Coherence is used to check the correlation between the output spectrum and the input spectrum. So you can estimate the power transfer between input and output of a linear system. It shows how well the input and output are related to each other.
Autospectrum is a function commonly explored both in signal and system analysis. It is computed from the instantaneous (Fourier) spectrum as:
Image 74: Autospectrum
There is a new, fundamental function - cross-spectrum - in the dual-channel processing. It is computed from the instantaneous spectra of both channels. All other functions are computed during post-processing from the cross-spectrum and the two auto spectrums - all functions are the functions of frequency.
Based on complex instantaneous spectrum A(f) and B(f), the cross-spectrum SAB (from A to B) is defined as:
Image 75: Cross-spectrum
The amplitude of the cross-spectrum SAB is the product of amplitudes, its phase is the difference between both phases (from A to B). Cross spectrum SBA (from B to A) has the same amplitude, but opposite phase. The phase of the cross-spectrum is the phase of the system as well.
Both auto spectra and cross-spectrum can be defined either as two-sided (notation SAA, SBB, SAB, SBA) or as one-sided (notation GAA, GBB, GAB, GBA). One-sided spectrum is obtained from the two-sided one as:
Image 76: Auto and Cross-spectrum
The cross-spectrum itself has little importance, but it is used to compute other functions. Its amplitude |GAB| indicates the extent to which the two signals correlate as the function of frequency and phase angle <GAB indicates the phase shift between the two signals as the function of frequency. The advantage of the cross-spectrum is that influence of noise can be reduced by averaging. That is because the phase angle of the noise spectrum takes random values so that the sum of those several random spectra tends to zero. It can be seen that the measured auto spectrum is a sum of the true auto spectrum and auto spectrum of noise, whilst the measured cross-spectrum is equal to the true cross-spectrum.
Image 77: Cross-spectrum
Coherence function Î³ indicates the degree of a linear relationship between two signals as a function of frequency. It is defined by two auto spectra (GAA, GBB) and a cross-spectrum (GAB) as:
At each frequency, coherence can be taken as a correlation coefficient (squared) which expresses the degree of the linear relationship between two variables, where the magnitudes of auto spectra correspond to variances of those two variables and the magnitude of cross-spectrum corresponds to covariance.
The coherence value varies from zero to one. Zero means no relationship between the input A and output B, whilst one means a perfectly linear relationship.
There are four possible relationships between input A and output B:
Perfectly linear relationship
A sufficiently linear relationship with a slight scatters caused by noise
Image 79: Sufficiently linear
Image 80: Non-linear
Image 81: No relationship
Low values indicate a weak relation (when the excitation spectrum has gaps at certain frequencies), values close to 1 show a representative measurement.
That means when the transfer function shows a peak, but the coherence is low (red circles in the picture below), it must not necessarily be a real resonance. Maybe the measurement has to be repeated (with different hammer tips), or you can additionally look for the MIF parameter.
Coherence is a Vector channel and therefore displayed with a 2D graph instrument.
The coherence is calculated separately for each point (e.g. Coherence_3Z/1Z, Coherence_4Z/1Z).
Image 82: Coherence displayed on a 2D graph in Dewesoft
In the case of no averaging, coherence is always equal to 1. In the case of averaging and samples, GAB influenced by noise, deviations in the phase angles cause that the resulting magnitude |GAB| is lower than it would be without the presence of noise (see the picture below). The presence of non-linearity has a similar influence.
Image 83: Averaging with and without noise