There are various channel equalization schemes that are applied to different channel conditions. We survey the representative equalization approaches for wireline and wireless communications in these few years. These papers treat of different channel equalization schemes for wireline band-limited channels, wireline severe ISI channels, and wireless fading channels, respectively.
The linear equalizers can recover the distorted data in wireline band-limited channels
[1], [2], [3]. For wireline severe ISI channels or wireless fading channels, the linear equalization schemes are unsuitable [4], [5].
In serve ISI channels, the DFEs [1], [2], [3], [5] can avoid the influence of the spectral nulls and outperform the LEs. For wireline high data rate communications, DFEs are applied to improve the data rate or reduce the error rate [25], [26], [27]. In practice circuits, the channel responses of different interconnect paths of parallel data I/O are different. The receiver must tolerate channel responses variance and sampling clock skew.
Besides, CCI makes the problem more severely.
The most popular training algorithm of DFEs is the least mean squares (LMS) algorithm, which is a minimum mean square error (MMSE) solution. One of the great methods for improving DFEs, support vector machines (SVM) based DFEs [28], [29], [30]
uses the minimum bit error rate (MBER) solution instead of the MMSE solution to enhance system performance, but requires the estimation of channel impulse response (CIR) to compute the weighting vectors. Although the performance of SVM DFE is better than LMS DFE, the complexity of SVM DFE is much higher due to the additional channel estimator.
The Viterbi Equalizer (VE) [31] that requires CIR estimation can also be used in severe ISI channels and achieve much better performance. However, the accuracy of CIR dominates the performance particularly, and a nonlinear distortion of received signal will cause significant performance degradation to VE.
Because feed-forward neural network based channel equalization schemes are the most suitable architectures for very large-scale integration (VLSI) implementation, we survey the several well-liked neural network models that contain single layer perceptron (SLP) neural networks [13], [14], [16], polynomial perceptron (PP) neural network [14], [16], functional-link (FL) neural networks [14], [16], [32], radial basis function (RBF)
neural networks [14], [15], [16], counterpropagation (CP) neural networks [14], [16], [33], and MLP/BP neural networks [12], [13], [14], [15], [16].
The single layer perceptron neural network is the simplest neural network model, but it can’t solve the linear non-separable problem. In wireline applications, SLP-based channel equalizers [34] are better than LMS-based linear equalizers.
The polynomial perceptron neural network uses a polynomial function to represent the input data and then a SLP neural network to combine these represented data and generate the output. By the input data represented, PP neural networks can solve linear non-separable problems. In severe ISI channels, PPNN-based channel equalizers outperform linear equalizers [35]. In multi-path fading channels, PPNN-based channel equalizers can suppress ISI and CCI [36], [37], simultaneously. The complexity of the PPNN-based channel equalization schemes is depended on tap number and polynomial degree.
Based on the same concept, the functional-link neural networks are proposed. The higher-order input terms of the FL neural networks can be generated by the expanded functions, which comprise polynomial functions, trigonometric functions, signum functions and other nonlinear functions. The PP neural network is a special case of FL neural network. In severe ISI channels, FLNN-based channel equalizers can recover severe distorted data [38], [39], [40]. In multi-path fading channels, FLNN-based channel equalizers can suppress ISI and CCI with better performance than LEs and DFEs [41], [42], [43].
Excluding above definitely defined functions, a set of radial basis functions, which paves the input space with overlapping receptive fields, can be taken as the functional expander of the RBF neural networks. The most frequently used radial basis function is the Gaussian function. The output of the radial basis function is maximized by minimized
the Euclidean distance between the input vector and the centroid. To find the correct centers of the radial basis functions is very important. Thus, the clustering technique is the key issue [44]. The RBF-based channel equalization schemes [45], [46], [47], [48], [49]
can be applied to wireline band-limited channels, severe ISI channels with or without nonlinearity, severe ISI channels with CCI, and wireless fading channels.
Overall, the architecture of PP neural networks, FL neural networks, and RBF neural networks consists of two main parts, the functional expander and the linear combiner. The functional expander, which performs nonlinear mapping for the input data, and make the linear non-separable problem become linear separable. Afterward, a SLP neural network, which is trained by the simple delta-learning rule, is taken as the linear combiner to associate the represented input data with the desired outputs. The pattern space separating boundaries of such neural networks are nonlinear.
The counterpropagation neural network is two-layer structure. The first layer is a winner-take-all network, and the second layer is perceptron-based architecture. The learning speed of CP neural networks is faster than MLP/BP neural networks, but the accuracy is worse. The CP-based channel equalizers outperform LEs under nonlinear channel characteristics [50].
Since late 1980s, the MLP/BP neural network is the most important and most popular neural network model [12], [13], [14], [15], [16]. The MLP neural network can be regarded as a separateness-summation modus operandi. Because the summation function of the MLP/BP neural network is a first order multivariable polynomial function, the boundaries of neighbors are linear or piecewise linear. Also, it is treated as continuous linear mapping processes.
In severe ISI channels, the MLP/BP-based feedforward equalizers [51], [52], and the MLP/BP-based decision feedback equalizers [53], [54], [55] have been widely used to
distorted signal recovery. The MLP/BP neural network combined decoder and equalizer [4] merges forward equalization and data decoder in an MLP/BP neural network. It can offer higher system integration and better performance than the traditional separate solutions. The MLP/BP DFE with lattice filter [56] uses a lattice filter to whiten its input signal. The lattice filter can reject a quantity of the noise and make the signal clear. The convergence rate of the neural network, the steady state mean square error, and the bit error rate of whole system can be improved in chorus.
For constant envelope signal processing, we can separate in-phase and quadrature-phase components and then the real-value activation functions can handle this problem. Besides, there are two main approaches for the development of a complex neural network. One looks for fully complex activation function [57], [58], and has been applied to distorted QPSK signal recovery [59]. Another has used split complex activation function [60], [61], and has been also employed to channel equalization [62].
For wireless communications, the MLP/BP-based DFEs are applied to indoor radio channels [63] and digital satellite channels [64]. In wireless applications, the length of the training symbols and the number of the training epochs are sternly limited. As well, the MLP/BP-based DFEs can be used to suppress not only ISI but also CCI that is due to other co-channel users [65], [66].
The MLP neural network with hierarchical backpropagation algorithm (HBP) combines the hierarchical approach and BP algorithm [67]. It can solve some problems of the local minimum in the BP algorithm and improve the system performance. Except MMSE based learning rule, the least relative entropy (LRE) [68], [69] based learning algorithm has been applied to SLP neural network based equalizers and MLP neural network based equalizers. The dynamics of the LRE based algorithm is better than that of the MMSE based learning rule. It means that the learning speed of the LRE based
algorithm is faster than the MMSE based one. Moreover, neural networks can be trained by fuzzy if-then rules [16]. For nonlinear channel equalization applications, an adaptive neural fuzzy filter provides good performance [70].