## 4 Adaptive Neuromorphic Joint FRESH Equalizer and Detector
### Sparse De-Noising Deep Auto-Encoders as Cyclostationary FRESH Equalizers  
A traditional auto-encoder neural network [27; 29; 30] is an unsupervised learning algorithm that works with the help of the backpropagation algorithm. It consists of an encoder and decoder. The encoder represents a deterministic mapping denoted by \(\mathrm{f_{0}}\) which transforms an n-dimensional  
Figure 1: Basic Block Diagram of RLS based BA-LCL-FRESH Filter. input vector \(\mathbf{x}\) into a hidden representation denoted by \(\mathbf{y}\). The hidden representation can be expressed by  
\[\mathbf{y}=\mathbf{f_{0}}(\mathbf{x})=\mathbf{s}\cdot(\mathbf{W}\cdot\mathbf{x}+ \mathbf{b}) \tag{26}\]  
where \(\theta=\{\mathbf{W},\mathbf{b}\}\) denotes the parameter set where \(\mathbf{W}\) is the d-by-d parameter matrix and \(\mathbf{b}\) is the offset vector of length 'd'. The decoder is responsible for mapping the hidden representation \(\mathbf{y}\), caused by an encoder operation, to a reconstructed vector \(\mathbf{z}\) of length d. This decoding operation can be represented mathematically by  
\[\mathbf{z}=\mathrm{g_{0^{\prime}}}(\mathbf{y})=\mathbf{s}\cdot\left(\mathbf{W }^{\prime}\cdot\mathbf{y}+\mathbf{b}^{\prime}\right) \tag{27}\]  
In the traditional auto-encoder, the output of the decoder is not an exact representation of the input vector. Rather, it can be represented in probabilistic terms as the parameters of a distribution \(\mathrm{P}(\mathrm{X}|\mathrm{Z}=\mathrm{z})\) that generates \(\mathbf{x}\) with high probability. As pointed out in Reference [27], the decoding operation involves optimization of a reconstruction error with respect to a loss function \(\mathrm{L}(\mathbf{x},\mathbf{z})\cong\left|\left|\mathbf{x}-\mathbf{z} \right|\right|^{2}\). The deep de-noising auto-encoder (DDA) is a variant of the traditional auto-encoder, where the corrupted input signal is mapped to a hidden representation, \(\mathrm{f_{0}}(\mathbf{x})=\mathrm{sigmoid}(\mathbf{W}\cdot\mathbf{x}+ \mathbf{b})\) using multiple layers of encoder and decoder. This can be achieved by joining the layers of the decoders to form a single decoder layer. The objective function in the DDA is the minimization of the squared error loss on the entire data set between the reconstructed output signal and the desired signal as mathematically expressed in Equation (28):  
\[\mathrm{E}=\frac{1}{\mathrm{N}}\sum_{i=1}^{\mathrm{N}}\left|\left|\mathrm{x_{i }}-\mathrm{z_{i}}\right|\right|^{2} \tag{28}\]  
The minimization of Equation (30) is done with respect to the weights using stochastic gradient descent approach. The concept of DDA can be extended to the area of cyclostationarity based adaptive interference cancellation using the concept of prediction filtering [25; 26]. The initial stage of FC-F-OFDM based SU signal extraction from the NOMCR signal is implemented using the DDA as shown in Figure 2.  
The DDA based FRESH filtering steps are tabulated below (Algorithm 2). As depicted in Figure 2, the error signal between the extracted PU-OFDM and the received NOMCR signal contains the FC-F-OFDM based SU signal. This is used for consecutive demodulation and channel decoding, as well as re-encoding and re-modulation of the estimated bit-stream of the  
Figure 2: Cyclic FRESH filtering using Deep De-noising Auto-Encoder (DDA) in NOMCR signal demodulation. PU-OFDM. The latter is fed back for cancellation from the received composite signal. The feedback and subsequent cancellation of estimated PU-OFDM from the received NOMCR, increases the correlation between the residual signal and its cyclic shifted versions, leading to improve desired signal estimation. As a consequence, the accuracy of estimation of the SU component in the NOMCR signal is improved. The details of auto-encoder operation are described in [27; 29; 30; 31]. Here, we provide the outline of the backpropagation-based auto-encoder algorithm for component layer extraction from the received signal.