Coherence (signal analysis)

The coherence function is a measure of the degree of the linear dependence of two time signals and over the frequency. From a mathematical point of view, it is nothing other than the square of the normalized mean cross power spectrum . It is calculated according to the equation ${\ displaystyle x (t)}$ ${\ displaystyle y (t)}$

{\ displaystyle \ gamma _ {XY} ^ {2} (f) = {\ frac {| \ left \ langle G_ {XY} (f) \ right \ rangle | ^ {2}} {\ left \ langle G_ { XX} (f) \ right \ rangle \ cdot \ left \ langle G_ {YY} (f) \ right \ rangle}} \,}

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When does the coherence become identical 1?

With complete linear dependence, the square of the mean cross power spectrum is the same as the product of the mean car power spectrum or . This results in the value 1 for the coherence in the entire frequency range. The level of the two signals does not matter, since the square of the magnitude of the normalized mean cross-power spectrum is considered. If there is no dependency, the cross-power spectrum and thus also the coherence function become zero. ${\ displaystyle | \ left \ langle G_ {XY} (f) \ right \ rangle | ^ {2}}$ ${\ displaystyle \ left \ langle G_ {XX} (f) \ right \ rangle}$ ${\ displaystyle \ left \ langle G_ {YY} (f) \ right \ rangle}$

Why is the coherence identical without averaging 1?

If one looks at the above expression without averaging and without forming the amount in the numerator

{\ displaystyle {\ frac {G_ {XY} (f)} {\ sqrt {G_ {XX} (f) \ cdot G_ {YY} (f)}}},}

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so the denominator is just the amount of the numerator and we thus get a standardized cross-power spectrum . The result of the cross spectrum is a complex number. And a complex number normalized by the amount is a point on the unit circle. The position of the complex number on the unit circle reflects the phase. As mentioned at the beginning, the denominator is the absolute value of the numerator. If you now also calculate the amount from the counter, then the amount of the counter divided by the same amount is now times 1.

What happens now with the averaging? If we assume that there is no complete linear dependence, it can be assumed that the individual cross-power spectra contain different phases, i.e. point in different directions in the space of complex numbers. In the worst case - all cross-power spectra point in different directions - the individual cross-power spectra average each other away, so that in the end a complex number with a small amount or even with the amount zero comes out. The car performance spectra are by definition positive, so they cannot average to zero. Thus, the coherence only makes sense when the averaging is used.

By the way, another interesting measure is the phase synchronization . Except for the averaging, the formula is the same as that for coherence.

Why does the coherence take on the value 1 in the entire frequency range when a sine with only one frequency is analyzed?

Imagine two signals and , each generated from a sine of the frequency . If the coherence is calculated from these signals, the value 1 is obtained over the entire frequency range . This would also be expected for the frequency itself. But why is this value also obtained for the remaining frequencies? Since there is no signal component in these frequencies, the individual components of the cross-power spectrum cannot average away from one another. Therefore, the amount in the numerator is equal to the amount in the denominator is zero. The limit value analysis shows that this quotient tends towards 1. This can be illustrated by sitting on the frequency introduced above and letting the amplitude of this frequency approach zero. Regardless of how small the amplitude is, the coherence remains equal to 1 for this frequency. The coherence therefore retains the value 1 for the limit of the amplitude towards zero. ${\ displaystyle x (t)}$ ${\ displaystyle y (t)}$ ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle f}$

Interpretation of a coherence spectrum

If the coherence between an input signal and an output signal of a vibration system in the frequency range of interest is not equal to 1, this is always an indication that a system identification (analysis of the system behavior) using the linear signal analysis theory is fraught with uncertainties. ${\ displaystyle x (t)}$ ${\ displaystyle y (t)}$

Reasons for coherences deviating from 1 can be given:

Uncorrelated noise in the measurement signals and / or ${\ displaystyle x (t)}$ ${\ displaystyle y (t)}$
Influence of the output signal through other, not correlated input signals ${\ displaystyle y (t)}$ ${\ displaystyle x (t)}$
Non-linear behavior of the system
Leak effects due to insufficient frequency resolution or similar (with digital signal analysis )

If there are several signal sources (so-called "multiple" input / output problem), the normal coherence function is no longer sufficient. For these cases, two further functions have to be defined, known as partial and multiple coherence.

The partial coherence describes the linearity between one of the input signals of the system and the output signal . It is always possible to calculate them if the input signal under consideration is not fully correlated with another and if all input signals are known. ${\ displaystyle x_ {ii} (t)}$ ${\ displaystyle y (t)}$ ${\ displaystyle x_ {ii} (t)}$

Completely independent of the degree of correlation between the inputs, statements about the common linear dependency between a number of input signals and the output signal can be obtained with the help of multiple coherence. This enables a check to be made as to whether all essential input signals have been recorded (assuming linear relationships between the recorded input signals and the output signal). The ordinary coherence function can be understood as a special case of the multiple coherence function with only one input signal.

literature

RB Randall: Frequency Analysis . 3. Edition. Brüel & Kjær, 1987, ISBN 978-87-87355-07-0 .