Classification Weighted Deep Neural Network Based Channel Equalization for Massive MIMO-OFDM Systems

346 L. GE, C. QI, Y. GUO, ET AL., CLASSIFICATION WEIGHTED DNN BASED CHANNEL EQUALIZATION FOR . . . Classification Weighted Deep Neural Network Based Channel Equalization for Massive MIMO-OFDM Systems Lijun GE 1, Changcheng QI 1, Yuchuan GUO 1, Lei QIAN 1, Jun TONG 2, Peng WEI 3 1School of Electronics and Information Engineering, Tianjin Key Laboratory of Optoelectronic Detection Technology and System, Tiangong University, Tianjin, 300160, China 2Electrial, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, NSW, 2522, Australia 3Department of Electronic Engineering, Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing, 100084, China g[email protected], [email protected], [email protected], [email protected], [email protected], [email protected] Submitted March 12, 2022 / Accepted July 9, 2022 / Online first August 3, 2022 Abstract. Massive multi-input multi-output (MIMO) has attracted significant interest in academia and industry, which can efficiently increase the transmission rate. However, the error rate of conventional channel equalizations in massive MIMO systems may be high owing to the dynamic channel states in practical conditions. To solve this problem, in this paper, we propose an improved channel equalization framework based on the deep neural network (DNN). Based on the analyzed relationship between the input and output of the DNN, the data can be recovered without the channel state information. Furthermore, aiming at reducing the convergence time and enhancing the learning ability of the DNN, a classification weighted algorithm is proposed to optimize the cost function of the DNN, which is named as classification weighted deep neural network (CW-DNN). Simulation results demonstrate that compared to conventional counterparts, the proposed CW-DNN based equalizer can achieve a better normalized mean square error (NMSE). Upon approximating the optimal neural network parameters with the significantly improved convergence speed and reduced training time of the network, under the condition of the fixed learning rate. Keywords Channel equalization, classification weighted, deep neural network, massive MIMO, optimization algorithm 1. Introduction Massive multi-input multi-output (MIMO) systems with tens or hundreds of antennas are one of the most promising transmission techniques in wireless communication systems, and they can considerably enhance spatial multiplexing gain and system capacity [1], [2]. However, due to the dynamic characteristics of the channel state information (CSI) and the user scheduling, limited radio frequency (RF) resources, and the pilot contamination [2–4], efficient channel equalization is required in wireless communication systems. Channel equalization strategies for massive MIMO orthogonal frequency division multiplexing (OFDM) systems have been extensively researched in [5–11], with linear and nonlinear methods being distinguished. The former contains equalizers with zero forcing (ZF) and minimum mean square error (MMSE) ,while the latter includes the maximum likelihood (ML) and lattice reduction-aided (LRA) equalizers. Compared to the nonlinear methods, the linear equalization methods have low complexity at the cost of degraded error performance. Therefore, significant efforts are dedicated to achieving a reasonable compromise between error performance and complexity. To improve the bit error rate (BER) performance for multi-user (MU) MIMO-OFDM systems, linear equalizers based on the least mean square (LMS) and recursive least square (RLS) algorithms were proposed by the authors of [5]. Besides, low-complexity algorithms for implementing the ML detector are considered in [6] and [7]. Full noun Abbreviation channel state information CSI multi-input multi-output MIMO normalized mean square eerror NMSE fast fourier transform FFT artificial neural network ANN multilayer perceptron MLP radial basis function RBF back propagation neural network BPNN deep neural network DNN uniform linear array ULA base station BS orthogonal frequency division multiplexing OFDM least mean square LMS recursive least square RLS Tab. 1. Abbreviations used in this article. DOI: 10.13164/re.2022.0346 RADIOENGINEERING, VOL. 31, NO. 3, SEPTEMBER 2022 347 However, for massive MIMO systems, since the number of antennas is tens or hundreds times of that in MIMO systems, the above algorithms still have high computation. To improve the efficiency of channel equalization for massive MIMO systems, an adaptive equalizer combined with a channel tracking method is proposed for the multi carrier-code division multiple access (MC-CDMA) system over the rapidly fading channel under colored noise in [8]. Furthermore, to enhance the stability of the adaptive channel equalizer, using adaptive generalized decision feedback equalization and ordered consecutive interference cancellation, the authors of [9] suggested an adaptive MIMO channel equalizer. In general, the above methods are classified as the model-driven channel equalization [10], which may suffer from severe error performance degradation in practical communication scenarios with unknown essential parameters, such as CSI, bandwidth, and pilot information. To break the model-driven frame, the artificial neural network (ANN) has been paid significant attention for channel equalization, owing to its strong capability of nonlinear mapping and the characteristic learning in classification or recognition [11]. In the ANN, multiple training sets of data are introduced to optimize the weights and bias until a stable network state [12]. In the last decade, various neural network models have been developed for machine learning-aided communication systems. The conventional multilayer perceptron (MLP) has been used to mitigate the inter-channel interference in OFDM systems in [13]. A conventional radial basis function (RBF) based neural network is developed in [14–16] to enhance channel estimation in pilot-aided OFDM systems. Although the RBF-based neural network has a stronger learning ability than the MLP based neural network, the radial base center in RBF needs to be determined artificially. Furthermore, the back propagation neural network (BPNN) is adopted in the space-time coded MIMO-OFDM systems for effectively estimating the channel correlation coefficients in [17]. Generally, the ANN-based channel equalization has good performance for systems with a few antennas. However,with the increase of the number of antennas, its performance decreases obviously. Therefore, the ANN-based channel equalization method is not suitable for massive MIMO systems. The deep neural network (DNN) with more layers and more neurons is significantly studied based on deep learning (DL) to obtain better feature extraction and generalization from a large amount of data [2, 18–20]. The authors of [2] proposed a novel channel prediction framework that integrates the imperfect channel estimation of the massive MIMO-OFDM into the DNN scheme and proves that DNN is an effective method. The work of this article is to perform channel estimation, which uses DNN to directly restore data without channel state information. Compared with [2], the number of layers of the DNN used, the number of neurons and the data processing method are very different. To solve the problems of channel equalization and signal detection in wireless communication systems, the DNN is leveraged in [21–25]. The authors of [21] proposed a nonlinear signal detector based on a DNN for the OFDM system, which achieves a good error performance with high complexity. Then, to reduce the complexity, a DNN based MIMO detector was designed by the gradient descent method in [23]. Furthermore, to obtain the error performance close to the ML detection, the authors of [24] proposed a dense-layer DNN for a single-path MIMO system. Aiming at improving the high error performance in a MIMO system, the cross-entropy modelled loss function of the DNN was optimized for the softly detected MIMO signal [26]. It shows that the selection of cost function has an important effect on DNN training. However, current signal detection methods cannot be easily adapted to DNN-based signal detection methods. In addition, there are also many researches focusing on the combination of DNN and current signal detection methods to further improve the error performance. For example, the iterative signal detection was projected into the DNN structure and realized by the DL method in [27–29] to improve the MIMO detection performance and robustness. Besides, an MU-SIMO channel detection scheme was proposed in [30], which combines the advantages of the feed-forward DNN and parallel interference cancellation to effectively eliminate the co-channel interference. Additionally, according to the idea of Residual Network (ResNet), a deeper DNN model was built to improve the performance of the channel estimation and equalization in [31]. Channel equalization first needs to obtain channel state information through channel estimation. The proposed CLMMSE algorithm [1] calculates the channel autocorrelation matrix by investigating the channel prior information based on compressive sensing (CS) theory. The channel state information is estimated by the algorithm, but it brings high complexity. Motivated by this problem, an optimized channel equalization method is proposed based on the classification weighted DNN (CW-DNN). Furthermore, the performance of the proposed method is verified by using a signal detection of the minimum distance-based symbolic slicer. We formulate the equalization of the massive MIMO channel as a classification problem using the DNN. In this paper, to solve the above problems, an optimized channel equalization method is proposed based on the classification weighted DNN (CW-DNN). The main contributions are summarized as follows. •In a typical massive MIMO-OFDM system, the input signal is classified into different categories according to the one-hot mapping for the constellation. We use a DNN to directly recover the data without channel state information and then recover the corresponding classes of transmitted symbols from the received signal. •Furthermore, in order to improve the performance of channel equalization, we specially design a new loss function. The classification weighting is adopted to optimize the cost function of the DNN. Without adjusting the learning rate, the proposed CW-DNN is capable of 348 L. GE, C. QI, Y. GUO, ET AL., CLASSIFICATION WEIGHTED DNN BASED CHANNEL EQUALIZATION FOR . . . enhancing the training speed of the DNN, while preventing the solutions from falling into the saddle point or the local minimum point, so as to obtain the optimal neural network parameters. •Finally, simulation results show that the proposed method using the classification weighted optimization of the cost function is an effective alternative compared to the methods using the classical cost function. Notation and symbols: Abbreviations used in this article in Tab. 1. Notation and symbols used in this article are shown in Tab. 2. The following is a summary of the remainder of this paper: Massive MIMO-OFDM system is presented in Sec. 2. Deep learning and optimization algorithm-based channel equalizations are analyzed in Sec. 3. Simulation results are discussed in Sec. 4, and a brief conclusion is summarized in Sec. 5. Symbol Definition/Explanation 𝑁tNumber of transmit antennas 𝑁rNumber of receive antennas (·)TTranspose operator (·)HHermitian operator k·k Frobenius norm operator E[·] Expectation operator (·)−1Inverse operator ⊗Kronecker product Vec Vectorization operator Tab. 2. Notation and symbols used in this article 2. Massive MIMO-OFDM System As shown in Fig. 1, the transmitter sends an information bit stream. The number of sub-carriers is 𝑁, after serial-toparallel conversion, it becomes a group of 𝑁parallel data. And then IFFT is performed, cyclic prefix is added, after quadrature amplitude modulation (16/32QAM) a set of parallel data after constellation mapping, modulate 𝑁subcarriers to obtain OFDM symbols. Then mix white Gaussian noise to the receiving end for demodulation. The model of a massive MIMO-OFDM system with 𝑁ttransmit antennas and 𝑁rreceive antennas. The frequency-domain symbols 𝑋𝑖(𝑘)on the subcarrier 𝑘at the 𝑖-th transmit antenna are modulated on 𝑁 subcarriers with 𝑘=0,1, . . . , 𝑁 −1. Thus, the time-domain signal at the 𝑖-th transmit antenna in time 𝑛is represented as 𝑥𝑖(𝑛)= 𝑁−1 ∑︁ 𝑘=0 𝑋𝑖(𝑘)ej(2𝜋𝑘𝑛/𝑁), 𝑛 =0,1, . . . , 𝑁 −1.(1) A uniform linear array (ULA) of 𝑁tantennas is assumed to be installed on base station (BS). The Rayleigh channel fading model can be given as ℎ(𝑡)=∑︁ 𝐿 𝑎𝐿exp [j(𝜙𝐿+2𝜋 𝑓 cos (𝛽𝐿)𝑡)] (2) where 𝑎𝐿is the amplitude of the 𝐿-th propagation path from the BS to the receivers. 𝜙𝐿and 𝛽𝐿are, respectively, the angle of arrival and random phase of the 𝐿-th path. The data stream at each transmit antenna can be expressed as 𝑥(𝑁𝑖). At the receiving end, the data stream of the massive MIMO-OFDM system can be expressed as 𝑦𝑗(𝑛)= 𝑁t ∑︁ 𝑁𝑖=1 𝐿−1 ∑︁ 𝑙=0 ℎ𝑗 𝑁𝑖(𝑛, 𝑙)𝑥𝑁𝑖(𝑛−𝑙) + 𝑤𝑗(𝑛).(3) Converting the signals from time domain to frequency domain by fast Fourier transform (FFT) operations, and then the obtained signals can be expressed as 𝑌𝑗(𝑘)= 𝑁t ∑︁ 𝑁𝑖=1 𝐿−1 ∑︁ 𝑙=0 𝐻𝑗𝑁𝑖(𝑛, 𝑙)𝑋𝑁𝑖(𝑛−𝑙) + 𝑊𝑗(𝑛).(4) We assume that in a time-invariant MIMO channel, the channel impulse response remains constant over the coherence time. Therefore, the received frequency-domain signal Y(𝑘)=[𝑌1(𝑘),𝑌2(𝑘), . . . ,𝑌𝑁r(𝑘)]Tat the 𝑘-th subcarrier is expressed as Y(𝑘)=H(𝑘)X(𝑘) + W(𝑘)(5) where W(𝑘)is the additive white Gaussian noise (AWGN). In most scenarios, H(𝑘)is often an underdetermined factor and X(𝑘)can be set as certain sequence which is known at both transmitting and receiving sides. Thus, when apply channel estimation algorithms to get the channel state information, (5) can be rewritten as Y(𝑘)=X(𝑘)T⊗𝐼Vec(H(𝑘)) + W(𝑘).(6) Here, ⊗represents Kronecker product, Vec is the vectorization operation which stacks the column vector by column of the matrix. In (6), we assume that the channel gain H(𝑘)and the constellation of the data X(𝑘)are unknown. Thus, the channel equalization aims at recovering X(𝑘)from the received signal Y(𝑘)at the receiver. Additionally, convert this system equation into a matrix form, the matrix form of (4) can be expressed as y=Hx +n(7) with the transmit data vector x, the massive MIMO channel impulse matrix H, the additive white Gaussian noise (AWGN) n, and the received signal vector y. RADIOENGINEERING, VOL. 31, NO. 3, SEPTEMBER 2022 349 ,))7&3  X ,))7&3  X ,))7&3 t N X ))75HPRYH&3&KDQQHO(TXDOL]DWLRQ))75HPRYH&3))75HPRYH&3  Y  Y r N Y Fig. 1. The block diagram of the massive MIMO-OFDM system. 3. Deep Learning and Optimization Algorithm Based Channel Equalizations The massive MIMO channel equalization based on DNN can be formulated as a classification problem, where the transmitted signal is classified into different groups according to the constellation. The DNN based equalizer solves the classification problem to recover the transmitted signal. The pilot symbols are used to train the DNN, and the variation of the pilot value is used to reflect the variation of the channel. 3.1 Algorithmically for Deep Learning DNN has the same structure as a traditional neural network, including hidden layers and neurons. Furthermore, its deep design facilitates the establishment of models, particularly in highly nonlinear contexts where processing is challenging. DNN learning is divided into two phases: training and testing. The network model is initially trained using the gradient descent approach to minimize the error between the output value and the real value before implementing the effective channel parameter estimation. Afterwards when, using partial derivatives of the cost function, the network weights and biases are maintained in real time. The proposed DNN-based equalizer uses a fully connected DNN with 𝐿layers, comprising one input layer, (𝐿−2) hidden layers, and one output layer, as illustrated in Fig. 2. Let 𝑤𝑙 𝑖 𝑗 represent the weight from the 𝑖-th neuron of the (𝑙−1)-th layer to the 𝑗-th neuron to the 𝑙-th layer, and 𝑏𝑙 𝑖represent the bias unit. Thus, the layer pre-activation is delivered by 𝑧𝑙 𝑖=∑︁ 𝑗 𝑤𝑙 𝑖 𝑗 𝑎𝑙−1 𝑗+𝑏𝑙 𝑖.(8) The activation of neuron output may then be rewritten as 𝑎𝑙 𝑖=𝑓(𝑧𝑙 𝑖)=𝑓 ∑︁ 𝑗 𝑤𝑙 𝑖 𝑗 𝑎𝑙−1 𝑗+𝑏𝑙 𝑖!.(9) … … … … … … Hidden layers Output layersInput layers … … … … XiOi X1 X2 O1 O2 Fig. 2. The illustration of the DNN model. 3.2 DNN Based Channel Equalization Framework This subsection introduces the massive MIMO channel equalization based on the DNN, where the non-sparse Rayleigh channel is considered. As shown in Fig. 3, different from the CSI requirement in conventional equalizer, the DNN equalizer removes the channel estimation and directly recovers the transmitted data. In the DNN model, for the transmitted data X= 𝑋1, . . . , 𝑋𝑁tTand the received data Y=𝑌1, . . . , 𝑌𝑁rT, we can rewrite them as XΔ =<( ˜ X) =( ˜ X) YΔ =<( ˜ Y) =( ˜ Y)(10) where <(·) and =(·) denote the real and imaginary parts, respectively, therefore, X∈𝑅2𝑁t, and Y∈𝑅2𝑁r. In this paper, we adopt the one-hot mapping to Re-parameterize the discrete constellations S={𝑠1, 𝑠2, . . . , 𝑠𝑀}, where 𝑀denotes digital modulation order. Corresponding to each 𝑠𝑚 for 𝑚=1, . . . , 𝑀, we define a unit vector u𝑖∈𝑅𝑀. The four-dimensional one-hot mapping of the actual component of 16-QAM constellations, is depicted as 350 L. GE, C. QI, Y. GUO, ET AL., CLASSIFICATION WEIGHTED DNN BASED CHANNEL EQUALIZATION FOR . . . &:'11%DVHG(TXDOL]HU5) Ċ ))763 ĊĊĊĊĊĊĊĊ 5HPRYH&3$'&$:*1  Y  Y U1< l  X l  X l W N X Fig. 3. Block diagram of the CW-DNN based receiver in the massive MIMO system. S1=−3↔u1=[1,0,0,0], S2=−1↔u2=[0,1,0,0], S3=1↔u3=[0,0,1,0], S4=3↔u4=[0,0,0,1]. (11) Then, we define this mapping as the function 𝑠=𝑓oh (u), so that 𝑠𝑖=𝑓oh (u𝑖)for 𝑖=1, . . . , 𝑀. Especially, when the inputs are not unit vectors, the function is defined as 𝑥=𝑓oh (𝑥oh)= 𝑀 ∑︁ 𝑖=1 𝑠𝑖[𝑥oh]𝑖.(12) The MIMO model involves a vector of 2𝑁tsymbols which is handled by stacking the one-hot mapping of each elements. In other words, a vector 𝑥oh ∈ {0,1}2𝑁t𝑀is mapped to 𝑥∈𝑀2𝑁t. In the training samples, the training set, the test set, and the validation set occupy 70%, 15%, and 15%, respectively. The input of the DNN based equalizer is Yint =[𝑦1,1, . . . , 𝑦𝑘,𝑛r]T, where 𝑦𝑘,𝑛rrepresents the data on the 𝑘-th subcarrier at the 𝑛r-th transmit antenna. The output of the DNN based equalizer is ⌢ Xout =[⌢ 𝑥1,1, . . . , ⌢ 𝑥𝑘,𝑛t]T, where ⌢ 𝑥𝑘,𝑛trepresents the estimated data on the 𝑘-th subcarrier from the 𝑛r-th transmit antenna. First, the DNN is trained with the known 𝑌and 𝑋to minimize the error between the input and output data. After the DNN based equalizer is trained, this paper uses the gradient descent algorithm to update the weight values and the hidden layer bais values. The symbolic slicer is a signal detection method based on minimum distance, divides the network output data through the DNN based equalizer into the correct class, and then restore the original signal. Let the output signal vector of the 𝑗-th estimation be ˆ𝑦𝑗 and the transmit signal vector of the 𝑗-th sample be 𝑥𝑗, the cost function of (20) is expressed as 𝐸=1 2 𝑃 ∑︁ 𝑗=1𝑥𝑗−ˆ𝑦𝑗2(13) where 𝑃denotes the total number of samples. The goal of the proposed DNN based estimation model is computing and updating the partial derivatives of weights and bias in DNN. In this paper, mini-Batch Gradient Descent (MBGD) is used, taking 𝑁(1≤𝑁≤𝑃)samples in each training batch, the cost function can be approximated as 𝐸=1 𝑁 𝑁 ∑︁ 𝑗=1𝑥𝑗−ˆ𝑦𝑗2.(14) The DNN updates the parameters from the output of each mini-batch, which can be expressed as 𝑊𝑙=𝑊𝑙−𝑢 𝑁 ∑︁ 𝑖=1 𝜕𝐸𝑖 𝜕𝑊𝑙,(15) 𝑏𝑙=𝑏𝑙−𝑢 𝑁 ∑︁ 𝑖=1 𝜕𝐸𝑖 𝜕𝑏𝑙(16) where 𝑢denotes the learning rate of the DNN. The corresponding input and output training data of the DNN based channel equalization is indexed by received data (include data symbols and pilot symbols) at the receiver and one-hot mapping of the original constellation symbols at the BS, respectively. The pilot symbols are uniformly distributed on the subcarrier of each OFDM symbol, and the pilot positions are fixed. The input of the DNN includes the received data symbols and the received pilot symbols. Since the change of the received pilot symbols value reflects the time-varying characteristics of the channel, the implicit estimation of the channel can be taken. 3.3 CW-DNN Based Channel Equalization and Signal Dection Due to the diversity and complexity of the antenna array, the DNN based equalizer requires the longer processing time. Furthermore, the convergence rate during the initial training process is slow. During the training of the DNN, it is difficult to find an appropriate learning rate, and the parameters of the weight and bias always approach to a local optimum instead of a global optimum. To solve the problem, this section analyzes the relationship between the input pattern sent RADIOENGINEERING, VOL. 31, NO. 3, SEPTEMBER 2022 351 by the known pilot and the network output. Then, aiming at reducing the convergence time and enhacing the learning ability of the neural network, a classification weighted algorithm is proposed by optimizing the cost function of the DNN, referred to as CW-DNN. The CW-DNN based equalizer is shown in Fig. 3. The new cost function by using classification weighted is proposed in this part. Finally, at a fixed learning rate, the original gradient is increased or decreased by changing the cost function to obtain better neural network parameters. In multipath channels, the transmitted signal reaches the receiver through different paths, which may enhance or attenuate. When the signal bandwidth is larger than the coherent bandwidth of the channel, the signal will be severely distorted. On the contrary, the distortion of the signal is very small. The purpose of equalization is to eliminate the distortion caused by the channel, and the elimination of inter-symbol interference is the result of equalization. The received signal in the frequency domain is the product of the transmitted signal and the frequency domain response of the channel. The frequency domain response of the channel can be directly obtained by the pilot inserted in each symbol, so that the equalization of the multi-carrier signal can be completed by a simple single-point equalizer. In this paper, DNN-based channel equalization is frequency-domain equalization. 𝑥𝑗and ˆ𝑦𝑗are mapped to −1 and 1. The scaling factor 𝑐is set by judging the positive or negative of the product of the transmitted data and the received data, and the cost function is improved to ensure the gradient of the cost function. It must be large enough and predictable enough to provide good guidance for the learning algorithm, maintain convergence and continue to search for the global optimum.The neural network can increase the training speed of the DNN, while preventing the solution from falling into a saddle point or a local minimum, so as to obtain the optimal neural network parameters. Then use the symbol limiter to restore the equalized signal to the corresponding constellation point. The output layer of the CW-DNN uses the tanh function instead of the traditional linear function. The output of the network is a one-dimensional real array distributed in [−1,1]. The output sample ˆ𝑦𝑗need to be transformed to ˆ𝑦𝑗/𝑐, where ˆ𝑦𝑗denotes the target data that the network tries to approximate. 𝑐(𝑐 > 0) is a scaling constant to make the range of all target data match the tanh activation function of the output layer. According to the characteristics of the pilot symbols combined with the network output, when the output value is positive, it is regarded as a mapping of 1. On the contrary, when the output data is negative, the approximated value is mapped to −1. If the digital modulation at the pilot frequency is BPSK, the input value of the CW-DNN will be a sequence of ±1. CW-DNN determines whether the received data is same as the original data by judging the 𝑥𝑗ˆ𝑦𝑗. The CW-DNN uses the characteristics of the symbolic function sign (𝑥): if 𝑥≥0, returns 1; if 𝑥 < 0, returns −1. Set two adjustable weight values 𝑐1and 𝑐2respectively. When 𝑥𝑗ˆ𝑦𝑗≥0, it indicates that the estimated data symbol is the same as the real sample symbol. For example, if the transmitted data is 1, the received data is also 1, or the transmitted data is −1, and the received data is also −1. At this time sign 𝑥𝑗ˆ𝑦𝑗=1, setting 𝑐2is relatively small, and the convergence can be maintained when the known pilot symbols and the output data through the CWDNN are the same, and continue to search for the global best. When 𝑥𝑗ˆ𝑦𝑗<0, it indicates the symbols are different. At this time sign 𝑥𝑗ˆ𝑦𝑗=−1, setting 𝑐1is relatively large, the penalty can be increased when the known pilot symbols and the output data through the CW-DNN are different. The CW-DNN based equalization can improve the convergence speed in the early stage of network training and continue to converge in the later stage of training, greatly reducing the training time of network parameters and improving learning efficiency. The original cost function of the DNN changes to (18) of the CW-DNN, the gradient descent algorithm is used to obtain partial derivatives of weights and offset values, the formula can be written as 𝐸∗𝑥𝑗,ˆ𝑦𝑗=1 𝑁 𝑁 ∑︁ 𝑗=1h𝑐𝑗𝑥𝑗−ˆ𝑦𝑗2i,(17) 𝑐𝑗=𝑐1,sign 𝑥𝑗ˆ𝑦𝑗=−1, 𝑐2,sign 𝑥𝑗ˆ𝑦𝑗=1.(18) Then use the symbol limiter to restore the equalized signal to the corresponding constellation point. Minimumdistance based signal detection is used to find the symbol having the minimum Euclidean distance between the constellation alphabet and the equalized symbol ˜ 𝑌𝑗. The corresponding symbol is taken as the hard detection result ˜ 𝑋𝑗, which can be expressed as ˜ 𝑋𝑗=arg min 𝑋∈𝐴˜ 𝑌𝑗−𝑋(19) where 𝐴is the constellation set, such as the 4-QAM modulation. In order to evaluate the performance of the proposed CW-DNN based channel equalization scheme, the normalized mean square error (NMSE) performance is used, which is defined as NMSE =𝐸          𝑁 ∑︁ 𝑘=1 k𝑥(𝑘) − ˆ𝑥(𝑘)k2 2 𝑁 Í 𝑘=1 k𝑥(𝑘)k2 2           (20) where 𝑁is the total number of the transmit data bits, 𝑥(𝑘) and ˆ𝑥(𝑘)are the transmit signal and the recovery signal, respectively. The proposed channel equalization method based on CW-DNN collects pilot position data through the massive MIMO channel and adopts offline model training to provide the better NMSE performance. Besides, the proposed CWDNN algorithm contains more hidden layers and the number of neurons, which can adjust the cost function adaptively, so 352 L. GE, C. QI, Y. GUO, ET AL., CLASSIFICATION WEIGHTED DNN BASED CHANNEL EQUALIZATION FOR . . . as to learn channel data information better. Via combining the characteristics of the pilot signal and the output data, the DNN based on the optimization of the classification weighting method is used. The specific process of the algorithm is shown in Apendix A. 4. Simulation Results In this section, the performance of the proposed CWDNN based channel equalization algorithm is simulated by the Monte Carlo method. For comparison, the ZF and BPNN based equalization algorithms are also simulated. System parameters are listed in Tab. 3. The channel adopts a Rayleigh distribution channel. The transmitted signal is multiplied by the Rayleigh fading coefficient, and then a Gaussian white noise signal with a different signal-to-noise ratio power is added to the signal. The multipath amplitude follows the Rayleigh distribution and the multipath follows a random distribution. Pilot insertion method is comb pilot insertion. In the process of channel data generation, the original channel data is updated by multiplying time expansion and multipath fading to obtain the original channel data at the current moment. The transmission and reception antenna expansion is used to convert the original channel data into a channel matrix. Then, the channel matrices on the path are combined to form a channel matrix on the distinguishable delay path. The simulation data is generated in MATLAB R2019B. Trainings are performed using TensorFlow 2.0, Keras and Pyhton. The experiments are performed on a computer with Intel Core i7 CPU (1.5 GHz). The CW-DNN pilot insertion rate of the CW-DNN channel equalization is set to 20%and the training rate is set to 0.001. In addition, 500000 data sample sets are randomly allocated through the massive MIMO channel, 70% of which are considered as the training set, 15%of which are considered as the test set, and 15%of which are as the verification set for training the neural network. To optimize the cost function, the weighted factor 𝑐1and 𝑐2are defined as 2 and 0.5, respectively. In addition, the entire CW-DNN is composed of 5 layers, including an input layer, 3 hidden layers and an output layer, where the hidden layer activation function is defined as the ReLU function, and the output layer activation function is the tanh function. Parameters Values Channel type AWGN, multi-path fading channels Number of transmit antennas 64 Number of receive antennas 64 Number of subcarriers 256 FFT size 256 Modulation 16, 32QAM Length of CP 64 Subcarrier spacing 15 KHz Tab. 3. Simulation parameters of the massive MIMO-OFDM system. The performance of BER under various modulation schemes is compared in Fig. 4. As predicted, the BER curve in Fig. 4 continues to have a clear declining trend as the SNR increases. In the simulated massive MIMO system, there is no FEC under the Rayleigh channel and multipath fading channels. The data transmission is directly resumed without channel state information obtained. As shown in figure 4, the system BER performance is affected. In Fig. 5, with various SNR levels, the relationship between the number of hidden layer neurons and BER performance in the massive MIMO-OFDM system is given. The SNR levels have been set at 15 dB, 20 dB, and 25 dB, respectively. The number of neurons in the hidden layer grows by 150 in each step, starting with 100. It can be seen in Fig. 5, the BER performance is not directly proportional to the number of neurons in the hidden layers. When the number of neurons in the hidden layers increases to 850, the BER performance improvement becomes small. While, the complexity of the algorithm also increases. Furthermore, when the number of neurons in the hidden layer increases to 1150, the BER performance is degraded, due to the phenomenon of “Overfitting”. 0 5 10 15 20 25 30 SNR(dB) 10-2 10-1 100 BER 32QAM 16QAM Fig. 4. BER performance of the CW-DNN under different modulation methods. 100 250 400 550 700 850 1000 1150 1300 1450 1600 Number of Hidden Neurons 10-3 10-2 10-1 BER SNR=10 SNR=15 SNR=20 Fig. 5. BER performance versus the number of hidden neurons with varying transmitted SNRs. RADIOENGINEERING, VOL. 31, NO. 3, SEPTEMBER 2022 353 Figure 6 provides the NMSE comparison of the proposed CW-DNN based scheme with batch size =10,30, and 50. It can be observed from Fig. 6 that as the SNR increases, the result of the NMSE value decreases. Furthermore, the NMSE performance is severely degraded by the small batch size. For example, when SNR =25 dB, the NMSEs are 0.5×10−3,1.1×10−3,1.5×10−3and 2.5×10−3for batch size =10,30,50, and 100, respectively. Besides, combined with the results in Fig. 7, it can be inferred that the small batch size can effectively alleviate the problem of the gradient dispersion. Figure 7 shows the NMSE of the DNN using the MSE cost function and the proposed classification weighted optimized cost function with the varying number of the training iterations. It can be seen from Fig. 7 that the proposed system using the CW-DNN converges faster at the initial stage of training, compared to conventional DNN. As the number of the training iterations increases, the NMSE of the proposed method will converge to a fixed value. We compared the DNN structure used in the literature [32]. Figure 8 and Figure 9 are the NMSE performance of the conventional ZF, BPNN, DNN [32], and the proposed CW-DNN based channel equalizations, with 16QAM and 32QAM, respectively. The number of the DNN training iterations is 1000, the number of hidden layer neurons is 850, and the batch size is 10. It can be seen from the Fig. 8 and Fig. 9 that with the increased value of the SNR, the NMSE performance of all equalization algorithms will be improved, where the performance of the proposed CW-DNN method is the best. Figure 10 shows the effect of different numbers of channel paths on BER performance. We used 16QAM modulation. It can be seen from Fig. 10 that as the number of channel paths increases, the BER performance does not improve significantly. Thus, the performance of the proposed method is independent of the number of paths in the multipath channel. 0 5 10 15 20 25 SNR (dB) 10-3 10-2 10-1 Normalized Mean Square Error Batch size=50 Batch size=30 Batch size=10 Fig. 6. The CW-DNN based scheme’s NMSE performance with batch sizes of 10, 30, and 50. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Number of Iterations 0 20 40 60 80 100 120 140 MSE DNN CW-DNN Fig. 7. Performance of a DNN-based equalizer in terms of convergence under various cost functions. 0 5 10 15 20 25 30 SNR(dB) 10-3 10-2 10-1 100 Normalized Mean Square Error =)%3'11&:'11 Fig. 8. NMSE performance when digital modulation is 16 QAM under different equalizers. 0 5 10 15 20 25 30 SNR(dB) 10-3 10-2 10-1 100 Normalized Mean Square Error ZF BP DNN CW-DNN Fig. 9. NMSE performance when digital modulation is 32 QAM under different equalizers. 354 L. GE, C. QI, Y. GUO, ET AL., CLASSIFICATION WEIGHTED DNN BASED CHANNEL EQUALIZATION FOR . . . 0 5 10 15 20 25 SNR (dB) 10-3 10-2 10-1 100 BER Channel Path=4 Channel Path=8 Channel path=12 Fig. 10. The BER performance of the CW-DNN based channel equalization in different channel paths. Operation Complexity MMSE O(𝑁3 r) ML O(𝑀 𝑁t) CW-DNN O(3𝑀 𝑁t) Tab. 4. Computational complexity of schemes. We compare the computation complexities of the proposed DNN-based channel Equalizations scheme and the conventional channel Equalizations scheme in Tab. 4. Assuming that the modulation method used in communication is M-QAM, the modulated signal reaches the receiving end through the channel, and the constellation diagram of the modulated signal is divided into M types. Note that, in our proposed scheme, we train a neural network offline using channel simulation values generated from specific channel scenarios. In the case of offline learning, computational complexity is a lesser concern, as the time required is usually not strictly limited. The computational complexity of the scheme in the test phase includes three fully connected operations. 5. Conclusions In this paper, we propose a DNN based improved channel equalization framework, which is formulated as a classification problem. Aiming at reducing the convergence time and enhacing the learning ability of the neural network, we also propose a classification weighted algorithm to optimize the cost function of the DNN. Simulation results demonstrate that under the condition of the fixed learning rate, the proposed CW-DNN based equalizer can obtain the approximated optimal neural network parameters with the significantly improved convergence speed and reduced training time. Compared to the conventional channel equalization methods of ZF and BPNN, the CW-DNN based channel equalization algorithm can provide better NMSE performance. Acknowledgments This work has been supported in part by the national natural science foundation of China (61302062, 61601325, 61901298), in part by the natural science foundation of Tianjin (13JCQNJC00900, 20JCQNJC00300) in China, and in part by Tianjin Key Laboratory of Optoelectronic Sensor and Sensing Network Technology. References [1] GE, L., ZHANG, Y., CHEN, G., et al. Compression-based LMMSE channel estimation with adaptive sparsity for massive MIMO in 5G systems. IEEE Systems Journal, 2019, vol. 13, no. 4, p. 3847–3857. 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