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Abstract
This article aims to present, analyze and evaluate a new equalizer architecture, inspired by the butterfly equalizer used in optical communication, based on Artificial Neural Networks (ANN) of the Multi-Layer Perceptron (MLP) type for nonlinear systems with two-dimensional modulation named the Butterfly Neural Equalizer (NE-Butterfly). The NE-Butterfly is intended to equalize any channel that has real or complex taps, whether linear or nonlinear. Simulation results are presented for different types of nonlinear fiber optic channels with complex and real taps, also containing inter symbolic interference and additive noise. The results are compared with other neural equalizers in the literature with the objective of validating the performance of the NE-Butterfly, which stands out as having the overall best performance against the ones it was compared to.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The first proposals for neural equalizers (NE) using the multilayer perceptron (MLP) were presented in [1–3], in which it was observed that the process of equalization could be understood as a problem of artificial neural networks (ANN) pattern classification. However, in these studies, only one-dimensional signals with 2-PAM (pulse-amplitude modulation) were used. Besides, the signal activation function was used, limiting its functionality to receivers with channel encoders associated with abrupt decision [4, 5]. Subsequently, the studies [6, 7] proposed an ANN with MLP for bi-dimensional signals, using M-PAM and M-QAM (quadrature amplitude modulation) [4, 5] and a complex signal modified version of the backpropagation (BP) algorithm, presented in [8]. One difference in these proposals is the modification of the activation function aimed to include a wider range of levels. In the neural equalizer proposed here there is no need to do this modification, since each MLP works in one dimension of the modulated signal, thus facilitating its normalization between −1 and 1. As presented in [8–10], the main challenges of a complex MLP are the adequacy of the BP algorithm and the choice of the activation function for the complex domain, in which its singular points must be evaded.
Recently, more studies were made on how to make neural equalizers that could work well with two dimensional modulations and nonlinear channels. In [11] the performance of BP and self organizing map (SOM) NE in optical orthogonal frequency division multiplexing (OFDM) using 4-QAM was evaluated, stating that the BP one outperforms the SOM one in high signal noise ratios (SNR) and both are able to improve the bit error rate (BER) performance. In [12], a radial basis function (RBF) ANN is applied to a coherent optical OFDM (CO-OFDM) and in [13] and [14], a MLP is used to the CO-OFDM. Both cases are using 16-QAM, which shows an increase in performance when compared with an usual ANN non-linear equalizer. The work presented in [15] applied a MLP for linear and non-linear impairment mitigation in intensity-modulated and direct-detection (IM-DD) Systems. There were also studies that focused on the training algorithm, such as [16], which used particle swarm optimization (PSO) in order to train a NE for non-linear channels, and [17], that uses direct search optimization (DSO) as a trainer to its NE and presented simulations that resulted in better performance than usual BP and PSO trained NEs.
One of the approaches that standouts the most are the bidimensional neural equalizer with a multilayer perceptron network (BNE-MLP-BP) and the BNE-MLP-BP with combined errors (called here as BNE-MLP-BP with the joint error or BNE-MLP-BP-JE), proposed by [18]. Where instead of modifying the activation function in order to include the complex domain of a two dimensional modulation, it uses two neural networks, one to analyze the signal in quadrature and the other to analyze its phase. Besides dropping the necessity to change the activation function, this approach also brought better results in the algorithm’s backpropagation training convergence in relation to the other evaluated implementations.
The BNE-MLP-BP brought a new concept regarding the implementation of neural equalizers, also achieving a performance that is superior to other equalizers in optical communication systems, as observed in [19–21]. There is however a big problem when the BNE-MLP-BP is used in channels that have complex taps, according to [21] none of the neural equalizers that were analyzed (including the BNE-MLP-BP) worked in those conditions and it is necessary to convert the channel with complex taps to a channel with real taps so they were able to work.
With that in mind, aiming to surpass this difficulty, the equalizer proposed in this article, named the Butterfly Neural Equalizer (NE-Butterfly), is composed with four MLP neural networks trained with the backpropagation algorithm and has the purpose of equalizing channels of any kind, be it with real or complex taps, be it linear or nonlinear. Its architecture is based in the butterfly equalizer used in optical equalization [22], which was needed in order to make it possible for the NE-Butterfly to deal with the phase shifts that occur due to complex taps in a channel. It receives its name due to the likeliness of the connections between its inputs and outputs with a butterfly with its wings spread open.
The results are presented using BER and mean square error (MSE) curves for the backpropagation training. In order to validate the NE-Butterfly, it was analyzed and compared with the BNE-MLP-BP, the BNE-MLP-BP-JE and a neural equalizer that uses a complex activation function, as presented by [23, 24], in order to work with a two dimensional modulated channel, which was made to simulante an Optical Channel in order to both compare it with the BNE-MLP previous results and also due to the increasing application of NEs in the optical medium because of its inherent non-linearity when converting the signals from and to the electrical medium.
2. Optical system characterization
Figure 1 depicts an optical fiber communication system where the complex multilevel signal a(n) = a(n)I + j · a(n)Q is used to modulate the intensity of the laser (by signal x(t)), generating the modulating signal g(t). The signal x(t) is characterized by
where p(t) is given byTs is the period of the pulse (symbol period), and the function ft (·) characterizes a low pass filter known as a transmission filter (pulse former filter) [5], which limits the bandwidth of the signal x(t) to the size of R = 1/Ts (Hz).
In the fiber, the signal is subjected to the chromatic and polarization mode dispersion effects, with output u(t), besides, there is also the noise n(t) generated by the optical amplifiers. The n(t) (called Amplified Spontaneous Emission noise or ASE noise) can be modeled as additive, circularly symmetric, white complex Gaussian over the spectral bandwidth of the transmitted optical signal. Finally, the signal z(t) passes through a photo detector that uses the squaring modulus of the received signal to generate the electrical signal r(t), which is then equalized and sent to its destination .
3. Optical channel model
Recently, researchers have been dedicated to exploring the optical interconnect solutions to achieve > 50 Gbps data rate and < 20 km distance to support rapid increases in networking traffic. For this case, the IM-DD scheme is still preferred [15, 25–27]. At present, most of the commercial optical communication systems use the IM-DD system. Therefore, this work had as target the IM-DD, and in this case one of the nonlinearity problems is applied to the photodetector, which is a square-law detector. However, the NE-Butterfly can be applied to the coherent systems, but for this, it is necessary to model the nonlinear fiber impairments. The simulated model and NE-Butterfly proposed also applied to visible light communication (VLC) [28] and Radio-over-Fiber (RoF) [29, 30].
From the definitions obtained in [31–34] the optical channel modeling can be made using the scheme of Fig. 2, where hx(t) and hy(t) represent the impulse response in each polarization of Chromatic Dispersion (CD), with transfer function given by
where f is the base band frequency of the signal, D is the chromatic dispersion constant of the fiber, also called material dispersion, L is the fiber length, λ is the signal wavelength and c is the speed of light [34, 35], and Polarization Mode Dispersion (PMD), with transfer function given by where f is the baseband signal frequency, β is the power splitting ratio indicating what fraction of total power is projected into each axis of polarization and τ represents the Differential Group Delay (DGD) between the two polarization-axes [34, 35], nx(t) and ny(t) represent the ASE noise (in each polarization) due to the amplifiers, the operator(·) represents the non-linear effect of the optical transmitter, and finally, the operator |(·)|2 represents the non-linear effect of the photodetector.As reported in [33] and [34] the impulse response functions hx(t) and hy(t) can be represented by the expressions
and where τx and τy represent the DGD of each polarization, βx and βy represent the fraction of the total power projected into each polarization, ρk is the k-th channel coefficient due to CD and N is the channel length. The coefficients ρk can be calculated using the expression presented in [33, 34] from the parameters D, λ, c and L.The output r(t) is given by
where andSubstituting Eqs. 8 and 9 in Eq. 7, gives
where it is observed that the received signal, r(t), has a non-linear response of the transmitted signal x(t).4. Adaptive equalizer
One of the main problems that affect communication systems is called inter symbol interference (ISI) [4]. This problem happens due to the superposition of symbols of the same transmitter in a receiver, a phenomenon that is caused due to band limitation and multi-path effects.
The multi-path phenomena are caused due to signal reflexions that occur during its transmission, creating a spreading of the signal in time, which is what originates the ISI [4, 5]. The use of equalizers aim to reduce the influence of this effect in current communication systems.
The main objective of equalization is to compensate non ideal responses of the channel with the objective of restoring the signal that was transmitted, specially when channel coding is not able to make this restoration efficiently.
There are channel where ISI can be very dynamic, for this kind of channel it is made necessary to make an equalization strategy using adaptive algorithms. This algorithms conveniently manipulate the amplitude and phase of the filter coefficients of the equalizers, aiming to attenuate the ISI [4, 5]. The basic scheme of equalization is presented in Fig. 3, where P is the number of inputs of the equalizer, τd is the delay, H(z) is the Z transform of the finite impulse response (FIR) of the channel, n is the additive white gaussian noise (AWGN), that is added to the sequence u(n) in order to obtain the observation sequence r(n). The equalizer must use the information of the samples r(n), r(n − 1),…, r(n − P − 1) to estimate the transmitted symbol a(n − τd) in the instant n. Using this symbol as a reference signal, which is compared with the equalizer output (n) in the training phase, which results in the error signal e(n) that is responsible for the update of the parameters of the training algorithm in the search of the minimum global error of the cost function.
5. Butterfly neural equalizer
The use of butterfly equalizers is recent and the inspiration for the creation of the NE-Butterfly came from this architecture in optical equalizers, which act directly in the optic fiber in optical communication systems, using algorithms like LMS through FIR filters, as observed in [22, 36–40].
Differently from the butterfly equalizers found in literature, which operate only in the optic fiber, the objective of the EN-Butterfly is to equalize any kind of channel, using information from its phase and quadrature to optimize the identification of nonlinearities that affect digital communication systems.
5.1. Architecture
The structure of the NE-Butterfly is illustrated in Fig. 4, which presents an architecture composed of four MLP neural networks, where K is the number of neurons on the hidden layer. These networks are organized in two groups, the former is made of two networks that contribute to the processing of the signal received in phase uI (n): MLP-II and MLP-IQ, whereas the latter is made of two networks that contribute to the processing of the signal received in quadrature uQ(n): MLP-QI and MLP-QQ. These groups operate independently from each other and all networks run in parallel. This grouping of the networks is made with the objective that each pair has input information from both the real and complex part of the signal and, by combining these inputs, they are able to better infer information about phase shifts due to complex taps in the channel. The detailed architecture of the MLP-II, MLP-IQ, MLP-QI and MLP-QQ networks is similar to the structure presented in Fig. 5. In each network, the hidden layer activation function φ(·) is the hyperbolic tangent function and the output layer function ϕ(·) is the linear function.
Thus, the estimated phase signal in the output for each of the respective neural network group in the NE-Butterfly can be written as
where and the estimated quadrature signal in the output for its neural network group is where where and are the signals estimates aI (n) and aQ(n) respectively.With the objective of respecting the hidden layer hyperbolic tangent activation function, the signals aI (n) and aQ(n) are normalized, where the symbols must obey a minimum distance dmin such that
where M is the modulation order.5.2. Training scheme
The training scheme proposed for the NE-Butterfly is shown in Fig. 6. It has two modes of training, supervised and non supervised. In the supervised mode, the error signal in phase and quadrature are given by
and where tr is the known training sequence and d is the equalization delay. In the non supervised mode, the decision directed algorithm is used, with signals in error and quadrature being respectively and where and are decision estimates of the received signal.The training algorithm used is the backpropagation, this algorithm approaches the global minimum through the gradient method, so it can be seen as a generalization of the least mean square algorithm, but applied to a more robust structure than a linear equalizer.
6. Simulations and results
The objective of the simulations is to validate the NE-Butterfly and evaluate its performance when compared to other proposals, such as the BNE-MLP-BP, BNE-MLP-BP-JE and a complex activation function neural network. In order to evaluate the simulations, bit error rate (BER) curves in function of Eb/N0 were traced, which denominates the relation between bit energy and power spectral density. The mean square error (MSE) curves of the backpropagation training for a Eb/N0 of 20 dB were also acquired in order analyze the difference in performance between the equalizers regarding the converging time in function of the quantity of frames transmitted, where 300 frames with 4096 symbols each were used for all equalizers and different values of Eb/N0. This specific value of Eb/N0 was chosen because it is easier to compare visually the difference of the constellations between the different equalizers.
Simulations were made for a 4-QAM (M = 4) digital communication system, with no channel encoders, using the optical channel model presented in Fig. 2, with five configurations for the hx(t) and hy(t) pair, which are given by
and where the values of a – g are shown in Table 1, βx = 0.8, βy = 1−0.8 and Ts is the symbol period of the transmitted signal x(t). The parameters used in the structures of the simulated equalizers, which were chosen after many simulations were made with different parameters, with changes made to each variable (P, K, d, µ, and α) while fixing the value on the others, until an optimal result was found for each equalizer. The variables µ and α are the backpropagation algorithm parameters, and they are called of the learning rate and momentum coefficient, respectively. Coincidently, the parameters chosen using this method were the same for all of the equalizers used in this simulation. That is, P = 16, K = 32, d = 12, µ = 0.0075 and α = 1 × 10−7.
Table 1. Channel taps for the different channels used in the simulations.
Regarding Channel 1, its BER curves are presented in Fig. 7a, its MSE curves are presented in Fig. 7b for Eb/N0 = 20 dB. Through the BER curves analysis it is possible to infer that, from 8 dB onward, the NE-Butterfly achieves a lower error rate than the other equalizers, matching the BNE-MLP-BP and BNE-MLP-BP-JE 20 dB performance at just 10 dB, and matching the NE-Complex 20 dB performance at around 15 dB. Looking at the MSE curves for 20 dB, one can also view that the NE-Butterfly is one of the fastest and from around 75 frames its learning increased drastically when compared to the other equalizers.
Fig. 7 Performance curves for 4-QAM system (without channel coding) using the optical channel model for Channel 1.
For Channel 2, its BER curves are presented in Fig. 8a, its MSE curves are presented in Fig. 8b for Eb/N0 = 20dB. Analyzing the BER curves, the NE-Butterfly achieves a drastically lower error rate when compared with the other equalizers at any Eb/N0 value. As stated previously, in this case, the BNE-MLP-BP and BNE-MLP-BP-JE are unable to equalize the channel, while the NE-Complex is able to reduce its error rate from 14 dB onwards, the NE-Butterfly can still match its 20 dB performance at only 14 dB. In the MSE curves for 20 dB, it is clear that the NE-Butterfly boasts the better performance, achieving the lowest error faster and with less standard deviation than the NE-Complex, it is also possible to once more confirm that the BNE-MLP-BP and the BNE-MLP-BP-JE are unable to equalize the channel.
Fig. 8 Performance curves for 4-QAM system (without channel coding) using the optical channel model for Channel 2.
The BER curves for Channel 3 are presented in Fig. 9a, its MSE curves are presented in Fig. 9b for Eb/N0 = 20 dB. In the BER curves, from 6 dB onward, the NE-Butterfly achieves a lower error rate than the other equalizers, matching the BNE-MLP-BP and BNE-MLP-BP-JE 20 dB performance at just 6 dB, and matching the NE-Complex 20 dB performance at around 17 dB. Analyzing the MSE curves for 20 dB, it is possible to view that the NE-Butterfly is the fastest to reduce the MSE value of its backpropagation training, with the NE-Complex catching up at around 25 frames, although it has a much higher standard deviation when compared to the more stable curve that the NE-Butterfly presents.
Fig. 9 Performance curves for 4-QAM system (without channel coding) using the optical channel model for Channel 3.
Figure 10a contains the BER curves and Fig. 10b presents the MSE curves with Eb /N0 = 20 dB for Channel 4. In the analysis of the BER curves, it becomes clear that the equalization of this channel is difficult for all evaluated equalizers, with the NE-Butterfly having the overall best performance, except for a point around 16 dB, where the NE-Complex has a better performance. Even with that, the NE-Butterfly matches the 20 dB performance of the BNE-MLP-BP at 10 dB, the BNE-MLP-BP-JE 20 dB performance at 15 dB and the NE-Complex 20 dB performance at 19 dB. The MSE curves for 20 dB shows that the difference in learning speed is drastic, with the NE-Butterfly outperforming the other equalizers by a very large amount of frames, matching the other equalizers 150 frames performance at less than 25 frames.
Fig. 10 Performance curves for 4-QAM system (without channel coding) using the optical channel model for Channel 4.
Channel 5 BER curves are presented in Fig. 11a, its MSE curves are presented in Fig. 11b for Eb/N0 = 20dB. Channel 5 is a reference channel with no complex taps, so through the BER curves analysis all equalizers have a very similar performance, with only the NE-Complex drifting a little from 18 dB onward, the NE-Butterfly achieves the same performance as the BNE-MLP-BP and BNE-MLP-BP-JE, matching the NE-Complex 20 dB performance at 18 dB. In the MSE curves for 20 dB, the NE-Butterfly and the BNE-MLP-BP basically follow the same curve, with NE-Complex having a very unstable learning curve and we have the lowest value on the BNE-MLP-BP-JE from the frame 200 onwards.
Fig. 11 Performance curves for 4-QAM system (without channel coding) using the optical channel model for Channel 5.
The results obtained in this work clearly indicate that the NE-Butterfly not only works as a neural equalizer should, it also has the overall best performance when compared with previously studied neural equalizers when applied to a channel with real or complex taps.
Analyzing the result obtained for Channel 5, which has only real taps, the performance of the NE-Butterfly is on par with those of the other equalizers, specially when compared with the BNE-MLP-BP and the BNE-MLP-BP-JE which were previously analyzed on [18–21], where they performed significantly better than all compared solutions. With that, then the NE-Butterfly proved to be at least as viable as them.
When we analyze the results for the channels with complex taps, then the NE-Butterfly presents a significant improvement on performance, making it clear that it is a better equalizer than the BNE-MLP-BP and BNE-MLP-BP-JE in those fronts. Also, when compared to another neural equalizer present in literature, the NE-Complex, at worst case the NE-Butterfly performs similarly to it and at best case it is far superior. One other important aspect to take note is the performance of the NE-Complex, which has quite some trouble adhering to the optimal drop curve of BER performance (Figures 7a, 8a, 9a and 10a) due to the way that the complex activation function introduces instability in its training algorithm when dealing with complex channels.
However, there are still improvements that could be made in order to improve the NE-Butterfly’s performance in certain channels, as seen on this work, Channel 1 and Channel 4 were still difficult for the equalizer to achieve a low bit error rate, in future works we aim to add a heuristic to its error calculation, similar to what the BNE-MLP-BP-JE has, in order to improve its learning potential. We also plan to implement the neural equalizer using other neural network techniques, such as radial base function (RBF) and deep learning techniques.
7. Conclusions
This paper presented a new strategy based in ANN for the implementation of an adaptive equalizer, called NE-Butterfly. As the name implies, four MLP neural networks are used, MLP-II, MLP-IQ, MLP-QQ and MLP-QQ, connected in an architecture that is similar to a butterfly with its open wings, thus the name. The objective of this equalizer is to perform the equalization in phase and quadrature of a modulated signal that runs through a channel that has real or complex taps. In its training scheme, the error is calculated independently for each MLP pair and can be supervised or not. The NE-Butterfly was used in a simulated optical channel which has the main problems found in the optical fiber transmission medium, which are the nonlinearities in the photo-electric converters, the chromatic and polarization mode dispersions. Through simulations it was possible to test the equalizers using different values of Eb/N0 and to compare them with other previously tested equalizers, such as the BNE-MLP-BP and the BNE-MLP-BP-JE, and a neural equalizer that has a complex activation function named NE-Complex, where the NE-Butterfly proved effective in equalizing the majority of the channels that were evaluated, making it a valid proposal to use in equalization for optical channels.
Funding
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) - Finance Code 001.
Acknowledgments
The authors wish to acknowledge the financial support of the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) for their financial support.
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