Application of multivariate uncertainty analysis to frequency response function estimates

Uncertainty estimation is an important part of any measurement but is often neglected for complex valued or multivariate data (e.g., vectors). This paper presents a methodology for estimating the uncertainty in multivariate experimental data and applies it to the measurement of the frequency response function obtained when using a periodic random input signal. This multivariate uncertainty method is an extension of classical uncertainty methods used for scalar variables and tracks the correlation between all variates along with the sample variance instead of just tracking the standard uncertainty. The method is used in this paper to propagate the sample covariance matrix from spectral density estimates to the uncertainty in the frequency response function estimate for two different system models. In the first model the case when only the output signal is corrupted by noise is considered, while in the second model both the input and output signals are corrupted by uncorrelated noise sources. The results for the single-noise model are verified by comparing them to published expressions in the literature, while the results for the two-noise model are verified by using a direct computation of the statistics. Finally, the method is applied to experimental data from two microphone measurements within an acoustic waveguide. The random uncertainty estimates in the frequency response function from the multivariate method agree well with the results from a direct computation of the statistics.