Stability analysis of an adaptive Wiener structure

In the context of digital pre-distortion, a typical requirement is to identify the power amplifier with stringently low computational complexity. Accordingly, we consider a simple gradient method which is used to adaptively fit a simplified Wiener model, i.e., a cascade of a linear filter followed by a memoryless nonlinearity, to a dispersive and saturating reference system which represents the power amplifier. For adaptation, the gradient method only relies on the difference between the output of the reference system and the Wiener model. We show that such a structure can be formulated as a proportionate normalised least mean squares (PNLMS) algorithm. As a consequence, conditions for stability in the mean square sense can be deduced. Although not proven in a strict sense, simulation results allow to conjecture robustness.