# Neural Network-Assisted DPD of Wideband PA Nonlinearity for Sub-Nyquist Sampling Systems
###### Abstract  
The design of conventional digital predistortion (DPD) requires an analogue-to-digital converter (ADC) with a sampling frequency that is multiple times the signal bandwidth, which is extremely challenging for sub-Nyquist sampling systems with undersampled signals. To address this, this paper proposes a neural network (NN)-assisted wideband power amplifier (PA) DPD method for sub-Nyquist sampling systems, wherein a dual-stage architecture is designed to handle the ambiguity caused by subsampled communications signals. In the first stage, the time-delayed polynomial reconstruction method is employed to estimate the wideband DPD nonlinearity coarsely with the undersampled signals with limited pilots. In the second stage, an NN-based DPD method is proposed for the virtual training of the DPD, which learns the up-sampled DPD behavior by taking advantage of the pre-estimated DPD model and the input data signals, which reduces the length of the training sequence significantly and refines the DPD behavior efficiently. Simulation results demonstrate the efficacy of the proposed method in tackling the wideband PA nonlinearity and its ability to outperform the conventional method in terms of power spectrum, error vector magnitude, and bit error rate. digital predistortion (DPD); undersampling; memory polynomial; attention mechanism model +
Footnote †: journal: Sensors  
0000-0002-4000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-0000-00000-  
Spectral extrapolation, a commonly used undersampling predistortion method proposed in [6], models the power amplifier (PA) under band-limited sampled data to recover or approximate the original output signal that should be sampled based on the estimated PA model. However, this approach has constraints that necessitate the inclusion of a band-limited filter in the feedback channel, which can be problematic in practical situations due to the performance of the band-limited filter and the challenges in its design. In [7], an undersampling recovery digital predistortion technique (USR-DPD) is proposed, which iteratively recovers the full-band output signal from the undersampled output signal, and then extracts the DPD parameter information using a memory model. However, this method requires the addition of extra DACs to the system, thereby increasing the system's complexity. Moreover, the method requires multiple internal and external iterations to obtain the final estimated DPD signal, which increases the computational cost. Ref. [8] proposed a forward behavior modeling approach that utilizes low-rate aliased PA output signals to estimate the DPD model coefficients. In this approach, the aliased low-rate PA output signals are employed to estimate the model parameters across the full frequency band of the PA. In [5], an undersampling digital predistortion method based on a multi-tone mixing feedback technique (MTM-DPD) is proposed for the processing of multi-tone mixing under undersampling conditions. In [9], a novel undersampled feedback signal iterative learning control (US-ILC-DPD) technique is proposed. This technique still exhibits good PA nonlinearity compensation even at low sampling rates. In [10], a random demodulation-based reduced sampling rate (RDRS) method is proposed to recover the full-band power amplifier (PA) output signal. This method uses only the I-way component of the band-limited feedback signal to reconstruct the full-band signal output. Although the method has a simple structure, it introduces a time alignment problem due to the additional random sequence generator and analog multiplier in the feedback path. This issue still needs to be addressed to ensure accurate signal reconstruction. In [11], a novel predistortion technique is proposed, which consists of a memoryless amplitude-to-amplitude (AM/AM) gain function that can be implemented in the analog domain and a nonlinear model with memory effects in the digital domain, reducing the sampling rate requirements of the forward and feedback paths.