Closed-Loop DPD

The defining architectural element of closed-loop DPD is the transmit observation receiver (TOR) . This dedicated feedback chain captures a coupled sample of the PA output, downconverts it, and digitizes it for comparison with the ideal reference signal. The error signal computed from this comparison drives the coefficient update algorithm, allowing the system to react to changes in temperature, bias drift, or channel loading within microseconds.

Unlike static open-loop systems, closed-loop architectures employ online training algorithms to continuously minimize a cost function—typically the Normalized Mean Squared Error (NMSE) between the desired input and the observed output. This adaptation compensates for:

• Thermal memory effects that shift the PA's optimal bias point
• Aging-related drift in transistor characteristics
• Load impedance variations from antenna mismatch The loop bandwidth must be sufficient to track envelope-frequency dynamics without introducing instability.

Closed-loop DPD can be implemented through two distinct training architectures. Direct Learning Architecture (DLA) minimizes the error between the predistorter input and the PA output directly, requiring a model inversion step. Indirect Learning Architecture (ILA) trains a postdistorter on the PA output and copies its coefficients to the predistorter, assuming the inverse is commutable. ILA is simpler to implement but can suffer from bias in the presence of measurement noise, while DLA provides theoretically optimal coefficient estimates.

Closed-loop systems introduce stability constraints that must be carefully managed. The loop gain, adaptation step size, and group delay through the feedback path all influence convergence behavior. Key design considerations include:

• Condition number of the data covariance matrix, which affects numerical stability
• Tikhonov regularization to prevent ill-conditioned solutions
• Misadjustment from gradient noise in stochastic updates Algorithms like QR-RLS are preferred over standard LMS when rapid convergence and numerical robustness are required for wideband signals.

Closed-loop DPD effectiveness is quantified through multiple complementary metrics measured at the PA output. Adjacent Channel Power Ratio (ACPR) verifies spectral regrowth suppression against regulatory masks. Error Vector Magnitude (EVM) assesses in-band signal fidelity critical for high-order QAM demodulation. Normalized Mean Squared Error (NMSE) provides a broadband measure of linearization accuracy. A well-tuned closed-loop system typically achieves ACPR improvements exceeding 20 dB and EVM below 1% for 5G NR waveforms.

The feedback path introduces loop delay that constrains the maximum correctable bandwidth. Total latency includes the analog feedback chain propagation, ADC conversion time, and digital processing for coefficient computation. For wideband 5G signals with instantaneous bandwidths exceeding 100 MHz, the loop delay must be kept below the inverse of the signal bandwidth to maintain causality. Sample-by-sample update strategies minimize latency but demand high-throughput FPGA implementations, while block update approaches trade adaptation speed for computational efficiency.

A closed-loop topology that directly estimates predistorter coefficients by minimizing the error between the desired input signal and the actual PA output. Unlike indirect methods, DLA solves for the predistorter in a single step by modeling the PA forward characteristic and then mathematically inverting it.

• Eliminates the assumption of commutation validity required by ILA
• Handles non-invertible PA characteristics more robustly
• Requires accurate PA behavioral modeling for successful inversion
• Often employs model inversion or iterative optimization techniques

A widely-used closed-loop architecture where a postdistorter is trained on the PA output to replicate the inverse transfer function. Once the postdistorter converges, its coefficients are copied directly to the predistorter placed before the PA.

• Assumes the commutation property holds for the nonlinear system
• Computationally simpler than DLA for many implementations
• Susceptible to bias when measurement noise is present in the feedback path
• Dominant architecture in early commercial DPD systems

The mathematical engine of closed-loop DPD that computes optimal predistorter parameters from observed input-output data. Common approaches include Least Squares (LS), Recursive Least Squares (RLS), and Least Mean Squares (LMS).

• LS: Batch solution minimizing squared error over a block of samples
• RLS: Recursive update with fast convergence, ideal for tracking time-varying PA behavior
• LMS: Low-complexity stochastic gradient method suitable for hardware-efficient implementation
• QR-RLS: Numerically stable variant using QR decomposition to avoid ill-conditioning

The dedicated feedback path that samples the PA output, downconverts it, and digitizes it for comparison with the reference signal. This path is critical to closed-loop operation and must maintain high linearity to avoid corrupting the error signal.

• Requires wider bandwidth than the transmit signal to capture spectral regrowth
• Must achieve higher linearity than the PA being corrected
• Introduces loop delay that must be compensated for proper time alignment
• Often includes IQ imbalance correction to prevent feedback-induced distortion

Two competing performance metrics in closed-loop DPD. Convergence rate measures how quickly coefficients reach steady-state, while misadjustment quantifies the excess error beyond the theoretical minimum caused by gradient noise.

• Faster convergence typically increases misadjustment due to larger step sizes
• Burst training updates coefficients only during idle intervals to avoid disrupting active transmission
• Sample-by-sample updates provide continuous tracking but introduce steady-state jitter
• Block updates balance latency and computational load by processing batches of samples

The mathematical objective minimized during closed-loop training. The most common cost function is the Normalized Mean Squared Error (NMSE) between the desired linear output and the actual PA output.

• NMSE: Standard metric representing error power normalized by input signal power
• ACPR-weighted cost: Incorporates adjacent channel leakage penalties for regulatory compliance
• EVM-aware cost: Prioritizes in-band signal quality for modulation accuracy
• Regularization terms: Added to prevent overfitting and coefficient drift due to noise amplification