# Coherence Formula Sheet

Download Markdown Download PDF

## Data Path - input stream: accepted RR intervals from the Heart Rate Measurement characteristic - physical H10: those RR intervals are device-derived from ECG beat timing - synthetic transport: the coherence solve still uses RR, while synthetic PMD ECG is emitted separately so the waveform view stays aligned to the same beat schedule ## Fixed Spectral Model The app keeps the paper-defined spectral constants fixed: - peak search band: `0.04-0.26 Hz` - peak integration window: `0.030 Hz` centered on the dominant peak - total-power band: `0.0033-0.4 Hz` These formulas operate on the RR-derived tachogram after the irregular beat-interval stream has been resampled into a continuous time series. In the notation below, `f` is frequency in hertz, `PSD(f)` is the power spectral density at that frequency, `f_peak` is the strongest oscillation in the allowed coherence band, `P_peak` is the power concentrated around that oscillation, and `P_total` is the broader band power used as the paper's reference total. ```latex f_{peak} = \arg\max_{f \in [0.04, 0.26]} PSD(f) ``` This step finds the dominant rhythm inside the coherence search band instead of across the entire spectrum. ```latex P_{peak} = \int_{f_{peak} - 0.015}^{f_{peak} + 0.015} PSD(f)\,df ``` This integrates a narrow window around the dominant peak so the metric rewards concentrated narrowband power rather than diffuse nearby energy. ```latex P_{total} = \int_{0.0033}^{0.4} PSD(f)\,df ``` This is the total reference power across the wider HRV band used by the cited method. ```latex \text{paper coherence ratio} = \left(\frac{P_{peak}}{P_{total} - P_{peak}}\right)^2 ``` This ratio rises when one narrow resonance-like peak dominates the spectrum and falls when power is spread across the rest of the band. ## UI-Facing Normalized Score The app also publishes a bounded score for the headline readout and summary charts: ```latex \text{normalized coherence}_{0..1} = \operatorname{clamp}\left(\frac{P_{peak}}{P_{total}}, 0, 1\right) ``` Here the app is simply reporting the fraction of tracked power that sits in the dominant peak window, then clamping that fraction to a display-friendly range. This normalized score is an app-side presentation choice. It is not the McCraty paper ratio itself. ## Runtime Choices Around The Paper Formula The cited source fixes the spectral constants above, but it does not define the operator-side runtime details used by this app. The current implementation uses: - a rolling RR window - cubic-spline resampling of the RR tachogram - a `128`-point Hann-windowed spectrum solve - a smoothed headline `0..1` display value - a separate confidence score based on RR count and window coverage Those runtime choices describe how the app gets from incoming RR intervals to `PSD(f)` and to a stable operator readout. They affect warmup, smoothness, and confidence behavior, but not the core paper ratio formula shown above. ## Alignment Status The implementation is aligned to the cited reference at the formula level for: - peak search band - peak-window width - total-band limits - paper coherence ratio The display normalization, smoothing, and confidence telemetry are explicit application-level additions. ## References - [Coherence Workflow](coherence-workflow.md) - [References](references.md) - McCraty, Atkinson, Tomasino, and Bradley, *The Coherent Heart* (2006)