Dieser Leitfaden ist derzeit auf Englisch verfügbar.
Technology / Technical guide
Frequency Coherence: Shared Content & Evidence Limits
Coherence describes a normalised relationship between two signals at each frequency. It can help distinguish shared spectral variation from unrelated contributions under a stated model. Its magnitude does not identify causal direction, and an estimate from insufficient averaging can look perfect even where that conclusion is unjustified.
Frequency-specific agreement is different from physical cause.
Relate the cross-spectrum to both individual spectra and the evidence used to estimate them.
01 / Definition
Cross-spectrum is normalised by both spectra.
Magnitude-squared coherence is Cxy(f) = |Sxy(f)|²/[Sxx(f)Syy(f)] where the denominator is nonzero.
Magnitude-squared coherence is Cxy(f) = |Sxy(f)|²/[Sxx(f)Syy(f)] where the denominator is nonzero. The auto-spectra describe each signal’s squared-frequency content; the cross-spectrum describes their frequency relationship under consistent conventions.
Its units cancel. For valid consistent joint spectra the result lies between zero and one. At a frequency with zero represented content in a denominator, division does not define an informative ordinary coherence value.
02 / Linear model
Unrelated output content changes the ideal value.
In a simple stationary model y = a x+v, suppose v is uncorrelated with x at the represented frequency and a is a real scalar gain.
In a simple stationary model y = a x+v, suppose v is uncorrelated with x at the represented frequency and a is a real scalar gain. The output spectrum contains a²Sxx+Svv, while its cross-spectrum with x contains the corresponding shared term.
Hypothetical spectral values: a = 2, Sxx = 1 and Svv = 4 in matching squared-unit densities give Syy = 8 and |Sxy|² = 4. Thus coherence is 4/(1×8) = 0.5. These values are invented.
03 / Meaning
The model fraction is not a general energy share.
In that particular one-input model, unrelated output content lowers coherence.
In that particular one-input model, unrelated output content lowers coherence. Outside its assumptions, a value of 0.5 does not universally mean half the mechanical power, half the energy or a fifty-percent causal contribution.
Nonlinear relationships and additional correlated inputs can produce different interpretations. A low value also does not establish that the two physical processes are independent. The statistic addresses their specified second-order frequency relationship, not every possible dependency.
04 / Finite estimation
One unsmoothed transform pair is misleading.
Using a single transform pair directly gives |X*Y|²/[|X|²|Y|²] = 1 wherever both are nonzero.
Using a single transform pair directly gives |X*Y|²/[|X|²|Y|²] = 1 wherever both are nonzero. That algebra alone is not an informative population estimate of coherence. The spectra need a justified estimation basis.
MathWorks describes segment-averaged estimation and notes the single-segment limitation. More plotted bins are not more independent averages. Overlapping segments can share samples, while nonstationary intervals can mix different relationships; the retained evidence matters beyond a nominal segment count.
05 / Causal boundary
Shared excitation can produce high coherence.
Two outputs driven by the same unobserved disturbance can be highly coherent without either causing the other.
Two outputs driven by the same unobserved disturbance can be highly coherent without either causing the other. Shared signal processing can also create dependence. Swapping the pair leaves the magnitude-squared relationship symmetric rather than revealing a causal arrow.
Likewise, coherent content does not by itself validate a transfer function. Input/output roles, measurement error and the complete model remain separate. The transfer-function guide establishes that boundary.
06 / Interpretation
Phase and confidence are additional quantities.
Coherence magnitude does not contain the cross-spectrum’s phase.
Coherence magnitude does not contain the cross-spectrum’s phase. Nor does a high finite estimate independently describe its statistical uncertainty. Compatible windows, periods and variable definitions belong to the represented relationship.
No MARSTON coherence result, vibration path or diagnostic threshold is supplied. The correlation guide addresses time-shift association; these principles provide no field test, spectral setting or causal diagnosis.
Related technical guides: Cross-Correlation: Lag Sign, Similarity & Time Delay · Power Spectral Density: Units, Integration & RMS · Dynamic States & Observability: What Outputs Can Reveal.
General background: MathWorks: Magnitude-squared coherence and averaging limitations. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Sxx and Syy | Consistently defined auto-spectra | Their nonzero content supports the denominator. |
| Sxy | Cross-spectrum of the declared pair | Its phase is not contained in coherence magnitude. |
| Cxy | |Sxy|²/(SxxSyy) | Dimensionless, with consistent spectral definitions. |
| Finite estimate | Averaged spectral relationship | One raw transform pair is insufficient evidence. |
| Causal claim | Physical direction or mechanism | Not established by high coherence alone. |
Engineering questions.
Does coherence have displacement or power units?
No. The consistently formed ratio is dimensionless.
Does 0.5 always mean half the output power comes from x?
No. That reading needs the particular one-input model and assumptions.
Can common excitation cause high coherence?
Yes, without either observed signal directly causing the other.
Does one transform pair prove perfect coherence?
No. Its algebra can give one without an informative estimate.
Does low coherence prove independence?
No. Other dependencies or measurement conditions can remain.
Is the example a turbine result?
No. Its spectra and scalar gain are hypothetical.
Technical enquiry
Discuss the technical question.
Coherence describes a normalised relationship between two signals at each frequency. Share the model and applicable operating or design information for an engineering discussion.