On the use of the Hs estimator for the experimental assessment of transmissibility matrices

• The concept of H2 and Hs estimators are applied to the experimental estimation of transmissibility matrices.
• It is shown theoretically that the Hs estimator tends to either H1 or H2, depending on scaling matrices.
• An indicator is proposed to assess the reliability of the estimation.
• The superiority of Hs over H1 or H2 is shown by means of numerical and experimental examples.

The experimental estimation of frequency response functions characterizing SISO linear systems is a well established topic. Several estimators are defined in the literature, each estimator being optimal depending upon the assumptions with respect to the balance of noise between the input and output of the system. H1 and H2 have to be used in case of presence of noise on output and input, respectively. The HV or Hs estimator is chosen if input and output are assumed to have equivalent SNR. These estimators are also established for MIMO linear systems, with additional difficulties due to the necessity of inversing cross spectral matrices. A transmissibility function is generally defined as a linear relationship between two outputs of a linear system. For MIMO systems, transmissibilities are not defined by the system only, it also depends on the input quantities. It is however possible to define a transmissibility matrix between two sets of outputs that is, under some assumptions, uniquely defined. This approach is especially the base of Operational Transfer Path analysis, an engineering method benefiting of a strong research effort in the last few years. This paper deals with the use of the application of MIMO system estimators to the experimental assessment of transmissibility matrices. Transmissibility matrices are generally estimated using a H1 like approach in the literature. The possibility of using H2 and Hs is presented in this work, from the theoretical point of view and with numerical and practical illustrations.