Volterra MIMO DPD

Unlike single-antenna Volterra models, MIMO formulations employ multidimensional kernels that capture interactions between input signals from different antenna branches. A p-th order kernel h_p(k1,...,kp) maps contributions from multiple transmit paths, enabling the model to represent cross-modulation products where nonlinearity in one PA is influenced by signals from adjacent chains. This structure is essential for modeling antenna mutual coupling and crosstalk-induced distortion that single-input models cannot capture.

The Volterra MIMO framework explicitly models nonlinear crosstalk through cross-kernel terms that couple the inputs of different transmitter branches. Key crosstalk mechanisms captured include:

• Linear crosstalk: Direct signal leakage between adjacent RF chains before the PA
• Nonlinear crosstalk: Intermodulation products generated when a coupled signal mixes with the main signal inside a nonlinear PA
• Antenna mutual coupling: Electromagnetic interaction at the radiating elements that reflects power back into neighboring PAs This comprehensive modeling prevents the crosstalk-induced degradation that plagues independent per-branch DPD approaches.

A full MIMO Volterra model grows exponentially with the number of antennas and nonlinearity order, making it impractical for massive MIMO arrays. Practical implementations apply basis function pruning to retain only physically meaningful terms:

• Near-neighbor pruning: Only kernels coupling physically adjacent antennas are retained, as crosstalk decays with distance
• Dynamic deviation reduction: Separates static nonlinearity from low-order memory effects, eliminating high-order dynamic cross-terms
• Sparse regression: LASSO or OMP algorithms identify the most significant kernels from measured data These techniques can reduce the coefficient count by 90% or more while maintaining linearization performance.

Volterra MIMO DPD requires joint estimation of all predistorter coefficients across the array rather than independent per-branch training. The estimation typically employs:

• Least squares (LS): Batch solution computing the pseudo-inverse of the composite basis function matrix from all antenna outputs
• Recursive least squares (RLS): Adaptive variant suitable for tracking time-varying crosstalk conditions during beam steering
• Indirect learning architecture: Swaps input and output to identify the post-distorter, then copies coefficients to the predistorter The joint approach ensures that cross-coupling cancellation is optimized simultaneously with per-branch linearization.

In phased-array systems, changing beamforming weights alters the effective load impedance seen by each PA, modifying its nonlinear characteristics. Volterra MIMO DPD addresses this through:

• Load-dependent kernel functions: Kernels parameterized by the instantaneous beamforming vector to track impedance variations
• Look-up table augmentation: Storing multiple coefficient sets indexed by beam angle or active reflection coefficient
• Online coefficient interpolation: Smoothly transitioning between pre-characterized DPD states as the beam scans This adaptation maintains ACLR compliance across the full beam-steering range without requiring continuous full re-training.

To further reduce complexity, the full Volterra MIMO model is often simplified to a MIMO Generalized Memory Polynomial (GMP) structure. This retains cross-branch memory terms while eliminating the most computationally expensive high-order cross-kernels. The MIMO-GMP formulation includes:

• Aligned memory terms: Standard memory polynomial contributions from each branch
• Cross-branch memory terms: Lagging and leading envelope couplings between adjacent transmitters
• Cross-modulation terms: Products of the complex baseband signals from different branches at various memory depths This structure provides an implementable compromise between full Volterra accuracy and FPGA/ASIC feasibility.

A signal processing method that directly mitigates the antenna mutual coupling and crosstalk effects modeled by the off-diagonal Volterra kernels. While the Volterra MIMO DPD model captures these interactions mathematically, cross-coupling cancellation implements the inverse operation to nullify unintended electromagnetic interference between adjacent elements. This is critical for maintaining array gain in dense massive MIMO deployments.

A complexity-reduction technique that identifies and selects only the most significant Volterra basis functions from a large candidate set. By applying sparse estimation algorithms such as LASSO or OMP, the number of active coefficients in a Volterra MIMO DPD model can be reduced by 70-90% without meaningful linearization performance loss. This directly addresses the exponential coefficient growth that limits practical Volterra MIMO implementations.

A MIMO predistortion training architecture where the post-distorter is identified by swapping the input and output signals during coefficient estimation. For Volterra MIMO DPD, the indirect learning architecture avoids the need to explicitly compute the inverse of the multidimensional Volterra model. Instead, it estimates the post-inverse directly using least squares, making it the most common training approach for Volterra-based array linearization.

A dimensionality reduction technique for massive MIMO linearization that decomposes the array's nonlinear distortion into dominant spatial modes. By applying PCA to the Volterra MIMO DPD coefficient space, only the principal components corresponding to the strongest distortion patterns are retained. This reduces the effective model dimension from scaling with the number of antennas to scaling with the number of significant spatial distortion modes.

A resource-efficient technique where a common set of Volterra kernel coefficients is applied across multiple antenna branches exhibiting similar nonlinear behavior. In a massive MIMO array, adjacent elements often share nearly identical PA characteristics and coupling environments. Coefficient sharing exploits this redundancy to dramatically reduce the total number of Volterra MIMO DPD parameters that must be stored and updated in real-time.