Wiener Model

The defining architecture places a linear dynamic filter before a static nonlinearity. This sequence models the physical reality where input matching networks and parasitic capacitances shape the signal's bandwidth before it reaches the transistor's nonlinear gate junction. The linear block is typically an FIR or IIR filter capturing memory effects, while the nonlinear block is a polynomial or look-up table representing AM-AM and AM-PM distortion.

Coefficients for both blocks are commonly extracted using least squares estimation on measured input-output data. The process is simplified because the intermediate signal between the two blocks is unobservable, requiring iterative optimization or specialized techniques like the Narendra-Gallman algorithm. This separates the linear and nonlinear identification problems, reducing computational complexity compared to full Volterra extraction.

For DPD applications, the Wiener pre-distorter is constructed by placing the inverse nonlinearity before the inverse linear filter. This directly compensates for the PA's cascade. The inverse nonlinear block pre-distorts the amplitude, while the inverse filter pre-equalizes the frequency response, effectively linearizing the entire transmitter chain when placed before the amplifier.

The Wiener model is a strict subset of the Volterra series. It represents only those Volterra kernels that can be factored into a product of linear filter coefficients and polynomial terms. This structural constraint makes it less general than a full Volterra model but significantly more parameter-efficient, requiring O(M+N) parameters instead of O(M^N) where M is memory depth and N is nonlinear order.

The Wiener model cannot capture memory effects that occur after the nonlinear distortion, such as thermal dynamics in the transistor's drain or bias network modulation at the output. For amplifiers where the dominant nonlinearity is at the input (e.g., gate-driven FETs), this limitation is negligible. For output-referred nonlinearities, the Hammerstein model (nonlinearity first, then filter) is more appropriate.

The Wiener structure excels at modeling amplifiers driven by wideband modulated signals (e.g., 100 MHz 5G NR carriers) where the input matching network's frequency response significantly colors the signal before nonlinear clipping occurs. The linear pre-filter accurately captures this frequency-selective memory, making it a preferred model for modern base station DPD where signal bandwidths continue to expand.

The structural inverse of the Wiener model, consisting of a static memoryless nonlinearity followed by a linear dynamic filter. This topology is effective for modeling systems where the input signal first undergoes nonlinear distortion—such as a saturating sensor—before experiencing linear dynamics. In power amplifier contexts, it represents a simplified Volterra subclass.

A generalized architecture comprising a bank of static nonlinearities operating in parallel, each followed by a dedicated linear filter. The outputs are summed to produce the final signal. This structure represents a specific subclass of the Volterra series and captures more complex nonlinear memory effects than a single Hammerstein branch, making it suitable for wideband power amplifier modeling.

A simplified Volterra model that retains only the diagonal terms of the Volterra kernels. It can be viewed as a parallel connection of nonlinear polynomial branches, each with a different memory tap. This structure effectively captures nonlinear memory effects while drastically reducing computational complexity compared to the full Volterra series, making it a workhorse for digital predistortion.

An enhanced memory polynomial that includes cross-terms between the signal and its lagging envelope values. By accounting for interactions between different time delays and nonlinear orders, it captures complex long-term memory effects more accurately than the standard memory polynomial. This model is particularly effective for Doherty and GaN-based power amplifiers exhibiting strong thermal trapping.

A Volterra model simplification technique that separates the static nonlinearity from low-order dynamic behavior. It assumes the system is weakly nonlinear with memory, drastically reducing the number of parameters needed. The model expresses the output as a static polynomial plus a sum of dynamic deviation terms, providing a compact representation for power amplifiers with mild memory effects.

The most general mathematical framework for modeling nonlinear dynamic systems with memory. It represents the output as a sum of multidimensional convolution integrals using Volterra kernels. While capable of modeling any fading memory system, its computational complexity grows exponentially with nonlinear order and memory depth, motivating the use of simplified block-structured models like Wiener and Hammerstein.