Dual-Band Volterra Series

The model is not an empirical fit but a rigorous mathematical derivation. It begins with the physical passband Volterra series and applies a dual-band input signal (two modulated carriers at ω₁ and ω₂). By expanding the nonlinear terms and selecting only the baseband components that fall within the two transmit channels, the model analytically derives the baseband equivalent nonlinear behavior. This preserves the physical link between circuit-level nonlinearity and the discrete-time model used for digital predistortion.

A defining characteristic is the inclusion of terms that model the interaction between the two carrier signals. The nonlinear output at band 1 depends not only on the signal at band 1 but also on the instantaneous envelope of the signal at band 2. Key interaction types include:

• Cross-modulation: The envelope of band 2 modulates the gain and phase of band 1.
• Intermodulation products: Third-order terms like 2ω₁ - ω₂ and 2ω₂ - ω₁ generate distortion that can fall directly into the receive or adjacent bands. These terms are essential for canceling cross-band distortion in a concurrent transmitter.

The model's kernels are structured around the complex baseband envelopes of the two signals, x₁(n) and x₂(n). A typical nonlinear term of order k involves a product of the form:

• x₁(n) · |x₁(n)|^(k-1-m) · |x₂(n)|^m This structure shows that the nonlinear response is a function of the complex envelope and the instantaneous powers of both bands. The index m controls the order of cross-band coupling. The model naturally separates into in-band (m=0) and cross-band (m>0) distortion components, allowing for targeted linearization strategies.

The dual-band Volterra series inherently captures memory effects through its convolutional structure. Unlike a static nonlinearity, the output depends on past samples of the input signals. The model includes:

• Linear memory: Taps of x₁(n-l) and x₂(n-l).
• Nonlinear memory: Products of delayed envelope powers, such as |x₁(n-l₁)|² · x₁(n-l₂).
• Cross-band memory: Terms where the envelope history of band 2 influences the current nonlinear output of band 1. This makes it suitable for wideband applications where thermal and electrical memory effects are significant.

The full dual-band Volterra series is the parent model from which simpler, more practical models are derived. By truncating the series and restricting the kernel structure, one obtains:

• 2D-DPD (Two-Dimensional DPD): A memoryless version that uses only the instantaneous magnitudes |x₁| and |x₂| as a 2D index.
• 2D Memory Polynomial (2D-MMP): Retains diagonal memory terms but drops off-diagonal memory cross-terms to reduce complexity.
• Multi-Band Generalized Memory Polynomial (MB-GMP): Adds envelope cross-terms between different lags to capture complex interactions with fewer coefficients than the full Volterra series.

The primary drawback of the full dual-band Volterra series is its exponential growth in coefficients with nonlinearity order and memory depth. For a 5th-order nonlinearity with M memory taps, the number of terms can reach thousands. Practical implementation requires:

• Kernel pruning: Removing terms that contribute negligibly to modeling accuracy.
• Symmetry exploitation: Using the fact that many cross-band terms are complex conjugates.
• Near-diagonal approximation: Assuming that the dominant memory effects occur when all lags are nearly equal. These techniques reduce the model to a computationally feasible size for real-time FPGA or ASIC implementation.