Error Signal

The error signal e(n) is computed as the complex baseband difference between the desired linear reference x(n) and the observed feedback signal y(n) after time alignment and gain normalization:

• e(n) = x(n) − y(n) — a complex-valued sequence capturing both magnitude and phase deviation
• Represents the exact distortion introduced by the PA at each sample instant
• Drives the coefficient update in gradient-based algorithms like LMS and NLMS
• The error signal's power is the quantity minimized by the cost function

In practice, any misalignment between x(n) and y(n) corrupts the error signal, making precise time alignment the single most critical preprocessing step.

The error signal contains two spectrally distinct components that map to different performance metrics:

• In-band error: Deviation within the occupied signal bandwidth, directly measured by Error Vector Magnitude (EVM). This component degrades modulation accuracy and bit error rate
• Out-of-band error: Spectral regrowth into adjacent channels, quantified by Adjacent Channel Leakage Ratio (ACLR). This component causes interference and regulatory non-compliance

A well-designed cost function may weight these components differently depending on whether the system prioritizes modulation fidelity or spectral mask compliance.

The error signal defines a multidimensional error surface over the coefficient space that the adaptive algorithm must navigate:

• For memory polynomial DPD, the error surface is quadratic with respect to the coefficients, guaranteeing a single global minimum for LMS and RLS algorithms
• The gradient of the squared error with respect to each coefficient determines the update direction
• Ill-conditioning of the basis function correlation matrix creates elongated error contours, slowing convergence along certain coefficient dimensions
• The minimum mean squared error (MMSE) floor represents the residual distortion that cannot be corrected, limited by feedback SNR and model order

The error signal is only as accurate as the feedback observation path that produces it:

• Feedback receiver noise adds random perturbation to e(n), increasing steady-state misadjustment and setting a floor on achievable linearization
• IQ imbalance in the feedback path introduces image-frequency artifacts that corrupt the error signal and bias coefficient estimates
• Phase noise from the local oscillator causes time-varying rotation of the error vector, degrading adaptation accuracy
• Quantization noise from the feedback ADC limits the dynamic range of observable distortion products

These impairments explain why feedback receiver design is as critical as the DPD algorithm itself.

The error signal plays fundamentally different roles depending on the learning architecture:

• Indirect Learning Architecture (ILA): The error is computed between the post-distorter output and the predistorter copy output in the feedback path. The PA model is bypassed entirely during coefficient estimation
• Direct Learning Architecture (DLA): The error is computed between the desired linear output and the actual PA output. The PA model is required to back-propagate the error gradient through to the predistorter coefficients
• DLA error signals directly represent the true linearization objective, while ILA error signals assume the predistorter copy accurately models the inverse, which may not hold under PA characteristic drift

The error signal's temporal evolution reveals the adaptation dynamics:

• Initial convergence: Large error magnitude during cold start as coefficients are far from optimal. The convergence rate determines how quickly the system reaches compliance
• Steady-state misadjustment: Residual error fluctuation around the MMSE due to gradient noise from stochastic updates. Controlled by the learning rate — smaller rates reduce misadjustment but slow convergence
• Tracking error: Error increase when PA characteristics change (temperature, aging, carrier frequency shift). The forgetting factor in RLS or adaptive learning rate in LMS determines tracking agility
• Coefficient freeze is triggered when the error signal falls below a threshold or when input power drops, preventing noise-driven coefficient drift

A critical in-band distortion metric that quantifies the deviation of received constellation points from their ideal reference positions. The error signal directly feeds EVM calculation.

• Measured as a percentage of RMS error relative to reference signal amplitude
• Directly correlates with Bit Error Rate (BER) in digital systems
• Typical 5G NR requirement: < 3.5% for 256-QAM
• The instantaneous error vector is the complex difference between the measured symbol and the ideal symbol location

The primary spectral regrowth metric that measures out-of-band distortion caused by PA nonlinearity. The error signal's spectral content outside the assigned channel directly determines ACLR performance.

• Defined as the ratio of in-channel power to adjacent channel power
• Typical 3GPP requirement: -45 dBc for adjacent channel
• Spectral regrowth results from intermodulation distortion products
• ACLR improvement is a direct optimization target for DPD adaptation loops

A stochastic gradient descent algorithm that updates predistorter coefficients proportionally to the instantaneous error signal. The simplest and most hardware-efficient adaptive filtering method.

• Update rule: w(n+1) = w(n) + μ · e(n) · x(n)
• μ (step size) controls convergence speed vs. steady-state error
• Computational complexity: O(N) per iteration, ideal for FPGA implementation
• Susceptible to slow convergence when input signals have high eigenvalue spread

An adaptive algorithm that minimizes a weighted least squares cost function of the error signal, offering significantly faster convergence than LMS at higher computational cost.

• Incorporates a forgetting factor (λ) to track time-varying PA characteristics
• Converges in approximately 2N iterations regardless of input conditioning
• Computational complexity: O(N²) per iteration
• Superior for tracking rapid changes in PA behavior due to thermal or bias drift

The mathematical objective that quantifies the aggregate error between desired linear output and actual PA output. The adaptation algorithm seeks to minimize this function.

• Common forms: Mean Squared Error (MSE), Weighted Least Squares
• MSE definition: J = E[|y_desired(n) - y_observed(n)|²]
• The error signal is the instantaneous sample of the cost function's argument
• Choice of cost function determines robustness to outliers and convergence properties

A dedicated observation receiver chain that captures a coupled sample of the PA output for error signal computation. Its linearity and dynamic range are critical to DPD performance.

• Must have wider bandwidth than the transmit signal (typically 3-5x for nonlinear observation)
• Requires higher linearity than the PA being linearized
• Includes down-conversion, filtering, and analog-to-digital conversion (ADC)
• Any distortion in the feedback path directly corrupts the error signal and limits DPD correction capability