Memory Polynomial (MP)

The MP model restricts the full Volterra series to only its diagonal terms, where all delayed samples share the same time index. This eliminates cross-terms between samples at different delays, dramatically reducing the number of coefficients from exponential to linear growth with memory depth and nonlinearity order.

• Volterra complexity: O(K^M) where K is nonlinearity order and M is memory depth
• MP complexity: O(K × M), making it feasible for real-time implementation
• Preserves the essential physics: envelope-dependent gain compression and short-term memory effects

The baseband MP model expresses the output as a sum over polynomial orders and memory taps:

y(n) = Σ_k Σ_m a_{km} · x(n-m) · |x(n-m)|^{k-1}

where x(n) is the complex baseband input, a_{km} are complex coefficients, k indexes odd nonlinearity orders, and m indexes memory delays. The term |x(n-m)|^{k-1} captures the envelope-dependent nonlinearity at each delay tap.

• Only odd-order terms are typically retained for bandpass nonlinearities
• The model is linear in its coefficients, enabling least-squares extraction

The GMP enriches the standard MP by adding lagging and leading cross-terms between the signal envelope and delayed samples:

• Lagging cross-terms: |x(n-m-l)|^{k-1} · x(n-m) captures envelope memory from earlier samples
• Leading cross-terms: |x(n-m+l)|^{k-1} · x(n-m) accounts for envelope effects from later samples
• These cross-terms model complex memory interactions that the diagonal-only MP cannot represent, such as thermal trapping effects in GaN HEMT amplifiers

In neural network DPD architectures, the MP model serves as an input transformation layer that expands the raw I/Q samples into a higher-dimensional feature space before feeding them into the network.

• The MP basis functions act as a physics-informed feature extractor, encoding known PA nonlinear dynamics
• This reduces the learning burden on the neural network, requiring fewer layers and training samples
• The combination of MP basis expansion with a shallow neural network is often called an Augmented MP model

Because the MP model is linear in its parameters, the optimal coefficients can be found using a closed-form least-squares solution:

a = (Φ^H Φ)^{-1} Φ^H y

where Φ is the regression matrix constructed from the MP basis functions and y is the measured PA output vector. This avoids iterative gradient descent and guarantees a global minimum.

• Enables fast model extraction from a single capture of input-output data
• The matrix inversion can be performed via QR decomposition or singular value decomposition for numerical stability

The MP model balances modeling accuracy against computational complexity for real-time FPGA or ASIC implementation:

• Memory depth (M): Typically 3-5 taps for capturing short-term memory effects in modern PAs
• Nonlinearity order (K): Usually 5-9 odd orders, sufficient for modeling gain compression up to 2-3 dB into compression
• Coefficient count: (K+1)/2 × (M+1) complex coefficients, typically 15-30 total
• The model can be implemented using look-up tables (LUTs) indexed by instantaneous power for each memory tap, avoiding real-time polynomial evaluation

An extension of the standard MP model that introduces lagging and leading envelope cross-terms to capture more complex nonlinear memory effects. While the MP model uses only diagonal Volterra kernels, the GMP adds off-diagonal terms that account for interactions between a signal sample and the envelope of a delayed sample.

• Key advantage: Superior modeling accuracy for wideband signals where memory effects span multiple symbol periods
• Trade-off: Significantly higher coefficient count than MP, increasing computational complexity
• Common use: Basis function for neural network predistorters requiring high fidelity

The parent mathematical framework from which the Memory Polynomial is derived. The Volterra series represents a nonlinear dynamic system as a sum of multidimensional convolution integrals. The MP model is a pruned Volterra series that retains only the diagonal kernel terms, dramatically reducing parameters while preserving essential memory and nonlinearity.

• Full Volterra: Complete but computationally intractable for real-time DPD
• MP simplification: Retains only terms where all delay indices are equal
• Relationship: MP = Volterra series with diagonal kernel restriction

A behavioral model that enriches the standard MP structure with envelope-dependent cross-terms between different time delays. These cross-terms model the interaction between the current signal's envelope and delayed signal samples, capturing nonlinear memory effects that the diagonal-only MP misses.

• Structure: Includes terms like x(n)|x(n-m)|^k where m ≠ 0
• Benefit: Improved modeling of strong memory effects in GaN and Doherty amplifiers
• Implementation: Often used as a feature expansion layer before a neural network predistorter

The most common architecture for training MP-based predistorters. In ILA, a postdistorter is first identified by swapping the input and output of the PA, then the trained coefficients are copied to the predistorter. This assumes the nonlinear blocks are commutable.

• Process: Train postdistorter on PA output → copy coefficients to predistorter
• Advantage: Avoids the need for real-time PA model inversion
• Limitation: Commutability assumption breaks down under strong nonlinearity
• MP role: The MP model serves as the coefficient structure for both postdistorter and predistorter blocks

A closed-loop predistorter training topology where the MP coefficients are updated by minimizing the error between the PA output and the desired linear signal. Unlike ILA, DLA directly optimizes the predistorter without assuming commutability.

• Mechanism: Error signal e(n) = y_PA(n) - G·x(n) drives coefficient updates
• Advantage: Theoretically optimal solution, no commutability assumption
• Challenge: Requires a PA model or gradient approximation for backpropagation
• Neural integration: DLA is the natural framework for training neural network predistorters with MP basis functions

A feedforward neural network that implements the MP model's tapped delay line structure on real-valued I/Q components. The RVTDNN decomposes the complex baseband signal into in-phase and quadrature branches, applies time delays, and feeds them through a neural network that learns the PA's inverse nonlinearity.

• Connection to MP: The delay taps and polynomial basis functions of MP are replaced by learned neural network layers
• Advantage: No need to pre-specify polynomial order; the network learns the optimal nonlinearity
• Implementation: Often uses vector decomposition to separate magnitude and phase information