Frequency response function estimation techniques and the corresponding coherence functions: A review and update

For a structural dynamics engineer assessing a system's vibration characteristics, the frequency response function (FRF) is indispensable, irrespective of whether the setup being tested is experimental, numerical or analytical. This paper outlines the different FRF estimation techniques that have been developed over the years. Algorithms that compute an ordinary least squares (OLS) estimate of the FRF, assuming uncorrelated measurement noise to be present either on the output (H-1) or the input (H-2) signals, have been compared with those that employ total least squares (TLS) equations by making use of an augmented input-output auto and cross power (G(FFX)/G(XFF)) matrix at every frequency such as the H-upsilon (using eigenvalue decomposition) and H-SVD (using singular value decomposition) algorithms. Another FRF estimation method based upon Cholesky decomposition (H-CD) is also discussed. Further discussion has been included of TLS algorithms computing the FRFs for one output at a time, as historically presented in the development of the H-upsilon algorithm, with a case that evaluates the FRF matrix computed for all the outputs simultaneously. The development of the corresponding coherence functions has been presented, highlighting the method dependent paradigms that led to concepts such as virtual coherence and partial coherence while underscoring their equivalence with ordinary and multiple coherence calculations. It has been shown that some of the existing conditioned coherence metrics are inconsistent from an input-output standpoint, for which the corrected interpretations have been subsequently described. This article is unique in that no previous work summarizes all of the previously developed FRF estimation and coherence algorithms and no previous paper explains the inconsistencies in the conditioned coherence functions with respect to multiple coherence.