Calculator
Phase coherence quantifies how tightly phase samples cluster on a circle. Higher coherence means stronger phase locking across oscillations or signals.
Formula Used
Phase coherence is commonly defined as the mean resultant length of phase unit vectors:
mean phase: φ̄ = atan2( Σ wk sinφk, Σ wk cosφk )
circular variance: V = 1 − r
circular std. dev.: σ = √( −2 ln r )
Here φk is each phase sample, wk is an optional weight, and r ranges from 0 to 1.
How to Use This Calculator
- Choose degrees or radians to match your data.
- Paste phase samples, or upload a CSV file.
- Optionally add weights with the same count.
- Enable wrapping if phases cross multiple cycles.
- Press Calculate to view coherence and summary.
- Use Download CSV for spreadsheets and reports.
- Use Download PDF to print a clean summary.
Tip: If your phases come from a time series, consider preprocessing to extract instantaneous phase first.
Example Data Table
This example uses degree phases that are moderately clustered. Load it into the form using the button above.
| Sample # | Phase (degrees) | Weight |
|---|---|---|
| 1 | 10 | 1.0 |
| 2 | 18 | 1.0 |
| 3 | 25 | 1.0 |
| 4 | 30 | 0.9 |
| 5 | 38 | 0.8 |
| 6 | 45 | 0.9 |
Phase Coherence in Physical Signals
1) What phase coherence measures
Phase coherence (often called the mean resultant length or phase-locking value) summarizes how consistently phase samples point in the same direction on the unit circle. A value near 1 indicates tight clustering (strong phase locking), while values near 0 indicate phases spread broadly across a cycle.
2) Why coherence is useful
Many oscillatory systems carry information primarily in timing rather than amplitude. Examples include coupled pendula, rotating machinery vibrations, and wave interference patterns. Coherence provides a compact statistic for comparing runs, tracking synchronization, and detecting transitions from ordered to disordered phase behavior.
3) Typical data sources
Phase samples can come from zero-crossing timing, Hilbert-transform instantaneous phase, Fourier component angles, or sensor arrays measuring periodic motion. When phases are computed from a time series, consistent sampling and stable filtering matter, because phase estimates become noisy near low amplitudes or sharp transients.
4) Interpreting r values
As a practical guide: r ≈ 0.2–0.4 often reflects weak alignment, r ≈ 0.5–0.7 indicates moderate locking, and r > 0.8 suggests strong locking across samples. Interpretation depends on sample count, extraction method, and whether phases represent repeated trials or consecutive time points.
5) Weighted coherence for confidence
This calculator supports optional weights. A common weighting strategy uses amplitude or signal-to-noise as wk, giving reliable phase estimates more influence. Weighting is especially helpful when some samples are derived from low-energy segments where phase is poorly defined.
6) Mean phase and spread
The mean phase φ̄ is computed from the average vector components Σw cosφ and Σw sinφ. The circular variance V = 1 − r increases as phases disperse. The circular standard deviation √(−2 ln r) provides an angle-like spread measure that grows rapidly when coherence is low.
7) Practical workflow tips
Start by wrapping phases into a single cycle to avoid misleading clustering across boundaries. Check for outliers caused by phase unwrapping jumps or sign flips. If you upload CSV data, keep phases in the first column and verify units. Increasing precision can help compare close conditions.
8) Limitations and good practice
Coherence alone does not reveal causality or frequency content. For time-evolving systems, compute coherence in sliding windows and report N and extraction settings. When comparing experiments, keep filtering, reference phase definition, and sampling strategy consistent to make coherence values comparable.
FAQs
1) What does r = 1 mean?
All phase samples align perfectly at the same angle, indicating ideal phase locking. In real measurements, values very close to 1 usually mean strong synchronization with minimal phase jitter.
2) Can r be negative?
No. This coherence definition is a magnitude (mean resultant length), so it ranges from 0 to 1. Direction information appears in the mean phase, not in the sign of r.
3) Degrees or radians—does it change results?
No, as long as the unit selection matches the input. The calculator converts internally and produces the same r. Only the reported mean phase value changes unit.
4) When should I use weights?
Use weights when some phase samples are more reliable, such as higher-amplitude segments or higher-quality trials. Weights reduce the impact of noisy or low-confidence phase estimates.
5) Why is wrapping helpful?
Wrapping maps phases into one cycle, preventing artificial gaps at boundaries like 359° and 1°. It makes clustering and mean phase estimates more stable for circular data.
6) What if my CSV has multiple columns?
This tool reads phases from the first column only. If your file includes additional columns, keep phase values in column one or export a simplified file containing only phases.
7) How many samples do I need?
More samples improve stability. For quick checks, 20–50 samples can work; for reliable comparisons, use 100+ and keep preprocessing consistent across conditions.