## 2 System Model and Problem Analysis
### Joint Effect of Group Delay Distortion and Nonlinear Distortion
Then, Equation (5) can be written as:  
\[z(r) =\frac{1}{N}\sum\limits_{n=0}^{N-1}y(n)\sum\limits_{k=0}^{N-1}A(k )e^{[\frac{2\pi}{N}k(r-n)+\varphi(k)]} \tag{7}\] \[=\frac{1}{N}\sum\limits_{n=0}^{N-1}y(n)\{A(0)+\sum\limits_{k=1}^{ N-1}A(k)e^{[\frac{2\pi}{N}k(r-n)+\varphi(k)]}+\sum\limits_{k=\frac{N}{2}+1}^{N-1}A(k )e^{[\frac{2\pi}{N}k(r-n)+\varphi(k)]}\}\] \[=\frac{1}{N}\sum\limits_{n=0}^{N-1}y(n)\{A(0)+\sum\limits_{k=1}^{ N-1}A(k)e^{[\frac{2\pi}{N}k(r-n)+\varphi(k)]}+\sum\limits_{k^{\prime}= \frac{N}{2}-1}^{N-1}A(k^{\prime})e^{[\frac{2\pi}{N}(N-k^{\prime})(r-n)+\varphi( N-k^{\prime})]}\}\] \[=\frac{1}{N}\sum\limits_{n=0}^{N-1}y(n)\{A(0)+\sum\limits_{k=1}^{ N-1}A(k)e^{[\frac{2\pi}{N}k(r-n)+\varphi(k)]}+\sum\limits_{k^{\prime}=1}^{N-1}A(k ^{\prime})e^{-j[\frac{2\pi}{N}k^{\prime}(r-n)+\varphi(k^{\prime})]}\}\]  
Apparently, \(\varphi(0)=\sum\limits_{p=1}^{p}c_{p}0^{p}=0\). Utilizing the transformation \(k=N-k^{\prime}\), and since \(k\) and \(k^{\prime}\) are operated in the same accumulation range, Equation (7) can be written as:  
\[z(r)=\frac{1}{N}\sum\limits_{n=0}^{N-1}y(n)\{A(0)+2\sum\limits_{k=1}^{N-1}A(k )\cos[\frac{2\pi}{N}k(r-n)+\varphi(k)]\} \tag{8}\]  
If the BPF has a linear phase response, \(\varphi_{ideal}(k)=\frac{2\pi}{N}c_{1}k\), the second term in Equation (8) becomes:  
\[\sum\limits_{k=1}^{N-1}A(k)\cos[\frac{2\pi}{N}k(r-n)+\varphi(k)]=\sum\limits_ {k=1}^{N-1}A(k)\cos[\frac{2\pi}{N}k(r-n+c_{1})] \tag{9}\]  
This means that for a fixed time \(r\), there exists \(n=r+c_{1}\) making:  
\[\sum\limits_{k=1}^{N-1}A(k)\cos[\frac{2\pi}{N}k(r-n+c_{1})]=\sum\limits_{k=1} ^{N-1}A(k) \tag{10}\]  
While \(n\
eq r+c_{1}\), the odd symmetry of the cosine function in the range of 0 to \(\pi\) makes this accumulation term equal to 0. Hence, we obtain:  
\[z(r)=\frac{1}{N}A(0)\sum\limits_{n=0}^{N-1}y(n)+\frac{2}{N}\sum\limits_{k=1}^{ N-1}A(k)y(n-c_{1}) \tag{11}\]  
On account of \(A(0)<<\sum\limits_{k=1}^{N-1}A(k)\) and the direct component having no variation in phase, the final expression is:  
Figure 2.0  
Equation (12) means that each frequency component in \(y(n)\) has an equal delay \(c_{1}\) when passing by the ideal BPF and they are outputted at the same time. However, if the phase variation \(\varphi(k)\) relative to frequency \(k\) is nonlinear, the manner that \(y(n)\) passes through the BPF is similar to the scattering of electromagnetic wave propagation. To be specific, \(Y(k_{1})\) will be outputted by the BPF after delay \(\tau_{k_{1}}\) which is different from \(\tau_{k_{2}}\), the delay of \(Y(k_{2})\), because \(\tau_{k}\) is related to a polynomial function of frequency \(k\). In this way, nonlinear products outputted at the same time \(n\) in \(y(n)\) are sampled at different moments of \(z\). Taking this effect into account when designing the nonlinearity identification criterion, modeling is expected to be more accurate and distortion will be mitigated more thoroughly. Thus:  
Figure 2.1  
To be exact, \(\tau\) should be a function of frequency. However, to make the realization easy, we choose a simplified method, in that:  
Figure 2.2  
Here, \(\tau_{best}\) is not a delay really existing in the RF front-end, but a sampling interval between the output of the nonlinear behavior model and the distortion component in \(z\). When the best delay is found, the minimum of the power residuals after optimization is expected to be more precise. In addition, the searching range to which the best delay belongs can be set according to prior knowledge of the \(S\) parameter of the RF front-end. Once the \(S\) parameter is obtained, it can be determined by the differential of the phase angle of \(S_{21}\) to the frequency.