Coherence is the measure of how much of a response is due to the (known) excitation. This is easily measured when the excitation is broadband and it is possible to average multiple captures of excitation and response. It is specifically calculated by dividing the product of the cross power spectrum and its conjugate by the product of the excitation and response power spectra, usually expressed as:

You can see that at any frequency, any energy in the response that is not causally related to the excitation will result in a coherence value less than 1 for that frequency. This is therefore a very useful measure of the quality of a transfer function measurement that uses the ensemble averaging of cross power and auto power spectra.

The problem with sine wave excitation – exciting one frequency at a time – is that any energy in the response not derived from the excitation is not separable from the excitation, thus making coherence measurements impossible. But there is a way, if you measure each frequency several times and arrange the phase of the excitation to be different for each measurement. The cross power spectrum average will then retain only the causally related part of the response allowing the coherence to be measured. Typically, this technique allows the measurement to discriminate against power frequency noise. The technique also avoids errors when measuring a transfer function at a resonant frequency of a system that is responding to an excitation which is already present in the environment.

All SignalCalc analyzers make use of this technique when performing transfer function measurements with stepped sine excitation.