I Introduction
[lines=2]New applications such as IoT and vehicular communications continuously increase the demands for higher data rate. To face this challenge, massive MIMO has become an essential factor in the design of future cellular systems [what5G]. In massive MIMO, the number of antennas in the BS scales up to serve several users and it was shown in [marzetta] that the effects of fast fading and interference vanish when increasing the number of antenna in the BS. Higher multiplexing and diversity gain are thus obtained with massive MIMO, which in turn results in a higher spectral efficiency and greater energy efficiency.
On the other hand, each antenna in a massive MIMO array requires a RF chain. Therefore, the power consumed by RF elements such as power amplifiers renders massive MIMO systems expensive and energy inefficient. To address this power consumption issue, reduce the costpower hardware overhead, and yet provide reasonable performance, the HBF technique was introduced [HBF_survey]. It consists of using a small number of analog beamformers deployed to drive multiple antennas to form a beam, each connected through a single RF chain to a digital precoder. This hybrid combination of phasebased analog and baseband digital beamformers reduce the number of transmission chains while keeping the sumrate to an acceptable level [Ayach2014, sohrabi_HBF]. In fact, hybrid beamforming techniques have been considered for fifth generation cellular network technology (5G) in the mmWave bands [Han2015]
. However, an explicit estimate of the mmWave channel is generally needed to design the hybrid beamforming matrices at the transmitter. Although several channel estimation techniques for hybrid beamforming have been proposed in the last few years
[cheHB], channel estimation remains a complicated task due to the hybrid structure of the precoding and to the imperfections of the RF chain. Among prior research, the authors in [HBF_OFDM1] designed hybrid beamforming by considering OFDMbased frequencyselective structures. The wideband mmWave massive MIMO systems was investigated in [widebandchHBF] to design the hybrid beamforming. In [unifiedHBF], the authors designed an analog beamformer based on the secondorder spatial channel covariance matrix of a wideband channel. Authors in [low_comp_HBF] assumed to have perfect knowledge of the CSI and proposed a lowcomplexity hybrid beamforming. All of the aforementioned methods strongly depend on full knowledge of the CSI or channel estimation by using pilots, which increases the signaling overhead and, therefore, reduces the spectral and energy efficiency of the system especially in FDD communication where CSI acquisition and feedback is a challenging task [Bjornson2016]. Therefore, in this paper we propose a system that, instead of considering the full knowledge of CSI or channel reciprocity, uses the RSSI to design the hybrid beamforming precoders. Unlike CSI, RSSI is a single real value which users readily measure from the received signal. By consequence, no explicit CSI feedback is required, which reduces the signaling overhead and increases the spectral efficiency of the system.Meanwhile, DL techniques have recently been applied widely to telecommunication system [DNN_WC_Survey] and it was demonstrated to be an outstanding tool for dealing with complex nonconvex optimization problems, thanks to its excellent classification and regression capabilities. Although the training process can be time consuming, the training is done in “offline” mode. Therefore, DL techniques are a promising approach to reduce the latency in cellular network. Several works have investigated the use of DNN to deal with difficult problems within the physical layer [dl_pl], including channel coding, channel estimation [DLChannelestimation1], detection [DL_detection2] and, beamforming [Framework, alkh1, HBF, alkh2, HBF_DNNApp]. In particular, the authors of [Framework] considered a multiinput singleoutput system and solved three optimization problems. By using DNN with FL and decomposition of the channel matrix in [HBF], the nearoptimal analog and digital precoders have been designed. In [alkh1, HBF]
, the authors propose a deep supervised learning based method to estimate the hybrid beamforming by knowing the full CSI. In
[alkh2], a supervised deep learningbased hybrid beamforming design for coordinated beamforming is proposed. The authors in [HBF_DNNApp] deployed a CNN to design the hybrid beamforming, by knowing the CSI. In all the mentioned literature, perfect knowledge of CSI is assumed to be available at the BS, and the DNN are trained using supervised learning. However, supervised learning requires the optimum targets to be known, and thus requires significant additional computing resources to find these targets using conventional optimization methods. In addition, in practical situations, the knowledge of the optimum hybrid beamforming structure is hard to obtain.Therefore, we introduce in this work a novel approach with RSSIbased unsupervised learning to design the HBF in massive MIMO system, improved from our first proposal where supervised deep learning was considered [hojatian2020rssi]. To the best of the authors knowledge, it is the first time that an unsupervised DL system is proposed for designing hybrid beamforming precoders in the context of massive MIMO systems. Furthermore, we train the DNN specifically for the area where the BS is located, so that the geometrical structure of the channel model can be learned. In this paper, this is done using a raytracing model [deepmimo]
. The same approach could also be used to train the DNN using direct measurements of the environment. The proposed DNN architecture is a multitasking CNN that generates both the analog and digital parts of the hybrid beamforming, enabling to reduce the computational complexity. Furthermore, a novel loss function based on sumrate is proposed. To train the model, we introduce methods to generate datasets and codebooks based on the deepMIMO channel model
[deepmimo]. Particularly, the synchronization signals transmitted by the BS are optimized in such a way that the RSSI measurements carry the maximum information about the CSI. Three different channel models have been examined to validate the reliability and robustness of the proposed method in different scenarios. These three scenarios have been chosen to cover different environments, received signal strengths and cell coverage. Moreover, we study the effect of RSSI quantization on the DNN’s performance. The simulation results show that the sumrate performance of the proposed RSSIbased model outperforms other stateoftheart fullCSI methods while the spectral efficiency, signaling overhead, training time and flexibility of the system are significantly improved.The major contributions of the paper can be summarized as follows:

a method to design the codebook for the phasebased AP;

a method to design synchronization burst sequences maximizing the channel information carried by the RSSI;

a procedures to generate the DNN datasets;

a low complexity approach for fully digital precoder and hybrid beamforming design; and

two unsupervised deep learning methods to directly design the hybrid beamforming.
The rest of the paper is organized as follows. Section II describes the system model, including beam training during IA, RSSI measurement and quantization. In Section III, the channel model and dataset generation for DNN training are presented, followed by the nearoptimal HBF solutions for massive MIMO. The proposed method for SS planning in IA and codebook design for the analog precoder are described in Section IV. In Section V, the DNN architecture and the unsupervised learning method are presented. Finally, in Section VI the proposed HBF methods are evaluated and compared with existing methods, and conclusions are drawn in Section VII.
Notation:
Matrices, vectors and scalar quantities are denoted by boldface uppercase, boldface lowercase and normal letters, respectively. The notations
, , , , , , and denote Hermitian transpose, transpose, MoorePenrose pseudoinverse, absolute value, norm, matrix inverse, real part, and imaginary part, respectively.Ii System Model
The considered system model consists of a massive MIMO BS in a singlecell system equipped with antennas and RF chains serving singleantenna users, as shown in Fig. 1. For both uplink and downlink transmission, HBF precoders are employed by the BS. We consider a fully connected architecture where each RF chain is coupled through 2bit phase shifters to all antennas at the BS. A 3step scenario similar to the one described in [hojatian2020rssi] is investigated in the following subsections.
Iia Step 1: SS Bursts Transmission
In the first step shown on the right side of Fig. 1, the BS transmits SS bursts, where each burst uses different 2bit phaseshift analog precoders . The SS are received by all users in the cell. By consequence, the received signal at the user for the burst can be written as
(1) 
where stands for the channel vector from the antennas at the BS to user and is the AWGN term.
IiB Step 2: RSSI Feedback
After receiving , the averaged RSSI value are measured by the user for the SS burst, which constitutes the second step. Therefore, we have
(2) 
where is the noise power. All RSSI values of each user are then transmitted to the BS through a dedicated errorfree feedback channel. These two first steps correspond to the establishment of the IA between the BS and the users.
In practical systems, due to the limited precision of the measurements and limitation in feedback channel, the RSSI must be quantized. Let us denote by the vector of all RSSI values obtained by the user, rescaled by a factor to ensure that . Then, we define as the quantized RSSI vector of user transmitted to the BS. Several quantization methods can be employed. We use linear quantization, given by
(3) 
where is the round operator and is the number of quantization bits. We study the effect of quantization on performance in Section VI.
IiC Step 3: Downlink Data Transmission
The last step corresponds to the downlink transmission where the BS transmits data to each user. The digital precoder matrix is where vector is designed to encode the data symbol of user index . The analog precoder is designed to transfer the output of the RF chain to antennas and applies to all users. To reduce the complexity of the HBF design, we consider that the analog precoder is chosen from a codebook composed of a set of analog beam codewords where is the analog precoder matrix of the codebook (the codebook design is discussed in Section IV).
The SINR of the user received signal for a given hybrid beamformer is then expressed as
(4) 
and the spectral efficiency of the system can be obtained by evaluating the sumrate expressed as
(5) 
Then, the HBF design consists of finding the digital precoder vectors and the analog pecoder matrix in the codebook that maximize the sumrate (5). This problem is however difficult to solve as, in our case, the BS does not have a direct knowledge of the channel coefficients and the noise power . The CSI is in fact partially embedded in the received RSSI. Therefore, we propose to employ DNN techniques to design the HBF precoders.
Iii Dataset Generation for DNN
To train the DNN, a dataset must be obtained beforehand. In practice, this dataset could be generated from channel measurements performed by the BS, while in this paper the channel measurements based on the system model described in Section II must be simulated. This section describes the channel model and dataset generation procedure followed by the nearoptimal fullCSI solution for the HBF and FDP techniques. The HBF and FDP described in this section are used as an upper bound to evaluate the unsupervised DNN performance.
Iiia Channel Model
The deepMIMO [deepmimo] millimeterwave massive MIMO dataset model is used to generate the channel coefficients for the train and test datasets. In this model, realistic channel information is generated by applying raytracing methods to a threedimensional model of an urban environment. It provides the channel vector (of length ) for each user position on a quantized grid. The considered set of channel parameters from this model are summarized in Table I. Scenario “O1” consists of several users being randomly placed in two streets surrounded by buildings. These two streets are orthogonal and intersect in the middle of the map.
System  Antennas  

Parameter  Value  Parameter  Value 
scenario  “O1”  num_ant_x  
bandwidth  GHz  num_ant_y  
num_OFDM  num_ant_z  
num_paths  ant_spacing 
A channel matrix entry in the dataset is obtained by concatenating channel vectors selected randomly from the available user positions of the considered area.
IiiB Dataset Generation Method
Two datasets are generated for training and testing the network. The first one, referred to as the core dataset, contains channel realizations. This dataset is used to design the codebook and the analog precoder of the SS burst . Furthermore, nearoptimal FDP and HBF precoder solutions are computed to compare the sumrate performance of the DNN. It is worth mentioning that, as described in Section V, the proposed DNN architecture is unsupervised, and does not exploit the nearoptimal solutions during training.
The second dataset contains channel realizations and their related RSSI measured from the burst generated from the core dataset. It is used to train and test the DNN. Note that we have as the core dataset requires resolving computationally heavy optimization problems, and we empirically found that samples is sufficient to obtain good codebook and SS design.
Fig. 2 shows the steps performed to generate both datasets, and the next sections provide more detail about the nearoptimal design of the full digital and HBF precoders, the codebook generation and the SS bursts design.
IiiC Fully Digital Precoder Design
Several FDP optimization techniques have been proposed in the literature to maximize the sumrate. Most of these techniques are computationally heavy when applied for massive MIMO cases, but the optimal FDP is needed to evaluate the performance of the DNN. The FDP design problem corresponds to solving the following optimization problem:
(6a)  
s. t.  (6b) 
where stands for the maximum transmission power, and
(7) 
The method we employed to find the optimal FDP is based on [BBO2014], where it is demonstrated that the optimal FDP vector of FDP matrix for user has the following analytical structure:
(8) 
where corresponds to the number of users, corresponds to the identity matrix, and are the unknown realvalued coefficients to be optimized, respectively corresponding to the beamforming power and Lagrange multiplier for the user . In addition, we have and .
Therefore, only realvalued coefficients must be evaluated to resolve the optimization problem, instead of the initial
complex coefficients. The particle swarm optimization (PSO)
[Trelea2003] algorithm can then be employed to obtain the optimal coefficients. However, we empirically found that nearoptimal solutions can be obtained by assuming that and by evenly distributing the power over users and setting for the remaining users. Therefore, solutions have to be evaluated to find the nearoptimal one.IiiD Hybrid Beamforming Design
When considering hybrid beamforming, (4) can be rewritten as
(9) 
From (9), , , can be seen as a virtual channel matrix with spatial paths. In this virtual system, is the FDP matrix of a transmitter with virtual antennas. Then, if is known, the optimization method presented in IIIC can be reused to find the optimal digital precoder . Therefore, the analog precoder must be designed first. We propose to find such that the channel capacity [T1999] of is maximized, giving the following integer nonlinear programming problem:
(10) 
where is the eigenvalue of , and is the waterfill level chosen to satisfy the following equation:
(11) 
with
corresponding to the SNR. To solve this problem, we used the genetic algorithm
[WD1994]. This iterative algorithm consists, for each iteration (or “generation”), of selecting, mutating and merging solutions from a set of candidate solutions (referred as “population”) who gives the best score with respect to the objective function (the sumrate in our case). At the first generation, candidates are generated randomly. The process of selection consists of keeping, for the next iteration, the solutions, referred as elites, which maximize the score. The mutation process makes small random changes on the population, excluding the elites. Finally, the merging process, called crossover, generate new solutions by mixing solutions from the previous population, with being the crossover factor. To optimize the analog precoder, we set the genetic algorithm parameters to , and . This hybrid beamforming design will be referred to as the HSHO method.It is worth noting that the design of the precoders (step ) has an impact on the amount of channel information carried by the RSSI (step ), which in turn affects the quality of the HBF solutions obtained by the DNN and the performance of uplink transmission in step . In addition, the codebook design can also greatly affect the performance of the DNN. Therefore, the codebook and SS for initial access need to be carefully planned.
Iv Synchronization Signal and Codebook design
In this section, we address the problem of the SS burst and analog beamformer codebook design with novel methods.
Iva Proposed SS Burst Design
As mentioned in Section IIC, the design of the SS burst can impact the HBF sumrate performance because it affects the amount of information that is revealed by the RSSI measurements about the CSI. The SSB length also impacts the system data rate in both downlink and uplink. Therefore, it is important to design the SSB so that the measured RSSI provide the maximum information about the CSI while minimizing the amount of data to transmit in the feedback channel.
To resolve this problem, we propose to find the SS sequences that maximize the mutual information between the channel matrices and the quantized RSSI matrix with being the number of SS burst
(12) 
where is the quantization function defined in (3), and where corresponds to the entropy of the variable and is the joint entropy of the variables , quantized on values, defined as
(13) 
with
denoting probability. Furthermore, we have
. It is worth mentioning that there is no need to include more than one user in the calculation of . In fact, user positions are selected randomly and independently, therefore knowing the RSSI of one user cannot provide any information for a second user.A straightforward computation of (12) can be computationally heavy, particularly when used in an optimization loop and in the case of massive MIMO. To reduce the complexity, we assumed that there exists a bijective function that links the channel matrix and the quantized user positions in the environment. In fact, this assumption is quite realistic due to the high channel diversity induced by massive MIMO systems. Under this assumption, we have . Finally, the genetic algorithm with the same parameters as described in Section IIID is used to design the SS burst sequences , by solving the following optimization problem:
(14) 
Note that gives the theoretical minimum number of bits required by the feedback channel to transmit all the information about the CSI. Since the user positions are selected randomly with uniform probability on a set of different locations, we also have .
IvB Proposed Codebook Design
To reduce the complexity of the AP design task for the DNN described in the next section, we propose to restrict the possible AP solutions to a subset (codebook) of solutions (codewords), directly chosen during the optimization phases when generating the core dataset. This is achieved through three successive steps described below.
The first step consists of generating a first codebook when the nearoptimal analog and digital precoder solutions presented in Section IIID are iteratively evaluated for each channel realization. Let us respectively denote and the AP and DP solutions related to the channel matrix of the core dataset. Each of these solutions has sumrate , which can be calculated from (5). As a reminder, denotes the analog precoder in the codebook. Then, is appended in the codebook if
(15) 
where is an arbitrary threshold value, set to in our setup, to decide if the analog precoder will be appended to the codebook, and is the DP solution for the channel matrix in the dataset when the analog precoder is chosen. Therefore, each analog precoder solution solved using the genetic algorithm is appended in the codebook if the obtained sumrate is higher than times the best sumrate obtained with the AP in the current codebook. Otherwise, the obtained HBF solution is replaced by the one in the codebook having the highest sumrate. To avoid high computational overhead, no AP are appended to the codebook when its size reaches AP.
Since the AP are iteratively appended in the first step, it may be possible that better AP solutions are found in the later iterations. Therefore, the sumrate could be improved for the first channel matrices in the dataset. To address this issue, the second step aims to update the AP solutions for each channel matrix in the dataset by choosing the AP in the codebook, obtained in the first step, that maximize the sumrate:
(16) 
The final step consists of reducing the size of the codebook while keeping an acceptable level of average sumrate over the whole dataset. We denote the set of AP index in the core dataset that use the AP label number () in the current codebook as solution. To reduce the codebook size , where represent the cardinality operator in this context, the following operations are iteratively executed:

The codebook is sorted in ascending order such that ,

All the APs in the core dataset using the AP codeword (indexed in ) are moved to other AP codewords giving the best sumrate:
(17) 
The AP codeword is then removed from the codebook.
Each time a codeword is removed, the average sumrate over the core dataset is reduced. To maintain good performance, the codebook size reduction is stopped when the average sumrate reaches % of the initial sumrate.
V RSSIBased Hybrid Beamforming with Deep Neural Network
In this section, we propose a novel RSSIbased method to design the HBF for massive MIMO systems. The RSSI measurements by the users was explained in Section IIB. The BS receives the quantized RSSI, through a dedicated errorfree feedback channel. Then, the BS designs the HBF matrices used to transmit data to users in the downlink step, as described in Section IIC. In supervised learning, a computationally heavy optimization problem must be solved not only for each sample in the dataset, but also for each BS in the cellular network because the dataset generation depends on the location of the BS. On the contrary, in unsupervised learning, the BS can be trained directly using data measured from the environment, thus significantly reducing the computational complexity of the training in the network.
Va Deep Neural Network Architecture
Two approaches are considered to obtain the HBF solution. In the first approach, we design a DNN, called “HBFNet”, to jointly predict the analog precoder and digital precoder . In the second approach, referred to as “AFPNet”, the DNN predicts the analog and fully digital precoders . The digital precoder is then computed using
(18) 
The AFPNet approach ensures that the alignment between analog and digital precoders is maintained. In both approaches, the DNN architecture is composed of a classification task for the analog precoder prediction, and of a regression task for the digital precoder prediction. To reduce the number of parameters and accordingly the computational complexity, we design a multitasking DNN [multitask] to perform both tasks in a single DNN. The proposed DNN architecture is shown in Fig. 3. Referring to Fig. 3, the input of the DL model is the quantized RSSI received at the BS followed by shared CL, FL layers and an output layer for each task. It is worth noting that for both HBFNet and AFPNet, the DNN has the same architecture, except for the dimension of the output layer. Since we separate the real and imaginary parts of the output, the dimension of the output layer for DP task in HBFNet is , and for FDP task in AFPNet. The dimension of the AP task is the same for both approaches and it is equal to size of the codebook (
). The CL use the “Same” convolution operators where the dimension of the inputs and outputs are preserved. After each layer, we use batch normalization to reduce the internal covariate shift and accelerate the learning. In fact, batch normalization is a kind of regularization technique which prevents overfitting because of its noise injection effect
[batchN2]. Since we have used batch normalization technique, the dropout probability for all layers is selected to a very small value () to avoid the over regularization problem. The dropout layers reduce the impact of the choice of initial weights [dropout]. In addition, we use the leaky ReLU activation function in each layer (except the output layers), to fix the
dying ReLU problem after batch normalization [leakyrelu]. This activation function with input and output is given by(19) 
This function does not have a zeroslope part [leakyrelu]
. For the output layer of the classifier or AP task, we use a conventional “Softmax” activation function defined as
(20) 
where is the value of the output of the DNN in AP task part and is the corresponding output of the Softmax activation function. Thus, the output vector of the classifier after activation function is where from Section IVB we consider that . Finally, we used the “Adam” algorithm as network optimizer and “ReduceLROnPlateau” to schedule the reduction of learning rate [Adam].
As shown in Fig. 3, the output of the FDP task is separated into real and imaginary parts. Some DL framework only support real algebra, and therefore cannot directly support complexvalued computation. Therefore, we separately deploy the real and imaginary part to compute the loss function. Then, to compute the pseudoinverse of all codewords , we use . To compute the , we define the square matrix and where and are purely real. We then have the following relations based on [matrix_comp_Inv]:
(21) 
Eq. (VA) is valid when the matrix is nonsingular, which holds in our case. In other cases where is singular, the corresponding equations are listed in [matrix_comp_Inv]. To obtain the predicted digital precoders in (18), we compute
(22) 
and then, is formed.
VB Unsupervised Training
We propose a novel loss function to train both AP task and DP or FDP tasks without any target. Further, as shown in Fig. 4, the full CSI is only used in training mode to calculate the unsupervised loss function, and in evaluation mode, only the RSSI information is used at the BS and the BS does not have access to CSI.
Since we design the hybrid beamforming with two different approaches, each approach requires its own loss function for training. As a reminder, the sumrate achieved by the HBF is given by (5), and based on (20), we define as the output vector of the AP task. We define below the unsupervised loss function for each approach.
VB1 HBFNet
The DNN in this approach is trained to jointly design the DP and AP directly from RSSI. For this approach we define the unsupervised loss function as
(23) 
where is the analog precoder of the codebook and is the output of the DP task. To satisfy we further normalize . The negative sign allows the sumrate to be maximized when the DNN is trained to minimize the loss function. Algorithm 1 summarizes the steps to train the HBFNet.
VB2 AFPNet
The loss function in this approach is different because here we aim to design the digital precoder from the FDP and AP. To do so, we first obtain the FDP and AP, and then by using (18), DP can be computed. If we define
(24) 
the loss function for the FDP task can be defined as
(25) 
where is the output of FDP task in AFPNet. This loss function results in the maximization of the FDP sumrate. Here again, we normalize to satisfying . Now, by knowing the FDP, given AP, we compute , where is the DP matrix obtained from (18) for the codeword in the codebook. The loss function for the AP task is defined as
(26) 
where unlike (23), this loss function is only tuning the Softmax outputs of the AP task. The final loss function for AFPNet is defined as
(27) 
so that the DNN maximizes the FDP sumrate while also maximizing the HBF sumrate by selecting the appropriate AP from the codebook. Algorithm 2 summarizes the steps to train the AFPNet.
It can be seen that AFPNet is more complex than HBFNet, which directly outputs the hybrid beamforming solution. However, as shown in Section VI, AFPNet achieves better sumrate performance by keeping the alignment between the analog and digital precoder.
VC Evaluation Phase
In the evaluation phase, a part of the DNN dataset is dedicated to test the network, as presented in Section IIIB. We used the sumrate metric to characterize the performance. For the evaluation phase, the maximum value of the “Softmax” output is selected in the classifier. So, we can compute the sumrate of HBF to evaluate its performance using , where is the analog precoder predicted by the AP task, expressed as where , and is the predicted digital precoder of HBFNet or AFPNet obtained from (18). To satisfy the power constraint we normalize the to have .
Likewise, to evaluate the DNN performance for FDP sumrate we compute , where is the predicted FDP matrix in FDP task of AFPNet. As discussed in the next section, we evaluate the DNN using different scenarios.
Vi Numerical Evaluation
In this section, the performance of the proposed DNN, implemented using the PyTorch DL framework, is numerically evaluated. To analyse how the DNN performance evolves with the environment, three types of datasets covering different areas are considered, as illustrated in Fig. 5:

Limited Area: small area of possible user positions in the main street, with base station number as the transmitter. The “active_user_first” (AUF) and “active_user_last” (AUL) deepMIMO parameters are set to and .

Extended Area: larger area than the limited one, with possible user positions in the main street, with , , and base station number as the transmitter.

Crossroad Area: area located at the intersection of the two streets, with users coming from every direction ( positions). Three deepMIMO channel environments have been generated and concatenated to obtain this last dataset. The first environment uses , , the second uses , and the third uses , . For all these models, the base station number corresponds to the transmitter.
The size of the DNN dataset is set to samples for each scenario, with 85% of the samples used for training as training set and the remaining ones are used to evaluate the performance as test set. Table II
shows the chosen hyperparameters which have been used for the proposed multitasking DNN.
Parameter  Set Value 

Minibatch size  500 
Initial learning rate  0.001 
ReduceLROnPlateau (factor)  0.1 
ReduceLROnPlateau (patience)  3 
Weight decay  
Dropout keep probability  .95 
Kernel size  3 
Zero padding 
1 
“” in BatchNorm (1D&2D) 
Via SumRate Evaluation for All Areas
We consider a system with users communicating with a BS equipped with antennas, RF chains, and full precision RSSIs are fedback to the BS. Fig. 6 shows the achievable sumrate for each considered area when considering different noise power values ranging from dBW to dBW. When considering the channel attenuation, the average SNR for dBW in Extended Area are dB, dB and dB, in Crossroad Area are dB, dB and dB, and in Limited Area are dB, dB and dB, respectively. It can be seen that the AFPNet has better performance when compared to the HBFNet. It is owing to the fact that in the AFPNet, the digital precoder is extracted from the predicted FDP and HBF based on (18). Therefore, the alignment between analog and digital precoder is preserved. In fact, the sumrate performance of the AFPNet is very close to the upper bound obtained by using the fullCSI HBF design method presented in Section IIID.
Furthermore, as shown in Fig. 6 the two proposed unsupervised HBF design methods presented in Section V have better sumrate performance, for all three areas, when compared with the PZF [low_comp_HBF] and the OMP [Ayach2014] full CSI techniques. The OMP technique uses as inputs the optimal fully digital precoder and the same AP codebook used by the DNN and designed according to the method proposed in Section IV. In fact, the PZF has very poor performance for the Limited Area, which is located far from BS, and therefore suffers at low SNR level. Unlike PZF which has good performance in high SNR, OMP achieves better sumrate in low SNR. However, our proposed nearoptimal HSHO solution and unsupervised learning DNN methods have stable performance in all SNRs. For instance, with dBW, our proposed method in AFPNet is better than PZF by 50%, 25% and 8733% for scenario (a), (b) and (c), respectively, and better than OMP by 66%, 59% and 75%. It is worth mentioning that both PZF and OMP techniques require perfect knowledge of the CSI, and therefore require a high bandwidth feedback channel to report the full CSI in FDD scenario, which penalizes the spectral efficiency of the system.
The “Random AP selection” curves in Fig. 6 show the achieved sumrate when the BS selects the AP from the codebook randomly, while obtaining the DP by substituting into (18) the random AP and the FDP predicted by AFPNet. The significant performance degradation observed for the Random AP selection confirms that the AP task of the DNN actually learns the system characteristics from the dataset.
Likewise, Fig. 7 shows that the sumrate performance of the FDP obtained with the proposed AFPNet method outperforms the ZF method, for all areas and different noise power. In fact, the proposed unsupervised learning in AFPNet has nearoptimal FDP sumrate performance in all selected areas. This make the proposed network a promising solution for FDP design in mmwave massive MIMO systems.
ViB Impact of The RSSI Length and Quantization
The effect of the number of SSB (also the number of RSSI values) on AFPNet is shown in Fig. 8, for the “Extended Area” scenario. Increasing increases the amount of CSI available to the DNN, and we see that the achievable rate indeed improves as increases. However, increasing also reduces the channel efficiency because of the need to send more SSB in Step . Furthermore we see that the performance of AFPNet with is close to the performance obtained with and .
In Fig. 9 we examined the proposed methods with different numbers of quantization bits for the RSSIs, computed using (3). The “Extended Area” is considered to evaluate the performance. In fact, there is a tradeoff between the spectral efficiency improvement of the system and the performance of the DNN when changing the number of bits. It can be seen in Fig. 9 that the sumrate performance of the DNN for to is very close to the DNN performance when considering full precision for the RSSIs.
ViC Impact of The Number of Users and Antennas
This section evaluates how the performance of the two proposed HBF designs scale when varying the number of users and the number of antennas . The number of RSSIs is set to , the noise power is fixed to dBW and the “Extended Area” is considered. In fact, the complexity of the DNN, measured in terms of the number of parameters, can be expected to depend on the complexity of the optimization problem. Thus increasing the number of antennas in BS or the number of users should require a more complex DNN. However, the results in Figures 10 and 11 show that the proposed architecture is complex enough to have excellent performance for a wide variety of number of antennas and number of users. Fig. 10 shows that although our method is RSSI based, it has better performance in comparison with other CSIbased methods and the sumrate performance scale with the number of users. For instance, our proposed method is better than PZF by 50%, 84% and 122% for , respectively. Meanwhile, the PZF and OMP method do not scale well when the number of users increases. This can be explained by the fact that increasing the number of user and fixing the number of antennas in BS will generate more interuser interference. Furthermore, PZF only provides nearoptimal results if the ratio between the number of antennas and the number of users is large enough in massive MIMO systems. By increasing the number of users, this ratio decreases and the sumrate performance of PZF collapses.
In Fig. 11, we evaluate our proposed method with different number of antennas in BS (). Again, we consider the “Extended Area” scenario where dBW, and . As we can see, the PZF method has poor performance when the number of antennas is low whereas the sumrate performance is nearoptimal when the number of antennas grows up to . Therefore, our proposed methods outperforms PZF for the , and antennas cases by 195%, 100% and 50%, respectively. Unlike OMP which has fair enough performance with a small number of antennas, the sumrate of PZF is almost the same as our proposal for higher number of antenna. However, it is worth mentioning that PZF and OMP require perfect knowledge of the CSI and by increasing the number of antennas in BS it reduces the spectral efficiency whereas our proposed methods only exploit RSSI measurements which we kept to and can therefore significantly improve the data rate of the system by reducing the signaling overhead. To clarify, in where the proposed method and PZF have almost same performance, our proposed method requires bits to feedback the RSSI where . On the other hand, PZF in same configuration requires to feedback the CSI to the BS where we find that to keep the sumrate same as fully precision. As a result, in same configuration and sumrate performance, PZF requires times more feedback bits in comparison with our proposed method.
Fig. 12 shows how the number of RSSI measurements affects the sumrate performance when the number of antennas is increased. It can be seen that the required number of RSSIs to achieve nearoptimal sumrate depends on the number of antenna. For instance, with using or even , the DNN has similar performance as larger number of RSSI such as or . However, for larger number of antennas, the DNN needs more information to design the HBF and as a result more RSSI transmission are required.
Vii Conclusion
Hybrid beamforming is an essential technology for massive MIMO systems that allows to reduce the number of RF chains and therefore increase the energy efficiency of the system. However, the design of digital and analog precoders is challenging, and the estimation of the CSI introduces important signaling overhead, especially in FDD communication. To alleviate this issue, we instead proposed to rely on RSSI feedback, improving the spectral efficiency of the communication system. The SS burst was efficiently designed so that the RSSIs provide maximum information about the CSI. We then proposed to design the hybrid beamforming by using unsupervised deeplearning methods. This unsupervised learning approach leads to decreased training time and cost since the system can be trained using only channel measurements without the costly need to obtain optimal solutions. The deeplearning methods select the AP from a codebook designed to reduce the complexity without sacrificing the sumrate performance. Finally, the performance of the proposed algorithm was evaluated using the realistic deepMIMO channel model. The results demonstrate that, despite not having access to the full CSI, the BS can be trained to robustly design the hybrid beamforming while achieving a similar sumrate as the HSHO nearoptimal hybrid precoder that was introduced as a benchmark. Moreover, the proposed method can be implemented in realtime systems thanks to its low computational complexity.
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