Envelope Memory Polynomial

Unlike standard memory polynomials that only consider the complex baseband signal's history, the EMP model introduces envelope memory terms that depend on past values of the signal magnitude |x(n-m)|.

• Captures bias circuit modulation effects where the PA's operating point shifts with signal envelope history
• Models thermal memory where die temperature varies with past power levels
• Essential for GaN and GaAs PAs where self-heating effects are pronounced
• The envelope path operates at baseband bandwidth, reducing sampling requirements compared to full RF memory modeling

The EMP model augments the standard memory polynomial with a parallel envelope-dependent branch:

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

• First summation: standard aligned memory terms (signal and envelope at same delay)
• Second summation: cross-terms between current signal and lagged envelope magnitudes
• The cross-term structure captures modulation of instantaneous gain by past envelope values
• Nonlinear order k is typically odd (3, 5, 7) to model odd-order intermodulation products

EMP excels at modeling memory effects with time constants ranging from microseconds to milliseconds:

• Bias circuit time constants: Decoupling capacitors and bias inductors create envelope-dependent memory spanning 1-100 μs
• Thermal time constants: Die-level heating responds to power envelope with 100 μs to 1 ms latency
• Trapping effects: Semiconductor charge traps in GaN HEMTs exhibit envelope-dependent time constants
• EMP requires fewer coefficients than a full Volterra series to capture these long-memory phenomena
• Memory depth M for envelope terms can be extended without the combinatorial explosion of cross-terms seen in GMP

The EMP model offers a favorable complexity-to-performance trade-off for FPGA and ASIC implementation:

• Coefficient count: O(K × M) for aligned terms plus O(K × M) for envelope cross-terms
• Significantly fewer coefficients than Generalized Memory Polynomial (GMP) which includes leading and lagging cross-terms
• Envelope computation requires magnitude extraction (CORDIC or sqrt(I²+Q²)) before the delay line
• Look-up table (LUT) based implementation can pre-compute polynomial terms indexed by quantized envelope magnitude
• Typical configuration: K=5 or 7 (nonlinear order), M=3-5 (memory depth) for envelope terms

EMP coefficients are extracted using standard linear estimation techniques due to the model's linear-in-parameters structure:

• Least Squares (LS) batch estimation solves for coefficients by minimizing the error between PA output and model prediction
• QR Decomposition (QRD) improves numerical stability when the data matrix becomes ill-conditioned
• Ridge regression adds regularization to prevent overfitting, critical when envelope memory depth is large
• The basis function matrix includes both aligned terms and envelope cross-terms as separate columns
• Orthogonalization of basis functions may be necessary if envelope and signal terms exhibit high correlation

EMP is particularly well-suited for specific PA architectures and signal conditions:

• Doherty power amplifiers: The auxiliary amplifier's bias modulation creates strong envelope-dependent memory
• Envelope tracking PAs: The dynamic supply voltage introduces envelope memory that EMP naturally captures
• Wideband signals (100+ MHz): EMP captures long-term memory without the bandwidth expansion of full Volterra models
• High-PAPR signals: OFDM and 5G NR signals with large peak-to-average ratios stress PA memory effects
• GaN-based transmitters: High power density creates significant thermal memory requiring envelope-aware modeling

An enhanced model that extends the EMP by adding cross-terms between the signal and its lagging or leading envelope samples. This captures complex memory effects that simpler models miss.

• Includes both signal and envelope lag/lead cross-terms
• Models nonlinearity with higher dimensional accuracy
• Higher complexity than standard MP or EMP
• Often used as a benchmark for neural network DPD comparisons

The EMP is specifically designed to capture long-term thermal memory and bias circuit dynamics in power amplifiers. These effects manifest as slow changes in gain and phase due to die heating.

• EMP's envelope-dependent terms model self-heating effects
• Captures video bandwidth limitations in bias networks
• Essential for GaN HEMT amplifiers with significant charge trapping
• Complements short-term memory models for complete linearization

A complexity reduction technique that removes insignificant kernels from a full Volterra series. When applied alongside EMP structures, it identifies which envelope-memory cross-terms are most critical.

• Uses significance metrics to rank kernel importance
• Retains only the most impactful distortion terms
• Reduces computational load for real-time FPGA implementation
• Can be combined with Orthogonal Matching Pursuit (OMP) for sparse model selection

The primary batch coefficient extraction algorithm used to solve for EMP parameters. It minimizes the sum of squared errors between the PA output and the model's prediction.

• Requires a matrix inversion of the basis function correlation matrix
• Works well for offline model extraction from captured IQ data
• Can become ill-conditioned with highly correlated EMP basis functions
• Often paired with QR decomposition for numerical stability

A numerical conditioning process that transforms correlated EMP basis functions into an orthogonal set. This dramatically improves coefficient estimation stability.

• Prevents ill-conditioned matrices during LS estimation
• Improves convergence speed for adaptive algorithms
• Common techniques include Gram-Schmidt and QR decomposition
• Essential when EMP terms have high cross-correlation due to envelope dependencies

A one-dimensional array containing the complex-valued weights for each EMP basis function. This vector fully defines the predistorter's linearization transfer characteristic.

• Each element corresponds to a specific nonlinear order and memory tap
• Must be updated adaptively as PA characteristics drift with temperature
• Size grows with nonlinear order × memory depth × envelope depth
• Stored in FPGA block RAM for real-time predistortion lookup