Coherence Function

The coherence function quantifies the linear relationship between two signals as a function of frequency. It is a fundamental quality metric in transfer function estimation, indicating how reliably the output signal can be predicted from the input signal at each frequency.

Definition

The magnitude-squared coherence (MSC) between input signal x and output signal y is defined as:

γ²_xy(f) = |S_xy(f)|² / (S_xx(f) · S_yy(f))

where S_xy is the cross-spectral density between x and y, S_xx is the auto-spectral density of x, and S_yy is the auto-spectral density of y. All spectral densities must be averaged over multiple blocks for the coherence to be meaningful.

Interpretation

The coherence value ranges from 0 to 1 at each frequency:

Coherence Value	Interpretation
γ² = 1.0	Perfect linear relationship. The output is entirely determined by the input through a linear system.
γ² ≥ 0.9	Excellent coherence. Transfer function estimate is highly reliable at this frequency.
0.7 ≤ γ² < 0.9	Good coherence. Results are usable but some caution is warranted.
0.5 ≤ γ² < 0.7	Marginal coherence. Transfer function estimate has significant uncertainty.
γ² < 0.5	Poor coherence. The transfer function at this frequency should not be trusted.

A single FFT block always produces γ² = 1.0 regardless of the actual relationship between the signals. At least 5–10 averages are needed for the coherence to provide meaningful information. More averages yield a more reliable estimate.

Requirements

For a valid coherence measurement, the following conditions must be met:

Simultaneous acquisition: Both channels must be sampled synchronously from the same data acquisition system.
Sufficient averaging: Multiple FFT blocks must be averaged. The number of independent averages directly affects the statistical reliability of the coherence estimate.
Stationarity: The signals should be approximately stationary over the measurement duration. Non-stationary signals will produce artificially low coherence.
Adequate SNR: Both channels must have sufficient signal-to-noise ratio. Noise on either channel reduces coherence.

Parameters

Parameter	Options	Description
Input Channel	Any signal channel	Reference (excitation) signal, typically the force input.
Output Channel	Any signal channel	Response signal, typically an accelerometer or microphone.
Spectrum Size	256 – 65536	FFT block size, determines frequency resolution.
Overlap	0% – 95%	Block overlap. 50% or 66.7% recommended with Hanning window.
Number of Averages	5 – unlimited	Number of spectral averages. More averages improve the statistical quality.
Window Type	Hanning, Hamming, etc.	Window function applied to each block.

Causes of Low Coherence

When coherence drops below acceptable levels, the following causes should be investigated systematically:

Measurement Issues

Noise contamination: Electrical noise, ambient noise, or sensor self-noise on either channel degrades coherence.
Poor excitation: Insufficient excitation energy at certain frequencies (e.g., anti-resonances or frequencies above the excitation bandwidth).
Sensor overload: Clipping or saturation on either channel introduces harmonics and reduces coherence at the fundamental frequency.
Cabling and grounding: Ground loops or intermittent connections introduce uncorrelated noise.

System-Related Issues

Nonlinearity: If the system under test is nonlinear, the coherence will drop because the linear model assumption is violated.
Multiple inputs: If the measured output is influenced by unmeasured inputs (e.g., background vibration), the coherence for the measured input-output pair will be reduced.
Leakage: Inadequate windowing or insufficient frequency resolution can reduce coherence, particularly near sharp resonance peaks.

Practical Tips

Always display the coherence alongside the transfer function to assess data quality.
Use force windows for impact testing and exponential windows for the response to minimize leakage and improve coherence.
If coherence is consistently low across all frequencies, check the channel configuration and verify that the correct input/output assignment is used.
For modal testing, aim for γ² ≥ 0.9 across the frequency range of interest. Accept lower coherence only at anti-resonances where it is expected.

High coherence does not guarantee that the measurement is correct — it only confirms a linear relationship between the two channels. Both channels could be picking up the same external disturbance rather than the intended excitation-response relationship.