Augmented Volterra for ET

The model accepts two independent inputs: the complex baseband RF signal and the dynamic supply voltage. This dual-input architecture explicitly captures supply-dependent nonlinearities that single-input models cannot represent.

• RF input terms: Standard Volterra kernels operating on past and present signal samples
• Supply input terms: Cross-kernels that multiply RF terms by supply voltage deviations
• Cross-memory terms: Products of delayed RF samples and delayed supply samples to capture dynamic supply-gain interactions

The kernel order is typically truncated to 3rd or 5th order for RF terms and 2nd or 3rd order for supply terms to balance accuracy with computational complexity.

A defining capability of the augmented Volterra model is its ability to characterize gain compression as a function of instantaneous drain voltage. As the supply modulator reduces voltage during low-envelope periods, the PA's gain characteristic shifts nonlinearly.

• Models the iso-gain contour behavior of the PA across its full supply voltage range
• Captures the nonlinear interaction between input drive level and supply voltage
• Enables the DPD to apply voltage-aware gain expansion to linearize the composite response
• Critical for ET systems where the PA operates across a wide range of supply-dependent bias points

Without this capability, traditional models fail to predict the AM-AM distortion that varies dynamically with the envelope tracking waveform.

The augmented Volterra model explicitly captures supply-induced phase modulation, a critical distortion mechanism in ET PAs where the dynamic drain voltage modulates the transistor's parasitic capacitances, causing input-phase-dependent output phase shifts.

• Supply-phase cross-kernels: Terms that multiply delayed phase inputs by supply voltage terms
• Models the voltage-dependent nonlinear input capacitance of GaN and LDMOS transistors
• Enables the DPD to apply phase pre-distortion that varies with both signal envelope and instantaneous supply voltage
• Essential for meeting EVM requirements in high-order QAM modulation schemes (64-QAM, 256-QAM)

This capability distinguishes augmented Volterra from memory polynomial models that assume constant phase response regardless of supply conditions.

Full augmented Volterra models suffer from exponential kernel explosion as memory depth and nonlinearity order increase. Practical implementations employ aggressive pruning strategies to reduce coefficients from thousands to hundreds.

• Near-diagonal pruning: Retains only kernels where RF and supply memory taps are close in time, reflecting the physical reality that cross-memory effects decay rapidly with temporal separation
• Dynamic deviation reduction: Separates the model into a static nonlinearity component and a low-order dynamic correction, dramatically reducing parameter count
• LASSO-based sparse regression: Uses L1-regularization during coefficient extraction to automatically identify and retain only statistically significant kernels
• Typical implementation: 50-200 coefficients for a 5G NR 100 MHz signal, down from 2000+ in the full model

The augmented Volterra framework can be extended to a joint model that simultaneously captures nonlinearities from both the power amplifier and the supply modulator. This unified approach eliminates the need for separate compensation stages.

• Modulator nonlinearity terms: Kernels that model supply voltage clipping, slew-rate limiting, and switching ripple artifacts
• Cascade nonlinearity representation: Captures the interaction where modulator distortion products intermodulate with the RF carrier through the PA's nonlinear transfer function
• Single-step linearization: The DPD pre-distorts the baseband signal to compensate for the entire modulator-PA cascade in one operation
• Reduces overall system complexity compared to cascaded compensation architectures that require separate modulator linearization and PA predistortion stages

Accurate augmented Volterra coefficient extraction requires training data that spans the full dynamic supply voltage range of the ET system. Standard single-supply training captures only one operating point and fails to generalize.

• Multi-level training: PA input-output data captured at multiple static supply voltages plus dynamic ET waveforms
• Envelope-statistics matching: Training signals must have the same complementary cumulative distribution function (CCDF) as the target modulation to ensure the model sees representative supply voltage distributions
• Temperature-stabilized capture: Thermal memory effects must be included by capturing data after the PA reaches thermal equilibrium at each operating condition
• Typical training dataset: 50,000-200,000 samples captured with a wideband observation receiver synchronized to both RF and supply voltage waveforms