What is the Relationship between FRF and Coherence?
The Frequency Response Function (FRF) describes how a system responds to an input at each frequency, while coherence quantifies how reliable that measured relationship is.
In short, FRF tells you “what the system does,” and coherence tells you “how much you can trust it.”
FRF (Frequency Response Function)
The FRF is defined as
- X(f) input spectrum
- Y(f) output spectrum
It measures and reveals
Magnitude-squared Coherence Function
Magnitude-squared coherence is defined as
where,
- Gxx Power Spectral Density (one-sided)
- Gxy Cross Power Spectral Density (one-sided)
It measures and indicates
Relationship Between FRF and Coherence
Magnitude-squared coherence Indicates FRF Reliability
- γxy2(f) ≈ 1 → FRF is reliable
- γxy2(f) ≪ 1 → FRF is unreliable
- High magnitude-squared coherence means the FRF accurately represents the system
Effect of Noise
- Output noise: contaminate Y(f), reduce magnitude-squared coherence
- Input Noise: contaminate X(f), biases FRF estimation
- Noise lowers magnitude-squared coherence value and degrades FRF accuarcy
Nonlinear Effects
- FRF assumes a linear system
- Nonlinearity introduces additional frequencies
- Magnitude-squared coherence decreases
- FRF no longer fully describes the system
Insufficient Averaging
- Random variations remain
- Cross-spectrum estimate unstable
- Lower magnitude-squared coherence value
- Noisy FRF
x(t) and y(t) signals for the analysis of FRF and coherence
Relationship between magnitude-squared coherence and FRF
Physical meaning at 100Hz
Around 100 Hz, the FRF exhibits a resonance peak accompanied by a rapid phase transition. This indicates the system’s natural resonance frequency, where energy is efficiently transferred and the vibration response becomes highly amplified.
The slight drop in coherence near 100 Hz suggests increased sensitivity to noise or nonlinear effects around the resonance region.
Key Takeaways
High magnitude-squared coherence ( ≈ 1 ) → FRF is trustworthy
- strong linear relationship
- low noise
- sufficient averaging
Low magnitude-squared coherence → FRF should be treated with caution
- noise present
- nonlinear behavior
- poor measurement conditions
Key Insight
The FRF describes the system behavior, while coherence indicates how accurately that behavior has been measured. Always interpret FRF together with coherence to ensure meaningful and trustworthy results.
Conclusion
The FRF and coherence are fundamentally linked in frequency-domain analysis.
- FRF provides the system response
- Magnitude-squared coherence evaluates its reliability
Suggested Further Reading
##You may also find these topics helpful:
What is the Relationship between FRF and Coherence?
The Frequency Response Function (FRF) describes how a system responds to an input at each frequency, while coherence quantifies how reliable that measured relationship is.
In short, FRF tells you “what the system does,” and coherence tells you “how much you can trust it.”
FRF (Frequency Response Function)
The FRF is defined as
It measures and reveals
resonance frequency, damping, stiffness / mass effects
Magnitude-squared Coherence Function
Magnitude-squared coherence is defined as
where,
It measures and indicates
indicates the quality and reliability of FRF measurements
Relationship Between FRF and Coherence
Magnitude-squared coherence Indicates FRF Reliability
Effect of Noise
Nonlinear Effects
Insufficient Averaging
x(t) and y(t) signals for the analysis of FRF and coherence
Relationship between magnitude-squared coherence and FRF
Physical meaning at 100Hz
Around 100 Hz, the FRF exhibits a resonance peak accompanied by a rapid phase transition. This indicates the system’s natural resonance frequency, where energy is efficiently transferred and the vibration response becomes highly amplified.
The slight drop in coherence near 100 Hz suggests increased sensitivity to noise or nonlinear effects around the resonance region.
Key Takeaways
High magnitude-squared coherence ( ≈ 1 ) → FRF is trustworthy
Low magnitude-squared coherence → FRF should be treated with caution
Key Insight
The FRF describes the system behavior, while coherence indicates how accurately that behavior has been measured. Always interpret FRF together with coherence to ensure meaningful and trustworthy results.
Conclusion
The FRF and coherence are fundamentally linked in frequency-domain analysis.
Suggested Further Reading
##You may also find these topics helpful: