Wiener Model

The defining architecture of the Wiener model is a linear time-invariant (LTI) dynamic filter cascaded with a static memoryless nonlinearity. The input signal first passes through the linear filter, which shapes the signal's temporal characteristics and introduces memory effects. The filtered output then feeds into the static nonlinear block, which applies amplitude-dependent distortion. This ordering is critical: it models power amplifiers where bias circuit dynamics and thermal effects create memory in the signal envelope before the transistor's nonlinear gain compression occurs.

The Wiener model is expressed mathematically as:

• Linear filter output: v(n) = Σ h(k) * x(n-k) for k=0 to M-1, where h(k) are the filter coefficients and M is the memory depth
• Nonlinear output: y(n) = f(v(n)), where f(·) is a static nonlinear function, typically a complex polynomial: f(v) = Σ a_p * v * |v|^(p-1) for odd p

The combined model captures frequency-dependent memory effects that manifest before the nonlinear gain compression, making it suitable for PAs with significant input matching network dynamics.

For digital predistortion, the Wiener predistorter is typically implemented as the inverse structure: a static nonlinearity followed by a linear filter (a Hammerstein model). This is because the predistorter must invert the PA's Wiener characteristic:

• The static nonlinear block pre-distorts the signal amplitude to counteract the PA's gain compression
• The linear filter then compensates for the PA's input memory effects

This inverse relationship is derived using the pth-order inverse theory, which provides a systematic method for computing the predistorter coefficients from the extracted PA model.

Coefficient extraction for the Wiener model typically employs batch least squares (LS) estimation on measured input-output data:

• The linear filter coefficients h(k) are first estimated by fitting the model to small-signal S-parameter measurements, where the PA operates linearly
• The static nonlinearity coefficients a_p are then extracted by applying a polynomial fit to the AM-AM and AM-PM characteristics measured under large-signal excitation
• Alternatively, iterative optimization can jointly estimate both blocks by minimizing the error between modeled and measured output waveforms

The Wiener model has specific limitations that define its applicability:

• Unidirectional memory: It only captures memory effects that precede nonlinearity, not cases where nonlinearity generates memory (e.g., dynamic thermal effects in the transistor channel)
• No cross-terms: Unlike the Generalized Memory Polynomial, it lacks envelope memory cross-terms that model interactions between the signal and its lagging envelope
• Best suited for: PAs with significant input matching network dynamics or bias tee filtering before the active device, where linear filtering dominates the memory behavior

For PAs where memory and nonlinearity are deeply intertwined, the Wiener-Hammerstein cascade or Parallel Hammerstein models offer superior accuracy.

The Wiener and Hammerstein models are complementary block structures distinguished by block ordering:

• Wiener: Linear filter → Static nonlinearity (memory before distortion)
• Hammerstein: Static nonlinearity → Linear filter (distortion before memory)

The choice depends on the physical origin of memory effects in the PA:

• Use Wiener when input-side parasitics and bias network filtering dominate
• Use Hammerstein when output-side thermal effects and drain bias modulation dominate
• For complex PAs exhibiting both, the Wiener-Hammerstein cascade combines both structures in series

A multi-branch architecture composed of multiple Hammerstein models operating in parallel, where each branch contains a unique static nonlinearity followed by a dedicated linear filter. This structure generalizes the Hammerstein model to represent complex nonlinear dynamics that cannot be captured by a single branch. It is functionally equivalent to a Volterra series with diagonal kernel structure and is widely used for modeling PAs with strong frequency-dependent nonlinear behavior, such as GaN HEMT amplifiers under wideband modulation.

An enhanced memory polynomial model that includes cross-terms between the signal and its lagging or leading envelope samples. These cross-terms capture complex memory effects that the standard MP model misses, such as those arising from bias circuit modulation and thermal dynamics. The GMP adds two types of cross-term clusters: lagging envelope terms (capturing long-term memory) and leading envelope terms (capturing non-causal effects from measurement alignment). It provides significantly improved modeling accuracy for Doherty and envelope tracking PAs.

A numerical conditioning process that transforms correlated polynomial basis functions into an orthogonal set to improve the stability and convergence speed of coefficient estimation. In Wiener and MP models, the polynomial terms of the input signal are highly correlated, leading to ill-conditioned data matrices. Orthogonalization techniques such as Gram-Schmidt or using orthonormal basis functions (Laguerre, Kautz) decorrelate the regressors, enabling faster and more robust least squares solutions during real-time DPD adaptation.