# Demodulation of EM Telemetry Data Using Fuzzy Wavelet Neural Network with Logistic Response
## 3 Methodology
### Estimating the Number of Wavelet Bases and the Pre-Selected Range for \(p_{ij}^{3}\)  
In general, fuzzy WNN methods combine wavelet theory with fuzzy logic and neural networks. The fuzzy models consist of a set of rules, and each rule acts like a \"local model\" by using a fuzzy set to partition the input space into local fuzzy regions. Each fuzzy rule corresponds to a sub-WNN consisting of wavelets with a specified dilation value (i.e., resolution). Thus, the sub-WNNs at different resolution levels are used to capture different behaviors (global or local) of the approximated function. Here, the role of the fuzzy set is to determine the contribution of the sub-WNNs to the output of the FWN. 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. In general, learning is fully automatic and does not require any external intervention, making these methods very useful in practical applications, for instance, during the gaming and field testing of sensors. Among the scaling function associated to these wavelets, the \(p_{ij}^{3}\) controls the shape of the membership function and, thus, determines the contribution of the sub-WNNs, with multiple resolutions to the output of the FWNN. Therefore, in a multi-frequency data/signal, frequency-focused output can be obtained within a regulated range of \(p_{ij}^{3}\). In other words, the value of \(p_{ij}^{3}\) affects the selective shape of the output signal and, in turn, the output resolution. Though functional approximators, other than wavelets, may have a universal approximation property, in general, they respond to the multi-resolution property of the sub-WNNs/wavelets. That is, the wavelets with coarse resolution can capture the global (low frequency) behavior easily, while the wavelets with fine resolution can capture the local behavior (higher frequency) of the function accurately. In general, the model is refined as more data are furnished to the system. With only a few data points, the information on the underlying surface is small, and a low-resolution description of the system is appropriate, while with an increasing number of data points, a higher resolution may be justified. Therefore, the number of wavelet bases (\"N\") used determines the level of variation and disparity in the wavelet function, and, in turn, increases either the smoothness or roughness or wavelength of the predicted output. Therefore, the change in the characteristic of the output signal with respect to the variation in the values of the number of wavelet bases and the pre-selected range for \(p_{ij}^{3}\) and the possible trade-off between the two properties is worth considering. In this section, we trained the FWNN using a range of values for the number of wavelet bases, n_range = 3:7, and \(p_{ij}^{3}\) as S_max = 3:7 and S_min = 1.5:5.5, using the same data as [38]. S_max and S_min represent the maximum and minimum values for \(p_{ij}^{3}\), respectively, within a single range. The number of features in the data is 5, meaning the common number of wavelet bases to be used will be 5. Furthermore, two properties obtained from the output signal (data)--(1) slope around the peaks across the data, which is a measure of the curvature, and (2) standard deviation of error (SDE) across the signal--were used to assess the possible trend in the output signal. Figure 5 shows the cross-plot of the above properties. The blue circle represents the slope around the peak across the data, while the red circle represents the SDE. The higher the value of the properties, the larger the radius of the circle. The following trends were observed in the cross-plot:  
Trend 1: Low SDE--High slope. This is a common response, with a low range of values for \(p_{ij}^{3}\), and the value along the plot of N equals 6, which is traced with a green marker.