# Extended Smoothing Methods for Sparse Test Data Based on Zero-Padding
## 3 Calculation Process of Smoothing Theory
### Zero-Padding  
The angular spectrum in the wavenumber domain is extended by zero-padding, and the angular spectrum value of a specific position is changed accordingly. This zero-paddingoperation does not add additional information about the wavenumber angular spectrum in the wavenumber domain. The specific steps are as follows. The generated angular spectrum \(P^{\prime}\) is acquired by zero-padding of the original angular spectrum \(P\). The data numbers of the original angular spectrum in \(k_{x}\) and \(k_{y}\) direction are \(N_{x}\) and \(N_{y}\), respectively, and those of the generated angular spectrum are \(2N_{x}\) and \(2N_{y}\), respectively. When the original angular spectrum is zero-padded and extended, the generated angular spectrum value is twice the original angular spectrum value at the following positions:  
\[ \begin{cases}P^{\prime}(n_{x}\Delta k_{x},n_{y}\Delta k_{y})=2P(n_{x}\Delta k_{ x},n_{y}\Delta k_{y})&n_{x}=0,\cdots,\frac{Nx}{2}-1;n_{y}=0,\cdots,\frac{Ny}{2}-1\\ P^{\prime}\big{(}(n_{x}+Nx)\Delta k_{x},n_{y}\Delta k_{y}\big{)}=2P(n_{x}\Delta k _{x},n_{y}\Delta k_{y})&n_{x}=\frac{Nx}{2}+1,\cdots,Nx-1;n_{y}=0,\cdots,\frac{ Ny}{2}-1\\ P^{\prime}\big{(}n_{x}\Delta k_{x},(n_{y}+Ny)\Delta k_{y}\big{)}=2P(n_{x}\Delta k _{x},n_{y}\Delta k_{y})&n_{x}=0,\cdots,\frac{Nx}{2}-1;n_{y}=\frac{Ny}{2}+1, \cdots,Ny-1\\ P^{\prime}\big{(}(n_{x}+Nx)\Delta k_{x},\big{(}n_{y}+Ny\big{)}\Delta k_{y}\big{)}= 2P(n_{x}\Delta k_{x},n_{y}\Delta k_{y})&n_{x}=\frac{Nx}{2}+1,\cdots,Nx-1;n_{y}= \frac{Ny}{2}+1,\cdots,Ny-1\end{cases}  \tag{4}\]  
At the following specified positions, the generated angular spectrum value remains the same as the original angular spectrum value, as shown in Equation (5):  
\[ \begin{cases}P^{\prime}(\frac{Nx}{2}\Delta k_{x},n_{y}\Delta k_{y})=P(\frac{Nx }{2}\Delta k_{x},n_{y}\Delta k_{y})&n_{y}=0,\cdots,\frac{Ny}{2}-1\\ P^{\prime}(\frac{3 Nx}{2}\Delta k_{x},n_{y}\Delta k_{y})=P(\frac{Nx}{2}\Delta k _{x},n_{y}\Delta k_{y})&n_{y}=0,\cdots,\frac{Ny}{2}-1\\ P^{\prime}(n_{x}\Delta k_{x},\frac{Ny}{2}\Delta k_{y})=P(n_{x}\Delta k_{x}, \frac{Ny}{2}\Delta k_{y})&n_{x}=0,\cdots,\frac{Nx}{2}-1\\ P^{\prime}((n_{x}+Nx)\Delta k_{x},\frac{Ny}{2}\Delta k_{y})=P(n_{x}\Delta k_{x}, \frac{Ny}{2}\Delta k_{y})&n_{x}=\frac{Nx}{2}+1,\cdots,Nx-1\\ P^{\prime}(n_{x}\Delta k_{x},\frac{3 Ny}{2}\Delta k_{y})=P(n_{x}\Delta k_{x}, \frac{Ny}{2}\Delta k_{y})&n_{x}=0,\cdots,\frac{Nx}{2}-1\\ P^{\prime}(\frac{Nx}{2}\Delta k_{x},\big{(}n_{y}+Ny\big{)}\Delta k_{y})=P( \frac{Nx}{2}\Delta k_{x},n_{y}\Delta k_{y})&n_{y}=\frac{Ny}{2}+1,\cdots,Ny-1 \end{cases}  \tag{5}\]  
At the following specified positions, the generated angular spectrum value is reduced to half of the original angular spectrum value, as shown in Equation (6):  
\[\begin{cases}P^{\prime}(\frac{Nx}{2}\Delta k_{x},\frac{Ny}{2}\Delta k_{y})= \frac{1}{2}P(\frac{Nx}{2}\Delta k_{x},\frac{Ny}{2}\Delta k_{y})\\ P^{\prime}(\frac{3 Nx}{2}\Delta k_{x},\frac{Ny}{2}\Delta k_{y})=\frac{1}{2}P( \frac{Nx}{2}\Delta k_{x},\frac{Ny}{2}\Delta k_{y})\\ P^{\prime}(\frac{Nx}{2}\Delta k_{x},\frac{3 Ny}{2}\Delta k_{y})=\frac{1}{2}P( \frac{Nx}{2}\Delta k_{x},\frac{Ny}{2}\Delta k_{y})\\ P^{\prime}(\frac{3 Nx}{2}\Delta k_Finally, the change of angular spectrum value in each region after zero-padded expansion is analyzed, and the division of each region is shown in Figure 3.  
Figures 2 and 3 clearly show the change of angular spectrum in the process of zero-padded expansion. Region A is the angular spectrum before smoothing. Region A\({}^{\prime}\) is the angular spectrum after smoothing, and so on. The region of the original angular spectrum is divided into three regions: A, B, and C. The generated angular spectrum region after zero-padded expansion includes four regions, A\({}^{\prime}\), B\({}^{\prime}\), C\({}^{\prime}\) and D\({}^{\prime}\), among which the newly added region D\({}^{\prime}\) is the zero-padded region. In addition, region A contains four small regions: A\({}_{1}\), A\({}_{2}\), A\({}_{3}\) and A\({}_{4}\).