## 4 An Efficient Convolutional Denoising Autoencoder Method for BDS NLOS Detection
### Proposed Denoising Autoencoder Module
#### 4.2.2 Dignoising Autoencoder  
For the autoencoder model, if the input contains a certain amount of noise, the reconstructed data cannot recover the input information completely. In other words, the features of the input data learned by the AE model are susceptible to noise. As a nonlinear feature extraction and dimensionality reduction technique, DAE [40] can eliminate noise and redundant information from high-dimensional spaces, thereby providing a useful low-dimensional input representation. Specifically, DAE is a deformed model that adds noise to the autoencoder. The dependence on the input signal is reduced by modeling the loss of information. The feature learning ability of the hidden layer is enhanced by introducing random noise in the input and forcing the autocoder to remove the random noise during the learning process, thus reducing its sensitivity to the input samples. The DAE improves the robustness of the encoder to complete the learning of input signals containing noise while improving its generalization ability and preventing training overfitting [41], the structure of which is shown in Figure 7. A denoising autocoder is similar to an autocoder and mainly consists of an encoder and a decoder. There is an encoder that creates a compressed representation. This representation goes into the decoder and outputs a reconstructed input [37; 38; 39]. In addition, the denoising autocoder adds extra noise to the input section to increase the robustness of the model. The DAE is mainly designed to learn more generalized features from input data containing noise, and its training process is as follows:  
Step 1: The process of destroying the initial input \(x\) by random mapping to obtain the datapoint \(\hat{x}\), which contains noise, can be expressed as follows:  
\[\hat{x}\sim q_{D}(\hat{x}\mid x) \tag{8}\]  
Figure 6: The overall structure of an autoencoder. Figure 7: The overall structure of a a denoising autoencoder. The purpose of adding noise is to hope that the model learns to withstand some random perturbations and become less sensitive to inputs. Reduced sensitivity usually means that the model is less dependent on the training set, which will help the model perform better on new data. Therefore, it helps to improve the generalization performance of the model. The probability density function of Gaussian noise follows a Gaussian distribution, which can be expressed as:  
\[f(x)=\frac{1}{\sigma\sqrt{2\pi}}exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right) \tag{9}\]  
where \(\mu\) is the expectation and \(\sigma\) is the standard deviation. Gaussian noise is used to better simulate unknown real noise. In a real environment, noise is often caused not by a single source but by a complex of many different sources of noise. If we consider real noise as the sum of many random variables with different probability distributions, each of them is independent. According to the central limit theorem, their normalized sum approaches a Gaussian distribution as the number of noise sources increases. \(S\) is the observed noise and can be expressed as  
\[S=\sum_{i=1}^{n}r_{i} \tag{10}\]  
where \(r_{i}\) is one of a large number of random noises. Step 2: The encoder maps the input values \(\hat{x}\) into a hidden representation \(h\) by means of an activation function. The process is identical to that of a conventional autoencoder, defined as follows:  
\[h=f_{\theta}(\hat{x})=s(W\hat{x}+b) \tag{11}\]  
where \(s\) is a nonlinear activation function and \(W\) and \(b\) are the weights and bias of the hidden layer, respectively. Step 3: The decoder uses an activation function to decode the hidden representation \(h\) to the output \(\hat{x}\), defined as follows:  
\[\hat{x}=g_{\theta^{\prime}}(h)=s\left(W^{\prime}h+b^{{}^{\prime}}\right) \tag{12}\]  
where \(s\) remains a nonlinear activation function, \(W^{\prime}\) and \(b^{{}^{\prime}}\) are the weights and bias of the output layer, respectively. The parameters of a DAE are \(\theta^{{}^{\prime}}=\{W,b,W^{\prime},b^{\prime}\}\).