The performance of the generalized MLP/BP-based DFEs is evaluated through the simulations for the (2-PAM sequences) NRZ signal recovery in a severe ISI channel (Channel 1). The training stability for different configurations of the traditional and the generalized MLP/BP neural networks is analyzed by the standard deviation (Std) of the mean square errors (MSE) of the training and the evaluation sets in numerous independent simulations with different random initiations. In addition, different levels of the nonlinear distortions in the channel that model the circuit saturation characteristics have been considered. These include the 0%, 10%, 20%, and 30% truncations of the output swing, denoted by “Sx=1.0”, “Sx=0.9”, “Sx=0.8”, and “Sx=0.7”, respectively. Furthermore, the proposed approach is applied to compensate the distorted NRZ signals in the several different ISI channels (Channels 2, 3, 4, 5, 6 and 7) and results in a significant performance improvement.
In a training procedure, the length of the training set is equal to 104 symbols and the total training epochs are 103. The two-phase learning is used with the learning rate of 0.5 when the mean square error of the training set is larger than 10-4, and the learning rate of 0.1, otherwise. When the training epochs exceed ninety percent of total epochs, the best parameters will be recorded to achieve the lowest mean square error of the training set in the last ten percent of training epochs. Hence the steady-state training results can be recognized. In fact, the simulations indicate no unstable problems as all training processes are converged.
Because different initial conditions lead to different effects, the non-training evaluation set that has 105 symbols is used to examine the training quality of numerous independent simulation outcomes. After numerous independent training and evaluation runs, those yielding better outcomes will be chosen to perform a long trial with the test set,
and then the best one will be the final test result. The length of the test set is 107 symbols, and the evaluation set is a subset of it.
At first, a severe ISI channel (Channel 1) described by the transfer function, H1=0.4575 + 0.7625z + 0.4575z-1 -2, is used to estimate the system performance of different equalization schemes, where the training noise and the evaluation noise are assumed to be SNR=15dB, and SNR of the test signal is between 10dB and 20dB.
Considering the distortions due to ISI and AWGN only, the training stability of different configurations is estimated by the standard deviation of MSEs of both the training and the evaluation sets. For each configuration, the results are analyzed according to 50 independent simulation outcomes.
Fig. 3-21 and Fig. 3-22 show the minimum MSE and the standard deviation of MSEs of the training and the evaluation sets for different hidden neuron multipliers and different summation function orders at SNR=15dB. Note that the increase of the neuron numbers in the hidden layer can decrease both the training error and the standard deviation of MSEs of the training set which can also be improved by increasing the summation function order.
Further improvements can be achieved by increasing both the neuron numbers in the hidden layer and the summation function order. However, the reciprocal advance for the evaluation set is difficult to obtain. An increase in the neuron numbers in the hidden layer can also lead to a decrease in both the training error and the standard deviation of MSEs of the evaluation set. When the summation function order is equal to 2 or 3, the evaluation error can be reduced efficiently. Beyond the order of 3 very little improvement can be observed. Therefore, the critical order can be defined as the order where the system has little enhancement with higher orders.
Although larger neuron number in the hidden layer increases the complexity, both the training error and the standard deviation of MSEs can be improved. In this work, we
select a larger scale MLP/BP neural network for better training stability and higher performance. However, in practical applications, a smaller size can be selected while considering the trade-off between performance and complexity.
In addition, there are no obvious improvements for the standard deviation of MSEs of the evaluation set with a larger summation function order. Since the training results may be located on a local minimum that leads to a limited performance, in most neural-based applications many independent runs are regularly carried out in search of the best outcomes. This demonstrates the reason for using non-training data to evaluate the training quality.
Fig. 3-23 illustrates the BERs of both the evaluation set and the test set for different hidden neuron multipliers and different summation function orders at SNR=15dB. The analogous trend in Fig. 3-23 that confirms our simulation assumption implies the estimation of training results with the evaluation set is feasible.
The BER performance for different levels of the nonlinearity with different hidden neuron multipliers at SNR=15dB is shown in Fig. 3-24. The critical order of the system with the nonlinearity is higher than the one without the nonlinearity. For “Sx=1.0” and
“Sx=0.9” with “Hx=4”, the critical order is 3, while for “Sx=0.8 and “Sx=0.7” with
“Hx=4”, the critical order increases to 4. This appearance is reasonable and confirms our expectation.
Based on the aforementioned results, the summation function order of the generalized MLP/BP-based DFEs is set to 3, and the hidden neuron multiplier is set to 4.
A comparison with the LMS DFEs, the ideal VE, the traditional MLP/BP-based DFEs and the generalized MLP/BP-based DFEs is shown in Fig. 3-25. With channel 1 and Sx=1.0, the proposed approach at BER=10-4 improves 0.8dB over the traditional MLP/BP-based DFE and 1.9dB over the LMS DFE. The proposed approach performs better than LMS
DFE and the traditional MLP/BP-based DFE in the severe ISI channel without the nonlinearity, but degrades 1.7dB as compared to the ideal VE. Nevertheless, at Sx=0.7 (30% truncations) with the BER is 10-3, the proposed scheme improves the traditional MLP/BP-based DFE by 0.8dB, the ideal VE by 2.2dB, and the LMS DFE by 4.4dB. As the distortions increase in the received signal, the proposed approaches achieve more improvement over the others.
Subsequently, several different ISI channels are described by transfer functions, which are H2(z) = 0.408 + 0.816z + 0.408z-1 -2, H3(z) = 0.3482 + 0.8704z + 0.3482z-1 -2, H4(z) = 0.227 + 0.460z + 0.688z + 0.460z + 0.227z-1 -2 -3 -4, H5(z) = 0.108 + 0.215 z-1 + 0.430 z-2 + 0.717 z + 0.430 z + 0.215 z + 0.108z-3 -4 -5 -6, H6(z) = 0.147 + 0.295 z + 0.590 z-1 -2 + 0.295 z-3 + 0.590 z-4 + 0.295 z-5 + 0.147z-6, and H7(z) =0.226 + 0.516 z + 0.645 z-1 -2 + 0.516 z-3, respectively. The time domain responses of these channels are symmetric, except that of channel 7 is asymmetric. The lengths of channel response in channels 1, 2 and 3 are the same, but the influences of ISI are different — channel 1 is the worst case, channel 2 is a better one and channel 3 is with the least interference. Moreover, channel 4, 5, 6, and 7 with longer ISI and lower signal-to-interference ratio results in a worse signal quality compared to others. Specially, the channel response of channel 7 is not symmetric.
These channels will be further used to evaluate the system performance of the equalization schemes.
The simulation results based on channels 2 and 3, where the simulation conditions are identical to those of channel 1, are shown in Fig. 3-26 and Fig. 3-27, respectively.
Note that the proposed approaches result in a better improvement over the LMS DFE and the traditional MLP/BP-based DFE in the severer ISI channel. In these three channels without nonlinearity, the ideal VE appear to outperform the traditional MLP/BP-based DFEs and the generalized MLP/BP-based DFEs. However, the accuracy of the CIR
estimator dominates the performance particularly, and thus the improvement of VE may be lower in real cases. Furthermore, the nonlinear distortions of the received signal will compromise the VE performance significantly. In the severer ISI channel with more nonlinearity, the traditional MLP/BP-based DFEs and the generalized MLP/BP-based DFEs outperform VE. The BER vs. SNR performance comparison with different equalizers for channels 1, 2, and 3 without truncations at BER=10-4 is illustrated in Tab.
3-4. That with 30% truncations at BER=10-3 is presented in Tab. 3-5.
Fig. 3-28 shows the simulation results with channel 4, where the training noise and the evaluation noise are assumed to be SNR=20dB and the SNR of the test signal is between 10dB and 25dB. Fig. 3-29, Fig. 3-30, and Fig. 3-31 show the simulation results with channel 5, 6, and 7, where the training noise and the evaluation noise are assumed to be SNR=15dB and the SNR of the test signal is between 10dB and 20dB. For these channels, the ideal VE is still the best method without nonlinear distortion in the received signal. Similarly, the performance of the LMS DFE and the ideal VE is limited due to large truncations. In these severe ISI channels, the MLP/BP-based DFEs and the GMLP/BP-based DFEs provide better robustness when large truncations present.
By above simulation results, the proposed GMLP/BP-based DFEs can yield a substantial improvement over the traditional MLP/BP-based DFE that performs better than the LMS DFE. In such applications, the proposed schemes can provide more improvement when larger truncation occurred. The proposed schemes present better capability of classifying the sampling patterns, and tolerate distortions.
Fig. 3-21: Minimum MSE and standard deviation of MSEs of the training set in Channel 1 at SNR=15dB: (a) Minimum MSE for different hidden neuron multipliers (Hx) and different summation function orders (Order), (b) Minimum MSE for different orders, (c) Minimum MSE for different multipliers, (d) Standard deviation of MSEs for different multipliers and different orders, (e) Standard deviation of MSEs for different orders, (f) Standard deviation of MSEs for different multipliers.
Fig. 3-22: Minimum MSE and standard deviation of MSEs of the evaluation set in Channel 1 at SNR=15dB: (a) Minimum MSE for different hidden neuron multipliers (Hx) and different summation function orders (Order), (b) Minimum MSE for different orders, (c) Minimum MSE for different multipliers, (d) Standard deviation of MSEs for different multipliers and different orders, (e) Standard deviation of MSEs for different orders, (f) Standard deviation of MSEs for different multipliers.
Fig. 3-23: BER of the evaluation set and the test set in Channel 1 at SNR=15dB: (a) BER of the evaluation set for different hidden neuron multipliers (Hx) and different summation function orders (Order), (b) BER of the evaluation set for different orders, (c) BER of the evaluation set for different multipliers, (d) BER of the test set for different multipliers and different orders, (e) BER of the test set for different orders, (f) BER of the test set for different multipliers.
Fig. 3-24: BER performance for different levels of the nonlinearity (Sx) with different hidden neuron multipliers (Hx) in Channel 1 at SNR=15dB.
Fig. 3-25: Channel 1 test results: (a) BER performance for different levels of the nonlinearity (Sx) at SNR=15dB, (b) BER performance for different summation function orders at SNR=15dB, (c) BER performance for different types of equalizers under training at SNR=15dB.
Fig. 3-26: Channel 2 test results: (a) BER performance for different levels of the nonlinearity (Sx) at SNR=15dB, (b) BER performance for different summation function orders at SNR=15dB, (c) BER performance for different types of equalizers under training at SNR=15dB.
Fig. 3-27: Channel 3 test results: (a) BER performance for different levels of the nonlinearity (Sx) at SNR=15dB, (b) BER performance for different summation function orders at SNR=15dB, (c) BER performance for different types of equalizers under training at SNR=15dB.
Fig. 3-28: Channel 4 test results: (a) BER performance for different levels of the nonlinearity (Sx) at SNR=18dB, (b) BER performance for different summation function orders at SNR=18dB, (c) BER performance for different types of equalizers under training at SNR=20dB.
Fig. 3-29: Channel 5 test results: (a) BER performance for different levels of the nonlinearity (Sx) at SNR=15dB, (b) BER performance for different summation function orders at SNR=15dB, (c) BER performance for different types of equalizers under training at SNR=15dB.
Fig. 3-30: Channel 6 test results: (a) BER performance for different levels of the nonlinearity (Sx) at SNR=15dB, (b) BER performance for different summation function orders at SNR=15dB, (c) BER performance for different types of equalizers under training at SNR=15dB.
Fig. 3-31: Channel 7 test results: (a) BER performance for different levels of the nonlinearity (Sx) at SNR=15dB, (b) BER performance for different summation function orders at SNR=15dB, (c) BER performance for different types of equalizers under training at SNR=15dB.
Table 3-4: The BER vs. SNR performance comparison with different equalizers for Channel 1, 2, and 3 without truncations at BER=10-4.
Channel ID
MLP/BP-based DFEs
GMLP/BP-based DFEs (Order=3) Sx Ideal VE LMS DFEs
1 1.0 14.0 dB 17.6 dB 16.5 dB 15.7 dB
2 1.0 13.8 dB 16.3 dB 15.9 dB 15.7 dB
3 1.0 12.9 dB 14.7 dB 14.7 dB 14.5 dB
Simulation Conditions for Neural-based Schemes:
Input Neural Number = 10 (Forward part: 5, and Feedback part: 5), Hidden Neural Number = 40 (Hx=4),
Output Neural Number = 1, Training Set = 104 symbols, Evaluation Set = 105 symbols, Test Set = 10 symbols, 7
Training Epoch = 103,
Learning Rate = 0.5 / 0.1 (Two Phase Learning, MSE Bound = 10 ), -3
Re-training Times = 50 Independent Runs.
Table 3-5: The BER vs. SNR performance comparison with different equalizers
Simulation Conditions for Neural-based Schemes:
Input Neural Number = 10 (Forward part: 5, and Feedback part: 5),
Re-training Times = 50 Independent Runs.
3-2-4 Summary
With multivariate power series as the summation function, the generalized MLP/BP neural network based decision feedback equalizer has been developed to compensate the distorted NRZ signal for severe ISI channels in wireline applications. In addition, the proposed approaches present continuous nonlinear pattern space mapping potential, leading to a better space mapping capability than the traditional MLP/BP neural networks in nonlinear applications. The simulation results show that the proposed equalizer can provide a significant improvement over the other schemes such as the LMS DFEs, the ideal VE, and the traditional MLP/BP-based DFEs when the received signal contains more distortions caused by ISI, AWGN and the nonlinearity.
CHAPTER 4 MIMO GMLP/BP-based DFEs
for Wireline Applications
Interconnect paths of parallel data I/O would cause the co-channel interference. The transmitted signals are tainted by the intersymbol interference that caused by the band-limited channel, the co-channel interference that caused by crosstalk between different channels, and background white noise. For recover the distorted data as well as suppress ISI, CCI and AWGN, a multi-input multi-output (MIMO) channel equalizer is essential. In wireline band-limited parallel channels, the MIMO MLP/BP-based DFEs and the MIMO GMLP/BP-based DFEs can suppress ISI, CCI and AWGN, simultaneously. By the simulations, the MIMO GMLP/BP-based DFEs can yield a substantial improvement over the MIMO MLP/BP-based DFEs that perform better than a set of the LMS DFEs.
This chapter is organized as follows. The MIMO MLP/BP-based DFEs are presented at the beginning. Afterward, the MIMO GMLP/BP-based DFEs are proposed.