# Power Amplifier Predistortion Using Reduced Sampling Rates in the Forward and Feedback Paths
## 3 Proposed Predistortion System
### Identification Process of the Proposed Predistortion System  
The flow chart of the proposed hybrid predistortion system is presented in Figure 5. First, the input and output waveforms to be used for the power amplifier characterization are acquired. The input waveform corresponds to the signal at the output of the DPD block (\(x_{out\_DPD}\)) while the output waveforms refers to the amplifier's output signal (\(x_{out\_PA}\)). Initially, the APD and DPD functions are bypassed, that is, their outputs are equal to their inputs. During this first characterization step, the forward path's DAC and feedback path's ADC operate at a sampling rate that is commensurate to \(K\times BW\), where \(BW\) represents the input signal's bandwidth, and \(K\) is a factor such that \(K<5\). The delay between the two waveforms is estimated and compensated for. Then, the analog predistortion function is identified. The analog predistortion function used in this work is a memoryless look-up table such that  
\[x_{out\_APD}(n)=x_{in\_APD}(n)\cdot G(|x_{in\_APD}(n)|), \tag{1}\]  
where \(x_{in\_APD}\) represents the input waveform of the APD's input signal, and \(G(|x_{in\_APD}|)\) is the memoryless gain function of the APD. Typically, look-up-table-based predistorters are complex-valued, in the sense that the gain \(G(|x_{in\_APD}|)\) has a magnitude and phase that are used to compensate for the amplitude modulation to amplitude modulation (AM/AM) and amplitude modulation to phase modulation (AM/PM) distortions of the PA, respectively. The AM/AM characteristic is obtained by computing the magnitude of the instantaneous gain of the device under test as a function of the instantaneous input power. The AM/PM characteristic corresponds to the phase of the instantaneous gain of the device under test as a function of the instantaneous input power. In the proposed system, the APD function is built using a real-valued memoryless look-up-table-based APD which only compensates for the AM/AM distortions. This function is obtained from the measured input and output waveforms using an exponentially weighted moving average algorithm [30]. The AM/PM distortions compensation is performed within the digital predistortion function. The rationale behind this design choice is twofold. First, the use of a real-valued LUT that compensates for memoryless AM/AM distortions only will reduce the implementation complexity of the APD function. Second, most of the power amplifiers employed in modern base stations exhibit mildly nonlinear AM/PM distortions that are mainly dominated by memory effects. Thus, AM/PM compensation can be embedded within the digital predistortion function only. The APD function is synthesized as a look-up table containing real-valued power-dependent gain values. This function can be implemented using a variable gain amplifier, a voltage-controlled attenuator, or using an RF vector multiplier. Such implementations  
Figure 5: Flow chart of the proposed hybrid DPD system identification and update. are preferred since they allow for the control of the nonlinearity profile in contrast with analog predistortion functions where the nonlinearity is implemented using diodes and intermodulation distortion generators. In this paper, the APD is emulated by applying its transfer characteristics in MATLAB (release R2022B) using a look-up table model. Following the identification of the APD function, this latter is applied to linearize the power amplifier. Therefore, the distortions as well as the spectral regrowth at the output of the power amplifier will be reduced during the second characterization step that will be used to derive the DPD function. Consequently, the impact of the low sampling rate in the feedback path will be reduced since the bandwidth of the signal at the output of the amplifier is narrower following the partial linearization performed by the APD function. Moreover, since the characterization data used to derive the DPD function includes the cascade made of the APD and the power amplifier, two important benefits can be achieved.