# 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  
In the direct learning method of DPD, establishing an accurate model of the power amplifier is essential for characterizing the relationship between its non-linear behavior and the input signal. This paper examines this relationship using a fully sampled analysis approach, which operates under the assumption that the system's bandwidth matches the sampling rate. Additionally, memory effects that the power amplifier's past states influence its current output must be considered. Consequently, precise modeling of RF PAs under fully sampled conditions is required. Generally, RF PAs with memory effects can be described by various models that accurately capture the dynamic characteristics of the amplifier under different operating conditions. Selecting an appropriate model is crucial, as it directly affects the accuracy and effectiveness of the pre-distorter design. Under full sampling conditions, any DPD model with memory capabilities can be applied to ensure that the non-linearity and memory effects of the amplifier are fully compensated during signal processing. Therefore, a memory polynomial is adopted as the basic model, as shown in Equation (1):  
Figure 1: Wideband DPD architecture based on edge signal correction. \[y_{h}(l)=\sum_{k=0}^{K-1}\sum_{m=0}^{M}a_{k,m}u_{h}(l-m)|u_{h}(l-m)|^{k} \tag{1}\]  
In Equation (1), \(a_{k,m}\) represents the pre-distortion model matrix parameters, where \(k\) denotes the order of the pre-distortion model, and \(m\) denotes the memory depth of the pre-distortion model. The output \(y(n)\) can be expressed as follows:  
\[y(n)=y_{h}(nF_{2})=\sum_{k=0}^{K-1}\sum_{m=0}^{M}a_{k,m}u_{h}(nF_{2}-m)|u_{h}( nF_{2}-m)|^{k} \tag{2}\]  
In Equation (2), \(F_{2}\) represents the sampling factor of the DAC and ADC. Since there is a relationship where \(u_{h}(nF_{2})=u(n)\), the relationship between \(u(n)\) and \(y(n)\) can be expressed as follows:  
\[y(n)=\sum_{k=0}^{K-1}\sum_{m=0}^{M}a_{k,m}u\left(n-\frac{m}{F_{2}}\right)\left| u\left(n-\frac{m}{F_{2}}\right)\right|^{k} \tag{3}\]  
Assuming \(q=\frac{m}{F_{2}}\), Equation (3) can be expressed as follows:  
\[y(n)=\sum_{k=0}^{K-1}\sum_{q=0}^{\frac{M}{F_{2}}}a_{k,qF_{2}}u(n-q)|u(n-q)|^{k} \tag{4}\]  
Let \(Q=\frac{M}{F_{2}}\), then Equation (4) can be revised to the following:  
\[y(n)=\sum_{k=0}^{K-1}\sum_{q=0}^{Q}c_{k,q}u(n-q)|u(n-q)|^{k} \tag{5}\]  
In Equation (5), \(c_{k,q}\) represents the model parameters, where \(k\) denotes the non-linear order of the model and \(q\) denotes the memory depth of the model. Equation (5) reveals the significant relationship between memory depth and sampling factors when constructing band-limited RF PA models and their corresponding DPD models. Since the band-limited characteristics of these models are closely related to the sampling rate, time-domain alignment techniques can be employed to model RF PAs and DPDs accurately. The key to this technique lies in ensuring consistency between the band-limited characteristics of the models and the sampling rate, thereby allowing the accurate capture of the signal's time-domain characteristics during modeling. By finely adjusting the memory depth and sampling factors, the performance of the models can be optimized to ensure the accuracy and efficiency of pre-distortion processing. Additionally, this method helps reduce the common computational burden in wideband signal processing by operating directly within the useful frequency band, thereby eliminating unnecessary high sampling rate processing. Next, the Least Squares (LS) algorithm is used to estimate the model parameters. This process involves precisely adjusting the model parameters to ensure that the pre-distortion processing effectively compensates for the non-linear characteristics of the power amplifier, ensuring that the amplification of in-band signals approaches ideal linear characteristics.