# Power Amplifier Predistortion Using Reduced Sampling Rates in the Forward and Feedback Paths
## 3 Proposed Predistortion System
### Identification Process of the Proposed Predistortion System
First, the DPD will be able to compensate for residual distortions due to the imperfections and inaccuracy of the APD function as well as the amplifier's memory effects. Second, because the cascade made of the APD and the power amplifier is equivalent to a mildly nonlinear system, the DPD function will be mildly nonlinear and hence, results in a diminished spectral regrowth at the output of the predistorter, as illustrated in the sketched spectrum of the signal \(x_{out\_DPD}\) in Figure 4. Therefore, it is possible to use reduced sampling rate in the signal's forward path DAC even when the digital predistortion function is applied. In the proposed hybrid predistorter, the digital predistortion function is implemented using a memory polynomial model. Hence, the output waveform of the digital predistorter is related to its input through the following:  
\[x_{out\_DPD}(n)=\sum_{i=1}^{N}\sum_{m=0}^{M}a_{im}x_{in\_DPD}(n-m)\cdot|x_{in \_DPD}(n-m)|^{i-1}, \tag{2}\]  
where \(x_{in\_DPD}\) and \(x_{out\_DPD}\) are the baseband complex waveforms at the input and output of the digital predistortion functions, respectively. \(N\) and \(M\) represent the nonlinearity order and the memory depth of the memory polynomial function, respectively. \(a_{im}\) represents the complex-valued coefficients of the memory polynomial function. For a given nonlinearity order \((N)\) and memory depth \((M)\), the total number of coeffcients in the memory polynomial function is given by \(N\times(M+1)\). When written in a matrix form, Equation (2) becomes  
\[x_{out\_DPD}(n)=\mathbf{X}_{in\_DPD}(n)\cdot\mathbf{A}, \tag{3}\]  
where  
\[\mathbf{A}= \begin{bmatrix}a_{10}&a_{11}&a_{12}&\cdots&a_{1m}&a_{20}&\cdots&a_{ N0}&a_{NM}\end{bmatrix} ^{T}, \tag{4}\]  
and  
\[\mathbf{X}_{in\_DPD}(n)= \begin{bmatrix}\mathbf{X}_{in\_DPD,1}(n)&\mathbf{X}_{ in\_DPD,2}(n)&\cdots&\mathbf{X}_{in\_DPD,N}(n)\end{bmatrix} , \tag{5}\]  
with  
\[\mathbf{X}_{in\_DPD,i}(n)= \begin{bmatrix}x_{in\_DPD}(n)|x_{in\_DPD}(n)|^{i-1 }&\cdots&x_{in\_DPD}(n-M)|x_{in\_DPD}(n-M)|^{i-1}\end{bmatrix} . \tag{6}\]  
In these equations, \( \begin{bmatrix}\end{bmatrix} ^{T}\) refers to the transpose function. The identification of the memory polynomial coefficient is performed by solving the linear system obtained when considering the \(L\) samples of the waveforms used to identify the coefficients of the polynomial model. In this work, 8000 samples were used to compute the polynomial coefficients using the Moore-Penrose pseudoinverse technique. Accordingly,  
\[\mathbf{A}=\mathit{pinv}(\mathbf{X}_{in\_DPD})\cdot\mathbf{X}_{out\_DPD}. \tag{7}\]In Equation (7), \(pinv(\mathbf{X}_{in\_DPD})\) is given by  
\[pinv(\mathbf{X}_{in\_DPD})=\Big{[}(\mathbf{X}_{in\_DPD})^{T}\mathbf{X}_{in\_DPD }\Big{]}^{-1}\cdot(\mathbf{X}_{in\_DPD})^{T}, \tag{8}\]  
where  
\[\mathbf{X}_{in\_DPD}=\big{[}\mathbf{X}_{in\_DPD}(s)\quad\mathbf{X}_{in\_DPD}(s +1)\quad\cdots\quad\mathbf{X}_{in\_DPD}(s+L-1)\big{]}^{T}, \tag{9}\]  
and  
\[\mathbf{X}_{out\_DPD}=\big{[}x_{out\_DPD}(s)\quad x_{out\_DPD}(s+1)\quad\cdots \quad x_{out\_DPD}(s+L-1)\big{]}^{T}. \tag{10}\]  
\(s\) refers to the starting index of the signal samples used for the coefficients identification. It is worth noting that the identification of both predistortion functions is performed using the indirect learning technique where the predistortion function is directly synthesized from the measured data without the need to derive a model of the device under test. Moreover, the digital predistortion function can be implemented either as a memory polynomial or any other function that compensates for memory effects such as the Volterra series. However, the digital predistortion function aims mainly at compensating the nonlinear distortions of the cascade made of the APD and the device under test, and this cascade behaves as a mildly nonlinear system with memory effects.