Volterra Series

The Volterra series represents a nonlinear system's output as an infinite sum of multi-dimensional convolution integrals. Each term involves a Volterra kernel of increasing order:

• First-order kernel: Linear impulse response (standard convolution)
• Second-order kernel: Quadratic nonlinearity with two-dimensional memory
• Third-order kernel: Cubic nonlinearity with three-dimensional memory

The output y(t) = h₀ + ∫h₁(τ)x(t-τ)dτ + ∫∫h₂(τ₁,τ₂)x(t-τ₁)x(t-τ₂)dτ₁dτ₂ + ...

This structure generalizes the Taylor series to systems with memory, making it the most general polynomial-based nonlinear model.

Volterra kernels exhibit triangular symmetry that reduces the number of unique coefficients without loss of generality:

• Symmetric kernels: h₂(τ₁,τ₂) = h₂(τ₂,τ₁) for second-order
• Permutation invariance: Higher-order kernels are symmetric under any permutation of their arguments
• Triangular domain: Only the region τ₁ ≥ τ₂ ≥ ... ≥ τₙ needs to be defined

This symmetry is exploited in pruned Volterra models to dramatically reduce computational complexity while maintaining modeling accuracy for power amplifier applications.

Practical implementations require finite memory length M and nonlinear order K:

• Memory depth M: Number of past samples contributing to the current output
• Nonlinear order K: Highest polynomial degree modeled (typically K=5 to K=9 for PAs)
• Total coefficients: Grows as O(M^K) without pruning

For a full Volterra model with M=5 and K=5, the coefficient count exceeds 3,000 terms. This exponential growth motivates simplified variants like the memory polynomial and generalized memory polynomial models.

The Volterra series has a powerful frequency-domain counterpart using multi-dimensional Fourier transforms of the kernels:

• Linear transfer function: H₁(ω) — standard frequency response
• Second-order nonlinear transfer function: H₂(ω₁,ω₂) — captures sum and difference frequency generation
• Third-order transfer function: H₃(ω₁,ω₂,ω₃) — critical for intermodulation distortion analysis

This representation directly predicts spectral regrowth and adjacent channel power from the kernel transforms, making it invaluable for ACPR prediction in wireless transmitter design.

The Volterra series converges only for systems with fading memory and inputs within a finite radius:

• Analytic nonlinearities: Required for series representation (no discontinuities or hysteresis)
• Input amplitude limits: Beyond the radius of convergence, the series diverges
• Practical implication: Power amplifiers driven near saturation may exceed convergence bounds

For strongly nonlinear systems like Class-F or Doherty amplifiers operating near compression, alternative representations such as neural network models or piecewise Volterra models may be required.

The Volterra series serves as the theoretical backbone for digital predistortion (DPD):

• Direct modeling: Volterra kernels model the PA's forward nonlinear behavior
• Inverse modeling: A Volterra-based predistorter approximates the PA inverse
• p-th order inverse: Theoretically exact inverse exists as another Volterra series under mild conditions

The indirect learning architecture uses Volterra-derived models to identify predistorter coefficients without requiring explicit PA inversion, enabling adaptive DPD systems that track changing amplifier characteristics in real-time.

A simplified Volterra series that retains only the diagonal terms of the Volterra kernels. This dramatically reduces computational complexity while still capturing essential nonlinear memory effects.

• Eliminates cross-terms between different time delays
• Complexity scales linearly with memory depth and nonlinear order
• Most widely used basis for digital predistortion in commercial systems

An extension of the memory polynomial that incorporates cross-terms between different time delays and nonlinear orders. Addresses the memory polynomial's limitation in capturing strong lagging and leading envelope effects.

• Adds terms like |x(n)|^{k-1} x(n-l) for l ≠ 0
• Improves modeling accuracy for Doherty and GaN amplifiers
• Bridges the gap between memory polynomial and full Volterra complexity

A block-structured behavioral model consisting of a linear time-invariant (LTI) filter followed by a memoryless nonlinearity. Represents a subset of the Volterra series with a specific kernel structure.

• Captures frequency-dependent gain variations before nonlinear distortion
• Effective for amplifiers where memory effects precede the nonlinear element
• Often paired with its inverse, the Hammerstein model, for predistorter design

A block-structured model with a memoryless nonlinearity followed by an LTI filter. Represents systems where the nonlinear distortion occurs before frequency-selective memory effects.

• Suitable for amplifiers where the input matching network introduces memory after the transistor nonlinearity
• Inverse structure (Wiener) often used for the corresponding predistorter
• Parameter extraction uses separable least-squares or iterative techniques

The dependence of a power amplifier's current output on past input values, causing frequency-dependent distortion that cannot be corrected by memoryless linearization alone.

• Electrical memory: Bias network impedance variations and trapping effects
• Thermal memory: Dynamic junction temperature changes affecting gain
• Manifests as asymmetric intermodulation distortion sidebands

The two fundamental nonlinear distortion mechanisms captured by Volterra models. AM-AM is the nonlinear relationship between input and output amplitude. AM-PM is the conversion of amplitude variations into phase shifts.

• AM-AM causes gain compression at high power levels
• AM-PM degrades EVM in phase-modulated signals like QAM
• Volterra kernels model both effects with memory dependence