Deep Learning based Joint Precoder Design and Antenna Selection for Partially Connected Hybrid Massive MIMO Systems
11 Deep Learning based Joint Precoder Design andAntenna Selection for Partially Connected HybridMassive MIMO Systems
Salman Khalid, Waqas bin Abbas, Farhan Khalid,
Member, IEEE
Abstract —Efficient resource allocation with hybrid precoderdesign is essential for massive MIMO systems operating inmillimeter wave (mmW) domain. Owing to a higher energyefficiency and a lower complexity of a partially connected hybridarchitecture, in this letter, we propose a joint deep convolutionalneural network (CNN) based scheme for precoder design andantenna selection of a partially connected massive MIMO hybridsystem. Precoder design and antenna selection is formulated as aregression and classification problem, respectively, for CNN. Thechannel data is fed to the first CNN network which outputs asubset of selected antennas having the optimal spectral efficiency.This subset is again fed to the second CNN to obtain theblock diagonal precoder for a partially connected architecture.Simulation results verifies the superiority of CNN based approachover conventional iterative and alternating minimization (alt-min)algorithms. Moreover, the proposed scheme is computationallyefficient and is not very sensitive to channel irregularities.
Index terms —
Millimeter Wave Communication, MassiveMIMO, CNN, Partially Connected Hybrid PrecoderI. I
NTRODUCTION
Researchers are exploring the millimeter (mmW) and ter-ahertz domain to meet the ever increasing demand of widerbandwidth and higher data rates [1]. The propagation environ-ment at such higher frequencies is suffered by severe path loss,scattering and penetration losses. Massive MIMO architecturewith precoding/beamforming gain is utilized to compensatefor the propagation losses [2]. Researchers are tilted towardsthe hybrid (analog coupled with digital) beamforming archi-tecture since it provides the gains of digital processing withlower power consumption [3][4]. Existing literature proposesmany techniques for both fully connected (where each RFchain is connected to every antenna) and partially connected(where each RF chain is connected to a subset of antennas)hybrid architectures [3]-[7]. Authors in [5] have utilized theorthogonal matching pursuit (OMP), a greedy algorithm, forthe computation of analog and digital precoders for a fullyconnected hybrid architecture by utilizing the array responsesof the transmitter and the receiver. [6] proposes the iterativesuccessive interference cancellation (SIC) algorithm for com-putation of hybrid precoder for an energy efficient partiallyconnected hybrid architecture. Authors in [7] have proposedthe manifold optimization (MO) and the phase extraction(PE) based alternating minimization (alt-min) techniques to
Salman Khalid (corresponding author, email: [email protected]),W. bin Abbas and Farhan Khalid are with the National University of Computerand Emerging Sciences (NUCES), Islamabad, Pakistan. compute the hybrid precoder for a fully connected architec-ture and the semi definite relaxation (SDR) based techniquefor a partially connected architecture. Alt-min approachesexplores the linkage between optimal and hybrid precodersto estimate the digital and analog precoder. The applicationof evolutionary algorithms for evaluation of hybrid precodersis demonstrated in [8][9]. For Massive MIMO systems, theresource allocation in terms of active antennas selection iscritical to ensure high energy efficiency. The spectral efficiencygain becomes constant beyond a certain number of antennas,hence to optimize the hardware, antennas experiencing goodchannel conditions should be selected. For antenna selectionproblem, authors in [10] and [11] have applied an iterativeevolutionary and estimation of distribution based algorithm.The above mentioned iterative, greedy and alt-min techniquesregarding antenna selection and precoding have drawbacks interms of computational time and achieving optimal solution interms of spectral efficiency.All above works on antenna selection and precoding givesthe sub optimal solution with considerable computationalcomplexity despite applying various optimization strategiesand selection criteria. Recently, the machine learning basedconvolutional neural network (CNN) methods have gainedinterest of researchers to solve the optimization problemsrelated to wireless communication. Well trained CNNs haveability to deduce features from a given set of observations withhigh efficiency and at a very low complexity as compared toconventional techniques. CNNs find its applications for solvingproblems such as channel estimation [12], interference coor-dination, beam management [13] and analog beam selection[14]. Very recently precoder design problem is also formulatedand solved using CNN [15]-[19]. However, all the research forprecoder and combiner design is limited to a fully connectedhybrid architecture and do not consider the partially connectedhybrid architecture which has proven ability of being energyefficient and bears reduced complexity [6].Keeping in view the low latency, low power consumptionand high energy efficiency requirements of future 5G and B5Gcommunication networks, in this letter, we propose a CNNbased joint antenna selection and precoder design for a par-tially connected hybrid structure. Two separate CNNs i.e., clas-sification and regression are trained for an antenna selectionand a precoding problem, respectively. Channel realizationsadded with noise are used to train the networks and estimatethe optimum antenna subset and precoder weights. The inputof the channel matrix is fed to the first stage CNN deployed a r X i v : . [ c s . I T ] F e b for an antenna selection. The reduced subset is further fedto second stage CNN which outputs the optimum analogprecoder. The training of both CNNs is performed offline,hence, all the computational overhead for the data generationand training is not present during online prediction, where thehybrid precoders prediction and antennas classification is doneby only feeding the channel matrix to the network.II. H YBRID M ASSIVE
MIMO S
YSTEM M ODEL
For joint estimation of precoder weights and antenna selec-tion, in this letter, we have considered a hybrid architecture in apartially connected configuration i.e, each RF chain energizesonly a subset of antennas M = N T / N RFT . We have consideredthe base station (BS) equipped with N T transmit antennas and N RFT
RF chains to transmit N S data streams. The user isconsidered to be equipped with N R receive antennas wherethe selection is performed to determine N r best antennas.The analog and baseband precoder at the transmiter arerepresented as F RF ∈ C N T × N RFT and F BB ∈ C N RFT × N S respectively. Therefore, the signal transmitted through BS isrepresented as x = F RF F BB s , where s is the N S × transmitted symbol vector. The analog precoder F RF is ablock diagonal matrix realized by phase shifters having equalmagnitude with variable phases and satisfying the powerconstraint (cid:107) F RF F BB (cid:107) F ≤ N RFT . The received signal y atthe user having N R antennas is expressed as y = (cid:112) P av HF RF F BB s + n (1)The average received power is P av , n ( CN (0 , σ ) ) is i.i.dcomplex Gaussian noise. H denotes the N R × N T full arraychannel between the transmitter and receiver. The clusteredgeometric Saleh-Valenzuela model [3] representing the lowrank mmW channel is used in this letter. H = (cid:115)(cid:18) N T N R (cid:15)K (cid:19) K (cid:88) k =0 η l a R ( µ k ) a HT ( θ k ) (2)where K is the number of paths, (cid:15) is the pathloss, the pathgain linked with the k th path is η k , the corresponding spatialsignatures of the receiver and the transmitter are a R and a T ,respectively, and µ k and θ k are the angle of arrival (AoA)and the angle of departure (AoD) of the k th path, respectively.Finally, the full array (without any antenna selection) spectralefficiency is defined as R = log( I N R + ρN s HF RF F BB F HBB F HRF H H ) (3)III. J OINT A NTENNA S ELECTION AND H YBRID P RECODER
Our first goal is to determine a subset of N r best antennasout of total available N R receive antennas. After the antennaselection, the RF and baseband precoders are determined usingreduced dimensions. The joint solution of antenna selectionand precoding have to satisfy following condition max Q , F RF , F BB log( I N r + ρN s H sel F RF F BB F HBB F HRF H Hsel ) (4) Here H sel = QH is a reduced dimension ( N r × N T )channel matrix obtained by performing antenna selection. Q is a ( N r × N R ) selection matrix with entries either or representing the antenna index. A. Antenna Selection
For antenna selection, picking N r antennas out of N R yields Q A = (cid:0) N R N r (cid:1) possible combinations. Hence, selectinga subset of antennas becomes a CNN classification problemwith Q A classes. Let q A th antenna subset configuration with q A ∈ Q A = { , ...., Q A } is selected, than the received signalvector with q A th selected subset with H q A ( N r × N T ) beingcorresponding channel matrix is expressed as y q A = (cid:112) P av H q A F RF F BB s + n q A (5)Similarly, the spectral efficiency with q A th selected subsetis expressed as R = log( I N r + ρN s H q A F RF F BB F HBB F HRF H Hq A ) (6)Note that R is dependent on q A through H q A . By maximiz-ing R for all combinations of antenna selection configurations,the best antenna subset is expressed as q A = arg max q A ∈Q A R ( q A ) (7) B. Partially Connected Hybrid Precoder Design
Let H q A is the reduced dimensions channel matrix obtainedafter performing antenna selection than the hybrid precoderproblem is defined as max F RF , F BB log( I N R + ρN s H q A F RF F BB F HBB F HRF H Hq A ) (8)We have considered a partially connected hybrid structurewhere each RF chain is connected to M = N T / N RFT numberof antennas. This implies that the structure of the RF precoder F RF must be a block diagonal with f RF i being the precodingvector for the i th RF chain only having M non zero elementsand is expressed as F RF = f RF · · ·
00 f RF · · · ... ... . . . ... · · · f RF Nrf (9)Hybrid precoder has to meet two constraints; C1: All non-zero elements of F RF must have the same amplitude and, C2:Meet the total power constraint (cid:107) F RF F BB (cid:107) F ≤ N RFT . For thecase of hybrid precoder, based on the proof [5], the Euclideandistance between the optimal unconstrained precoder and thehybrid precoder should be minimized. In other words, thehybrid precoder design problem is rewritten as arg min F RF , F BB (cid:107) F opt − F RF F BB (cid:107) F (10) Fig. 1. Joint CNN Architecture for Antenna Selection and Precoding
The optimal solution for the above mentioned optimizationproblem can be obtained using the singular value decom-position performed on the channel matrix, which can beused to generate the labels for output layer of regressionCNN network. Even with the memory-friendly approaches,it is computationally complex to enumerate over all possibleantenna selection subsets in real time. Also the optimal designof hybrid precoders requires iterations and extensive computa-tions. In order to address this issue, we have formulated a deepCNN based solution where the networks are trained offline andperform computations for antenna subset configuration andoptimal precoder. Afterwards, the trained network can simplybe deployed as a classification and regression network to selectantennas and estimate hybrid precoders.IV. CNN T
RAINING AND D ATASET G ENERATION
Our proposed deep neural network consists of two separateCNNs (Fig. 1) to perform antenna selection and precoding.The input to the first CNN AS is the full dimension channelmatrix which selects the best antenna subset q A . The secondCNN RF accepts the reduced dimensions channel matrix withonly selected rows corresponding to the selected antennas andestimates the RF precoder at its output. For both architectures,the training data is generated using channel realizations whichare further assigned with corresponding output class/label forantenna selection and hyrid precoding.Let the input data X be a N R × N T × c = 3 channels.The first channel of input is the absolute value of imperfectchannel matrix ˜ H whereas the real and imaginary part ofchannel matrix are stored in second and third channels, re-spectively. For data generation, N different channel matrixrealizations are generated. Afterwards for each realization, L noisy channel matrices are created with synthetic noise whichis added element wise. Hence, the total size of training inputdata becomes N R × N T × × N L . In order to obtain the outputlabels, for antenna selection CNN RF the best antenna subset isselected and afterwards for second CNN RF , F RF is obtainedby performing SVD operation on reduced dimension channelrealizations. Hence, the input/output pairs are established. Thetraining process of both CNNs is identical but with differentinput dimensions.The input sizes of CNN AS and CNN RF is N R × N T × N r × N T ×
3, respectively. Each CNN is composed of 14 layers.Input layer being the first layer of corresponding input datasize. The convolutional layers are second, fourth and sixthwith 64 filters of dimensions 2 ×
2. Eighth and eleventh layers are fully connected layers with 512 nodes. The tenth andthirteenth layers are dropout layers with 50% probability. TheRELU activation function is utilized. Finally, the output layerof CNN AS is a classification layer with softmax function tooutput the antenna subset class which gives maximum spectralefficiency and output layer of CNN RF is a regression layer ofdimensions N T × F RF . After estimating the non zeros elements of F RF , theblock diagonal structure is obtained by appending zeros atappropriate locations. CNN RF is used to predict the F RF andthe F BB is obtained using equivalent channel approach.V. N UMERICAL S IMULATIONS AND R ESULTS
In this section, we evaluate the performance of our proposedapproach. The performance of CNN based hybrid precoderis evaluated against state of the art SDR alt-min and SICalgorithms. Uniform planner array with N T = 36 or 144 and N R = 16 for the transmitter and receiver respectively, aregenerated. For antenna selection, the N r is kept as 8. TheRF chains at the transmitter N RFT and at the receiver N RFR are kept as 4. For CNNs, the training data is generated for N = L = 100 realizations. The proposed network is trained usingMATLAB as a simulation environment. SGD algorithm is usedfor network parameters with learning rate of 0.005 and mini-batch size 500 with 200 epochs. The cross entropy function isused as the loss function. During the training phase, 30% and70% of all data is divided into validation and training datasets,respectively. Finally the validation data is used to verify theperformance of the proposed architecture in the simulationsfor 100 Monte Carlo trials.Fig. 2 and Fig. 3 with N T as 36 and 144 respectively,shows the spectral efficiency for different algorithms withdeep antenna selection (DAS). The N TRF and N S are con-sidered equal and set to 4, the N R is set as 16 and DASCNN network selects N r = 8 antennas. After performingDAS, the hybrid precoders are determined. CNN RF is usedto determine the F RF whereas the F BB is obtained usingequivalent channel approach. The CNN based hybrid precoderis outperforming the SDR alt-min and SIC algorithms. TheCNN based hybrid precoder is efficiently predicting the RFprecoder which contributes towards maximization of spectralefficiency. To evaluate the antenna selection technique, theperformance of CNN based antenna selection is compared withrandom antenna selection (RAS) applied with precoding. It isevident that spectral efficiency using RAS algorithm is trailingbehind DAS algorithm irrespective of precoding technique. -20 -15 -10 -5 0 5 10 15 20 SNR, [dB] S pe c t r a l E ff i c i en cy [ b i t s / s / H z ] DAS + OPTRAS + OPTDAS + DHBRAS + DHBDAS + SDR Alt-MinRAS + SDR Alt-MinDAS + SICRAS + SIC
Fig. 2. Spectral Efficiency with N T =36, N R =16, N r =8, N TRF =4 -20 -15 -10 -5 0 5 10 15 20 SNR, [dB] S pe c t r a l E ff i c i en cy [ b i t s / s / H z ] DAS + OPTRAS + OPTDAS + DHBRAS + DHBDAS + SDR Alt-MinRAS + SDR Alt-MinDAS + SICRAS + SIC
Fig. 3. Spectral Efficiency with N T =144, N R =16, N r =8, N TRF =4 The computation time of CNN based precoder, SDR alt-minand SIC algorithms are also computed for N T = 144. The CNNbased precoder only required 0.01s, SDR alt-min requires 1.6sand SIC based precoder requires 0.02s for computations. TheCNN based hybrid precoder is outperforming all algorithmsin terms of spectral efficiency. Hence, both the computationalefficiency and spectral efficiency of the proposed CNN basedhybrid precoder is established.VI. C ONCLUSIONS
This letter presents the CNN based solution for joint hybridprecoder design of a partially connected mmW massive MIMO system with antenna selection. The proposed novel techniqueoutperforms the existing algorithms for a partially connectedhybrid precoder design both in terms of spectral efficiencyand computational complexity. Moreover the antenna selectionenables efficient resource allocation for systems with largeantenna arrays. R
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