# Demodulation of EM Telemetry Data Using Fuzzy Wavelet Neural Network with Logistic Response
## 4 Examples
### Comparison of the FWNN with BP Using Synthetic Data with Three Noise Variants  
In this section, synthetic data with defined noise characteristics added to the modulated signal are demodulated to check the robustness of the FWNN and its filtering capability. The test system is composed of data resampling and demodeling. Data reduction is the first part of the system that leads to the derivation of the more distinguishable attributes of the transmitted signal. Similarly, prior to the training of the FWNN system, the number of wavelet bases \"N\" and the pre-selected range for \(p_{ij}^{3}\) to be used were selected within a range of [5:20] and [1.5:5.5; 3:7], respectively. This was done using 352 data sets with multiple noise characteristics such as white noise, red noise, and blue noise. Figure 7 shows the representative of the pre-selected values with their respective test results. The results are presented in sections, with 101 samples per section. The test set result in green represents the noisy EMT data, while the test set prediction in red is the FWNN output signal. In the general application of the FWNN, Figure 7, with the matching test set prediction, would be the required result, meaning N equals 8 and \(p_{ij}^{3}\) ranges from 5.5 to 7 would be our selected values. However, in EMT demodulation, Figure 7,b has the best representation of the required denoised signal, similar to Figure 4 Type 2. Therefore, in this study, a value of 8 was selected for the number of wavelet bases (\"N\") and an adopted range of [4 6.5] was pre-selected for \(p_{ij}^{3}\). The value of \(p_{ij}^{3}\) is allowed to change freely within the given range during the iteration process. Using the selected values, we trained the FWNN with a logistic response using 680 randomly selected datasets with varying noise types. The training time was approximately  
Figure 7: Fuzzy wavelet neural network test result for “N” and \(p_{ij}^{3}\) range selection in EMT/EM MWD response demodulation with a logistic response; (**a**) N equals 5 and \(p_{ij}^{3}\) ranges from 2.5 to 4, (**b**) N equals 8 and \(p_{ij}^{3}\) ranges from 5 to 6.5, (**c**) N equals 20 and \(p_{ij}^{3}\) ranges from 5 to 6.5, and (**d**) N equals 8 and \(p_{ij}^{3}\) ranges from 5.5 to 7. 8 h. Thereafter, we evaluated our method on a randomly selected test set with 54 datasets. The recorded cost was 0.02, and the prediction result accuracy varied between 92.5% to 100% accuracy, with the SNR ratio varying from \(-\)20 to 10 dB. Figures 8 and 9 show part of the results for the retrieved code. The recovered code in Figure 8 was a 100% match with the transmitted code, as against that of Figure 9, where the latter codes were a mismatch due to the low signal-to-noise ratio. Figure 8: Description of the low-noise EM MWD response demodulation result using an FWNN with a logistic response. (**a**) The generated code, representing transmitted information from BHA to the surface; (**b**) the transmitted EM MWD signal based on the code in (**a**); (**c**) transmitted signal with noise; (**d**) post-processed transmitted signal with reduced data samples and more defined signal characteristics; and (**e**) recovered signal code using an FWNN with a logistic response. Figure 9: Description of high-noise EM MWD response demodulation result using an FWNN with a logistic response. (**a**) The generated code, representing transmitted information from BHA to the surface, (**b**) the transmitted EM MWD signal based on the code in (**a**); (**c**) transmitted signal with noise; (**d**) post-processed transmitted signal with reduced data samples and more defined signal characteristics; and (**e**) recovered signal code using an FWNN with a logistic response.