# Error Correction Method of TIADC System Based on Parameter Estimation of Identification Model
## 2 Research on the Calibration Method
### Frequency Characteristic Analysis of the TIADC System Identification Model  
In fact, identification is the process of extracting mathematical models from a system's input and output data. When the order of differential equations is higher, it is very heavy work to decompose denominator polynomials without the aid of computers. Therefore, after establishing a mathematical model of a complex field, the solution to the equation is generally not required. One of the advantages of the Laplace transform is that it can use a graphic method to predict system performance. To facilitate the estimation of parameters of the TIADC system identification model, the transfer function in Equation (8) can be transformed into the following form:  
\[G(s)=\frac{n_{0}+n_{1}(s)+n_{2}(s)^{2}+\cdots+n_{k}(s)^{k}}{1+m_{1}(s)+m_{2}(s) ^{2}+\cdots+m_{k}(s)^{k}} \tag{9}\]  
In Equation (9), \(m_{1}\), \(m_{2}\),..., \(m_{k}\), \(n_{1}\), \(n_{2}\), \(n_{3}\) and \(n_{k}\) represent the parameters to be identified, and \(k\) is the system order-number. For different applications, the order of the identification system may be even or odd. The transformation from Equation (8) to Equation (9) is a completely mathematical process. Figure 5: Equivalent model of a single sub-channel ADC in a TIADC system with channel mismatch. The frequency characteristics of the system can be expressed as follows:  
\[G(j\omega)=\frac{n_{0}+n_{1}(j\omega)+n_{2}(j\omega)^{2}+\cdots+n_{k}(j\omega)^{k} }{1+m_{1}(j\omega)+m_{2}(j\omega)^{2}+\cdots+m_{k}(j\omega)^{k}} \tag{10}\]  
If \(k\) is odd, define the parameter vectors and information vectors in Equation (10) as follows:  
\[\alpha_{1}=[m_{1},m_{3},m_{5},\cdots,m_{k}]^{T}\in R^{(k+1)/2} \tag{11}\]  
\[\alpha_{2}=[m_{2},m_{4},m_{6},\cdots,m_{k-1}]^{T}\in R^{(k-1)/2} \tag{12}\]  
\[\beta_{1}=[n_{1},n_{3},n_{5},\cdots,n_{k}]^{T}\in R^{(k+1)/2} \tag{13}\]  
\[\beta_{2}=[n_{2},n_{4},n_{6},\cdots,n_{k-1}]^{T}\in R^{(k-1)/2} \tag{14}\]  
\[\varphi(\omega)=\left[\omega,-\omega^{3},\omega^{5},\cdots,(-1)^{(k-1)/2} \omega^{k}\right]^{T}\in R^{(k+1)/2} \tag{15}\]  
\[\psi(\omega)=\left[-\omega^{2},\omega^{4},-\omega^{6},\cdots,(-1)^{(k-1)/2} \omega^{k-1}\right]^{T}\in R^{(k-1)/2} \tag{16}\]  
If \(k\) is even, define the parameter vectors and information vectors in Equation (10) as follows:  
\[\alpha_{1}=[m_{1},m_{3},m_{5},\cdots,m_{k-1}]^{T}\in R^{k/2} \tag{17}\]  
Figure 5:0  
Figure 5:1  
Figure 5:2  
Figure 5:3  
Figure 5:4  
Whether \(k\) is odd or even, the transfer function of the system can be expressed by these parameter vectors and information vectors. They can be transformed into the following forms:  
\[ \begin{array}{rl}G(j\omega)&=\frac{[n_{0}+\psi^{T}(\omega)\beta_{2}]+j\varphi ^{T}(\omega)\beta_{1}}{[1+\psi^{T}(\omega)\alpha_{2}]+j\varphi^{T}(\omega) \alpha_{1}}\\ &=\frac{[n_{0}+\psi^{T}(\omega)\beta_{2}+j\varphi^{T}(\omega)\beta_{1}]\left[1 +\psi^{T}(\omega)\alpha_{2}-j\varphi^{T}(\omega)\alpha_{1}\right]}{\left[1+ \psi^{T}(\omega)\alpha_{2}\right]^{2}+\left[\varphi^{T}(\omega)\alpha_{1} \right]^{2}}\\ &=\text{Re}[G(j\omega)]+jIm[G(j\omega)]\end{array}