Autoencoder based Robust Transceivers for Fading Channels using Deep Neural Networks Sandesh Rao Mattu1, Lakshmi Narasimhan T2, and A. Chockalingam1 1Department of ECE, Indian Institute of Science, Bangalore 560012 2Department of EE, Indian Institute of Technology, Palakkad 678623 Abstract—In this paper, we design transceivers for fading the codewords [3]. For given dimensions, the block codes channels using autoencoders and deep neural networks (DNN). which maximize the Hamming distance separation between ¹ ; º Specifically, we consider the problem of finding n k block their codewords achieve the best performance. The minimum codes such that the codewords are maximally separated in terms of their Hamming distance using autoencoders. We design an distance decoder (MDD) [3] based on the Euclidean distance is encoder and robust decoder for these block codes using DNNs. the conventionally used decoder for block codes. It can also be Towards this, we propose a novel training methodology for the shown that MDD is the optimal maximum likelihood receiver DNN that attempts to maximize the minimum Hamming distance when the noise in the communication channel is additive white between codewords. We propose a loss function for this train- Gaussian noise (AWGN). A disadvantage of MDD is that its ing which has stable weight updates during back propagation compared to other loss functions reported in the literature. The computational complexity is exponential in the dimension of block codes learned using the proposed methodology are found to the input message. achieve the maximal Hamming distance separation that is known In the literature, neural networks have been employed to in theory. We also propose two different receiver architectures build decoders for block codes [4]. In [2], [5], autoencoders based on fully connected deep neural network (FCDNN) and bi- were employed to design FEC codes and corresponding en- directional recurrent neural network (BRNN) that are suited for complex fading channels. The proposed DNN based receiver is coders and decoders. The design of machine learning based shown to achieve significantly better error performance when transceivers for AWGN channels was studied in [2]. The use compared to their classical counterparts in the presence of of convolutional neural networks (CNN) and recurrent neural channel model mismatches. In the presence of model mismatches networks (RNN) for digital demodulation was studied in [6]. such as imperfect channel knowledge and noise correlation, the Demodulation of signals through a fading channel using neural proposed DNN based transceiver is shown to offer increased reliability and robustness than the conventional transceiver. networks was reported in [7], [8]. Further, the design of con- Keywords – Deep neural networks, linear block codes, autoen- stellations for communication in AWGN channels using DNNs coder, correlated noise, Gaussian noise, fading, transceiver. was studied in [2], [9]. Deep learning based demodulation for MIMO fading channels has been reported in [10]. However, I. INTRODUCTION to the best of our knowledge, the design of robust transceivers Recently, deep neural networks (DNN) have shown promis- with block codes using DNNs for fading channels has not been ing performance in inference tasks in several fields [1]. A DNN reported in the literature so far. Our new contributions in this needs to be trained before it can be employed for inference paper can be summarized as follows. tasks. A trained DNN requires relatively less computational • complexity for performing the inference tasks compared to We propose an autoencoder based DNN to design block conventional optimal statistical inference methods. DNNs have codes, encoder, and decoder. • been used to build efficient wireless communication systems We propose a novel training methodology for the DNN [2]. With the advent of sophisticated software tools and to obtain block codes with maximal Hamming distance optimized hardware for machine learning, the computational separation that meets the theoretical upper bound. We complexity of machine learning algorithms have become prac- propose a loss function for this training. • tical. Current generation mobile phones and computational We propose two receiver architectures based on fully con- devices are built with fast general purpose neural networks in nected deep neural network (FCDNN) and bi-directional the hardware, which can be configured to perform signal pro- recurrent neural network (BRNN) for fading channels. • cessing tasks for communication in real-time. In this context, Finally, we show that the proposed DNN based we study the design of a transmitter and receiver using DNN transceiver is robust and it outperforms the conventional whose performance is robust to channel model mismatches. transceiver in the presence of channel model mismatches. Two primary model mismatches that often occur in practice are II. PROBLEM FORMULATION imperfect channel state information (CSI) and colored noise. Consider a point-to-point wireless communication system In the design of the transceiver, we focus on improving the with a single-antenna transmitter and receiver. At the trans- reliability of the system under practical conditions. To this mitter, the input message m of k bits is encoded using a block end, we use DNNs to design: ¹iº block codes over binary field code to a codeword c of length n bits (n > k). The encoded bits for forward error correction (FEC), and ¹iiº receivers that are are modulated using a constellation A (e.g., QAM, PSK) to robust to imperfect CSI and noise correlation. obtain the transmit signal x. The symbols in x are transmitted The metric used in the literature to quantify the performance serially over the wireless channel. Let h be the fading channel of block codes is the Hamming distance separation between gain of the wireless channel. The received signal is given by This work was supported in part by the J. C. Bose National Fellowship, = Department of Science and Technology, Government of India, and the Intel y hx n, where n is the noise vector and h is assumed to India Faculty Excellence Program. be constant over the transmission period of a codeword. Conventionally, the noise is modeled as independent and The number of errors that can be corrected by a code is given − = b dmin 1 c ¹Cº identically distributed zero mean complex Gaussian random by t 2 . Hence, it is desirable to maximize dmin variable with some variance denoted by σ2. However, in to improve the performance of the code. We shall employ practice, the noise in devices can become colored or correlated autoencoders to design block codes with maximum dmin. [11]. In [12], the authors show that the auto-correlation of such Autoencoders consist of two neural networks connected colored noise, referred to as flicker noise, is given by back to back, and they are traditionally used to obtain lower 8 dimensional representation of the input data. In traditional ><> A cτ t 2 ¹0; Toº & 1 cos 1 if τ 2 ¹− ; º autoencoders, the first neural network represents or encodes Rnn¹t; τº = ¹1−t/T º α ¹1−t/T º α To To ; (1) > o o the input data into a lower dimensional output and the second :0 elsewhere neural network decodes this compressed data to recover the where To is the period of observation, c is the lowest fluctu- original data [13]. In the proposed setup, the first neural α ation frequency, and A are positive constants that depend network encodes or represents the input signal with a high on the hardware device characteristics. For practical values of dimensional output, subsequently channel distortions are in- these parameters, it can be seen that the correlation of noise troduced, and the second neural network maps this high samples over the period of a codeword remains almost constant dimensional signal after channel distortions to the input signal [12]. Thus, the covariance matrix of the correlated noise n can space. The input of the autoencoder is the k length message. ¹ − ρº ρ ρ be given by 1 I 1, where is the correlation coefficient The output of the autoencoder are 2k length one-hot vectors computed from (1), I is an identity matrix and 1 is a matrix [14] corresponding to the input vectors. The encoder and ρ = of all ones. For 0, we get the case of the i.i.d. AWGN. decoder networks are a sequence of fully connected layers. We assume a quasi-static flat fading channel. The fade co- The autoencoder is trained in the following manner. The 2k efficients h are modeled to be a complex circularly symmetric possible messages are input to the encoder neural network in normal random variable CN¹0; 1º. To recover x from y, the a random order. The encoder neural network layers are built conventional receiver computes with the tanh activation function (c.f. (5)). The output from the h∗y h∗n = x ; (2) encoder neural network is passed through the wireless channel. jhj2 jhj2 The output of the wireless channel is fed to the decoder neural ∗ where h is the complex conjugate of h. The receiver employs network. We use the softmax activation function (σ¯ ¹zºi, where MDD to obtain the transmitted message from (2). Hence, the i = 1;:::; n) at the final layer of the decoder network. The knowledge of channel gain (CSI) is required for decoding at activation functions are the receiver. However, in practice, perfect CSI may not be ¹ º − ¹− º ¹ º exp x exp x exp zi n available. The channel gains are estimated before the decoding tanh¹xº = ;σ ¯ ¹zºi = Í ; z 2 R : (5) exp¹xº exp¹−xº n exp¹z º operation using pilot information. The error in CSI can be j=1 j modeled using the distribution of the noise at the receiver. This training is performed for several instances of CN¹0; 1º In a practical receiver, the conventional assumption of r.v. (fading case), and training is carried out with decreas- AWGN and perfect CSI may not be true.