Multi-Dimensional DPD

The predistorter uses a multi-dimensional indexing structure where the correction signal is a function of the instantaneous envelope magnitudes of all concurrent bands. For an N-band transmitter, the DPD function becomes:

• f(|x₁(n)|, |x₂(n)|, ..., |xₙ(n)|)
• Each dimension represents the amplitude of one carrier signal
• Cross-terms capture inter-band modulation dependencies
• The dimensionality grows linearly with the number of bands

This generalized framework naturally extends 2D-DPD concepts to tri-band, quad-band, and higher-order concurrent transmission scenarios.

Multi-dimensional DPD explicitly models cross-band nonlinear interactions that occur when multiple carriers pass through a common power amplifier:

• Cross-modulation products: Envelope transfer between bands
• Inter-band IMD: Distortion products falling between or on top of active carriers
• Cross-band memory effects: Past envelope history in one band affecting current behavior in another

The model includes cross-term basis functions that couple the instantaneous amplitudes of different bands, capturing the full nonlinear mixing behavior within the amplifier.

The multi-dimensional DPD model is mathematically derived from a passband Volterra series expansion, analytically describing baseband nonlinear behavior for N concurrent signals:

• Multi-dimensional Volterra kernels capture nonlinearity order and memory depth across all bands
• Pruned basis functions reduce complexity by selecting only physically relevant cross-terms
• Multi-Band Generalized Memory Polynomial (MB-GMP) provides a practical, simplified implementation
• Includes both diagonal terms (self-distortion) and off-diagonal terms (cross-band distortion)

This rigorous foundation ensures the model can theoretically cancel all nonlinear products of a specified order.

All predistorter coefficients—including cross-band terms—are estimated simultaneously through joint coefficient extraction:

• Multi-Band Indirect Learning Architecture (MB-ILA) identifies a post-distorter from attenuated PA output
• Least squares estimation solves for all coefficients in a single optimization step
• The estimation problem scales as O(K³) where K is the total number of basis functions
• Block-wise estimation can reduce computational burden for high-dimensional models

Joint estimation ensures that cross-band cancellation terms are optimally aligned with the self-distortion terms for each band.

Multi-dimensional DPD faces significant implementation challenges as the number of bands increases:

• Basis function count grows combinatorially with dimensionality
• Look-up table size expands exponentially for multi-dimensional LUT approaches
• Sampling rate requirements must satisfy Nyquist for the full composite bandwidth
• FPGA resource utilization scales with the number of complex multipliers and memory blocks

Practical implementations often use pruned basis sets, multi-rate processing, or subband decomposition to manage complexity while maintaining linearization performance.

Multi-dimensional DPD is essential for 3GPP carrier aggregation scenarios in 5G NR and LTE-Advanced:

• Intra-band contiguous CA: Multiple component carriers within a single band
• Intra-band non-contiguous CA: Carriers within same band but separated in frequency
• Inter-band CA: Carriers in different frequency bands sharing a common PA
• Dual connectivity (EN-DC): Simultaneous LTE and NR transmissions

The framework supports mixed numerology scenarios where different carriers have independent bandwidths, modulation schemes, and resource block allocations.

The direct precursor to multi-dimensional DPD, 2D-DPD uses a two-dimensional indexing structure based on the instantaneous magnitudes of two concurrent baseband signals to synthesize the correction signal. It establishes the core principle that predistortion in a concurrent multi-band transmitter requires knowledge of the envelope of all active carriers, not just the band being linearized. The 2D look-up table or 2D memory polynomial serves as the baseline upon which higher-dimensional models are built.

The primary impairment that multi-dimensional DPD is designed to cancel. When three or more carriers interact in a nonlinear power amplifier, they generate intermodulation products that fall into and around each desired transmit band. These products include:

• Cross-modulation: envelope transfer between bands
• Inter-band IMD: products falling in frequency gaps
• In-band distortion: spectral regrowth within each carrier Multi-dimensional DPD models these interactions explicitly through cross-term coefficients.

The MB-GMP is the mathematical workhorse for implementing multi-dimensional DPD. It extends the generalized memory polynomial by incorporating cross-band envelope coupling terms and sample-crossing terms that capture the nonlinear interaction between three or more concurrent signals. The model includes:

• Diagonal terms for self-distortion per band
• Off-diagonal cross-terms for inter-band effects
• Memory depth parameters for each interaction path This structure balances modeling accuracy with computational tractability.

A parameter identification technique that simultaneously estimates all coefficients of a multi-dimensional predistorter—including cross-band terms—in a single optimization step. Unlike sequential or band-by-band extraction, joint estimation accounts for the coupled nature of multi-band distortion. The process typically uses least-squares or recursive algorithms operating on a composite error vector that spans all transmit bands, ensuring that correcting distortion in one band does not inadvertently degrade linearization in another.

The most common instantiation of multi-dimensional DPD, Tri-Band DPD linearizes a power amplifier simultaneously transmitting three independent carrier signals. The predistorter becomes a three-dimensional function of the instantaneous amplitudes |x₁|, |x₂|, and |x₃|. This architecture is increasingly relevant for carrier aggregation scenarios in 5G NR where three component carriers are aggregated through a single transmitter chain, requiring explicit modeling of all three-way cross-band interactions.

The MB-ILA provides the closed-loop adaptation mechanism for multi-dimensional DPD. A post-distorter model is identified from the attenuated PA output and then copied to the predistorter in the forward path. For multi-dimensional systems, the post-distorter training uses a multi-input error signal that captures residual distortion across all bands simultaneously. This architecture avoids the instability risks of direct learning while enabling continuous adaptation to changing PA characteristics due to temperature, aging, or load variation.