Hammerstein Model

The first stage applies a memoryless polynomial to the input signal magnitude. This block captures AM/AM and AM/PM distortion—the instantaneous gain compression and phase shift that depend solely on the current input envelope. Common implementations use:

• Odd-order polynomial terms (3rd, 5th, 7th order)
• Complex-valued coefficients to model phase distortion
• Look-up table (LUT) indexing for efficient hardware realization

The absence of memory in this block means it cannot model thermal or bias-related hysteresis effects independently.

The second stage is a linear time-invariant (LTI) filter that models the frequency-dependent memory effects of the PA. This filter shapes the output of the static nonlinearity, capturing:

• Short-term memory from impedance matching networks
• Dispersion effects in the signal path
• Frequency-selective behavior of bias circuits

Typically implemented as a finite impulse response (FIR) or infinite impulse response (IIR) filter. The linear nature of this block makes parameter estimation straightforward using standard system identification techniques.

The Hammerstein model assumes a cascaded architecture where nonlinearity strictly precedes memory. This is valid for PAs where:

• The active device generates distortion before matching network filtering
• AM/PM conversion occurs primarily in the transistor's nonlinear transconductance
• Memory effects are dominated by output-side impedance interactions

For PAs where memory effects precede nonlinearity (e.g., input matching network filtering before the transistor), the Wiener model (LTI filter followed by static nonlinearity) is more appropriate. The Wiener-Hammerstein cascade generalizes both structures.

Coefficient estimation for the Hammerstein model typically uses least squares (LS) estimation in a two-step process:

1. Decouple the blocks: Use correlation-based techniques to separate the nonlinear and linear contributions
2. Sequential fitting: Fit the static nonlinearity first, then identify the LTI filter from the residual

Alternative approaches include iterative separable least squares and prediction error methods. The block-structured nature enables more efficient extraction than a full Volterra series while maintaining good accuracy for many PA classes.

For digital predistortion, the Hammerstein model requires inversion to create the predistorter. The inverse of a Hammerstein system is a Wiener system (LTI filter followed by static nonlinearity). This means:

• The DPD implementation uses a different block order than the PA model
• The static predistorter nonlinearity approximates the inverse of the PA's AM/AM and AM/PM curves
• The LTI filter compensates for frequency-dependent memory

This inversion property makes Hammerstein-based DPD particularly suitable for indirect learning architectures where the predistorter is trained as the post-inverse of the PA.

The Hammerstein model is a special case of the memory polynomial (MP) model. Key differences:

• MP model: Includes cross-terms between different delays and nonlinear orders, capturing interactions between memory and nonlinearity
• Hammerstein: Restricts to separable nonlinearity and memory, reducing coefficient count
• Accuracy trade-off: Hammerstein uses fewer parameters but cannot model delay-dependent nonlinear behavior
• Complexity advantage: Lower computational load makes it suitable for FPGA implementation with limited DSP resources

For wideband signals where memory-nonlinearity interactions are significant, the Generalized Memory Polynomial (GMP) or Parallel Hammerstein structures offer better accuracy.

The structural inverse of the Hammerstein model, consisting of a linear time-invariant (LTI) dynamic filter followed by a static memoryless nonlinearity. This topology is appropriate for power amplifiers where memory effects precede nonlinear distortion, such as those with significant input matching network dynamics. The Wiener model is often paired with the Hammerstein model to create more complex cascaded structures.

A three-block model that sandwiches a static memoryless nonlinearity between two linear time-invariant filters. This structure captures complex PA dynamics where memory effects occur both before and after the nonlinear distortion. Key characteristics:

• Input filter models source impedance and matching network effects
• Output filter captures bias network and thermal memory
• Provides superior accuracy for GaN Doherty amplifiers with complex electro-thermal interactions

An architecture composed of multiple Hammerstein branches operating in parallel, where each branch has a distinct static nonlinearity followed by a linear filter. This structure models complex nonlinear dynamics by decomposing the overall behavior into parallel subsystems. Each branch can be tuned to capture different nonlinear orders or memory time constants, making it effective for wideband signals where frequency-dependent nonlinearities are significant.

A foundational behavioral model that uses a polynomial with tapped delay lines to capture both nonlinear distortion and memory effects simultaneously. Unlike the Hammerstein model's sequential structure, the MP model applies polynomial terms directly to delayed signal samples. This model serves as the baseline for more advanced structures like the Generalized Memory Polynomial (GMP) and is widely used due to its simple implementation and linear-in-parameters estimation.

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 interactions that the standard MP model misses. The GMP structure is particularly effective for:

• Envelope-dependent memory effects
• Asymmetric intermodulation distortion
• Amplifiers with significant bias modulation effects

A model variant that incorporates memory effects of the signal's envelope magnitude rather than just the complex baseband signal. This structure effectively captures long-term thermal and bias-related memory in power amplifiers by modeling how the envelope history influences current gain. The envelope memory polynomial is particularly useful for amplifiers exhibiting slow electro-thermal dynamics that cannot be captured by short-term memory models alone.