Cross-Term Memory Polynomial

The defining feature of the CTMP is the inclusion of terms of the form:

• x(n-m) · |x(n-m-l)|^k

These terms multiply a delayed signal sample by the envelope power of another delayed sample, capturing the interaction between the signal's instantaneous value and its past or future amplitude. This allows the model to represent nonlinear memory effects where the distortion at time n depends on the signal's envelope at a different time instant.

CTMP introduces two distinct cross-term types:

• Lagging cross-terms (l > 0): The envelope term is further in the past than the signal term, modeling how past envelope values influence current distortion.
• Leading cross-terms (l < 0): The envelope term is closer to the present than the signal term, capturing precursor memory effects often observed in wideband power amplifiers.

This bidirectional temporal modeling significantly improves accuracy over the unidirectional memory polynomial.

The CTMP is a specific formulation within the broader Generalized Memory Polynomial framework. The GMP extends the memory polynomial by adding:

• Type 1 terms: Signal and envelope delays are independent.
• Type 2 terms: Envelope of the delayed signal multiplied by a further delayed signal.

The CTMP typically focuses on a subset of these cross-terms to balance modeling accuracy with computational complexity, making it suitable for real-time digital predistortion applications.

Despite its nonlinear structure, the CTMP is linear in its coefficients. This means the predistorter coefficients can be extracted using the Least Squares (LS) algorithm:

• Construct a basis function matrix where each column represents a specific cross-term evaluated over the training data.
• Solve for the coefficient vector using the pseudo-inverse of the basis matrix.

This closed-form solution avoids the iterative training and local minima issues associated with neural network-based predistorters.

The primary benefit of CTMP's enhanced modeling capability is superior Adjacent Channel Leakage Ratio (ACLR) improvement. By accurately capturing the nonlinear memory mechanisms that cause spectral regrowth, a CTMP-based predistorter can:

• Achieve 5-10 dB better ACLR than a standard memory polynomial for wideband signals.
• Effectively linearize Doherty and envelope tracking PAs that exhibit strong memory effects.
• Maintain performance across varying signal bandwidths without retraining.

The improved accuracy of the CTMP comes at the cost of increased basis function count. The number of terms grows with:

• Memory depth (M): Number of taps in the delay line.
• Nonlinearity order (K): Highest polynomial order.
• Cross-term span (L): Range of lagging/leading indices.

Careful selection of these parameters is critical for FPGA implementation, where the number of multiply-accumulate operations per sample must fit within the device's DSP slice budget and timing constraints.