Complex Baseband Volterra

Operates on the complex baseband equivalent of the RF signal, where the carrier frequency is removed. The input is a complex-valued signal x̃(n) = I(n) + jQ(n), and the model directly produces the complex output envelope. This formulation inherently captures both AM-AM distortion (gain compression through magnitude changes) and AM-PM distortion (phase shift through angle changes) in a single unified framework, eliminating the need for separate amplitude and phase models.

Only odd-order nonlinear terms (3rd, 5th, 7th order) are retained in the baseband formulation. Even-order distortion products fall at harmonics of the carrier frequency and are filtered out by the bandpass nature of the PA and matching networks. The baseband kernel is constructed from terms of the form:

• x̃(n) · |x̃(n)|² for 3rd-order
• x̃(n) · |x̃(n)|⁴ for 5th-order
• x̃(n) · |x̃(n-m)|² · |x̃(n-k)|² for cross-memory terms This pruning reduces the parameter count by approximately half compared to passband models.

By working with the complex envelope rather than the RF carrier, the model operates at the signal's baseband bandwidth rather than the Nyquist rate of the carrier frequency. For a 100 MHz bandwidth signal at a 3.5 GHz carrier, the baseband model requires sampling at ~200-300 MHz instead of >7 GHz. This makes real-time DPD implementation feasible on current FPGA and ASIC hardware, where clock rates and power budgets are constrained.

Unlike real-valued Volterra models, the complex baseband formulation includes conjugate signal terms x̃*(n) to capture asymmetric distortion spectra. These terms model the interaction between positive and negative frequency components, which is essential for accurately reproducing spectral regrowth that is not symmetric around the carrier. The full baseband kernel includes both x̃(n-m₁) · x̃*(n-m₂) and x̃(n-m₁) · x̃(n-m₂) type products to capture this asymmetry.

The full complex baseband Volterra model grows exponentially with nonlinear order and memory depth, becoming computationally intractable. Practical implementations apply aggressive pruning:

• Memory Polynomial: Retains only diagonal terms x̃(n) · |x̃(n-m)|^(k-1)
• Generalized Memory Polynomial: Adds cross-terms between the signal and lagging envelope values
• Dynamic Deviation Reduction: Separates static nonlinearity from low-order dynamics
• LASSO Regularization: Forces insignificant coefficients to exactly zero during estimation

Model coefficients are estimated using complex-valued least squares or adaptive algorithms operating directly on I/Q data. The estimation problem is formulated as y = X · h, where y is the measured complex output, X is the regressor matrix of basis functions, and h is the vector of complex kernel coefficients. Key considerations include:

• Condition number of the regressor matrix, which degrades with correlated wideband signals
• Regularization to prevent overfitting to measurement noise
• Cross-validation across different signal types to ensure generalization

The complex baseband Volterra model inherently captures both AM-AM (amplitude-dependent gain compression) and AM-PM (amplitude-dependent phase shift) distortion through its complex-valued coefficients.

• AM-AM: Models the nonlinear relationship between input envelope magnitude and output envelope magnitude.
• AM-PM: Models the phase rotation introduced as a function of instantaneous input power.
• Complex Gain: The combined effect is represented as a single complex-valued gain term that varies with signal envelope.

The Memory Polynomial is the most widely adopted simplification of the full complex baseband Volterra model, retaining only the diagonal terms of the Volterra kernels.

• Diagonal Terms: Only terms where all memory delays are equal (e.g., x(n)x(n-1)x*(n-1)) are kept.
• Complexity Reduction: Reduces parameters from O(M^K) to O(M×K), where M is memory depth and K is nonlinear order.
• Trade-off: Sacrifices the ability to model certain cross-memory effects but captures the dominant nonlinear dynamics in most power amplifiers.

Extracting the complex baseband Volterra coefficients from measured input-output data is a system identification problem typically solved using least squares estimation.

• Linear-in-Parameters: Despite modeling nonlinear behavior, the Volterra series is linear with respect to its coefficients, enabling efficient batch estimation.
• Least Squares (LS): Minimizes the squared error between the model's predicted complex baseband output and the measured PA output.
• Regularization: Techniques like LASSO regression are applied to prune insignificant kernel terms, creating a sparse model suitable for real-time DPD implementation.

The Indirect Learning Architecture is the dominant method for extracting DPD coefficients using a complex baseband Volterra model without requiring a direct PA inverse.

• Post-Distorter Identification: A Volterra model is first trained to act as a post-distorter, estimating the inverse of the PA's nonlinear behavior.
• Copy to Pre-Distorter: The identified post-distorter coefficients are directly copied to the pre-distorter, assuming the inverse is commutative.
• Practical Advantage: Avoids the numerical challenges of directly inverting a nonlinear PA model, making it the preferred architecture in commercial DPD systems.

The Generalized Memory Polynomial extends the standard memory polynomial by adding cross-terms between the signal and its lagging envelope values, capturing more complex memory effects.

• Envelope Memory Terms: Includes terms like |x(n)|^p · x(n-m) that model the interaction between current envelope power and delayed signal samples.
• Lagging Cross-Terms: Adds x(n) · |x(n-m)|^p terms to capture the effect of past envelope values on the current output.
• Wideband Performance: Significantly improves modeling accuracy for wideband signals (e.g., 100 MHz 5G NR carriers) where simple diagonal models fail.

LASSO (Least Absolute Shrinkage and Selection Operator) regression is applied to complex baseband Volterra models to automatically select only the most significant kernel terms.

• L1-Norm Penalty: Adds a penalty proportional to the absolute value of coefficients, forcing many to exactly zero.
• Automatic Pruning: Eliminates redundant or noise-fitting terms, reducing a model from thousands of coefficients to dozens.
• FPGA Implementation: The resulting sparse model dramatically reduces the number of multiply-accumulate operations required in hardware, enabling real-time DPD on resource-constrained FPGAs.