Multi-Band Memory Polynomial

The defining characteristic of the MB-MP model is its inclusion of cross-band envelope coupling terms. These terms model how the instantaneous envelope power of one carrier band modulates the nonlinear distortion experienced by another band.

• Captures intermodulation distortion (IMD) products that fall within or near the transmit bands
• Models the physical mechanism where the total instantaneous power drives amplifier compression
• Uses terms of the form x₁(n)|x₂(n-m)|ᵏ where the signal in band 1 is multiplied by envelope powers of band 2
• Essential for carrier aggregation scenarios where bands are widely spaced

The MB-MP is derived from the full passband Volterra series by analytically selecting only the distortion terms that fall within the linearization bandwidth of each carrier.

• Begins with the general Volterra series and applies bandpass filtering assumptions
• Retains only baseband terms that alias into the transmit band after nonlinear mixing
• Eliminates terms that produce distortion at frequencies far from the carriers, reducing coefficient count by orders of magnitude
• Results in a model with polynomial order × memory depth × band count complexity rather than exponential growth

The MB-MP extends static polynomial models by incorporating memory taps that capture the amplifier's dependence on past input samples.

• Each band includes its own memory polynomial terms: xᵢ(n-m)|xᵢ(n-m)|ᵏ
• Cross-band memory terms model how the past envelope of one band affects current distortion in another
• Captures short-term memory effects from bias circuit impedance and matching network dynamics
• Memory depth typically ranges from 2–5 taps, balancing accuracy with FPGA implementation cost

The MB-MP offers a middle-ground complexity between the full Volterra series and simpler single-band memory polynomials.

• Coefficient count scales as O(B² × M × K) where B is band count, M is memory depth, and K is nonlinearity order
• For a dual-band system with M=3, K=7: approximately 100–200 complex coefficients
• Significantly fewer coefficients than the full 2D-DPD model which includes all cross-sample interactions
• Implementable on modern FPGA fabric with dedicated DSP slices for real-time 5G NR bandwidths
• Matrix conditioning during extraction is manageable with standard least-squares solvers

The MB-MP model structure is well-suited for direct learning architecture (DLA) adaptation, where coefficients are updated by minimizing the error between the predistorter output and the observed PA output.

• The model is linear in its coefficients, enabling closed-form least-squares estimation
• Supports online training with block-based or recursive least-squares (RLS) algorithms
• Compatible with indirect learning architecture (ILA) as a post-distorter model
• Enables joint coefficient estimation where all band and cross-band parameters are extracted simultaneously
• Convergence is typically achieved within 5–10 iterations for stationary PA characteristics

The MB-MP inherently accounts for the composite crest factor of multi-band signals, which is a critical driver of PA nonlinearity.

• The instantaneous composite envelope √(|x₁|² + |x₂|²) determines the amplifier's operating point on its AM-AM/AM-PM curves
• Cross-band envelope terms in the MB-MP approximate this composite envelope effect without requiring explicit signal combination
• Enables accurate prediction of peak compression events when both carriers simultaneously reach maximum amplitude
• Works in conjunction with multi-band crest factor reduction (MB-CFR) to optimize overall linearization performance

The direct two-dimensional extension of the memory polynomial, forming the mathematical core of many MB-MP implementations. It models the nonlinear output in one band as a function of its own baseband signal and the envelope magnitude of the concurrent band's signal.

• Cross-Term Structure: Includes terms like x1(n-m) * |x2(n-m)|^k to capture cross-band modulation
• Indexing: Uses a 2D address space based on |x1| and |x2| for look-up table implementations
• Complexity Scaling: The number of coefficients grows quadratically with nonlinearity order, requiring careful pruning for FPGA deployment

Nonlinear interference products generated when multiple carrier signals interact within a shared power amplifier. These fall into three critical categories that the MB-MP must explicitly model:

• Inter-Band IMD: Distortion falling in the frequency gap between transmit bands, often requiring dedicated cancellation
• Cross-Modulation: The envelope of a strong carrier in Band 1 modulating the amplitude and phase of a weaker carrier in Band 2
• In-Band Cross-Distortion: IMD products from the interaction of two bands that fall directly on top of one of the desired transmit channels, degrading EVM

An enhanced variant that extends the standard MB-MP by incorporating sample-crossing terms between memory taps of different bands. This captures more complex nonlinear interactions at the cost of increased parameter count.

• Cross-Band Memory: Terms like x1(n-m) * |x2(n-l)|^k where m ≠ l model how the past envelope of Band 2 affects the current nonlinear behavior in Band 1
• Lead/Lag Envelope Coupling: Includes both aligned and misaligned envelope samples to capture dispersive effects in the PA bias network
• Use Case: Essential for wideband carrier aggregation where memory effects span multiple symbol periods across bands

A parameter identification technique that simultaneously solves for all MB-MP coefficients—including cross-band terms—in a single least-squares optimization. This contrasts with sequential estimation where each band's predistorter is extracted independently.

• Coupled Matrix Formulation: Constructs a block-structured regression matrix where off-diagonal blocks represent cross-band coupling
• Numerical Conditioning: Requires regularization (e.g., Tikhonov) to handle ill-conditioned matrices caused by high correlation between basis functions
• Convergence Guarantee: Joint estimation avoids the error propagation problem inherent in sequential band-by-band extraction

The dominant closed-loop adaptation framework for deploying MB-MPD in real-time systems. A post-distorter model is identified from the attenuated PA output and then copied to the predistorter in the forward transmission path.

• Training Signal Flow: PA output is attenuated, downconverted, and fed to a post-distorter block whose coefficients are estimated to minimize the error between its output and the original PA input
• Coefficient Copy: The converged post-distorter parameters are transferred directly to the predistorter, assuming the nonlinearity is invertible
• Stability Consideration: Requires gain normalization to prevent the post-distorter from attempting to invert an ill-conditioned PA characteristic at compression

A hardware-efficient implementation of the MB-MP where complex gain correction values are stored in a two-dimensional memory addressed by the instantaneous magnitudes of both concurrent baseband signals.

• Address Generation: |x1(n)| and |x2(n)| are quantized to form a 2D index, retrieving a complex gain word applied to the composite signal
• Interpolation: Bilinear interpolation between adjacent LUT entries prevents spectral discontinuities caused by coarse quantization
• Adaptation: LUT contents are updated via LMS or RLS algorithms operating on the error signal, enabling tracking of thermal and aging effects
• FPGA Footprint: A 64×64 2D-LUT with 16-bit complex entries consumes approximately 16 KB of block RAM, making it viable for Zynq UltraScale+ implementations