Applications of the Complex Modulation Transfer Function on SEA-systems
Abstract: The SEA coupling and dissipation loss factors are determined in-situ from Complex Modulation Transfer Functions, CMTF:s, based on measured impulse-responses. A quotient of CMTF:s is least-square curve-fitted to a SEA model and the SEA loss factors are determined from the results of the curve-fit. A two subsystem SEA model with time-varying power is considered. In the frequency domain, power-energy transfer functions can be formed. In a power-energy transfer function in the model with two poles and one zero, the zero and one pole are almost equal, if the coupling is weak. When trying to determine the poles and the zero from the CMTF curve one then runs into problems since the zero is so close to the pole. The situation with one pole is easier to handle. Therefore instead a quotient of two SEA power-energy transfer functions with the same input power was taken. The result is a model with one pole leading to a robust evaluation of the loss factors. The result is the quotient of the two subsystem energies where the input power does not enter and thus need not be known. The virtual boundary condition using this new transfer function model is given by putting the denominator energy equal to zero, meaning that the corresponding subsystem is energy- earthed. Even for SEA models with more than two subsystems, a model with one pole can be derived. In the physical system the power can be time-varied by letting the system excitation signal consist of random noise modulated with a deterministic time-varying function. However, since the ensemble average of the squared response is proportional to the squared impulse-response convolved with the squared modulating function, random excitation is avoided and replaced by impulse-response measurements. The Fourier transform of the low-pass filtered squared impulse- response is the CMTF. A CMTF curve for a short distance between source and observation position does not show simple low-pass character, but has a "floor". A model for the modulation transfer function could therefore consist of a modulation frequency-independent part diminishing with source- distance and originating from the direct field and a low-pass character source-distance independent part originating from the reverberant field. At a certain distance from the source the magnitudes of the two parts are equal. The distance is here called the modulation direct field radius in analogy with the ordinary direct field radius. The modulation direct field radius is monotonically increasing with the modulation frequency. Thus the distance affected by the direct field is larger for the modulation than for the non-modulated, stationary, case. Experiments were carried out on two plates connected at a point by a spring and in two rooms divided by a wall.
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