Questions: Coherence and Cross-Spectral Density

Question 1

An engineer simultaneously measures vibration at two locations on a machine. The coherence between the two signals is 0.92 at 60 Hz and 0.04 at 400 Hz. What does this indicate?

• A. The machine vibrates more strongly at 60 Hz than at 400 Hz — coherence reflects signal amplitude
• B. The two measurement points share a common vibration source or propagation path at 60 Hz, but are largely independent or driven by different uncorrelated sources at 400 Hz
• C. The 400 Hz signal is contaminated by sensor noise and needs recalibration before analysis
• D. The cross-spectral density magnitude is 23 times larger at 60 Hz than at 400 Hz

Answer: (B)

Coherence indicates the degree of linear dependence between two signals at each frequency — it is independent of signal amplitude. High coherence (0.92) at 60 Hz means the two sensors' signals at that frequency are strongly linearly related, suggesting a common source or transmission path. Low coherence (0.04) at 400 Hz means the signals are nearly independent at that frequency, which could indicate different local excitation sources, nonlinear behavior, or uncorrelated noise. Coherence does not measure signal strength; two weak signals can be perfectly coherent.

Question 2

What information is preserved in the cross-spectral density Sxy(f) that is lost if you simply compute the product of the individual power spectral densities Sxx(f) · Syy(f)?

• A. The absolute power level of each signal at each frequency
• B. The phase relationship (time delay or lead/lag) between the two signals at each frequency
• C. Whether each signal is stationary or non-stationary over the measurement period
• D. The signal-to-noise ratio of each individual measurement channel

Answer: (B)

The cross-spectral density Sxy(f) is complex-valued: its magnitude captures the correlation strength between x and y at frequency f, and its phase captures the time delay between the two signals at that frequency — a constant propagation delay appears as a linear phase increase with frequency. The product Sxx(f)·Syy(f) uses only real-valued magnitudes and discards all phase information. Coherence, which is |Sxy(f)|²/(Sxx(f)·Syy(f)), also sacrifices phase to achieve the [0,1] normalization, but the cross-spectral density itself retains it.

Question 3

Coherence is bounded between 0 and 1 and can be interpreted as a frequency-resolved squared correlation coefficient — analogous to R² in linear regression, but evaluated independently at each frequency.

• T. True
• F. False

Answer: True

True. This analogy is exact. Coherence Cxy(f) = |Sxy(f)|²/(Sxx(f)·Syy(f)) at each frequency f. Like R², a value of 0 means no linear relationship, and a value of 1 means perfect linear predictability. At a given frequency, coherence tells you what fraction of the signal power in y can be linearly predicted from x. This frequency-resolved interpretation is what makes coherence so powerful: rather than one global correlation coefficient, you get a complete picture of linear dependence across the spectrum.

Question 4

A coherence of exactly 1.0 between two measured signals at a given frequency proves that one signal is the sole linear cause of the other, with no noise or third-party influences at that frequency.

• T. True
• F. False

Answer: False

False. Coherence of 1.0 means the two signals are perfectly linearly related at that frequency — but it does not identify causality or rule out a common third source driving both. If a third unobserved variable causes both x and y at 60 Hz in a perfectly linear way, coherence will be 1.0 even though neither signal causes the other. Additionally, in practice, coherence estimates with finite data never reach exactly 1.0 due to statistical variability; the apparent ceiling depends on the number of spectral averages used.

Question 5

Why is coherence more useful for diagnosing the relationship between two signals than simply examining the magnitude of the cross-spectral density?

Model answer: The cross-spectral density magnitude |Sxy(f)| is not normalized — it depends on the absolute power of both signals. A large magnitude could reflect a strong relationship or simply high signal power in both channels. Coherence divides by the individual PSDs, normalizing the result to [0,1] regardless of signal amplitude, making it directly interpretable: coherence 0.9 at 400 Hz means 90% of signal y's power at that frequency can be linearly predicted from x, in the same way R² = 0.9 means 90% of variance is explained. Without normalization, comparing relationships across frequencies, experiments, or different sensor sensitivities is ambiguous.

This is why coherence is the standard diagnostic for frequency response function validity in structural dynamics: before computing an FRF (output/input in the frequency domain), engineers check coherence to identify frequency bands where the estimate is unreliable (low coherence), which indicates extraneous noise, nonlinearity, or signal clipping at those frequencies.