# Complex-Valued Phase Transmittance RBF Neural Networks for Massive MIMO-OFDM Receivers
## 1 Introduction
Along with the application of nonlinear filters designed for specific problems in telecommunications, artificial neural networks (ANNs) have been extensively studied in various challenging areas of digital communications, including soft and hard fault detection, channel estimation, equalization, and beamforming [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. Neural networks can operate like nonlinear filters, in a structure that can be modeled by nonlinear activation functions, as in multilayer perceptrons (MLPs), or by Gaussian neurons in radial basis function neural networks (RBFNN) [35]. RBFNN Gaussian neurons have two free parameters, namely the Gaussian centers and the variances. Moreover, there is a linear free parameter vector of weights, which linearly weighs the neuron outputs to yield the network output [35; 41]. With these three independent sets of parameters, RBFNNs are able to represent high-order nonlinear surfaces without increasing the number of layers, thereby reducing complexity compared with deep neural networks [11; 35]. In this context, this work proposes a novel complex-valued RBF neural network architecture: a multiple-input multiple-output phase transmittance RBF (MIMO-PTRBF) neural network for channel estimation and symbol detection in massive MIMO-OFDM communication systems. The PTRBF neural network model was chosen due to its lower computational complexity when compared with deep neural networks and due to its crucial role in avoiding the phase invariance which occurs in a standard complex-valued RBFNN [11; 35; 41]. The proposed MIMO-PTRBF is an extension to multiple outputs of the single-output PTRBF neural network presented in [41]. This work is based on [35], in which the authors redesigned the PTRBF neural network of [41] to obtain a low-complexity MIMO beamforming transmitter with the complex-MIMO RBF (CMM-RBF). In the present paper, using the Gaussian neuron output bounds presented in [35], we prove convergence in the mean of the PTRBF neural network, relaxing the condition of the trans-dimensional transformation of the complex-valued Gaussian neuron layer. Preliminary results indicate that the proposed architecture competes quite favorably with the conventional MIMO quasi-orthogonal space-time block coding (QOSTBC) with maximum likelihood (ML)decoding based on linear processing [11]. In addition, an STBC coding algorithm with full-rate and half-diversity was developed for massive MIMO coding with square matrices. However, because of its high computational complexity, the ML decoding may be unfeasible for mMIMO with a large number of antennas, as opposed to the MIMO-PTRBF neural network proposed here. Although other decoding techniques with lower computational complexities could be taken into account, for performance comparison, we chose ML decoding because of its optimal performance in linear channels. For example, the sphere decoding has lower computational complexity compared to the ML decoding, but its performance is only lower-bounded by the ML decoding [42]. Furthermore, the MIMO-PTRBF is able to both estimate the channel and decode the received signal, relying on a training sequence. Simulation results show that the proposed architecture achieves significantly improved BER figures when compared with MIMO QOSTBC in an equal scenario, either with linear or with nonlinear impairments under 5G channels. This paper is an extension of [PERSON] MSc. dissertation developed at the School of Electrical and Computer Engineering, University of Campinas, in the area of Telecommunications and Telematics [11]. In addition to the complex-valued RBF-based MIMO system proposed in the dissertation [11], in this paper, the MIMO-PTRBF receiver is further elaborated with additional results and with the use of 5G channel models. The remainder of this work is organized as follows. In Section 2, a brief review of multi-antenna systems is presented. The proposed STBC coding scheme and the MIMO-PTRBF for channel estimation and symbol detection in massive MIMO-OFDM systems are presented in Section 3.