Multi-Band PA Modeling

Unlike single-band models, multi-band behavioral models must explicitly account for cross-modulation and intermodulation distortion (IMD) products generated by the nonlinear mixing of concurrent carriers. The model structure includes cross-terms that depend on the instantaneous envelope magnitudes of all active bands. For example, in a dual-band scenario, the output in Band 1 is a function not only of Band 1's input but also of Band 2's envelope, capturing effects like cross-band gain compression and envelope-dependent phase shifts.

The complexity of multi-band models scales nonlinearly with the number of concurrent bands. A 2D-DPD model for dual-band operation uses a two-dimensional indexing structure based on the magnitudes of both baseband signals. For tri-band or higher-order concurrent transmission, the model becomes a multi-dimensional function where the number of cross-term combinations grows combinatorially. This dimensionality explosion drives the need for pruning strategies that eliminate statistically insignificant cross-terms while retaining modeling fidelity.

Multi-band PAs exhibit both intra-band memory effects (within a single carrier) and cross-band memory effects (where the past envelope of one band influences the current nonlinear behavior of another). The 2D Memory Polynomial (2D-MMP) extends the classical memory polynomial by including lagging cross-terms that depend on the delayed envelope magnitudes of both bands. Thermal memory effects are particularly challenging in multi-band operation because the composite signal's time-varying peak-to-average ratio creates complex, band-dependent thermal dynamics.

The Dual-Band Volterra Series provides the rigorous analytical basis for multi-band behavioral modeling. Derived from the passband Volterra series through careful frequency-domain analysis, it yields baseband equivalent models that separate in-band distortion from cross-band distortion products. Key simplifications include:

• Multi-Band Generalized Memory Polynomial (MB-GMP): Retains critical cross-band envelope and sample-crossing terms while pruning higher-order Volterra kernels
• Multi-Band Memory Polynomial: A further simplified structure balancing accuracy against coefficient count for real-time implementation

Implementation constraints on FPGAs and ASICs drive the adoption of 2D Look-Up Table (2D-LUT) architectures for multi-band predistortion. Rather than computing polynomial cross-terms in real time, a 2D-LUT stores pre-computed complex gain correction values indexed by the quantized instantaneous magnitudes of both input signals. This approach trades memory for computational complexity, enabling high-bandwidth multi-band DPD on resource-constrained hardware. Adaptive LUT update mechanisms must handle the multi-dimensional interpolation required when input magnitudes fall between table entries.

Joint coefficient estimation simultaneously extracts all model parameters—including cross-band terms—in a single optimization step, typically using least-squares or iterative learning algorithms on the composite multi-band waveform. This approach preserves the statistical coupling between bands but requires solving larger linear systems. Alternatively, separate identification estimates per-band and cross-band coefficients in sequential stages, reducing computational burden at the potential cost of suboptimal cross-band cancellation. The Multi-Band Indirect Learning Architecture (MB-ILA) provides a closed-loop framework for adaptive coefficient updates.

Nonlinear interference products generated by the interaction of multiple carrier signals within a single power amplifier. These unwanted components fall on top of or near the desired transmit bands.

• Inter-band IMD falls in the frequency gap between transmit bands.
• Cross-modulation transfers the modulation envelope of one signal onto another.
• Requires dedicated cross-band predistorter blocks for cancellation, as single-band DPD cannot address these products.

A rigorous mathematical model derived from the passband Volterra series that analytically describes baseband nonlinear behavior in dual-band transmitters.

• Provides the theoretical foundation for all simplified multi-band models.
• Explicitly includes cross-band kernel terms that capture the physical origin of intermodulation.
• While too complex for real-time implementation, it serves as the gold standard for model validation and coefficient extraction benchmarking.

A hardware-efficient predistorter implementation where complex gain correction values are indexed by a two-dimensional address derived from the instantaneous magnitudes of two concurrent input signals.

• Eliminates the need for real-time polynomial evaluation, reducing FPGA DSP slice consumption.
• Requires sophisticated adaptive table update mechanisms to track changing PA characteristics.
• The 2D indexing structure directly maps to the envelope coupling terms in the 2D-MMP model.

The signal processing procedure for estimating multi-band DPD model parameters from observed PA input and output waveforms.

• Joint coefficient estimation solves for all parameters, including cross-band terms, in a single optimization step.
• Indirect learning architecture (MB-ILA) identifies a post-distorter from the attenuated output and copies it to the forward path.
• Requires precise time alignment and phase synchronization between multi-band capture streams.

A long-term memory phenomenon where the nonlinear behavior in one frequency band is influenced by the past envelope history of a signal in a different band.

• Caused by thermal coupling and shared bias network impedance across frequency paths.
• Cannot be captured by single-band memory models; requires explicit cross-band envelope lag terms.
• Particularly significant in GaN Doherty amplifiers operating at high power with closely spaced carriers.