Coherence (signal processing)

The coherence function is a measure of the degree of linear dependence between two time signals and the frequency. It is mathematically nothing else than the normalized mean square value of the cross-power spectrum. It is calculated according to the equation

When will the coherence identical one?

With full linear dependence the magnitude square of the average cross power spectrum is equal to the product of the average or auto power spectra. This results in consistency in the entire frequency range of the value 1, the level of the two signals is irrelevant, since the absolute square of the normalized mean cross power spectrum is considered. No dependency exists, then the cross-power spectrum and therefore the coherence function to zero.

Why is the coherence without averaging identical one?

Considering the above expression without averaging and without education amount in the numerator

So the denominator is just the sum of the numerator and thus we obtain a normalized cross-power spectrum. The result of the cross-spectrum is a complex number. And a normalized by the amount of a complex number is a point on the unit circle. The position of the complex number on the unit circle reflects the phase. As mentioned above, the denominator is just the sum of the counter. If we now also from the counter the amount, then the amount of the counter divided by the same amount is now time 1

What happens in the averaging? Suppose that there is no complete linear dependence exists, then it must be assumed that the individual cross-power spectra contain different phases, ie show in the space of complex numbers in different directions. In the worst case - all cross power spectra show in different directions - forward, the individual cross-power spectra each other away, so that at the end of a complex number comes out with a small amount or even the amount is zero. The auto power spectra are positive by definition so that it can not aufmitteln zero. Thus, the coherence renders only through the application of the averaging sense.

Another interesting measure is the way, the phase synchronization. The formula to the average is equal to the coherence.

Why the coherence across the frequency range takes the value 1 if a sinus is analyzed with only one frequency?

Consider two signals and which are generated in each case from a sine wave of the frequency. If one calculates from these signals, the coherence, we obtain over the entire frequency range the value 1 for the frequency itself, one would expect the same. But why do we obtain this value for the remaining frequencies? Since no signal component is present in these frequencies, the individual units of the cross power spectrum can not path means mutually exclusive. Therefore, the amount in the numerator is equal to the sum in the denominator equal to zero. The limit analysis shows that this ratio tends to 1. One can understand clearly, by sitting on the introduced above frequency and the amplitude of this frequency can be close to zero. No matter how small the amplitude, the consistency for this frequency is equal to 1 Thus, the coherence remains 1 for the limit of the amplitude to zero.

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 equal to 1, so this is always an indication that a system identification (analysis of system behavior ) is afflicted by means of the linear signal analysis theory with uncertainty.

The reasons for deviating from 1 coherences can be mentioned:

• Influencing the output signal by others, not with correlated input signals
• A non-linear behavior of the system
• Leakage effects due to low frequency resolution, etc. ( in digital signal analysis)

If multiple sources are present (so-called " multiples" Eingangs-/Ausgangsproblem ), the approach of the ordinary coherence function is no longer sufficient. For these cases, two functions must be defined, which are known under the names and multiple partial coherence.

The partial coherence describes the linearity between the input signals of the system and the output signal. Their calculation is always possible, when the input signal is considered not fully correlated with one another and if all input signals are known.

Totally independent of the degree of correlation between the inputs may be obtained by means of the multiple coherence statements on the common linear relationship between a plurality of input signals and the output signal. This allows to check whether all the essential input signals have been detected ( linear relationships between the detected input signals and the output signal required). The ordinary coherence function can be considered as a special case of multiple coherence function with a single input signal.