## 3 Dynamic Flexible Processing of Sampled Signals
### Flexible Processing of Denoised Signal  
After undergoing dynamic denoising, the denoised processed signal makes it difficult to retain all the useful signal features. Therefore, we propose the concept of spatial information entropy, and from this, the denoised signal is further flexibly processed. The purpose of flexible processing is not only to compensate for the signal feature loss due to dynamic denoising but also to characterize the \(\hat{y}_{i}\) before recovering the support set. The flexible processing facilitates the subsequent recovery of the support set. Therefore, we need a criterion to determine whether the denoised signal needs to be processed or not. For the OMP algorithm, the reconstructed signal's value is the least squares solution corresponding to the support set \(\Psi\). When the \(\Psi\) is determined, its reconstruction signal is defined. Before the \(\Psi\) is recovered, there are multiple possibilities for the reconstructed signal, which make up the range of possible values for the reconstructed signal. When the signal is distorted, the uncertainty of the \(\Psi\)'s value increases. The smaller the value range of the \(\Psi\) is, the more beneficial it is for recovering the support set. Therefore, when the value range of the reconstructed signal increases, it is necessary to linearly compensate the distorted signal. In this paper, the spatial information entropy \(H(X)\) is used to reflect the solution space size of the reconstructed signal. \(H(X)\) is defined as follows. \[H(X)=\sum\limits_{i=1}^{n}-x_{i}\log(x_{i}) \tag{15}\]  
\(x_{i}\) is the result of the normalization of the signal. Where \(x_{i}\geq 0\) and \(\sum\limits_{i=1}^{n}x_{i}=1\). In information theory, when a certain event occurs with probability \(q_{i}\), the information quantity \(-\log(q_{i})\) will be obtained. There are various outcomes from the development of event \(X\). The information entropy is defined as the expectation of the amount of information brought by these outcomes. Borrowing the concept of information entropy, we define the size for the least squares solution space of the signal mapping as the spatial information entropy \(H(X)\). Higher \(H(X)\) represents more uncertainty in the \(\Psi\)'s value fetch and the reconstructed signal solution space is enlarged. Firstly, \(y_{i}\) and the denoised signal \(\hat{y}_{i}\) are normalized in their absolute value to obtain \(y_{i}^{\prime}\) and \(\hat{y}_{i}^{\prime}\). These two processed signals are divided into multiple signal fragments of length \(R\), respectively. Compensation is performed for each fragment. \[y_{i}^{\prime}=[V_{1},V_{2},\ldots V_{N}](n=1,2,\ldots N),\sum\limits_{n=1}^{ N}v_{n}=1 \tag{16}\]  
\[\hat{y}_{i}^{\prime}=[U_{1},U_{2},\ldots U_{N}](n=1,2,\ldots N),\sum\limits_{n =1}^{N}u_{n}=1 \tag{17}\]  
where \(V_{n}=[v_{n1},v_{n2},\ldots v_{nR}]\) and \(U_{n}=[u_{n1},u_{n2},\ldots u_{nR}]\). The spatial signal entropy corresponding to these two signals is as follows:  
\[ \begin{split} H(y_{i})&=\sum\limits_{n=1}^{N}H(V_{n })\\ &=\sum\limits_{n=1}^{N}\sum\limits_{r=1}^{R}-v_{nr}\ln(v_{nr}) \\ H\left(\hat{y}_{i}^{\prime}\right)&=\sum\limits_{n=1 }^{N}H(U_{n})\\ &=\sum\limits_{n=1}^{N}\sum\limits_{r=1}^{R}-u_{nr}\ln(u_{nr}) \end{split}  \tag{18}\]When \(H\big{(}\hat{y}_{i}^{\prime}\big{)}>H(y_{i})\), it represents that the signal in that fragment needs to be compensated. The signal segment \(U_{n}\) of \(\hat{y}_{i}^{\prime}\) is amplified \(k_{n}(k_{n}\geq 1)\) times, and the amplified signal is normalized again.