Dual-Input Behavioral Model

Unlike conventional single-input models that map only |x(n)| to output, dual-input models construct a 2D nonlinear basis indexed by instantaneous input magnitude and supply voltage V_DD(n). This captures the supply-dependent gain surface where amplifier behavior shifts as the drain voltage modulates. Common basis functions include products of monomials: |x(n)|^k · V_DD(n)^m, creating a rich function space that spans the full operating envelope.

Dual-input models explicitly capture cross-memory between the RF and supply paths—a phenomenon absent in single-input frameworks. When the supply modulator's bandwidth is limited, the actual V_DD(t) lags the ideal envelope, creating memory in the supply dimension. The model includes terms like:

• x(n-l) · V_DD(n-m) for l ≠ m
• |x(n-l)|^k · V_DD(n-m)^p These cross-tap products characterize the interaction between RF memory and supply memory, critical for wideband signals where both paths exhibit significant dynamics.

The Augmented Volterra for ET extends the classical Volterra series by introducing a second input dimension for the supply voltage. The general form is:

y(n) = Σ h_k,m(l₁,...,l_k; q₁,...,q_m) · Π x(n-l_i) · Π V_DD(n-q_j)

where kernels h_k,m weight products of k RF terms and m supply terms at various delays. Truncation to low nonlinearity orders (typically k+m ≤ 7) and short memory depths keeps the model computationally tractable while maintaining excellent prediction accuracy for ET PAs.

For memoryless or weakly-memory dual-input predistortion, a 3D LUT provides an efficient hardware implementation. The table is indexed by:

1. Instantaneous input power |x(n)|² (or magnitude)
2. Instantaneous supply voltage V_DD(n)

Each table entry stores a complex gain correction G(|x|², V_DD). The predistorted output is x_DPD(n) = x(n) · G(|x(n)|², V_DD(n)). Bilinear interpolation between table entries ensures smooth correction across the continuous 2D operating space. This structure maps efficiently to FPGA block RAM with dual-port addressing.

Model coefficients are extracted from iso-gain contour measurements taken across the PA's 2D operating plane. The characterization process sweeps:

• Input power from small-signal to deep compression
• Supply voltage from minimum to maximum rated V_DD

At each (P_in, V_DD) point, complex gain is measured, creating a gain surface. Least-squares fitting then extracts the dual-input model parameters. This data-driven approach captures the supply-dependent AM-AM and AM-PM characteristics without requiring detailed physics-based PA models.

Advanced dual-input models decompose the PA behavior into:

• Static nonlinearity: A memoryless 2D function f(|x(n)|, V_DD(n)) capturing instantaneous compression
• Dynamic nonlinearity: Linear filters or low-order Volterra terms capturing memory effects in both dimensions

This Hammerstein-Wiener inspired decomposition reduces parameter count while maintaining accuracy. The static block handles the dominant supply-dependent gain compression, while dynamic blocks model thermal trapping and modulator bandwidth limitations. The structure enables independent optimization of each sub-model during coefficient extraction.