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
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  crossspectrum  in the dualchannel processing. It is computed from the instantaneous spectra of both channels. All other functions are computed during postprocessing from the crossspectrum and the two auto spectrums  all functions are the functions of frequency.
Cross spectrum
Based on complex instantaneous spectrum A(f) and B(f), the crossspectrum SAB (from A to B) is defined as:
Image 75: Crossspectrum


The amplitude of the crossspectrum 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 crossspectrum is the phase of the system as well.
Both auto spectra and crossspectrum can be defined either as twosided (notation SAA, SBB, SAB, SBA) or as onesided (notation GAA, GBB, GAB, GBA). Onesided spectrum is obtained from the twosided one as:
Image 76: Auto and Crossspectrum
The crossspectrum 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 crossspectrum 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 crossspectrum is equal to the true crossspectrum.
Image 77: Crossspectrum
Coherence
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 crossspectrum (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 crossspectrum 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 
Image78: Linear
 Image 79: Sufficiently linear

Nonlinear relationship  No relationship 
Image 80: Nonlinear  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 nonlinearity has a similar influence.
Image 83: Averaging with and without noise