# A Wideband Digital Pre-Distortion Algorithm Based on Edge Signal Correction
## 2 Wideband DPD Model Based on Edge Signal Correction
### Wideband DPD Algorithm Based on Edge Signal Correction
The model described in Equation (5) can be represented in matrix form as follows:  
\[\hat{\mathbf{Y}}=\mathbf{\Psi}\hat{\mathbf{C}} \tag{6}\]  
\[\hat{\mathbf{Y}}=\left[\hat{y}(n),\hat{y}(n+1),\ldots,\hat{y}(n+N-1)\right]^{T} \tag{7}\]where \(\mathbf{\hat{Y}}\) is the estimated output signal matrix at a low sampling rate, and \(\mathbf{\Psi}\) represents an \(N\times K(Q+1)\) basis function matrix, where each row contains only the non-linear basis functions from Equation (5). Using the least squares method, the band-limited power amplifier model parameters \(\hat{C}\) are expressed as follows:  
\[\hat{C}=\left(\mathbf{\Psi}^{H}\mathbf{\Psi}\right)^{-1}\mathbf{\Psi}^{H}Y \tag{8}\]  
Combining the obtained DPD model parameters \(\mathbf{\hat{C}}\) with the input signal \(u(n)\), the output signal \(\hat{y}(n)\) can be estimated. To improve the model's accuracy, this paper further explores the estimation process of the DPD model. It was found that the difference between the source signal and the actual amplified signal of the RF PA can directly quantify the deviation between the actual amplified signal and the ideal amplified signal. Processing this difference signal can effectively enhance the estimation accuracy of the model. By accurately calculating this error metric, the effectiveness of the pre-distortion technique can be evaluated. In this case, the DPD model can be adjusted and improved to ensure that the final amplification performance is closer to the desired linearity. This method effectively enhances the calibration capability of the DPD system, reduces signal distortion, and thereby improves the overall system performance. Additionally, this error-feedback-based iterative optimization process provides a systematic means for fine-tuning the DPD model, allowing the model to better adapt to various signals and operating conditions. The input signal matrix for the pre-distorter can be expressed as follows:  
\[\mathbf{X}=\left[x(n),x(n+1),\ldots,x(n+N-1)\right]^{T} \tag{9}\]  
Thus, the system error function is the following:  
\[y_{h}(l)=\sum_{k=0}^{K-1}\sum_{m=0}^{M}a_{k,m}u_{h}(l-m)|u_{h}(l-m)|^{k} \tag{1}\]0  
\[y_{h}(l)=\sum_{k=0}^{K-1}\sum_{m=0}^{M}a_{k,m}u_{h}(l-m)|u_{h}(l-m)|^{k} \tag{1}\]1  
In Equations (10) and (11), \(\mathbf{\widetilde{e}}\) represents the model error function, \(\mathbf{\Delta W}_{l}\) is the parameter iteration weight, and \(\mathbf{\Omega}\) represents the basis function matrix, which can be expressed as follows:  
\[\mathbf{\Omega}= \begin{bmatrix}\Omega_{0,0}(n)&\Omega_{1,0}(n)&\ldots&\Omega_ {K-1,Q}(n)\\ \Omega_{0,0}(n+1)&\Omega_{1,0}(n+1)&\ldots&\Omega_{K-1,Q}(n+1)\\ \vdots&\vdots&\ddots&\vdots\\ \Omega_{0,0}(n+N-1)&\Omega_{1,0}(n+N-1)&\ldots&\Omega_{K-1,Q}(n+N-1)\end{bmatrix}  \tag{12}\]  
\[\Omega_{k,q}(n)=x(n-q)|x(n-q)|^{k} \tag{13}\]  
Furthermore, the cost function can be modified as follows:  
\[J=\left\|\mathbf{\varepsilon}\right\|^{2}=\left\|\mathbf{e}-\mathbf{\widetilde {e}}\right\| \tag{14}\]  
From Equations (10) and (11), the parameter iteration weight \(\mathbf{\Delta W}_{l}\) can be derived as follows:  
\[\mathbf{\Delta W}_{l}=\left(\mathbf{\Omega}^{H}\mathbf{\Omega}\right)^{-1} \mathbf{\Omega}^{H}\big{(}\mathbf{\hat{Y}}-\mathbf{X}\big{)} \tag{15}\]  
\[\mathbf{W}_{l}=\mathbf{W}_{l-1}-\mu\mathbf{\Delta W}_{l} \tag{16}\]where the step size factor \(\mu\) satisfies \(\mu\leq 1\), and \(\mathbf{\Omega}^{H}\) represents the conjugate transpose of \(\mathbf{\Omega}\). First, the basic parameters of the pre-distortion model are obtained using the above algorithm. Then, considering the influence of dynamic cross-terms, the model parameters are optimized.