Coherence in Random Vibration

What Is It?

Coherence is the normalised measure of the linear relationship between two random signals in the frequency domain. It is derived from the cross-spectral density and the auto-spectral densities (PSDs). Coherence ranges from 0 (no linear correlation) to 1 (perfect linear correlation) at each frequency. It provides a frequency-dependent measure of how strongly two signals are related.

Why It Matters

Coherence is the key to determining whether multiple inputs can be treated as independent or whether their correlation must be accounted for. It is also used in experimental structural dynamics to assess measurement quality and to identify resonant frequencies. Understanding coherence is essential for multi-input random vibration analysis and for interpretation of vibration test data.

Coherence quantifies the correlation between two signals at each frequency. It determines whether multi-input analysis must include cross-spectral terms or whether inputs can be treated as independent.

Coherence Function

The coherence function γ²(f) is the ratio of the squared cross-spectral density magnitude to the product of the two auto-spectral densities. It is always between 0 and 1. A value of 0 means the signals are uncorrelated at that frequency. A value of 1 means they are perfectly correlated (linearly related). Values between 0 and 1 indicate partial correlation.

Coherence function:

γ²(f)  =  |G_xy(f)|²  /  ( G_xx(f) · G_yy(f) )

where:
G_xy(f)  =  cross-spectral density between x and y
G_xx(f)  =  auto-spectral density (PSD) of x
G_yy(f)  =  auto-spectral density (PSD) of y

Range:  0  ≤  γ²(f)  ≤  1

γ² = 0:  uncorrelated at frequency f
γ² = 1:  perfectly correlated at frequency f
0 < γ² < 1:  partially correlated

Correlated Excitation

When the coherence between two inputs is 1 (or close to 1) at a given frequency, the inputs are correlated at that frequency. They move together — the same source or the same structural motion drives both. In this case, cross-spectral terms must be included in the multi-input analysis. Treating correlated inputs as independent produces incorrect results.

• γ² ≈ 1 — inputs are correlated at this frequency
• Same source or same structural motion drives both inputs
• Cross-spectral terms MUST be included in analysis
• Treating as independent produces incorrect results

Uncorrelated Excitation

When the coherence is 0 (or close to 0) at a given frequency, the inputs are independent at that frequency. They are driven by different sources with no relationship. In this case, cross-spectral terms are negligible and the inputs can be treated as independent. The response is the SRSS combination of the individual responses.

• γ² ≈ 0 — inputs are independent at this frequency
• Different sources, no relationship between inputs
• Cross-spectral terms are negligible
• Response = SRSS combination of individual responses

Partial Correlation

When the coherence is between 0 and 1, the inputs are partially correlated. Some of the motion at one input is related to the other, and some is independent. Partial correlation is the most common case in practice — for example, two support points on a structure share some common motion (from the same source) but also have some independent motion. Partial correlation requires the full cross-spectral treatment.

Interpretation

Coherence varies with frequency — two signals may be correlated at some frequencies and uncorrelated at others. This frequency dependence is important. For example, two support points may be perfectly correlated at low frequencies (rigid-body motion of the supporting structure) but uncorrelated at high frequencies (independent local vibration). The coherence function should be examined across the full frequency range, not just at a single frequency.

Coherence varies with frequency. Two signals may be correlated at some frequencies and uncorrelated at others. Always examine the coherence function across the full frequency range.

Coherence in Measurement

In experimental structural dynamics, coherence is used to assess measurement quality. A coherence near 1 between input and response at a resonant frequency indicates a clean measurement — the response is well explained by the input. A coherence below 1 may indicate noise, nonlinear response, or unmeasured additional inputs. Low coherence at a resonance suggests the measurement or the model needs investigation.

• γ² ≈ 1 at resonance — clean measurement, response well explained by input
• γ² < 1 at resonance — possible noise, nonlinearity, or unmeasured inputs
• γ² < 1 away from resonance — normal, response is low relative to noise
• Coherence is a key measurement quality indicator

Coherence and Number of Averages

Coherence estimated from measured data depends on the number of averages used in the PSD computation. With too few averages, the coherence is biased high — it appears closer to 1 than it truly is. With sufficient averages (typically 50+), the bias is small. This is a practical consideration when interpreting measured coherence — low coherence with few averages is reliable; high coherence with few averages may be biased.

Measured coherence is biased high with too few averages. Low coherence with few averages is reliable. High coherence with few averages may be an artefact of insufficient averaging.

Key Takeaways

• Coherence γ²(f) ranges from 0 (uncorrelated) to 1 (perfectly correlated) at each frequency
• γ² ≈ 0: inputs independent — SRSS combination, cross terms negligible
• γ² ≈ 1: inputs correlated — cross-spectral terms essential
• Partial correlation (0 < γ² < 1) is the most common practical case — requires full analysis
• Coherence varies with frequency and is used for both multi-input analysis and measurement quality