Post-Distortion Error

Post-distortion error is the complex-valued difference between the desired ideal linear output and the actual measured amplifier output after predistortion is applied. Mathematically, it is expressed as e(n) = y_ideal(n) - y_measured(n), where y_ideal(n) is the linearly scaled input signal and y_measured(n) is the captured PA output. This error vector is the primary cost function minimized by adaptive DPD algorithms like the Indirect Learning Architecture and Direct Learning Architecture.

In the frequency domain, post-distortion error manifests as residual spectral regrowth in adjacent channels. While a perfect DPD would eliminate all out-of-band emissions, practical systems leave a measurable error floor. Key metrics include:

• Adjacent Channel Leakage Ratio (ACLR) improvement limits
• Residual spurious emissions in the transmit noise floor
• In-band error vector magnitude (EVM) degradation This residual spectrum is the ultimate indicator of linearization quality for regulatory compliance.

The instantaneous error signal exhibits distinct temporal patterns tied to amplifier physics. Peak error events typically coincide with signal envelope peaks where the PA is driven deep into compression. Memory effects cause error persistence that spans multiple samples, visible as temporal correlation in the error sequence. Analyzing the amplitude-to-amplitude modulation (AM/AM) and amplitude-to-phase modulation (AM/PM) characteristics of the residual error reveals whether the DPD model order is sufficient to capture the PA's nonlinear dynamics.

Post-distortion error serves as the real-time feedback mechanism for adaptive DPD coefficient updates. In Least Mean Squares (LMS) and Recursive Least Squares (RLS) algorithms, the error directly drives coefficient adjustment. Key monitoring behaviors include:

• Monotonic error power decrease during initial convergence
• Steady-state error floor indicating the best achievable linearization
• Error divergence signaling model instability or PA characteristic drift Tracking the error power over time is essential for robust field deployment.

Not all post-distortion error originates from PA nonlinearity alone. A comprehensive error budget includes:

• Modeling residual: Error from insufficient DPD model complexity or order
• Estimation noise: Coefficient inaccuracies from noisy observation receivers
• Time misalignment: Residual error from imperfect loop delay estimation
• IQ impairments: Uncorrected IQ imbalance in the modulator or demodulator
• Quantization noise: Finite precision effects in the digital predistortion path Isolating these components is critical for targeted system improvement.

NMSE is the standard scalar metric for quantifying post-distortion error magnitude. It normalizes the mean squared error by the reference signal power, expressed in dB. Typical DPD systems target NMSE values below -35 dB to -45 dB for wideband signals. NMSE correlates strongly with ACLR improvement and EVM performance. It is computed over a validation dataset distinct from training data to assess generalization and detect overfitting in the extracted behavioral model.

The primary figure of merit for quantifying post-distortion error. NMSE calculates the power of the residual error signal normalized by the power of the ideal reference signal, typically expressed in decibels (dB).

• Formula: 10*log10(mean(|y_ideal - y_measured|^2) / mean(|y_ideal|^2))
• A lower (more negative) NMSE indicates superior linearization performance.
• Values below -40 dB are generally considered excellent for wideband signals.
• NMSE is sensitive to both in-band distortion (EVM degradation) and out-of-band distortion (spectral regrowth).

A time-domain metric that directly measures the deviation of the actual transmitted symbol constellation from the ideal reference points. EVM is a direct consequence of residual post-distortion error corrupting the modulated signal.

• EVM is expressed as a percentage of the peak or average symbol power.
• Standards like 3GPP mandate strict EVM limits (e.g., 3.5% for 256-QAM in 5G NR).
• While NMSE captures total error power, EVM isolates the in-band distortion that degrades bit error rate (BER).
• Poor EVM despite good NMSE often indicates IQ imbalance or phase noise, not just nonlinearity.

The ratio of power transmitted in an adjacent frequency channel to the power in the intended carrier channel. Post-distortion error manifests as residual spectral regrowth that elevates ACLR.

• Regulatory bodies like the FCC and ETSI impose strict ACLR masks to prevent interference.
• A well-tuned DPD system can improve ACLR by 15-25 dB compared to an unlinearized amplifier.
• ACLR is primarily sensitive to odd-order nonlinearities (IM3, IM5) in the PA.
• Monitoring ACLR alongside NMSE provides a complete picture of both in-band and out-of-band error.

The dominant closed-loop architecture for DPD coefficient estimation. In ILA, a post-distorter model is trained to minimize the error between the PA output and the desired linear output, then the coefficients are copied to the predistorter.

• The key assumption is that the post-inverse equals the pre-inverse, which holds for systems with mild memory effects.
• Post-distortion error is the explicit training signal for the post-distorter block.
• ILA is computationally efficient because it avoids solving a direct inverse problem.
• However, measurement noise in the feedback path can bias the coefficient estimate, increasing residual error.

An adaptive architecture that directly minimizes the post-distortion error by iteratively updating predistorter coefficients based on the difference between the ideal input and the measured PA output.

• DLA does not rely on the post-inverse equals pre-inverse assumption, making it theoretically more robust than ILA.
• Requires a nonlinear optimization solver (e.g., Gauss-Newton or Levenberg-Marquardt) at each iteration.
• Better suited for strong memory effects and complex Doherty amplifier topologies.
• The convergence speed and final residual error depend heavily on the accuracy of the PA model used in the optimization loop.

The minimum achievable post-distortion error limited by physical phenomena that a DPD model cannot compensate for. This floor sets the ultimate performance boundary for any linearization system.

• Contributing factors:
  • Thermal noise in the feedback receiver chain.
  • Quantization noise from the ADC in the observation path.
  • Unmodeled dynamics such as charge trapping in GaN HEMT devices.
  • Modulator impairments (IQ imbalance, LO leakage) that are not corrected.
• Understanding this floor prevents over-engineering the DPD model complexity beyond what the hardware can exploit.