv layer here is not only Conv,but a combination of Conv(·),BN(·),and ReLu(·).Similarly,a DeConv layer is a combination of Deconv(·),BN(·),and ReLu(·).Also,we add the skip connections as shown in Fig.2(a).The output of the convolutional layer will be added to the deconvolutional layer to increase the feature learning ability to make the result more robust.Each convolutional layer and deconvolutional layer apply a filter with the size of 3×3 and 1×1 respectively to extract channel features from the received data.We change the dimension of the input data as shown in Fig.2(b) by setting the value of out_channels and padding as ‘same’.It changes from low-dimensional to high-dimensional and then to low-dimensional.This process reduces the computational cost of the Linear layer by changing out_channel,and the use of multiple convolutional and deconvolution layers improves the non-linearity of the HNN.This also makes it possible for HNN to more accurately approximate mapping functions and thus solve more complex problems.

The loss function considered in HNN is the MSE,and the formula is as follows:

4.Simulation results and analysis

In this section,we describe the simulation settings detailedly,including data set generation,HNN parameters,and simulation results.The experimental environment is based on Python 3.6.2 and Pytorch 1.9.0.The computer is equipped with 8 Inter i7-10700 CPU Cores,Nvidia GeForce GTX 1660 SUPER GPU,and 64 GB memory.The data generation and processing algorithm is written using Matlab,and the HNN is written using Python.In the simulation experiment,the data generation and results analysis are as follows.

4.1 Simulation settings

We adopt the mmWave channel and system model in Section 2,where we set data streamsas a random vector to facilitate the experiment.Actually,the data streamscould be any vector,butsin the train andsin the test should be the same vector.We apply the hybrid beamforming algorithm in [24] to obtainFbbandFrf.The data set contains a total of 30 000 samples,with a ratio of 6:2:2 for training,validation and test set.Each sample consists of a received signal and the ideal channel matrix that corresponds to it.In the HNN,we set the learning rate to 0.001 and use the Adam optimizer [25] with the momentum of 0.9.A batch size of 64 and the loss function in (3) are applied to update the network weights.The network is trained with 500 epochs.Since the neural network cannot process complex data,we extract the real,imaginary,amplitude,and phase information of the received data as well as the real,imaginary information of the ideal channel matrix,respectively.Then we perform the splicing operation to form two real matrices:

as the input and label of the HNN.The output of the HNN is the estimated channel matrixThen the output data is concatenated to obtain a complex channel estimation matrix with dimension

[:] means the matrix interception operation

4.2 Analysis of simulation results

In this subsection,we evaluate the performance of the proposed DL-based channel estimation method.To evaluate the performance of HNN,we apply MSE which is defined as

We test the MSE performance of HNN at the different numbers of scattering paths,as shown in Fig.3.For simplicity,we fix the scattering pathNrayin each cluster and observe the performance change by changing the number of scattering clustersNcl.As can be seen from Fig.3,the performance of the proposed method in mmWave massive MIMO channel estimation gradually improves with the increase ofNcl.The reason is that with the increase of the total number of scattering paths,the channel information features covered in the received data will also increase.Then its estimation ability will be better after learning by HNN.

Fig.3 Performance of HNN trained with samples with different scattering paths and cluster Nt=Nr=64

Fig.4 shows the MSE performance of HNN with the different numbers of antennas at BS and UE.With fixed scattering clustersNcl=5 and the scattering pathsNray=10in each cluster,we test the performance of MSE with 4,16,and 64 antennas at BS and UE.As shown in Fig.4,the performance of 64 antennas is significantly better than that of 16 and 4 antennas.The reason is that the input received data dimension will be smaller,although the estimated channel matrix dimension becomes smaller when the number of antennas decreases.There will be very little feature information that can be extracted from it,which makes channel estimation more difficult.

Fig.4 Performance of HNN trained with samples with different numbers of Nt and Nr when Ncl=5 and Nray=10

Furthermore,we also consider the performance comparison between the proposed method and the existing algorithm as shown in Fig.5.Compared with the methods in the listed references,it can be seen that the HNNbased channel estimation method proposed in this paper has better performance and does not require iterative online calculation.

Fig.5 Performance of HNN trained with samples with different methods when Ncl=5 and Nray=10

In addition,it can be seen from Fig.3 and Fig.4 that HNN also performs well under the condition of low SNR.

4.3 Complexity analysis

Table 1 Parameters for HNN complexity ( Nt=64)

5.Conclusions

In this paper,we propose a DL-based method to solve the channel estimation problem in the mmWave massive MIMO system.We develop a network called HNN with two modules,namely,a feature learning module and a channel estimation module.Instead of dimensionality reduction of data in the feature learning module,we fully learn the received data through several convolutions and deconvolutions.In the channel estimation module,two fully connected network layers are employed to perform channel estimates.The proposed HNN can estimate the channel fastly and effectively.The proposed method is suitable for the scenarios with quasi-static or static channels and the system environment that change infrequently.The application of transfer learning [30] and meta-learning [31] currently makes it possible to extend the proposed approach to scenarios where the environment is easily changed.The existing channel estimation based on DL almost all relies on the data obtained by simulation,and there is not a large amount of actually collected data set yet.We plan to collect real data for experiments in future work and try to effectively associate DL with mobile devices.