A global error estimator for the uncertainty of a multi-channel spectral analysis

• An estimator of the uncertainty of the average frequency response function in multi-channel spectral analysis is proposed.
• Experimental validations are given.
• It is demonstrated that the analysis record length should at least be equal or longer than the reverberation time.
• For a given signal recording, a compromise needs to be found between the record length and the number of averages.
• Using the global estimator the optimal record length can be determined.

A global estimator of the uncertainty of the average frequency response function in multi-channel spectral analysis measurements is proposed. The proposed global estimator is a generalization of the random error estimator of the frequency response function magnitude of a single-input–single-output system. In principle, the signal-to-noise ratio (and thus the quality of the frequency response function estimation) is increasing with increasing number of averages M  , according to M. However, in the situation that, for practical reasons, there is a maximum imposed upon the total measurement time TmaxTmax, it is clear that there is a trade-off between the number of averages M and the record length T (s) that is used to obtain an estimate of a single-average-frequency-response-function. There is a choice between a few long records or many short records, with the requirement that, assuming zero overlap, the number of averages M times the record length T   may not exceed the total available measurement time, i.e. M×T⩽TmaxM×T⩽Tmax. In addition to the existence of such an optimum, a minimum record length is required as well which is related to the reverberation time of the system. The newly proposed global estimator is used to determine the optimal record length of a multi-channel system, such that a minimum error of the average frequency response function is obtained. It is also shown by experimental results that indeed the minimum allowable record length is related to the reverberation time of the system being measured.