In the field of wideband communication systems, the continuous expansion of bandwidth has led to a drastic increase in the computational burden for estimating DPD model parameters. This challenge becomes particularly prominent in ultra-wideband signal processing, demanding higher precision in parameter estimation. The complexity of wideband signals is characterized by irregularities and significant amplitude fluctuations between adjacent frequency components. Accurately characterizing the non-linear characteristics of RF PAs requires the introduction of more parameters and the consideration of higher-order effects. Unfortunately, these effects lead to a sharp rise in complexity. To address these issues, this paper proposes a wideband DPD algorithm based on edge signal correction. By only collecting the signals within the central frequency band and using the filtered frequency domain information for DPD model parameter estimation, the algorithm significantly reduces data processing complexity. Furthermore, by incorporating cross-terms into the model structure and continuously correcting the entire model using edge signals, the fitting accuracy of the model is markedly improved. The core of this method lies in its ability to effectively reduce computational complexity while accurately capturing the non-linear characteristics of RF PAs. Consequently, this algorithm can substantially lower computational demands without compromising accuracy.
2.1. Wideband DPD Architecture Based on Edge Signal Correction
Figure 1 illustrates the wideband DPD architecture based on edge signal correction designed in this study. The model design is inspired by the direct learning method [
14], which learns and corrects non-linear distortions directly from the differences between the input and output signals, thereby effectively reducing or eliminating distortions caused by the power amplifier.
The source signal
passes through a bandpass filter to generate a band-limited input signal
. The signal
is then processed by a band-limited pre-distorter to produce the pre-distorted signal
. The band-limited pre-distorter applies non-linear compensation to the input signal, thereby linearizing the in-band output of the power amplifier. The signal
is converted into an analog pre-distortion signal via a Digital-to-Analog Converter (DAC). Before being sent to the power amplifier, it undergoes up-conversion to align with the amplifier’s center frequency. After passing through the power amplifier, the signal’s bandwidth becomes five times wider than that of the source signal [
15]. Directly processing this wideband signal would significantly increase data rates and computational complexity. Therefore, a bandpass filter is employed to selectively capture feedback signals in the vicinity of the center frequency. The feedback signal is first down-converted and then digitized through an Analog-to-Digital Converter (ADC) to obtain an estimated input signal
. The difference between the estimated input signal
and the band-limited input signal
is calculated. This difference is then processed through parameter calculations to determine the parameters of the pre-distortion model. To enhance the accuracy of the pre-distortion model, edge frequency signals
are collected and fed into a parameter calibrator to compute the model’s cross-term parameters, thereby finalizing the calibration of the pre-distortion model. From the acquisition of the pre-distortion model parameters, through parameter calibration, a wideband digital pre-distortion model is ultimately generated, marking the completion of one iteration. Through multiple iterative updates, the inverse model of the power amplifier’s non-linear effects is progressively approximated, which compensates for non-linear distortions and thereby improves the quality and stability of communication transmission.
2.2. 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):
In Equation (1),
ak,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
can be expressed as follows:
In Equation (2),
represents the sampling factor of the DAC and ADC. Since there is a relationship where
, the relationship between
and
can be expressed as follows:
Assuming
, Equation (3) can be expressed as follows:
Let
, then Equation (4) can be revised to the following:
In Equation (5), ck,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.
The model described in Equation (5) can be represented in matrix form as follows:
where
is the estimated output signal matrix at a low sampling rate, and
represents an
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
are expressed as follows:
Combining the obtained DPD model parameters with the input signal , the output signal 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:
Thus, the system error function is the following:
In Equations (10) and (11),
represents the model error function,
is the parameter iteration weight, and
represents the basis function matrix, which can be expressed as follows:
Furthermore, the cost function can be modified as follows:
From Equations (10) and (11), the parameter iteration weight
can be derived as follows:
where the step size factor
satisfies
, and
represents the conjugate transpose of
.
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. The corrected power amplifier output signal after modification can be expressed as follows:
In Equation (17), bk represents the non-linear polynomial parameters, and dq represents the envelope parameters, both of which are related to the cross-terms.
Cross-terms are incorporated to improve the model’s fitting accuracy by utilizing edge signal data and calculating the cross-term parameters through the construction of a cost function.
In Equation (18), represents the association expression for . The corresponding to the minimum value of is designated as . Here, denotes the i-th iteration of , and represents the signal difference at index L. indicates the complex conjugate.
Taking the partial derivative of
J with respect to
dq results in the following:
In Equation (19), step size factor and represents the gain parameter.
In summary, the goal of training the pre-distortion model is to iteratively optimize the pre-distortion processing parameters and cross-term parameters. This algorithm focuses on the signal within the central frequency band, using only the filtered frequency domain information for DPD model parameter estimation. Additionally, by incorporating cross-terms into the model structure and continuously calibrating the entire model using edge signals, the fitting accuracy of the model is significantly enhanced. The design of this algorithm reduces computational resource consumption while maintaining pre-distortion effectiveness, thereby providing a more efficient signal processing solution for wideband wireless communication systems. This method is expected to achieve a better balance between performance and resource usage in practical applications. Simulations and experiments will provide empirical evidence to validate the feasibility and benefits of this theory in actual wireless communication environments.