## 2 Principle
### Linear Situation  
Firstly, the simplest condition of RPC process, the linear situation should be discussed. The expression of linear system is shown below:  
\[r=h(d)\xrightarrow{Fourier\ transform}R=HD \tag{1}\]  
where the \(d\), \(r\) are the time series of drive signals and response signals of the system, \(h(\ )\) is the system TF in time domain, equivalently, the \(R\), \(D\) are the Fourier transform of drives and responses, \(H\) is the system TF in frequency domain.  
Figure 2: RPC iteration block diagram.  
It assumes that the system is linear and invertible, error and noise are not considered. \(R_{0}\) is the desired response, \(D_{0}\) is the initial drive, \(\Delta_{i}\) is the \(i\)th response error, \(\Delta_{Di}\) is \(i\)th drive correction, \(D_{i}\) is the drive after \(i\) times of correction, \(H_{L}^{-1}\) is the inverse linear TF. The linear TF estimation of the system is expressed below:  
\[H_{L}=R_{s}/D_{s} \tag{2}\]  
where the \(D_{s}\) is the Fourier transform of standard road and the \(R_{s}\) is the response of SUT while the drive is \(D_{s}\).  
\(HH_{L}^{-1}\) is set as \(P\) which represents TF deviation ratio. Then the initial drive is:  
\[D_{0}=H_{L}^{-1}R_{0} \tag{3}\]  
The first response error is:  
\[\Delta_{1}=R_{0}-HD_{0}=(I-P)R_{0} \tag{4}\]  
The first drive correction is:  
\[\Delta_{D1}=H_{L}^{-1}\Delta_{1}=H_{L}^{-1}(I-P)R_{0} \tag{5}\]  
The first drive is  
\[D_{1}=D_{0}+\Delta_{D1} \tag{6}\]  
The first drive is the sum of initial drive and first drive correction. When \(k\geq 1\), the response error, drive correction and drive after k times iteration are:  
\[\Delta_{k}=R_{0}-HD_{k-1}=(I-P)\Delta_{k-1} \tag{7}\]  
\[\Delta_{Dk}=H_{L}^{-1}\Delta_{k} \tag{8}\]  
\[D_{k}=D_{k-1}+\Delta_{Dk} \tag{9}\]  
The \(k\)th drive is the sum of \(k-1\)th drive and \(k\)th drive correction. So the expressions of \(i\)th drive correction and drive after \(k\) times of correction are:  
\[\Delta_{Di}=H_{L}^{-1}(I-P)^{i}R_{0},\;i\geq 1 \tag{10}\]  
\[D_{k}=D_{0}+\sum_{i=1}^{k}\Delta_{Di} \tag{11}\]  
In the linear situation, the drive correction is geometric progression, so the \(k\)th drive is the sum of the geometric progression. For real number, if the ratio of the geometric progression is less than 1, the progression converges. For the complex number, the convergence condition is that the magnitude of the ratio less than 1. And for the complex number vector which is the situation in this paper, then the convergence condition is that the magnitude of the complex vector of the response error ratio \((I-P)\) is less than 1.  
The convergence condition is expressed as \(|I-P|<I\). If the convergence condition satisfies, then \(i\)th drive correction trends to zero.  
\[\lim_{i\rightarrow\infty}\Delta_{Di}=0 \tag{12}\]  
The series of the drive converges, then the drive after infinite steps is:  
\[D_{\infty}=D_{0}+\sum_{i=1}^{\infty}\Delta_{Di}=H_{L}^{-1}\Biggl{(}\sum_{i=0 }^{\infty}(I-P)^{i}\Biggr{)}R_{0}=H_{L}^{-1}P^{-1}R_{0}=H^{-1}R_{0} \tag{13}\]  
The actual drive for the desired response is:\[D_{R}=H^{-1}R_{0} \tag{14}\]  
The result after infinite times of iteration shows that the reproduced drive and response meet the desired drive and response.