Forward Path Modeling

Forward path models operate as black-box behavioral models, meaning they replicate the input-output relationship of a power amplifier without requiring knowledge of internal transistor physics. The model is trained solely on observed data—typically complex baseband IQ samples—to learn the mapping from input envelope to output distortion. This approach enables rapid prototyping across different amplifier architectures, including Doherty, envelope tracking, and GaN-based designs, without customizing the model structure for each topology.

The model must accurately reproduce AM-AM distortion (gain compression/expansion) and AM-PM distortion (phase shift variation) as functions of instantaneous input amplitude. These static non-linearities are typically represented through:

• Polynomial basis functions (memory polynomial, Volterra kernels)
• Look-up tables indexed by input magnitude
• Neural network activation functions (RVTDNN architectures)

Accurate static modeling is the prerequisite before addressing temporal dependencies.

Forward path models must capture memory effects—the dependence of current output on past input values. These arise from:

• Thermal dynamics: Transistor junction temperature changes with signal envelope
• Bias network impedance: Low-frequency envelope currents modulate supply voltage
• Trapping effects: Charge capture/release in semiconductor defects

Models incorporate memory through tapped delay lines, finite impulse response filters, or recurrent neural network connections that span multiple symbol periods.

The Generalized Memory Polynomial (GMP) serves as the workhorse forward path model, extending the basic memory polynomial with cross-terms between the signal and its lagging/leading envelope values. The GMP equation includes:

• Aligned memory terms: x(n-m)|x(n-m)|^k
• Lagging cross-terms: x(n-m)|x(n-m-l)|^k
• Leading cross-terms: x(n-m)|x(n-m+l)|^k

This structure captures complex interactions between instantaneous power and recent signal history that simpler models miss.

Real-Valued Time-Delay Neural Networks (RVTDNN) process I and Q components separately through:

• Input tapped delay lines spanning memory depth
• Multiple hidden layers with non-linear activation functions (tanh, ReLU)
• Dual output neurons for I and Q predistorted components

Compared to polynomial models, neural forward models offer superior generalization to unseen signal distributions and can naturally capture higher-order non-linearities without exponential growth in coefficient count.

Forward path model accuracy is quantified using:

• Normalized Mean Squared Error (NMSE): Measures time-domain waveform fidelity between modeled and measured output. Values below -35 dB indicate excellent modeling
• Adjacent Channel Error Power Ratio (ACEPR): Evaluates spectral regrowth prediction accuracy in adjacent channels
• EVM contribution analysis: Isolates the portion of total EVM attributable to modeling inaccuracy versus measurement noise

Validation must span multiple signal types (LTE, 5G NR, WLAN) and power levels to ensure generalization.

The fundamental physical phenomenon that forward path modeling seeks to replicate. As a PA approaches its saturation point, its gain compresses and phase shifts, creating a non-linear transfer function. This behavior is the root cause of spectral regrowth and in-band distortion in wireless transmitters. Accurate modeling of this non-linearity—including its dependence on input amplitude, frequency, and temperature—is the primary objective of behavioral modeling.

A critical complexity in forward path modeling where the PA's current output depends not only on the instantaneous input but also on past signal values. These effects arise from:

• Thermal dynamics: Transistor junction temperature changes with signal envelope
• Bias network impedance: Low-frequency resonances in the DC supply
• Trapping effects: Charge capture and release in semiconductor materials Models that ignore memory effects, such as static AM-AM/AM-PM curves, fail to linearize wideband signals. Volterra series and memory polynomials explicitly account for these temporal dependencies.

A widely adopted behavioral model structure that extends the basic memory polynomial by including cross-terms between the signal and its lagging or leading envelope values. The GMP captures complex memory effects with fewer coefficients than a full Volterra series, making it computationally tractable for real-time DPD. Its equation includes:

• Aligned signal and exponentiated envelope terms
• Lagging cross-terms (signal * delayed envelope)
• Leading cross-terms (signal * advanced envelope) This structure serves as a strong baseline before exploring neural network-based forward models.

The rigorous mathematical foundation for modeling non-linear dynamic systems with memory. A Volterra series represents the PA output as a sum of multi-dimensional convolution integrals, where each order captures progressively higher-order non-linear interactions. While theoretically complete, the exponential growth of coefficients with memory depth and non-linearity order makes it impractical for wideband signals. Pruned Volterra models and their simplified derivatives—like the memory polynomial—are the practical implementations used in forward path modeling.

The two fundamental distortion components that a forward model must characterize:

• AM-AM (Amplitude-to-Amplitude): The non-linear relationship between input amplitude and output amplitude, manifesting as gain compression or expansion
• AM-PM (Amplitude-to-Phase): The non-linear phase shift introduced as a function of instantaneous input amplitude These are typically measured using a vector network analyzer or extracted from IQ data captures. Static AM-AM/AM-PM curves provide a memoryless baseline, while dynamic models extend these into surfaces that vary with signal history.

The black-box methodology underlying forward path modeling. Rather than deriving equations from semiconductor physics, behavioral modeling treats the PA as an unknown system and identifies its input-output relationship purely from observed data. This approach:

• Requires no knowledge of internal transistor geometry or biasing
• Uses system identification techniques to fit model parameters
• Can be implemented with polynomials, neural networks, or look-up tables Behavioral models are validated by comparing their output to measured PA data using metrics like NMSE (Normalized Mean Square Error).