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Optimal Calibration of the Endpoint-corrected Hilbert Transform

This paper presents a principled, closed-form analysis of the endpoint-corrected Hilbert transform (ecHT) that derives an optimal scalar calibration to eliminate systematic phase and amplitude biases while characterizing irreducible error floors, thereby enabling accurate, low-latency phase estimation for real-time applications.

Original authors: Eike Osmers, Dorothea Kolossa

Published 2026-06-26

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Eike Osmers, Dorothea Kolossa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to listen to a specific instrument in a busy orchestra, but you only have a very short window of time to hear it. You want to know exactly when that instrument is playing its note (its "phase") so you can clap in perfect rhythm with it. This is the challenge faced by scientists trying to control brain stimulation or synchronize machines in real-time.

The paper introduces a solution to a specific problem with how we currently listen to these signals. Here is the breakdown:

The Problem: The "Glitch" at the Edge

The standard tool for listening to these rhythms is called the Hilbert Transform. Think of it like a camera that takes a snapshot of a sound wave. To work, the camera assumes the sound wave repeats itself perfectly forever, like a looped video.

However, in the real world, the sound doesn't loop perfectly. When the camera cuts off the snapshot at the very end (the "endpoint"), it creates a sudden, jarring jump in the data. In the paper, this is compared to a Gibbs phenomenon, which is like a camera flash causing a blinding white ring around a subject. This "ringing" creates a massive error right at the very last second of the snapshot—the exact moment the computer needs to make a decision.

The Previous Fix: The "Endpoint-Corrected" (ecHT)

Scientists previously tried to fix this by adding a "smoothing filter" (the ecHT). Imagine putting a soft, fuzzy lens over the camera to blur out that jarring jump. This helped reduce the noise, but it introduced a new problem: it consistently shifted the timing of the note. It was like the camera lens was slightly tilted, making the note sound like it happened a fraction of a second too early or too late. This "bias" meant the system was always slightly out of sync.

The New Solution: The "Calibrated" (cecHT)

The authors of this paper realized that this shift wasn't random noise; it was a predictable, mathematical error.

They treated the signal like a recipe. They discovered that the output of the filter is actually made of two parts:

The Good Part: The actual rhythm you want, just slightly scaled up or down and shifted in time (like a song played at the wrong speed).
The Bad Part: A tiny bit of "leakage" from other frequencies that creates a wobble you can't fully remove.

The Breakthrough:
The authors derived a simple mathematical "correction factor" (a single number) that acts like a tuning knob.

If the camera lens is tilted, this knob turns the image back straight.
If the volume is too loud or quiet, this knob fixes the gain.

By applying this "calibration" (which they call cecHT), they can mathematically undo the systematic shift. The result is that the timing error drops from a noticeable mistake (several degrees of phase error) to practically zero.

Why It Matters

It's a "Drop-in" Upgrade: You don't need to rebuild the whole system. You just add this one calculation step to existing tools.
It's Fast: It doesn't slow down the computer, which is crucial for real-time applications like stopping a tremor in a patient's hand the instant it starts.
It's Honest: The authors admit that while they fixed the predictable shift, there is still a tiny, unavoidable "wobble" caused by the fundamental limits of looking at a short slice of time. However, they proved this wobble is so small it doesn't matter for most practical uses.

The Real-World Test

The team tested this on two types of signals:

Brainwaves (Alpha rhythms): When people close their eyes, their brain produces a steady rhythm. The new method removed the timing bias, making the "clap" perfectly synchronized with the brainwave.
Tremors: For patients with shaking hands (essential tremor), the method corrected the timing error significantly, allowing for more precise stimulation to stop the shaking.

In short: The paper takes a tool that was "almost right" but consistently off-beat, analyzes exactly why it was off, and provides a simple, free "tuning screw" to make it perfectly on-beat, without slowing anything down.

Technical Summary: Optimal Calibration of the Endpoint-corrected Hilbert Transform

Problem Statement
Accurate, low-latency estimation of the instantaneous phase of oscillations is critical for closed-loop sensing and actuation, particularly in phase-locked neurostimulation and real-time control systems. The standard approach involves constructing an analytic signal via the Hilbert Transform (HT) on a sliding window. However, discrete-time implementations using the Discrete Fourier Transform (DFT) assume periodicity within the window. When the signal is not perfectly periodic across the window boundaries, this assumption creates discontinuities that manifest as Gibbs ringing, causing significant amplitude and phase distortions at the window edges. Since real-time systems rely on the most recent sample (the endpoint), these boundary artifacts directly corrupt the critical phase estimate.

While the Endpoint-corrected Hilbert Transform (ecHT) was introduced to mitigate these issues by applying a causal, narrow-band filter to the analytic spectrum, its systematic endpoint distortions have lacked a principled, closed-form analysis. Previous evaluations were largely empirical, leaving the nature of the residual bias and the optimal strategy for correction undefined.

Methodology
This study provides an analytical, DFT-based characterisation of the ecHT. The authors derive the endpoint operator analytically for a narrow-band single-tone signal, decomposing the ecHT output into two distinct components:

A deterministic complex gain ( $G_+$ ): A positive-frequency term that induces a calibratable amplitude and phase bias.
A residual leakage term ( $G_-$ ): A negative-frequency term that escapes the analytic mask and filter, creating a phase-dependent ripple that varies with the instantaneous phase.

Based on this decomposition, the paper derives a Mean-Squared-Error (MSE)-optimal scalar calibration. The authors prove that for any ecHT implementation, there exists a unique complex scalar $C_{opt}$ that minimizes the error between the calibrated output and the ideal analytic signal. This scalar is derived analytically as:
$C_{opt} = \frac{E[\hat{Z}^* Z]}{E[|\hat{Z}|^2]}$
where $Z$ is the true endpoint and $\hat{Z}$ is the ecHT estimate. Geometrically, this represents an orthogonal projection (a scalar Wiener filter) that extracts the maximally linearly recoverable information. The authors demonstrate that $C_{opt}$ depends only on the positive-frequency gain and negative-frequency leakage, which can be pre-computed for any window length, filter bandwidth, order, and center frequency without data-driven fitting.

Key Contributions

Analytical Decomposition: The paper establishes that ecHT errors consist of a correctable systematic bias (determined by $G_+$ ) and an irreducible variance floor (determined by $G_-$ ).
Optimal Calibration (cecHT): The authors introduce the "calibrated ecHT" (cecHT), which applies the derived scalar $C_{opt}$ to eliminate the systematic phase bias. This results in near-zero mean phase error under stable conditions.
Design Rules: The study provides explicit bounds for endpoint phase and amplitude errors and relates practical design parameters (window length, filter bandwidth/order, and center-frequency mismatch) to residual bias via an endpoint group delay.
Computational Efficiency: The calibration adds negligible overhead (a single complex multiplication) to the existing $O(L \log L)$ FFT-based ecHT pipeline, preserving real-time compatibility.

Results
The efficacy of the cecHT was validated through simulations and real-world data:

Simulations: In single-tone simulations, the cecHT consistently reduced the mean phase error from approximately 8.53° (uncalibrated ecHT) to 0.40°, with a maximum error reduction from 11.70° to 1.23°. The calibration rendered the algorithm virtually independent of parameter choices (bandwidth, filter order, frequency drift), allowing parameters to be tuned for other constraints (e.g., noise robustness) without sacrificing phase accuracy.
EEG Alpha Waves: Applied to the EEG Alpha Waves Dataset, the cecHT reduced the mean phase error from 4.9° to 0.0° ( $p < 10^{-6}$ ) while preserving the Phase Locking Value (PLV) and significantly reducing the Phase Lag Index (PLI), indicating a removal of phase asymmetry.
Essential Tremor: In tremor data analysis, calibration reduced the mean phase error from -15.9° to 5.2°. When combined with periodic updates of the center frequency ( $f_0$ ) to track spectral drift, the mean error was further reduced to 0.0°, demonstrating that residual errors were primarily drift-induced rather than intrinsic to the calibration method.

Significance and Claims
The paper claims that the cecHT offers a "practical drop-in replacement" for existing ecHT pipelines, providing a statistically robust estimator with clear design rules. By separating correctable bias from intrinsic leakage, the framework offers both a practical tool for immediate improvement and a diagnostic language to understand the limits of phase estimation.

The authors emphasize that while calibration eliminates the systematic bias encoded in the filter's phase response, it cannot remove the irreducible error floor caused by spectral leakage and non-stationarity. Therefore, the method achieves unbiased estimates under stable conditions, with remaining jitter reflecting either estimator limitations or genuine signal instability. The study positions the cecHT as an intermediate solution between simple causal filters and complex adaptive models, suitable for real-time neural signal processing provided that frequency stability and artifact robustness are managed.