Parallel Hammerstein

The model consists of multiple parallel branches, each containing a static memoryless nonlinearity followed by a linear time-invariant (LTI) filter. The outputs of all branches are summed to produce the final output. This structure explicitly separates the nonlinear distortion generation from the frequency-dependent memory effects, making it physically intuitive for modeling power amplifiers where the input signal first experiences nonlinearity at the transistor before encountering the matching network's filtering.

• Branch 1: Static nonlinearity → Linear filter
• Branch 2: Static nonlinearity → Linear filter
• Branch N: Static nonlinearity → Linear filter
• Output: Sum of all branch outputs

The Parallel Hammerstein model is mathematically equivalent to a Volterra series with diagonal kernel structure. Each branch corresponds to a specific nonlinear order, and the linear filter in that branch represents the diagonal elements of the Volterra kernel for that order. This restriction significantly reduces the number of coefficients compared to a full Volterra series while retaining the ability to model frequency-dependent nonlinear behavior.

• Full Volterra kernels: O(M^K) parameters where M is memory depth and K is nonlinear order
• Parallel Hammerstein: O(K × M) parameters
• Trade-off: Reduced complexity with minimal accuracy loss for many PA architectures

Each branch employs a memoryless nonlinear function that operates on the instantaneous input sample. Common choices include polynomial basis functions, spline interpolations, or look-up tables. For power amplifier modeling, odd-order polynomials are typically sufficient due to the differential structure of push-pull amplifiers canceling even-order distortion products at the output.

• Polynomial: y = a₁x + a₃|x|²x + a₅|x|⁴x + ...
• Spline: Piecewise polynomial segments with smooth junctions
• LUT: Directly tabulated complex gain values indexed by input magnitude
• Basis pursuit: Can use orthogonal basis functions to improve numerical conditioning

The linear filters following each nonlinearity capture the frequency-dependent memory effects of the power amplifier. These are typically implemented as finite impulse response (FIR) filters, allowing direct estimation of coefficients using linear regression techniques. The filter length determines the memory depth captured by the model, with longer filters required for wideband signals or amplifiers with significant thermal and trapping memory effects.

• FIR structure: y[n] = Σ b_k · x[n-k] for k = 0 to M-1
• Memory depth M: Typically 3-10 taps for most wireless applications
• Complex coefficients: Capture both magnitude and phase memory
• Adaptive estimation: Coefficients updated via LMS or RLS algorithms

The Parallel Hammerstein model benefits from linear-in-parameters structure, enabling efficient coefficient extraction using least squares estimation. The static nonlinearities are applied first to generate intermediate signals, which are then filtered by the linear blocks. Since the overall output is a linear combination of these filtered signals, the estimation problem reduces to solving a standard linear regression.

• Step 1: Apply static nonlinearities to input data
• Step 2: Construct regression matrix from filtered nonlinear outputs
• Step 3: Solve normal equations: w = (X^H X)^(-1) X^H y
• Advantage: Guaranteed global optimum with no local minima issues

The Parallel Hammerstein generalizes the single-branch Hammerstein model (static nonlinearity → linear filter) by using multiple parallel paths. Unlike the Wiener model (linear filter → static nonlinearity), the Hammerstein structure places nonlinearity before filtering, which better matches power amplifiers where the transistor's nonlinearity precedes the output matching network's frequency response.

• Hammerstein: Single branch, limited frequency-dependent nonlinearity
• Wiener: Filter-first structure, less common for PA modeling
• Parallel Hammerstein: Multiple branches, rich frequency-dependent behavior
• Wiener-Hammerstein: Cascaded filter-nonlinearity-filter for maximum flexibility

The foundational single-branch block-structured model consisting of a static memoryless nonlinearity followed by a linear dynamic filter. Unlike the Parallel Hammerstein, which uses a bank of parallel branches, the standard Hammerstein model captures only one nonlinear path. It represents a simplified Volterra structure where the nonlinearity precedes the memory, making it suitable for power amplifiers where input distortion dominates. The model's parameters are typically estimated using least squares or iterative methods.

The structural dual of the Hammerstein model, composed of a linear dynamic filter followed by a static memoryless nonlinearity. This topology models systems where linear filtering occurs before nonlinear distortion, such as power amplifiers with significant input matching network effects. When combined with the Hammerstein structure, it forms the more general Wiener-Hammerstein model, which can represent a broader class of nonlinear dynamic systems than either model alone.

The general mathematical framework from which the Parallel Hammerstein model is derived. A Volterra series represents the system output as a sum of multidimensional convolution integrals with Volterra kernels. The Parallel Hammerstein corresponds to a diagonal Volterra structure, retaining only the diagonal terms of the kernels. This simplification dramatically reduces the number of coefficients from exponential to linear growth with memory depth, making it practical for real-time DPD implementation.

A closely related simplified Volterra model that also retains only diagonal kernel terms. The key distinction: the Memory Polynomial applies a single polynomial nonlinearity with tapped delay lines, while the Parallel Hammerstein uses separate static nonlinearities per branch followed by individual linear filters. Both models capture nonlinear memory effects efficiently, but the Parallel Hammerstein offers greater flexibility in modeling frequency-dependent nonlinear behavior through its branch-specific filter design.

An enhanced model that extends the Memory Polynomial by including cross-terms between the signal and its lagging envelope values. These cross-terms capture more complex memory effects that the standard Parallel Hammerstein may miss, such as those arising from bias network modulation and thermal dynamics. The GMP can be viewed as a superset that combines diagonal Volterra terms with off-diagonal envelope couplings, offering higher accuracy at the cost of increased parameter count.

A Volterra simplification technique that separates the model into a static nonlinearity and low-order dynamic corrections. This approach is philosophically similar to the Parallel Hammerstein's decomposition into static and dynamic blocks but uses a different mathematical basis. DDR models are particularly effective for weakly nonlinear systems where high-order dynamics are negligible, offering a compact alternative when the full Parallel Hammerstein structure proves unnecessarily complex.