*2.2. J-DBN Nonlinear Equalizers*

There are two main NN-based equalization schemes, namely the blind NN equalization and adaptive NN equalization algorithms. To avoid the residual error for equalizing

nonlinear channels, the CMA equalizer can be combined with the NN algorithm. Meanwhile, DD-LMS combined with the NN equalization algorithm can further accelerate the convergence speed and ensure the optimum nonlinear decision. Our proposed J-DBN method, including an adaptive DBN-1-based equalizer and a blind DBN-2-based equalizer, is mainly based upon CMA and DD-LMS blind equalizers, as shown in Figure 2a.

**Figure 2.** The proposed J-DBN equalizers: (**a**) schematic diagram; (**b**) architecture.

Figure 2b shows the detailed architecture of our proposed J-DBN equalizer, including two sequential DBNs. Each link between one visible layer and multiple hidden layers in J-DBN is associated with the weight value *w<sup>k</sup> ij*, where *k* denotes the *k*-th hidden layer, and *i* and *j* represent the *i*-th node in the visible layer and the *j*-th node in the current hidden layer, respectively. The output of nonlinear neurons is summed as *h<sup>k</sup> <sup>j</sup>* = *σ* ∑*S <sup>i</sup>*=<sup>1</sup> *<sup>w</sup><sup>k</sup> ijxi* , where *S* is defined as the length of input samples, and *σ*(·) denotes the nonlinear active function between multiple hidden layers. The selection of a matching activation function is an important part of DBN construction.

During the feedforward training process, the output of the *j*-th neuron in the other (*k +* 1)-th visible layer can be calculated as *vk*+<sup>1</sup> *<sup>j</sup>* = *σ* ∑*<sup>N</sup> <sup>i</sup>*=<sup>1</sup> *<sup>w</sup>k*+<sup>1</sup> *ij <sup>v</sup><sup>k</sup> j* , where *v<sup>k</sup> <sup>j</sup>* is the output of the *j*-th neuron cell in the *k*th hidden layer, and the total number of cells in this layer is *N*. When *k* is increased as *λ*, we define *hL*+<sup>1</sup> *<sup>j</sup>* as the final output signal calculated from the output layer. The DBN-1 and DBN-2 equalizers update the weights according to their respective cost functions, guaranteeing the optimal solution of the whole equalization process.
