Nonlinear Order

The nonlinear order (typically denoted as K) is the maximum exponent of the input signal envelope magnitude in a polynomial model. In a memory polynomial, the output is a sum of terms like x(n-m)|x(n-m)|^(k-1), where k ranges from 1 to K (odd orders only for bandpass signals). This parameter directly determines the model's ability to capture gain compression and AM-AM/AM-PM distortion near the amplifier's saturation point.

A nonlinearity of order K generates intermodulation distortion (IMD) products up to the K-th order. For example:

• 3rd-order nonlinearity produces IMD3 products that fall close to the carrier, directly causing adjacent channel interference.
• 5th-order nonlinearity generates IMD5 products, which can fold back into adjacent channels in wideband signals.
• 7th-order and higher terms capture severe compression near saturation but contribute less energy. Selecting K too low leaves residual distortion; selecting K too high introduces numerical instability with minimal accuracy gain.

For bandpass communication signals, only odd-order nonlinear terms (3rd, 5th, 7th, etc.) generate distortion products that fall within or near the transmission band. Even-order terms produce distortion at DC and harmonics of the carrier frequency, which are filtered by the output matching network. Consequently, memory polynomial models typically include only odd orders: k = 1, 3, 5, ..., K. The 1st-order term represents the linear gain component and is always included.

The number of basis functions grows multiplicatively with nonlinear order and memory depth. For a memory polynomial with memory depth M and nonlinear order K (odd terms only), the total coefficient count is M × (K+1)/2. Doubling K roughly doubles the coefficient vector size, increasing:

• Matrix dimensions in least squares estimation
• Multiply-accumulate operations per sample in real-time DPD
• FPGA resource utilization (DSP slices, block RAM)

Practical systems often limit K to 7 or 9 for wideband signals, balancing ACLR improvement against implementation cost.

Power amplifiers driven deep into compression for efficiency enhancement (e.g., Doherty PAs at peak efficiency) exhibit sharp nonlinearity that requires higher-order terms to correct. A PA operating at its 1 dB compression point (P1dB) may need K=5; a PA driven 3 dB beyond P1dB may require K=9 or K=11. GaN HEMT amplifiers with their sharp saturation characteristics particularly benefit from higher nonlinear orders to capture the abrupt gain roll-off.

High-order polynomial terms create ill-conditioned data matrices because |x|^k values span many orders of magnitude. This degrades the accuracy of least squares (LS) estimation and slows recursive least squares (RLS) convergence. Mitigation strategies include:

• Basis function orthogonalization to decorrelate polynomial terms
• Ridge regression with a regularization parameter to stabilize inversion
• Principal component analysis (PCA) to discard low-variance high-order components
• Orthogonal matching pursuit (OMP) to greedily select only the most significant orders