Hammerstein Model

The Hammerstein model consists of two distinct blocks connected in series:

• Static Nonlinearity (F): A memoryless nonlinear function that maps the instantaneous input amplitude to an intermediate signal. This block captures AM-AM distortion and AM-PM distortion without considering history.
• Linear Time-Invariant Filter (G): A dynamic linear filter that processes the output of the nonlinear block, introducing memory effects through its impulse response.

The overall output is given by: y(n) = Σₖ h(k) · F(x(n-k)), where h(k) represents the filter coefficients and F(·) is the static nonlinear function.

A key advantage of the Hammerstein structure is the separation of nonlinear and linear parameters, which simplifies identification:

• The static nonlinearity can be extracted using single-tone AM-AM and AM-PM measurements
• The linear filter can be identified using small-signal S-parameter measurements or system identification techniques
• This decoupling reduces the dimensionality of the estimation problem compared to full Volterra series models
• Enables modular implementation where the nonlinearity can be implemented as a Look-Up Table and the filter as an FIR structure

The most common implementation uses a polynomial basis for the static nonlinearity:

• F(x) = Σₚ aₚ · x(n) · |x(n)|^(p-1) for odd-order nonlinearities
• Combined with an FIR filter: y(n) = Σₖ Σₚ h(k) · aₚ · x(n-k) · |x(n-k)|^(p-1)
• This formulation reveals that the Hammerstein model is a restricted subset of the Volterra series containing only diagonal kernel terms
• The restriction significantly reduces complexity while maintaining good accuracy for amplifiers with weak memory effects

The Hammerstein model assumes that memory effects are linear and independent of the signal amplitude, which breaks down for:

• GaN HEMT amplifiers with significant trapping-induced nonlinear memory
• Doherty power amplifiers where the load modulation creates amplitude-dependent memory dynamics
• Wideband signals where frequency-dependent nonlinear behavior becomes pronounced

For these cases, extensions like the Augmented Hammerstein model or Generalized Memory Polynomial are preferred, as they introduce cross-terms between nonlinear orders and different time delays.

The first stage of the Hammerstein model, representing the static AM-AM and AM-PM distortion. This block maps instantaneous input amplitude to output amplitude and phase without any dependence on past signal values.

• Typically implemented using a polynomial or look-up table (LUT)
• Captures gain compression and phase shift at high power levels
• Does not account for frequency-dependent behavior

The second stage that follows the static nonlinearity, introducing memory effects into the model. This filter shapes the frequency response of the distortion products generated by the preceding nonlinear block.

• Usually implemented as a finite impulse response (FIR) or infinite impulse response (IIR) filter
• Models electrical memory from bias networks and matching circuits
• The filter order determines the memory depth of the overall model

The dual architecture of the Hammerstein model, consisting of an LTI filter followed by a memoryless nonlinearity. The order of operations is reversed, making it suitable for different physical phenomena.

• Models systems where linear dynamics precede nonlinear distortion
• Often used when the PA input matching network dominates memory
• Hammerstein and Wiener models can be cascaded to form more complex structures

The process of determining the polynomial coefficients and filter taps from measured input-output data. Extraction typically uses least-squares estimation on a training dataset.

• Requires time-aligned complex baseband waveforms
• Cross-validation with independent test signals prevents overfitting
• The normalized mean square error (NMSE) quantifies extraction quality
• Numerical stability depends on the condition number of the regression matrix

The Hammerstein model can be expressed as a special case of the memory polynomial when the nonlinearity is polynomial-based. The LTI filter effectively weights the polynomial terms across time delays.

• A Hammerstein model with a polynomial nonlinearity and FIR filter is equivalent to a diagonal Volterra series
• The generalized memory polynomial extends this by adding cross-terms
• Hammerstein models typically require fewer coefficients than full Volterra models for comparable accuracy

Hammerstein models serve as inverse models for digital predistortion (DPD). The predistorter is often implemented as a Hammerstein structure placed before the PA to linearize the overall cascade.

• The indirect learning architecture extracts a postdistorter and copies it as the predistorter
• Hammerstein-based DPD is well-suited for FPGA implementation due to its separable structure
• Effective for PAs where static nonlinearity dominates over dispersive memory effects