ET-DPD Joint Model

The defining structural characteristic of an ET-DPD joint model is its dual-input topology. Unlike conventional single-input DPD models that only accept the RF baseband signal, this architecture explicitly accepts both the complex baseband input (I/Q) and the instantaneous supply voltage as independent variables.

• Input 1: Complex baseband envelope signal
• Input 2: Dynamic drain/collector voltage from the supply modulator
• Output: Predicted complex baseband output, capturing supply-dependent distortion

This dual-input structure is essential because the PA's gain and phase response shift dynamically with the supply voltage. A single-input model cannot distinguish between distortion caused by RF compression and distortion caused by supply modulation, making it fundamentally incapable of linearizing an ET system.

Joint models must capture cross-term interactions between the RF envelope and the supply voltage. These are nonlinear products where the distortion depends on both signals simultaneously, not simply the sum of their individual effects.

Key cross-term phenomena include:

• Supply-dependent AM/AM: Gain compression that varies with instantaneous drain voltage
• Supply-dependent AM/PM: Phase shift that changes as the supply modulates
• Intermodulation products: Mixing between the RF carrier and supply modulator switching ripple

Mathematically, this requires model structures with bivariate kernels—terms that multiply functions of the RF input with functions of the supply voltage. The Volterra series is extended to include these cross-kernels, dramatically increasing model complexity compared to single-input DPD.

An effective ET-DPD joint model must account for three distinct categories of memory effects operating on different timescales:

• RF memory effects: Caused by matching network impedance and bias circuit dynamics, typically spanning nanoseconds to microseconds
• Supply modulator memory: Introduced by the limited bandwidth and non-flat frequency response of the DC-DC converter, affecting the envelope tracking path
• Thermal memory: Long-term drift in PA characteristics due to self-heating, which interacts with supply modulation to create slow-varying distortion

Joint models incorporate memory through tapped delay lines on both the RF and supply inputs, with cross-memory terms that capture interactions between delayed versions of both signals. The memory depth required for the supply path is often longer than for the RF path due to the slower dynamics of power converters.

The primary engineering challenge of ET-DPD joint models is the exponential growth in model coefficients as nonlinearity order and memory depth increase across two input dimensions.

Consider a practical comparison:

• A single-input memory polynomial with nonlinearity order 7 and memory depth 3 requires approximately 28 coefficients
• An equivalent dual-input model with the same parameters requires approximately 196 coefficients (7×7×4 combinations)

This dimensionality explosion creates three critical problems:

• Coefficient extraction becomes computationally intensive and numerically ill-conditioned
• Real-time implementation demands significant FPGA resources for coefficient storage and multiplication
• Overfitting risk increases, requiring careful regularization during training

Pruning strategies, such as near-diagonal kernel restriction and principal component analysis, are essential for practical deployment.

The defining operational advantage of a joint model is that it produces a single predistorted signal that simultaneously compensates for both PA and supply modulator nonlinearities. This eliminates the need for separate correction stages.

Signal flow in a joint ET-DPD system:

1. Baseband I/Q signal enters the predistorter
2. The shaping function generates the target supply voltage from the signal envelope
3. Both the I/Q signal and the target supply voltage feed into the joint predistorter model
4. The model applies a complex gain correction that pre-distorts the I/Q signal
5. The predistorted I/Q drives the RF path while the shaped envelope drives the supply modulator
6. The combined nonlinearities of both subsystems cancel at the PA output

This unified approach ensures that the predistortion accounts for the compounded distortion at the point where RF and DC power converge—the PA transistor.

Training an ET-DPD joint model requires specialized measurement data that captures the PA's behavior across its full two-dimensional operating space. Standard single-supply characterization is insufficient.

Essential training data characteristics:

• The PA must be exercised with varying supply voltages synchronized to the RF envelope
• The excitation signal must cover the full dynamic range of both input dimensions
• Time-aligned captures of RF input, supply voltage, and RF output are mandatory
• Data must include representative envelope tracking waveforms, not just static supply points

Measurement campaigns typically use a two-dimensional grid sweep: the PA is characterized at multiple fixed supply voltages, and interpolation fills the gaps. Advanced approaches use actual ET waveforms with wideband modulated signals to capture dynamic supply-RF interactions directly. The dataset must be large enough to prevent ill-conditioning during least-squares coefficient extraction.

The foundational mathematical structure for an ET-DPD Joint Model. Unlike a standard single-input model, this framework accepts two independent variables: the complex baseband RF input signal and the instantaneous dynamic supply voltage. This allows the model to capture the nonlinear interaction where the PA's gain and phase shift are functions of both input drive and drain bias. It is essential for predicting the supply-dependent gain compression that a joint predistorter must invert.

An extension of the classical Volterra series specifically designed for the ET-DPD Joint Model. It introduces additional kernel terms that are functions of the dynamic supply voltage. This structure mathematically captures the complex cross-memory effects between the RF envelope and the supply modulation. For example, a term might describe how the PA's response to a past RF sample is scaled by the current supply voltage, enabling the model to represent the compounded nonlinear dynamics of the entire transmitter chain.

A memoryless implementation of the joint model, ideal for compensating static nonlinearities. This structure is indexed by two dimensions: instantaneous input power and instantaneous supply voltage. The output is a complex gain correction factor. It directly maps the iso-gain contours of the PA, providing a fast, hardware-efficient method to linearize the ET-induced AM/AM and AM/PM distortion without solving complex polynomials in real-time.

The most critical physical prerequisite for a valid joint model. The RF signal path and the supply voltage path must be precisely time-synchronized at the transistor's drain. A timing mismatch of even a fraction of a nanosecond causes the model to learn a false correlation between the wrong input sample and the wrong supply voltage, leading to severe model error and a complete failure of the predistortion. Sub-nanosecond alignment is non-negotiable.

A source of error that a robust joint model must inherently account for. The supply modulator is not an ideal voltage source; it introduces its own distortions like slew-rate limiting, switching ripple artifact, and a non-flat frequency response. A comprehensive ET-DPD model treats the modulator and PA as a single cascaded nonlinear system, learning the combined transfer function so that the predistorter can pre-compensate for both the modulator's and the amplifier's imperfections.

The coefficient extraction process unique to the joint model. Training data must be collected while the PA is operating under dynamic supply modulation, sweeping through its full range of voltages and input powers. This ensures the model learns the PA's behavior not just at a fixed bias, but across the entire iso-gain contour map. The resulting predistorter is then valid for all tracking conditions, not just a single static point.