Loop Delay Estimation

Loop delay comprises an integer component (whole sample periods) and a fractional component (sub-sample misalignment). Integer delay is resolved via cross-correlation peak detection, while fractional delay requires interpolation-based refinement using sinc or Lagrange filters. Even a 0.1-sample offset introduces phase distortion that corrupts coefficient estimation, particularly in wideband signals where the fractional error translates to significant phase rotation at band edges.

The primary integer delay estimation method computes the cross-correlation between the transmitted reference and the observed feedback signal. The lag index corresponding to the maximum correlation value indicates the bulk propagation delay. This technique is robust to noise and nonlinear distortion because the linear component of the amplifier output retains strong correlation with the input. For periodic training waveforms, circular cross-correlation eliminates boundary effects.

Fractional delay can be estimated by examining the phase slope of the cross-spectrum between reference and feedback signals. A pure time delay manifests as a linear phase ramp in the frequency domain. By computing the unwrapped phase difference across the signal bandwidth and fitting a straight line, the fractional delay is derived from the slope. This method is particularly effective for OFDM signals where the frequency-domain representation is naturally available.

For high-precision applications, an iterative loop alternates between delay estimation and model extraction:

• Initial coarse alignment via cross-correlation
• Preliminary model coefficient estimation
• Residual delay error computed from the post-distortion error signal
• Fine adjustment using gradient-based delay optimization This closed-loop approach converges to sub-0.01-sample accuracy, essential for mmWave DPD systems where fractional delay errors severely degrade linearization performance.

Loop delay is not static. Sources of variation include:

• Temperature-dependent analog group delay in filters and amplifiers
• Clock drift between transmit and observation path ADCs/DACs
• Carrier frequency changes altering RF path electrical length
• Power amplifier bias shifts modifying device transit time Online delay tracking using recursive estimation with a forgetting factor compensates for slow drift without requiring periodic recalibration sequences.

Delay misalignment directly degrades coefficient estimation fidelity. A timing error of even 0.5 samples causes the regression matrix to misalign basis functions with their corresponding output samples, introducing systematic bias in least-squares solutions. The resulting predistorter model compensates for a distorted version of the amplifier characteristic, leading to residual spectral regrowth and degraded adjacent channel leakage ratio. Accurate delay estimation is therefore a prerequisite for all subsequent model extraction steps.

The critical pre-processing step of synchronizing the input reference waveform with the captured observation feedback waveform to sub-sample accuracy. Without precise time alignment, even a perfect behavioral model will fail to linearize the amplifier.

• Compensates for propagation delay through the transmit chain and feedback path
• Fractional delay interpolation achieves alignment finer than one sampling period
• Misalignment by even a fraction of a sample introduces significant modeling error

A robust method for estimating the integer-sample delay between reference and feedback signals by finding the lag that maximizes their cross-correlation. This technique is resilient to noise and nonlinear distortion.

• Computes the cross-correlation sequence between the ideal input and captured output
• The index of the peak correlation value gives the integer delay estimate
• Often combined with fractional delay interpolation for sub-sample refinement

After integer-sample alignment, residual sub-sample delay must be corrected using interpolation filters. Common implementations include Farrow structures and Lagrange interpolators.

• Farrow structure: Efficient variable fractional delay filter with polynomial coefficients
• Lagrange interpolation: Classic polynomial-based approach for arbitrary delay values
• Residual fractional delay causes dispersion in the predistorter model

A closed-loop DPD parameter estimation structure where a post-distorter model is trained on the amplifier's output and then copied to the predistorter. Time alignment between the PA input and the post-distorter reference is essential.

• Avoids the need for a direct inverse model of the amplifier
• Delay mismatch between paths causes the copied predistorter to be suboptimal
• Requires precise loop delay compensation in both training and deployment paths

The receiver chain that captures the power amplifier's output, downconverts it, and digitizes it for comparison with the reference signal. The propagation delay through this path is a primary component of the total loop delay.

• Includes analog components: couplers, mixers, filters, and ADCs
• Group delay variation across frequency can complicate wideband alignment
• Calibration of the feedback path itself is sometimes required for high-accuracy systems