Memory Polynomial (MP)

The MP model is architecturally equivalent to a Parallel Hammerstein configuration. Each tap in the delay line feeds a distinct static polynomial nonlinearity. The outputs of all branches are summed to produce the final model output. This structure elegantly separates the modeling of nonlinearity and memory, allowing each delay tap to have its own unique polynomial coefficients. The model is expressed as:

• K: Nonlinear order (highest polynomial power)
• M: Memory depth (number of delay taps)
• a_km: Complex coefficients for each order k and tap m

This formulation captures how the PA's current output depends on both the instantaneous input envelope and its past values.

Memory effects are introduced via a tapped delay line at the model input. The current complex baseband sample, x(n), and its M previous samples, x(n-1) through x(n-M), are stored. Each delayed sample is processed by its own dedicated polynomial function. This structure captures short-term memory effects caused by impedance matching networks, bias circuits, and filter group delay. The delay M must be chosen to span the duration of the PA's impulse response to fully capture its dynamic behavior.

In practice, the MP model is often restricted to odd-order nonlinear terms only (e.g., 3rd, 5th, 7th order). Even-order distortion products typically fall far outside the band of interest and are filtered out by the PA's matching networks. Removing even-order terms significantly reduces the number of coefficients without sacrificing in-band and adjacent-channel modeling accuracy. This simplification is a key reason for the MP model's computational efficiency compared to a full Volterra series.

A critical advantage of the MP model is that it is linear with respect to its coefficients. The model output is a weighted sum of basis functions. This allows the coefficient vector to be extracted using robust, closed-form Least Squares (LS) estimation. Given a matrix of basis function outputs and a vector of measured PA outputs, the optimal coefficients are found via the pseudo-inverse. This guarantees a global minimum for the error surface, unlike neural network approaches that may converge to local minima.

The raw polynomial basis functions of the MP model are highly correlated, leading to an ill-conditioned data matrix during coefficient estimation. This causes numerical instability and slow convergence in adaptive systems. To mitigate this, basis function orthogonalization is applied. Techniques like Gram-Schmidt or using orthogonal polynomials (e.g., Chebyshev) transform the correlated regressors into an orthogonal set. This dramatically improves the condition number of the matrix, enabling faster and more stable coefficient extraction with lower precision arithmetic.

An enhanced model that extends the standard memory polynomial by adding cross-terms between the signal and its lagging or leading envelope samples. These cross-terms capture complex memory effects that simpler models miss, such as those arising from bias network dynamics and thermal trapping in GaN amplifiers. The GMP structure includes both signal-and-envelope and envelope-and-signal cross products, significantly improving modeling accuracy for wideband signals at the cost of increased coefficient count.

The highest power of the input signal envelope considered in a polynomial model, typically limited to odd orders (3rd, 5th, 7th, 9th) because even-order distortion products fall out of band in narrowband systems. The nonlinear order directly determines the model's ability to capture gain compression and high-order intermodulation products. Higher orders improve accuracy near saturation but increase computational complexity and risk numerical ill-conditioning during coefficient extraction.

The number of past input samples used in a model with memory, defining the temporal span over which the power amplifier's history influences its current output. Memory depth is measured in sample periods and must be sufficient to cover the longest significant thermal time constant or bias circuit transient. Typical values range from 3 to 10 taps for short-term memory effects, while long-term thermal memory may require depths of 50 or more samples.

A batch coefficient extraction algorithm that minimizes the sum of squared errors between the power amplifier output and the model's prediction. The LS solution is computed via the pseudo-inverse of the basis function matrix: w = (X^H X)^-1 X^H y. This method provides the optimal coefficients for a given dataset but can suffer from ill-conditioning when basis functions are highly correlated, requiring orthogonalization or regularization for stable implementation.

A numerical conditioning process that transforms correlated polynomial basis functions into an orthogonal set to improve the stability and convergence speed of coefficient estimation. Techniques include Gram-Schmidt orthogonalization and QR decomposition. Orthogonalization reduces the condition number of the data matrix, preventing large coefficient swings and improving the robustness of the extracted predistorter model, especially when using fixed-point arithmetic in FPGA implementations.

A key linearity metric defined as the ratio of power leaked into an adjacent frequency channel to the power in the main channel, typically expressed in dBc. ACPR quantifies the effectiveness of a digital predistorter by measuring spectral regrowth suppression. Regulatory standards such as 3GPP specify maximum ACPR limits (e.g., -45 dBc for LTE), and a well-designed memory polynomial DPD can achieve improvements of 15-25 dB in ACPR performance.