# Demodulation of EM Telemetry Data Using Fuzzy Wavelet Neural Network with Logistic Response
## 2 ANN Architecture  
A neural network can be classified as either a static or dynamic network. The most common form of static network is the static feed-forward network. The output is calculated directly from the input through feed-forward connections, without feedback elements and delays such as backpropagation and cascade BPNN [13] (Figure 1). The result forms the argument of an activation function, \(\varphi\), which acts as a filter and is responsible for the resulting neuron's response as a single number [14].  
\[Y_{K}(t)=\varphi\left(\sum_{j=1}^{n}w_{kj}(t)x_{j}(t)+b_{k}(t)\right) \tag{1}\]  
Here, \(x_{j}\) (\(t\)) is the input value of parameter \(j\) at time-step \(t\); \(w_{kj}(t)\) is the weight assigned by neuron \(k\) to the input value of parameter \(j\) at time \(t\); \(\varphi\) is a non-linear activation function; \(b_{k}\) (\(t\)) is the bias of the \(k\)-neuron at time \(t\), and \(Y_{K}(t)\) is the output signal from neuron \(k\) at time \(t\).  
Dynamic networks, on the order hand, depend on both the current input to the network and the current or previous inputs, outputs, or states of the network. Examples of this are the recurrent dynamic network, with feedback connections enclosing several layers of the network, and the wavelet neural network, which is commonly used in time-series modeling [15; 16; 17].  
Wavelet neural networks (WNNs), at their inception, attracted great interest because of their advantages over radial basis function networks as they are universal approximators but achieve faster convergence and are capable of dealing with the so-called \"curse of dimensionality\" [18; 19; 20; 21]. The main characteristic of the wavelet NN is that wavelet functions are used in place of the sigmoid function as the non-linear transformation function in the hidden layer. Incorporating the time-frequency localization properties of wavelets, this technique has shown its advantages over the regular methods for complex non-linear system modeling.  
There are two major kinds of WNN structure: the first has fixed wavelet bases, that is, the dilation and translation parameters of the wavelet basis are fixed while the output layer weights are adjustable; the second type is the variable wavelet bases, with adjustable dilation and translation parameters and output layer weights. In capturing the importance of the adjustable dilation parameter, which has an explicit physical concept, i.e., resolution, as it plays a significant role in wavelet analysis and approximation of a given function, the fuzzy wavelet network (FWNN) is adopted. This improves function approximation accuracy in terms of the dilation and translation parameters of wavelets while not increasing the number of wavelet bases. Compared with fuzzy neural networks, the FWNN is the second half of embedding wavelet neural networks into fuzzy neural networks. Here, the role of the fuzzy set is to determine the contribution of the sub-WNNs to the output of the FWNN [22]. As a result, the difficulties of selecting wavelets are reduced; additionally, wavelets with different dilation values under these fuzzy rules are fully utilized to capture various essential components of the system.