## 4 Denoising Methods  
### BP Denoising  
BP denoising (BPDN) is now mainly applied to image processing denoising, signal recovery, and reconstruction [39]. It is a typical noise reduction method based on the sparse signal model. It is defined by the minimization of a convex cost function that contains a quadratic data truth term and a nondifferentiable convex penalty term, whose penalty term is a compound term of a linear operator and a L1 norm [40]. When the signal or its differential signal is sparse, the noise in the signal can be effectively suppressed, which can be regarded as a convex optimization problem with quadratic data fidelity. The BPDN algorithm extracts a subset \(y\) from the columns of \(M\), and its goal is to minimize the number of extracted columns or minimize the number of non-zero items in the vector \(x\). The \(x\) is reconstructed from the noisy data \(y\). The BPDN algorithm transforms Equation (8) into the following optimization problem:  
\[\min\frac{1}{2}\left\|y-Mx\right\|_{2}^{2}+\lambda\left\|x\right\|_{1} \tag{9}\]  
In the BPDN model, the signal can be regarded as a linear combination of multiple basis functions. Most of the basis function coefficients are zero, while only a few of them have non-zero coefficients. The basis function here is called the atom, and all the combinations of atoms are called dictionaries. When the number of atoms in the dictionary is greater than the signal dimension, the dictionary is overcomplete or redundant. Sparse representation is to find the optimal solution in the whole signal space by redundant dictionary according to the sparse metric standard. The typical sparse metric is L0. The commonly used measurement matrix is the Gaussian random matrix. It has been proved that the high probability of such a matrix satisfies the RIP property [41]. Therefore, when the Gaussian random matrix is used to observe the original signals, the original signals can be accurately recovered from the measured value with a very high probability through the reconstruction algorithm. The measurement matrix plays an important role in CS and signal reconstruction. The reconstruction error is smaller if the performance of the measurement matrix is better [42]. The measurement matrix \(M\) makes the dimension of ECG signals reconstructed from L measured values such as Figure 1. The measurement matrix \(M\) can be established according to the length of ECG signals \(N\) and the number of measurements L. So \(M\) is a matrix of \(L\times N\). In this model, considering the periodic pulse characteristics of the ECG signal on the time series and the need for real-time performance, an analysis dictionary is selected and constructed using short-time Fourier transform (STFT) [43]. \[y=Mx=My \tag{10}\]  
where \(M\psi\) satisfies the RIP feature. That is, for any sparse signal \(x\) and constant \(\xi\), \(\xi\in(0,1)\), then \(1-\xi\leq\left\|Mx\right\|_{2}^{2}/\left\|x\right\|_{2}^{2}\leq 1+\xi\). The initial observation matrix \(M\) is a full-rank matrix. The row vectors of the measurement matrix \(M\) meet the orthogonal characteristics, and the energy is equal. That is  
\[\sum_{k=1}^{N}M_{i,k}M_{j,k}=\left\{ \begin{array}{ll}0,&i\
eq j\\ \text{E},&i=j\end{array} \right. \tag{11}\]  
The M column vector of the measurement matrix is normalized, namely:  
\[\sum_{i=1}^{L}M_{i,k}^{2}=1.\ k=1,2,......L \tag{12}\]  
Figure 2 shows the BP signal reconstruction and BPDN denoising process. Figure 2a shows the original ECG signals. Figure 2b is the signal reconstruction through BP algorithm. It can be seen from Figure 2c that the measurement process amplifies the variance of the  
Figure 1: Measurement matrix construction. original noise component, which is the phenomenon of noise folding. This is mainly because the compression measurement process aliases the noise corresponding to the zero component of the sparse signal into the signal, making the noise multiplied.