Hammerstein Model

The defining architecture of the Hammerstein model is its cascade of two distinct blocks: a static memoryless nonlinearity followed by a linear time-invariant (LTI) dynamic filter. This separation assumes that the system's nonlinear behavior is instantaneous and occurs before any linear memory effects. The input signal first passes through a polynomial nonlinearity, and the resulting distorted signal is then shaped by the linear filter's impulse response. This structure is a special case of the Volterra series where the kernels are constrained to be separable products of the nonlinear coefficients and the linear filter taps.

The first block applies an instantaneous, amplitude-dependent distortion without any dependence on past inputs. This is typically modeled as a polynomial function of the input magnitude:

• Odd-order terms dominate in differential and push-pull amplifier configurations
• Even-order terms appear in single-ended designs and contribute to spectral regrowth at baseband
• The nonlinearity captures AM-AM distortion (gain compression/expansion) and AM-PM distortion (phase shift as a function of amplitude)
• Common implementations use a look-up table (LUT) indexed by instantaneous input power for efficient real-time execution

The second block is a linear time-invariant filter that models the frequency-dependent memory effects of the system. This filter shapes the spectrum of the distorted signal produced by the static nonlinearity. Key characteristics include:

• Implemented as a finite impulse response (FIR) or infinite impulse response (IIR) filter
• Captures electrical memory effects caused by bias network impedance and matching circuits
• Models thermal memory effects through low-frequency poles in the filter response
• The filter order determines the memory depth of the model, with higher orders capturing longer-duration transient effects

Hammerstein model coefficients are typically estimated using least squares estimation in a two-step or iterative procedure. The separable structure allows for efficient identification:

• Iterative Narendra-Gallman algorithm alternates between updating the nonlinear and linear blocks
• Over-parameterization method treats the combined parameters as a bilinear problem, solving via singular value decomposition
• Prediction error minimization jointly optimizes all parameters using gradient descent
• The model requires significantly fewer coefficients than an equivalent Volterra series, reducing the risk of overfitting and improving numerical conditioning

The Hammerstein model is a constrained subset of the Volterra series where the Volterra kernels are restricted to be separable. Specifically:

• The nth-order Volterra kernel is expressed as the product of the nth-order nonlinear coefficient and the linear filter's impulse response
• This constraint eliminates all cross-term kernels that involve interactions between different time delays at different nonlinear orders
• The model is exact for systems where the nonlinearity is strictly at the input and the linear dynamics are strictly at the output
• For power amplifiers, this assumption holds when the dominant nonlinearity is in the input capacitance or gate-source junction of the transistor

The Hammerstein model's primary limitation is its inability to capture output-dependent nonlinearities or nonlinear memory effects where the distortion characteristics change with signal history. Specific constraints include:

• Cannot model systems where linear filtering precedes nonlinear distortion (use the Wiener model instead)
• Fails to capture nonlinear memory effects such as trapping-induced dispersion in GaN HEMT amplifiers
• Inaccurate for Doherty power amplifiers where the load modulation creates complex interactions between linear and nonlinear dynamics
• The Parallel Hammerstein model extends the structure by using multiple parallel branches to capture more complex behaviors while maintaining block-structured simplicity

The Hammerstein model serves as a foundational behavioral model for power amplifiers in DPD systems. Its structure—a memoryless nonlinearity followed by a linear filter—accurately captures PAs where the dominant nonlinear distortion occurs at the input stage, before the matching network's filtering effects. This makes it computationally efficient for real-time predistorter coefficient extraction.

• Models AM-AM and AM-PM distortion as a static nonlinear block
• Captures frequency-dependent memory effects via the linear dynamic block
• Enables direct inversion for feedforward predistorter design

Satellite communication channels often exhibit nonlinear amplification in the traveling wave tube amplifier (TWTA) followed by dispersive filtering in the output multiplexer. The Hammerstein model naturally represents this cascade, where the TWTA's soft-limiting nonlinearity precedes the multiplexer's bandpass filtering.

• Models TWTA saturation and phase conversion
• Accounts for group delay distortion from output filters
• Used for end-to-end link budget simulation and predistortion

For amplifier designs where the transistor's input capacitance creates a dominant nonlinearity before the device's internal parasitics filter the signal, the Hammerstein model provides a physically motivated representation. This is common in Class AB and Class B amplifier stages.

• Static block represents transconductance nonlinearity
• Dynamic block models bias network and thermal impedance
• Validated for GaN HEMT and LDMOS technologies

The Hammerstein model effectively captures frequency-dependent IQ modulator impairments where the nonlinear mixing process in the quadrature modulator precedes the analog reconstruction filters. The static nonlinearity models mixer compression, while the linear filter captures the frequency response mismatch between I and Q paths.

• Models local oscillator leakage and gain imbalance
• Compensates for frequency-selective quadrature error
• Enables joint DPD and IQ correction in a unified framework

In envelope tracking transmitters, the supply modulator's nonlinear dynamics interact with the RF PA's input nonlinearity. A Hammerstein model can represent the composite nonlinear behavior where the envelope path's static distortion precedes the PA's intrinsic filtering, enabling coordinated linearization.

• Captures supply modulator slew-rate limitations
• Models interaction between drain bias and RF gain
• Supports dual-input predistortion architectures

Directly modulated laser diodes exhibit a nonlinear current-to-light conversion (static nonlinearity) followed by chromatic dispersion in the optical fiber (linear dynamic filtering). The Hammerstein model accurately represents this electro-optical cascade for dispersion pre-compensation and nonlinearity mitigation.

• Models laser threshold and saturation nonlinearity
• Captures fiber dispersion as a linear all-pass filter
• Enables electronic predistortion at the transmitter