Channel Optimization in Mode Division Multiplexing Using Neural Networks

X 1 Z 1 Z h X i 1 O r Feed-Forward Feed Backward Input Layer Hidden Layer Output Layer U 0 1 U 0h U 1 1 U 1h U i h U i 1 V 1 r V jr 1 V 0 r MMF VCSEL Data Neural Network Bloack LG 11 LG 12 Feed backward Photodetector (a)(b) Input channels Output channels Fig. 1 MDM model with Neural Network equalization Channel Optimization in Mode Division Multiplexing Using Neural Networks Yousef Fazea*, Mohd Samsu Sajat, Amran Ahmad, Mustafa Muwafak Alobaedy 1 InterNetWorks Research Laboratory, School of Computing, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia yosiffz@internetworks.my Abstract—Mode division multiplexing (MDM) has emerged as a new multiplexing paradigm for enhancing the bandwidth by leveraging the orthogonal modes as a parallel channel for transferring information. Although capacity gains theoretically increase in relation to the number of modes in MDM, mode coupling inevitably causes modes to interchange power randomly, leading to channel degradation from different arrival mode delay and inter-symbol interference (ISI). Hence, this paper demonstrates a new neural network feed-forward and back propagation equalizer to mitigate pulse broadening caused by mode-coupling. Keywords—Channel modeling; Channel estimation; Deep learning; Feed-forward back propagation algorithm I. INTRODUCTION The information revolution has enhanced the exponential growth in network traffic in optical backbones. Nevertheless, current long-haul optical networks with single mode fiber backbones are approaching their capacity limits due to nonlinear propagation effects [ 1]. Consequently, new technologies have emerged to enhance the bandwidth of a single optical fiber through advanced multiplexing schemes [2-4]. Mode division multiplexing (MDM) has occurred as a new multiplexing paradigm for enhancing the bandwidthdistance product by exploiting the orthogonality of modes [ 5]. In MDM, modes are used as a parallel channel for transferring information. MDM has made significant strides experimentally and numerically through the development of the new spatial encoding schemes[6], fabrication of few mode fibers (FMF) [7, 8], backlighting for improving mode extinction ratio, new wave fronts with spiral phase distributions and radially offset launches for exciting specific mode groups[9- 11]. Although the capacity gains theoretically improved in relation to the MDM number of the modes, mode coupling due to manufacturing effects, bends, splices and connectors inevitably causes modes to interchange power randomly between each other, thus resulting in a differential mode delay (DMD) [ 12-14]. To alleviate the detrimental effects of mode coupling, the channel impulse response of MDM systems have been filtered through equalization techniques in conjunction with few mode fiber, multimode fiber, and spatial light modulators [ 15, 16]. Nevertheless, most of these equalization techniques are based on tapped filters, either in the time domain or in the frequency domain. Therefore, in this paper, the equalization based on mode excitation in a vertical cavity surface-emitting laser (VCSEL) is proposed. In this paper, a novel feature introduced is the application of an artificial neural network (ANN) as an equalizer for mitigating mode coupling and DMD by redistributing the power of modes. The ANN equalizer is placed at the receiver of the MDM system and acts as an adaptive filter to compensate for the effects of the varying channel distortion. This paper is proceed as Section I critically review the literature review and the related work on MDM equalization techniques. Section II presents the ANN model. Section III investigates the effects of the NN equalizer-based feed-forward propagation algorithm on the pulse shape. Section IV presents the paper’s conclusion. II. NEURAL NETWORK EQUALIZER FOR MDM 2018 IEEE 14th International Colloquium on Signal Processing & its Applications (CSPA 2018), 9 -10 March 2018, Penang, Malaysia 978-1-5386-0389-5/18/$31.00 ©2018 IEEE173 Fig. 3 Target channel impulse response for both channels Fig. 2 Input channel impulse response for both channels Fig. 4 Output channel impulse response MDM transmission was modelled and simulated in Synopsis Optsim [17] to transmit two data channels on modes LG 11 (Channel 1) and LG 12 (Channel 2), as shown in Fig. 1 (a). The ANN equalizer extracted the power coupling coefficients from the photodetector output from each channel within a predefined time and applied the feed-forward back propagation algorithm to shape the channel impulse response for each channel. The ANN equalizer achieved the desired channel impulse response by applying feed-forward back propagation algorithm to minimize the error between the equalizer output and the target output. The ANN equalizer was developed in MATLAB. The ANN equalizer is classified into three parts as shown in Fig. 1 (b), namely the input layer, the hidden layer with two nodes, and the output layer with a neuron. Two biases were used; one is at the hidden layer and the other is at the output layer. The input channel impulse response for both channels is shown in Fig 2. The target output is shown in Fig 3. The feed-forward back propagation algorithm is mathematically described as follows. In the feed-forward phase, the channel impulse response from each channel in the photodector x i will be sent to each hidden layer z h then to the output layer. This is expressed as: 1 n hohi ih i zuxu = = + ∑ (1) where the hidden layer z h has a weight u ih between neuron i in the previous layer and neuron h in the current hidden layer; u oh is the weight of the bias for neuron h, x i is the input channel Impulse response. A sigmoid transfer function was used as an activation function for each hidden layer that is expressed as: −   −   = +   = +    ∑ 1 0 1 ) 1 h n i hih i uxu ze (2) Since only one hidden layer is present in the neural network model, the hidden layer output was sent directly to the output layer o i . The input to the output layer is described as: 0 1 p irh hr j o vzv = = + ∑ (3) where v 0r is the weight of the bias for neuron r, v hr is the weight between the neuron h from hidden layer z and neuron r from the output layer. The back-propagation algorithm was applied on the output of the feed-forward mechanism, () ri o fo= (4) where o r is the output of the back-propagation a lgorithm at neuron r at the output layer. The goal of back propagation algorithm is to minimize the error between the target channel impulse response and the actual channel impulse response. The back propagation retrieved the feed-forward output in Eq (3) as an input to the hidden layer. The adjustments of the weight between each layer is calculated using v oj = -η v hj (t) and the bias was computed as ∆v oj = -ηδ j where η is the training rate and δ j is the error of the back propagation at neuron j. The channel error is sent to the hidden layer, whereby the weights were calculated as: δδ = = ∑ 1 b hr hr r v (5) The local gradient of the hidden layer z h (expressed in terms of x i ) was calculated as: ( )(1) hh h fz zz ′ = − (6) where f’(z h ) is the differentiation of the activation function for the hidden layer z. Then, u ih was updated using the weight correction as follows: ηδ∆=− ihh i ux (7) 2018 IEEE 14th International Colloquium on Signal Processing & its Applications (CSPA 2018), 9 -10 March 2018, Penang, Malaysia 174 Fig. 5 Training Performance The weights of all layers were updated simultaneously. III. RESULTS AND DISCUSSION Fig. 4 shows the output pulse of Channel 1 and Channel 2 after the neural network equaliztaion.The mean squared error (MSE) is calcuated in terms of the target as [18]: 2 0.5() i kk k MSEto=− ∑ (8) where t k is the target impulse response and o k is the equalizer output for each channel. The calculated MSE for both, the training and validation, are shown in Fig. 5 which shows no indication of over-fitting. The validation curve is almost similar to the testing curve. The maximum MSE for Channel 1 and Channel 2 is 0.018185 and 0.018254, respectively. Thus the width of the impulse response has been reduced from 0.217 ps to 0.040 ps. IV. CONCLUSION The feed-forward feedback neural network equalizer has successfully redistributed the power of modes in a 2-mode MDM model and reduced the width of the channel impulse response by 81.56%. Neural network equalization is also viable for compensation of the other channel distortions, such as chromatic dispersion and scattering. References [1] R.-J. Essiambre, "Nonlinear Capacity Limit to Optical Communications," in Nonlinear Optics, Kauai, Hawaii, 2015, p. NTu2A.3: Optical Society of America. [2] M . Ye et al., "SOI based Photonic Interconnection for Multi- Dimensional Multiplexed System," in Optical Fiber Communication Conference, Los Angeles, California, 2015, p. W1A.6: Optical Society of America. [3] Y. Fazea, A. Amphawan, and H. Abualrejal, "Wavelength division multiplexing-mode division multiplexing for MMF in access networks," Advanced Science Letters, vol. 23, no. 6, pp. 5448- 5451, 2017. [4] Y. Fazea, A. Amphawan, and A. 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