Neural Network Aided Computation of Generalized Spatial Modulation Capacity Anxo Tato * , Carlos Mosquera * , Pol Henarejos † , Ana P´ erez-Neira †‡ * atlanTTic Research Center, Universidade de Vigo, Galicia, Spain † Centre Tecnol` ogic de Telecomunicacions de Catalunya (CTTC), Castelldefels, Spain ‡ Dept. of Signal Theory and Communications, Universitat Polit` ecnica de Catalunya (TSC-UPC) Email: {anxotato, mosquera}@gts.uvigo.es, {pol.henarejos, ana.perez}@cttc.cat Abstract—Generalized Spatial Modulation (GSM) is being con- sidered for future high-capacity and energy efficient terrestrial networks. A variant such as Polarized Modulation (PMod) has also a role in Dual Polarization Mobile Satellite Systems. The implementation of adaptive GSM systems requires fast methods to evaluate the channel dependent GSM capacity, which amounts to solve multi-dimensional integrals without closed-form solutions. For this purpose, we propose the use of a Multilayer Feedforward Neural Network and an associated feature selection algorithm. The resulting method is highly accurate and with much lower complexity than alternative numerical methods. Index Terms—Index Modulations, Generalized Spatial Mod- ulation, Polarized Modulation, Machine Learning, Multilayer Feedforward Neural Network. I. I NTRODUCTION The family of Index Modulations (IM) schemes [1] is gaining traction for next generation terrestrial and satellite networks. Among others, we can cite Generalized Spatial Modulation (GSM), its more simpler variant Spatial Modulation (SM) and Polarized Modulation (PMod). In all of them, part of the information is encoded in the selection of the building blocks, antennas in the case of SM and GSM, or polarizations in the case of PMod. SM and GSM have been proposed for future 5G networks [1], since they increase the spectral efficiency compared with single antenna systems with simpler hardware requirements as compared with other multi-antenna techniques, reducing the power consumption. On the other hand, PMod allows to increase the spectral efficiency of the scarce spectrum mobile satellite systems, through the use of Dual Polarization and Multiple-Input-Multiple-Output (MIMO) signal process- ing techniques [2]. Moreover, some works [3] highlight PMod as a means to improve satellite coverage in remote areas to serve the vast number of Machine-to-Machine (M2M) devices. The capacity calculation of any modulation scheme is not only interesting from a theoretical point of view, but it also has a This work was funded by the Xunta de Galicia (Secretaria Xeral de Universidades) under a predoctoral scholarship (co-funded by the European Social Fund). It is also funded by projects MYRADA (TEC2016-75103-C2- 2-R) and TERESA (TEC2017-90093-C3-1-R). practical interest. For example, the application of Adaptive Coding and Modulation (ACM) requires the evaluation of the instantaneous capacity to select the proper Modulation and Coding Scheme (MCS). In [4] the authors provide some analytical approximations to the Mutual Information (MI), i.e., the capacity constrained to specific constellations, of SM systems. However, similar approximations for GSM are not found in the literature to the best of our knowledge; the expression to compute the true GSM capacity is presented in [5], a multi-dimensional integral which does not admit a closed-form solution. In this work we evaluate the capacity of a GSM link by using a very simple neural network. Namely, we use a Multi- layer Feedforward Neural Network (MFNN) with some input features properly selected by using an algorithm which pre- process the channel matrix and the Signal to Noise Ratio (SNR). Although in the last years several works applying Machine Learning to Communications have appeared, see [6], the application of neural networks to obtain the capacity of a non-conventional modulation scheme is something new. Simulation results show that neural networks can compute successfully the capacity of SM/PMod and GSM with a very low error and moderate complexity, significantly lower than that of methods such as Monte Carlo. Thus, link adaptation methods in both terrestrial and satellite systems can track more accurately the channel capacity on the fly. This paper is structured as follows. Section II explains our system model and introduces GSM briefly. Then, in Section III the integral expressions to compute the true capacity of GSM are given. Later, Section IV provides an overview of MFNNs and it also details the algorithm to select the neural network input features. Lastly, Section V contains the main simulation results before the presentation of the main conclusions. II. SYSTEM MODEL Generalized Spatial Modulation (GSM) is a family of multi- antenna modulation schemes where information is transmitted not only by modulating the amplitude, phase and/or frequency of a sinusoidal carrier, but also by selecting the group of