Deep Learning for Joint MIMO Detection and Channel Decoding

Transcription

1 Deep Learning for Joint Detection an Decoing Taotao Wang, Lihao Zhang an Soung Chang Liew Many etection schemes have been propose [12] Linear etection can first be use to cancel multipleantenna interference with low complexities; after that channel ecoing is performe [6] [8] In these schemes, linear etection an channel ecoing operate in a sequential manner Since linear etection introuces noise amplification an correlation, such sequential linear etection an channel ecoing schemes typically result in large perforarxiv: v1 [csit] 17 Jan 2019 Abstract We propose a eep-learning approach for the joint etection an channel ecoing problem Conventional receivers aopt a moel-base approach for etection an channel ecoing in linear or iterative manners However, ue to the complex signal moel, the optimal solution to the joint etection an channel ecoing problem (ie, the maximum lielihoo ecoing of the transmitte coewors from the receive signals) is computationally infeasible As a practical measure, the current moel-base receivers all use suboptimal ecoing methos with afforable computational complexities This wor applies the latest avances in eep learning for the esign of receivers In particular, we leverage eep neural networs (DNN) with supervise training to solve the joint etection an channel ecoing problem We show that DNN can be traine to give much better ecoing performance than conventional receivers o Our simulations show that a DNN implementation consisting of seven hien layers can outperform conventional moel-base linear or iterative receivers This performance improvement points to a new irection for future receiver esign I INTRODUCTION Multiple-antenna technology, also nown as multiple-input multiple-output (), is one of the most important techniques for avance wireless communications systems It has alreay been incorporate into many wireless stanars, eg, 80211n/ac [1] an LTE 4G [2] It has also been shown theoretically that can increase spectrum efficiency linearly with the numbers of transmit an receive antennas [3] Of much interest are low-complexity functional units that have goo performance A transmitter transmits multiple ata streams, one on each transmit antenna A receiver receives a multiplexe copy of the multiple ata streams plus noise on each receive antenna A etector emultiplexes an ecoes the multiplexe ata on all the receive antennas into the originally transmitte multiple ata streams plus noise an interference To achieve near-capacity performance, avance channel coing schemes, such as LDPC an polar coes, have been suggeste for 5G systems [4], [5] These channel coes protect the ata streams against channel faing, interference, an noise The output of a etector consists of a noisy version of the coewor transmitte by the transmitter The T Wang is with College of Information Engineering, Shenzhen University an the Department of Information Engineering, The Chinese University of Hong Kong (ttwang@szueucn) Lihao Zhang an Soung Chang Liew are with the Department of Information Engineering, The Chinese University of Hong Kong (zl018@iecuheuh, soung@iecuheuh) function of channel ecoing is to map the noisy coewor to the original information bits at the transmitter For optimal ecoing, etection an channel ecoing nee to be performe in a joint manner The conventional ecoing schemes all use a moel-base approach However, ue to the complex signal moel, the optimal solution to the joint etection an channel ecoing problem (ie, the maximum lielihoo ecoing of the transmitte coewors from the receive signals) is computationally infeasible As a practical measure, the current moel-base receivers all use suboptimal ecoing methos with afforable computational complexities For example, instea of joint etection an channel ecoing, [6] [8] propose to perform etection an channel ecoing sequentially an separately, where etection is realize by linear equalizations with zero forcing (ZF) or minimum mean square error (MMSE) criteria By contrast, [9] [11] propose to perform ecoing an channel ecoing iteratively with soft information exchanges between the two components Thus, etection an channel ecoing are performe in a joint manner However, to contain complexity, the original signal moel has been relaxe an replace by an approximate moel (ie, it separately moels the signal an the channel coe) As a result, the solutions are still suboptimal This leaves a gap for further performance improvement with better ecoer esigns To narrow the performance gap, this wor applies the latest avances in eep learning for the esign of receivers In particular, we leverage eep neural networs (DNN) with supervise training to solve the joint etection an channel ecoing problem We show that DNN can be traine to give much better ecoing performance than conventional receivers o Our simulations show that a DNN implementation consisting of seven hien layers can outperform conventional moel-base linear or iterative receivers A Relate Wor

2 mance loss ue to the mismatch between the noise moels at the output of the etector an the input of the channel ecoer To enhance the performance of etection, nonlinear etectors have also been propose, eg, etectors base on sphere ecoing [13] [15], semi-efinite relaxation [16], [17], an lattice reuction [15], [18] Unfortunately, these nonlinear etectors can only output har estimates of channel symbols, maing them incompatible with moern channel ecoers that require soft input to achieve superior ecoing performance Sphere ecoing an list ecoing algorithms were use for soft etection [9] [11], [19] that prouces soft output This soft information can then be fe to a channel ecoing Moreover, information exchange can be performe iteratively between soft etection an channel ecoing to improve the overall performance of ecoing Although these iterative ecoing schemes have better performance than the sequential schemes, their solutions are still approximate an suboptimal, ue to the mismatch between the noise moel of the soft output of the etector an the assume noise moel at the input of channel ecoer Furthermore, iterative information exchange introuces large ecoing latencies Unlie the above moel-base approaches, [20] propose a eep learning approach for etection Specifically, the metho approximates etection using eep neural networs (DNN) The metho progressively improves the approximation by ajusting the weights of a DNN base on a series of training signals Compare with moel-base etection, eep-learning etection achieves similar etection accuracies with faster etection spee However, this eep-learning etection scheme can only perform har etection an cannot be combine with a soft channel ecoing scheme DNN is use to perform channel ecoing for the first time in [21], followe by further wor in [22], [23] It was shown that DNN channel ecoing can approach the MAP performance with lower ecoing latency than traitional channel ecoing Wor [24] employe a neural networ constructe by unfoling the factor graph of linear coes to improve the performance of belief propagation ecoing when the factor graph of the linear coes contains many samll loops Wor [25] investigate DNN-base joint equalization an channel ecoing problem for non- systems A survey on the applications of eep learning to wireless systems can be foun in [26] The remainer of this paper is organize as follows Section II presents the system moel of systems Section III reviews the existing moel-base receivers Section IV presents our eep learning receiver Section V provies the simulation results Finally, Section V conclues the paper II SYSTEM MODEL This section presents the system moel of systems an the format of the receive signals Consier a system where the transmitter is equippe with M T antennas an the receiver is equippe with M R antennas The channel between each transmit-receive antenna pair is assume to incur frequency-flat faing an the channel state remains constant within one transmitte pacet We assume M T M R an M T parallel ata streams are transmitte, one on each transmit antenna Figure 1 shows the bloc iagram of the transmitter At the transmitter sie, a vector of K information bits, b = [b 1, b 2,, b K ] T, is first channel-encoe into a coewor vector c = [c 1, c 2,, c N ] T of length N = K/R, where R is the coe rate The vali set of coewors is enote by C an thus c C The coe bits in vector c are moulate to a vector of complex ata symbols, x = [ ] T x 1, x 2,, x N/B, where B is the number of coe bits per complex ata symbol The moulation constellation is scale so that the moulate symbols in x have unit average power Through serial-to-parallel conversion, the vector x is partitione into L = N/(BM T ) consecutive ata vectors of length M T, {x 1, x 2,, x L }, ie, we have x = [ x T 1, x T 2,, xl] T T Then, L pilot vectors of length M T, {p 1, p 2,, p L }, are prepene to the ata vectors {x 1, x 2,, x L } to form an M T (L + L) signal matrix X = [X p, X ], where X = [x 1, x 2,, x L ] is the M T L ata matrix that contains the ata vectors, an X p = [p1, p 2,, p L ] is the M T L pilot matrix that contains the pilot vectors We assume L M T to facilitate the channel estimation [27] The signal matrix X represents one transmitte pacet The M T symbols of the t-th column vector in the signal matrix X are simultaneous transmitte on the M T transmit antennas in the t-th time slot At the receiver sie, the receive signals are written into an M R (L + L) matrix, Y = [y 1, y 2,, y L +L], where the t-th vector y t contains the receive signals on the M R receive antennas in the t-th time slot The receive signal matrix can be written as 1 Y = HX + W (1) M T where H is an M R M T complex channel matrix with zero-mean an σ 2 -variance inepenent complex Gaussian entries, an W is the M R (L + L) aitive white Gaussian noise (AWGN) matrix that has zero-mean an unit-variance inepenent complex Gaussian entries We also ivie the receive signal matrix an the AWGN matrix into two subparts: Y = [Y p, Y ], W = [W p, W ], where Y p is the M R L matrix that contains the receive signal vectors for the transmitte pilot vectors, Y is the M R L matrix that contains the receive signal vector for the transmitte ata vectors, an W p, W, are the matrices containing the noise components in Y p, Y, respectively The aim of the receiver is to ecoe the transmitte information bits in b from the receive signal matrix Y For comparison with our propose receiver, in Section 3 we review some conventional moel-base receivers

3 b c x x x x Encoing Fig 1 QAM Moulation S/P Convertion,,, L 1 2 Pilot Insertion X p, p,, p, x, x,, x 1 2 L 1 2 L Xp X The bloc iagram for the transmitter Y p Y Estimation Fig 2 Detection Y bˆ bˆ ˆ ˆ 1, b2,, b K Decoing The bloc iagram for the linear receiver T III MODEL-BASED RECEIVERS Traitional receivers have been extensively stuie in the literature an implemente in real systems This section gives a brief overview of these receivers A symbol-wise optimal receiver ecoes each information bit, b, from the receive signal matrix Y by minimizing the symbol error probability or equivalently maximizing the a posteriori probability (APP): ˆb = arg max b p (b Y, H, C ) (2) where ˆb enotes the estimate of the information bit b, an {1, 2,, K} The problem as expresse in (2) is in fact a joint etection an channel ecoing problem, since ata symbol etection an the channel ecoing are implicitly performe in (2) We point out that joint etection an channel ecoing as in (2) require the nowlege of the channel matrix H In practice, the channel matrix is typically estimate from the receive pilot signals Y p, eg, the least square (LS) estimate of the channel matrix is given by: = ( ) M T Y p X H p Xp X H 1 p [27]; then, the channel matrix estimate is substitute bac to (2) to replace the real channel matrix H Even with the above approximation ( which replaces ) H by, Y the exact computation of APP, p b,, C, is ifficult an highly complex The computation ifficulty is ue to: i) the correlation among the ata symbols introuce by channel encoing; ii) the parallel signal interference cause by the channel Therefore, suboptimal etection an channel ecoing schemes with manageable implementation complexities are typically use in practice We overview two suboptimal schemes in the following A Linear Receivers One suboptimal etection an channel ecoing approach is to cancel the parallel signal interference with a linear etection first an then perform channel ecoing next We refer to this approach as linear receivers For example, the zero-forcing (ZF) etection [6] removes the interference by Ỹ = M T ( H) 1H Y = X + W (3) where ( Ỹ is ) the post-cancellation signals an W = MT H H H 1W is the post-cancellation noise Since parallel signal interference is alreay remove in (3), the postcancellation signals, Ỹ, can be fe to a traitional channel ecoer to recover ata symbols Figure 2 shows the bloc iagram for this linear receiver There is no loss of information in (3) since one can get bac Y from Ỹ The suboptimality in the linear ecoing arises from the fact that the traitional channel ecoer assumes the transforme noise W is white, but it is actually not after the transformation in (3) Although the complexity of this linear receiver is low, its performance is far from optimal B Iterative Receivers The secon etection an channel ecoing approach performs iterative soft-in soft-out etection an channel ecoing Using, Y an the prior information about the ata symbols, a soft etector computes the extrinsic information about the ata symbols [9] an elivers the soft information to a soft channel ecoer The soft channel ecoer then computes the new extrinsic information about the ata symbols an sen the compute new extrinsic information bac to the soft etector for further iteration In the next roun of iteration, the soft etector replaces the prior information about the ata symbols with the information sent from the soft channel ecoer an recomputes its extrinsic information about these ata symbols again Several rouns of such iterations are performe to ensure the convergence of the overall etection an channel ecoing process We refer to such iterative etection an channel ecoing schemes as iterative receivers It yiels an approximate solution to the joint etection an channel ecoing problem expresse in (2) Figure 3 shows the bloc iagram for the iterative receiver The soft etection often use is the sphere algorithm [11] an the soft channel ecoing often use is the belief propagation algorithm The complexity of the iterative receiver is much higher than that of the linear receiver Although the iterative receiver has better performance than the linear receiver oes, there is still a large performance gap with respect to the optimal receiver Moreover, the iterative information exchange introuces large ecoing latency IV DEEP-LEARNING RECEIVERS We propose to employ eep neural networs (DNN) to solve the joint etection an channel ecoing problem

4 Y p Y Estimation Fig 3 Soft Sphere Detection Decoing Joint Detection an Decoing T b1 b2 b K bˆ ˆ, ˆ,, ˆ The bloc iagram for the iterative receiver state in (2) with the goal of improving performance The DNNs are traine uner the framewor of supervise learning We consier the training of DNN at the receiver after the channel matrix estimate has alreay compute from the receive pilot signals Using this channel matrix estimate at the receiver, we generate a set of training signals to train a DNN to solve the joint etection an channel ecoing problem (2) uner the framewor of supervise learning The training an eployment framewor of DNN for is illustrate in Figure 4 We escribe the associate proceures in the following The receiver generates the training ata by calling a functional bloc that mimics the operation at the transmitter Specifically, for training purposes, the receiver ranomly generates many length-k binary vectors, b (i), i = 1, 2,, Z Each binary vector b (i) is transforme into a ata matrix X (i) using the functional bloc of the transmitter as escribe in Section II Then, with the channel matrix estimate given by the channel estimator, the receiver generates a training signal by multiplying with X (i) followe by aing AWGN: Y (i) = 1/M T X (i) + W (i) where Y (i) is the i-th training signal an W (i) is the corresponing ) generate { AWGN } The training set is given Z by D ( =, b(i), where Y (i) is the i-th Y (i) training signal an b (i) is the corresponing label for Y (i) We emphasize that the training set is epenent on the channel matrix estimate We use the generate training set to train a DNN, f θ ( ), that approximates the solution to problem (2), where θ is the set containing all the weights { of the } eges in the DNN When Z we fee the training signals to the inputs of the i=1 Y (i) DNN, we also fee the channel matrix estimate to the DNN (as illustrate in Figure 4) We optimize the DNN weights by miming the cross entropy loss function [28]: ( )) L θ, D ( Z K [ = 1 ZK i=1 =1 b (i) ln (ˆb(i) ) ( + i=1 1 b (i) ) ( ln 1 )] (i) ˆb (4) where b (i) {0, 1} is the -th target information bit of the i-th label vector b (i) (i), ˆb is the soft estimate of b (i) {0, 1} given by the DNN The training algorithm use to minimize (4) for DNN is the so calle stochastic graient escent (SGD) algorithm [28] After the training is finishe, the weights of the DNN are fixe to ˆθ an we can use the traine DNN fˆθ ( ) to ecoe the receive signals as ˆb = fˆθ (Y ) We have the following remars on this DNN for : leftmargin=*,labelsep=58mm The variables of interests to the DNN are the ata symbols in X (i) The size of the variable space is thus 2 K, where K is the length of b (i) (Note that we have the one-to-one mapping: b (i) X (i) ) Accoring to the results shown in [21], if the DNN can see all possible coewors, the ecoing performance of the DNN is the best Lie the investigation in [21], we also aopt short coes an train the DNN with all ifferent coewors The training of the DNN is quite time-consuming Therefore, the training proceure will introuce a large ecoing latency an it cannot be eploye for applications with stringent latency requirements, such as voice transmissions; it is, however, suitable for ata transmissions with relaxe latency requirements V SIMULATION RESULTS In this section, we present simulation results for the evaluation of the propose DNN receiver The moulations use are BPSK an QPSK The channel coe use is the polar coe [5] with coe rate 1/2 We assume that that each pacet consists of K = 16 bits in the simulations The aoption of the short pacet length is ue to the exponential training complexity when DNN is use to perform channel ecoing [21] 1 Pacets of short length are of interest in some practical systems such as the internet of things (IoT) After channel encoing an moulation, K = 16 information bits are transforme to 32 BPSK symbols or 16 QPSK symbols Our simulations assume matrices of imensions M R M T = 2 2, 4 4 an 8 8 We implement a DNN consisting of one input layer, six hien layers an one output layer using the eep-learning software toolit of Keras The nonlinear activation function at the neurons of the input layer an the hien layers is the Rectifie linear unit (ReLu) function [28] The input layer is a ensely-connecte layer Each hien layer is a enselyconnecte layer with batch normalization (BN) operations before the operations by ReLu The output layer is a enselyconnecte layer with the sigmoi activation functions The architecture of the DNN is illustrate in Figure 5 We train our NN over several epochs In each epoch, the graient of the loss function is compute over the entire training set using Aam, a metho for stochastic graient escent optimization [29] Our training set contains all ifferent 2 K coewors, K is the length of information bits Setting the number of learning epochs to 10 5, we train the DNN with atasets of ifferent training s (from 0 B to 6 B) After the training 1 The extension of exten DNN channel ecoing to long pacet length can follow the solution of [23] We will consier how to incorporate the solution of [23] into our DNN joint etection an channel ecoing scheme in future wor

5 Generating Training Data Y p Ranom Source Bits Generator i b Estimation Transmitter X i AWGN Generator i Y ˆ i b Receive Signals From Transmitter Y DNN Joint Detection an Decoing ˆb Fig 4 The training an eployment framewor of DNN for is finishe, the traine DNN is use to ecoe the receive signals Fig 5 The architecture of the aopte DNN consisting of six hien layers with 512, 356, 128, 64, 32, an 16 neurons respectively For comparison, we treat the following two traitional receivers as our benchmars: i) the linear receiver that employs ZF etection followe by the MAP polar ecoing of [5], ii) the iterative receiver that iterates between the sphere etection of [11] an the MAP polar ecoing of [5] We investigate the performance of receivers with perfect nowlege as well as with imperfect nowlege of the channel matrix For the latter, we assume LS estimation [27] is use to estimate the channel matrix For a fixe, we evaluate the average BER results of the receivers over 100 ifferent channel realizations Figure 6 an Figure 7 show the BER of the receivers with perfect nowlege of the channel matrix for BPSK an QPSK, respectively We can observe that our DNN receiver can inee outperform the linear an iterative receivers in terms of BER For example, the DNN receiver has aroun 1 B an 35 B gain over the linear an iterative receivers, respectively, at the BER of 10 4 for BPSK an 8 8 channels Figure 8 an Figure 9 show the BER of the receivers with imperfect nowlege of the channel matrix for BPSK an QPSK, respectively For the channel matrix estimation, we place a Haamar matrix at the beginning of the pacets as pilots an use the LS estimation base on the receive pilots to estimate the channel matrix at the receivers In general, the performance tren for the cases of perfect an imperfect channel estimates are the same The only ifference between them is that for the cases of imperfect channel estimates, the gain obtaine by our DNN receiver is even larger For example, the DNN receiver now has aroun 2 B an 10 B gain over the linear an iterative receivers at the BER of 10 4 for BPSK an 8 8 channels VI CONCLUSIONS This wor use a eep-learning tool, eep neural networ, to evelop a new solution to the problem of joint etection an channel ecoing Conventional receivers perform etection an channel ecoing in a sequential or an iterative manner The algorithms of these conventional receivers relax the signal moel of coe As a result, they are suboptimal solutions to the joint etection an channel ecoing problem, leaving the possibility for further improvement Our eep learning solution uses a DNN for joint etection an channel ecoing uner the framewor of supervise learning The eep-learning receiver oes not separate the etection an channel ecoing into two parts an oes not perform sequential or iterative operations on them It treats the etection an channel ecoing as a joint ecoing process an employs a single DNN to approximate the joint ecoing process This joint process improves the overall ecoing performance In our simulations, we traine a DNN consisting of six hien layers to ecoe signals The simulation results emonstrate notable gains obtaine by our eep-learning receiver over the conventional linear an iterative receivers A rawbac of the current propose eep-learning receiver is that the DNN nees to be traine for each ifferent channel matrix, introucing a large ecoing latency In general, to train the same DNN for ecoing with

6 BER BER BER BER 2X2 Linear Rx 2X2 Iterative Rx 2X2 DNN Rx X4 Iterative Rx 8X8 Linear Rx 8X8 DNN Rx X2 Linear Rx 2X2 Iterative Rx 2X2 DNN Rx 4X4 Iterative Rx 8X8 Linear Rx 8X8 DNN Rx Fig 6 BER of the receivers with perfect nowlege of the channel matrix for BPSK Fig 8 BER of receivers with imperfect nowlege of the channel matrix for BPSK X2 Linear Rx 2X2 Iterative Rx 2X2 DNN Rx 4X4 Iterative Rx 8X8 Linear Rx 8X8 DNN Rx Fig 7 BER of receivers with the perfect nowlege of the channel matrix for QPSK X2 Linear Rx 2X2 Iterative Rx 2X2 DNN Rx 4X4 Iterative Rx 8X8 Linear Rx 8X8 DNN Rx Fig 9 BER of receivers with imperfect nowlege of the channel matrix for QPSK ifferent channel matrices is challenging, since the space of all possible channel matrices is huge It is impossible to let the DNN see all the channel realizations In [20], a scheme to construct one DNN for etection with ifferent channel matrices is given However, it is not clear how to exten the associate DNN to solve the problem of joint etection an channel ecoing A DNN for joint etection an channel ecoing that can hanle ifferent channel matrices with one training (ie, no nee to train an reajust the weights in the DNN for each ifferent channel matrix) is an interesting irection for further investigations REFERENCES [1] O Bejarano, E W Knightly, an M Par, Ieee ac: from channelization to multi-user mimo, IEEE Communications Magazine, vol 51, no 10, pp 84 90, 2013 [2] A Ghosh an R Ratasu, Essentials of lte an lte-a Cambrige University Press, 2011 [3] A Golsmith, S A Jafar, N Jinal, an S Vishwanath, Capacity limits of mimo channels, IEEE Journal on selecte areas in Communications, vol 21, no 5, pp , 2003 [4] T Richarson an S Kuear, Design of low-ensity parity chec coes for 5g new raio, IEEE Communications Magazine, vol 56, no 3, pp 28 34, 2018 [5] E Arian, polarization: A metho for constructing capacityachieving coes for symmetric binary-input memoryless channels, IEEE Transactions on Information Theory, vol 55, no 7, pp , 2009 [6] T Haustein, C Von Helmolt, E Jorswiec, V Jungnicel, an V Pohl, Performance of mimo systems with channel inversion, in Vehicular Technology Conference, 2002 VTC Spring 2002 IEEE 55th, vol 1 IEEE, 2002, pp [7] K R Kumar, G Caire, an A L Moustaas, Asymptotic performance of linear receivers in mimo faing channels, arxiv preprint arxiv: , 2008 [8] A Heayat an A Nosratinia, Outage an iversity of linear receivers in flat-faing mimo channels, IEEE Transactions on Signal Processing, vol 55, no 12, pp , 2007 [9] B M Hochwal an S Ten Brin, Achieving near-capacity on

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