Direct learning adaptation of power amplifier pre-distortion based on Wirtinger calculus

To improve efficiency of power amplifier (PA), linearity characteristics is often compromised when targeting lower power consumption (class B). Moreover, sophisticated PA efficiency improvement schemes such as envelope tracking tend to further boost the nonlinear characteristics of the PA. Digital pre-distortion (DPD) is a technique to improve the linearity of a power amplifier (PA) at expense of extra processing in the base-band. Derivation of optimal DPD adaptive learning involves optimization of real-valued objective functions of complex variables, whose derivative or gradient does not exist in the standard complex-analysis sense, due to non-holomorphic nature of the function. This is often overlooked in the literature and results in erroneous results. For instance, the methodology presented in [8] computes the gradient with respect to the variable to compute the updates. However, as discussed in [3] and [1], it is the gradient with respect to the conjugate of the variable (and not the variable) that leads to the nonpositive increment of the objective function. We resort to the theory of Wirtinger calculus to derive the proper first-and second-order derivatives (gradient and Hessian operators) of the non-holomorphic objective function and extend the results to optimization methodologies such as Newton, Gauss-Newton, and their quasi-variants. Results are assessed through experimental validation of the proposed methods on WLAN PAs.