Optimal ecHT calibration slashes phase error for real-time neurostimulation

In a new arXiv preprint, researchers Eike Osmers and Dorothea Kolossa tackle a long-standing issue with the endpoint-corrected Hilbert transform (ecHT): systematic distortions at the signal boundary that degrade instantaneous phase estimates. The ecHT is widely used in real-time closed-loop systems—especially phase-locked neurostimulation—to reduce boundary artefacts, but its endpoint behavior lacked a rigorous theoretical foundation. The authors derive the ecHT endpoint operator analytically, showing that the output can be split into a desired positive-frequency term (with a deterministic complex gain that introduces a calibratable amplitude/phase bias) and a residual leakage term that sets an irreducible variance floor.

From this decomposition, they provide explicit characterizations and bounds for endpoint phase and amplitude error, leading to a mean-squared-error-optimal scalar calibration. The calibrated ecHT achieves near-zero mean phase error while remaining computationally lightweight for real-time pipelines. The paper also offers practical design rules relating window length, filter bandwidth and order, and center-frequency mismatch to residual bias via an endpoint group delay. The authors have released code and analyses to accompany the work, making this a significant step forward for precise, low-latient phase estimation in neural and signal processing applications.