2D Memory Polynomial (2D-MMP)

The core innovation of 2D-MMP is its basis function, which is a product of delayed samples and envelope-dependent cross-terms. The model output is a function of x1(n-m) * |x1(n-m)|^k * |x2(n-m)|^j and x2(n-m) * |x2(n-m)|^k * |x1(n-m)|^j.

• Cross-band memory: Captures how the past envelope of Band 2 affects the current distortion in Band 1.
• Nonlinearity order: Controlled by the polynomial orders k and j for self and cross terms.
• Memory depth: Defined by the maximum lag M, representing the duration of memory effects.

For a dual-band input with baseband signals x1(n) and x2(n), the 2D-MMP output for Band 1 is:

y1(n) = Σ_{m=0}^{M} Σ_{k=0}^{K} Σ_{j=0}^{J} a_{m,k,j}^{(1)} * x1(n-m) * |x1(n-m)|^k * |x2(n-m)|^j

• Coefficients: a_{m,k,j} are the complex-valued model parameters.
• Symmetry: A structurally identical model with coefficients a_{m,k,j}^{(2)} is used for Band 2.
• Truncation: The double summation over k and j is truncated to a finite set of orders to manage complexity.

A key advantage of 2D-MMP over two independent memory polynomials is its ability to model cross-band memory effects. This is the phenomenon where the thermal or electrical memory of the power amplifier causes the distortion in one band to depend on the past envelope power of the other band.

• Mechanism: The |x2(n-m)|^j term explicitly makes the Band 1 output a function of the delayed Band 2 envelope.
• Importance: Critical for wideband concurrent signals where long-term thermal time constants cause inter-band dynamic interactions.
• Result: Significantly improves modeling accuracy and DPD linearization performance compared to band-ignorant models.

The 2D-MMP is linear in its coefficients, allowing for robust extraction using Least Squares (LS) estimation. However, the total number of coefficients grows rapidly.

• Coefficient count: (M+1) * (K+1) * (J+1) per band. A model with M=3, K=5, J=2 has 72 coefficients.
• Estimation: A single LS matrix solve can extract all coefficients simultaneously from a captured input-output data record.
• Trade-off: The model offers high fidelity but can become computationally heavy for hardware implementation if not pruned.

The 2D-MMP is a pruned subset of the full Dual-Band Volterra series. It retains only the terms that are aligned in time delay across the signal and its envelope cross-products.

• Volterra: Includes terms with different delays for each component, e.g., x1(n-m1) * x1*(n-m2) * x2(n-m3).
• 2D-MMP: Restricts all delays to the same index m, drastically reducing the number of terms.
• Rationale: This alignment captures the dominant memory effects while discarding less significant, computationally expensive cross-delay terms.

In a DPD application, the 2D-MMP model is used in the reverse configuration as a predistorter. The input signals x1 and x2 are the desired transmission signals, and the model output is the predistorted signal sent to the DAC.

• Architecture: Typically used within an Indirect Learning Architecture (ILA) for coefficient adaptation.
• Hardware: Can be implemented on FPGAs using a combination of LUTs and multipliers, where the 2D index is (|x1|, |x2|).
• Challenge: The high dimensionality of the basis function requires careful resource management and coefficient quantization.

The overarching linearization architecture that 2D-MMP serves. This technique compensates for nonlinear distortion in a single power amplifier that is simultaneously transmitting two or more widely spaced carrier signals. Unlike single-band DPD, it must account for cross-band intermodulation products that fall both in-band and out-of-band. The predistorter generates a composite correction signal that pre-inverts the PA's multi-band nonlinear characteristic.

The primary impairment that 2D-MMP is designed to model and cancel. These are nonlinear interference products generated by the interaction of multiple carrier signals within a power amplifier. Key types include:

• Inter-band IMD: Products falling in the frequency gap between transmit bands
• Cross-modulation: Envelope transfer from one band onto another
• In-band IMD: Products overlapping with the original transmit signals 2D-MMP captures these through cross-terms dependent on the envelope magnitudes of both bands.

A broader class of predistortion models that use a two-dimensional indexing structure to synthesize the correction signal. The two dimensions are typically the instantaneous magnitudes of two concurrent baseband signals: |x₁(n)| and |x₂(n)|. 2D-MMP is a specific polynomial-based implementation of 2D-DPD. Other implementations include 2D Look-Up Tables (2D-LUT), which offer hardware-efficient alternatives by storing pre-computed complex gain corrections indexed by quantized magnitude pairs.

An extension of the generalized memory polynomial that incorporates more complex cross-band interactions than 2D-MMP. While 2D-MMP includes cross-terms dependent on envelope magnitudes, MB-GMP adds:

• Cross-band sample-crossing terms: Products of delayed samples from different bands
• Higher-order cross-envelope terms: More complex envelope coupling functions MB-GMP offers higher modeling accuracy at the cost of significantly more coefficients, making 2D-MMP the preferred choice when computational complexity is constrained.

The signal processing procedure for estimating 2D-MMP parameters from observed PA input-output waveforms. This typically employs least-squares estimation on a matrix constructed from basis functions of the dual-band input signals. Key considerations include:

• Joint coefficient estimation: Solving for all coefficients including cross-terms in a single optimization step
• Condition number management: Ensuring the basis matrix is well-conditioned for stable inversion
• Data length requirements: Sufficient samples to capture the full memory depth and nonlinear order

The closed-loop adaptation method most commonly used to train 2D-MMP predistorters. In MB-ILA:

1. A post-distorter model is identified from the attenuated PA output
2. The post-distorter coefficients are copied directly to the predistorter in the forward path
3. This assumes the post-inverse equals the pre-inverse, which holds for most PA nonlinearities MB-ILA avoids the need for a PA model and converges robustly in real-time adaptive systems.