Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method | EURASIP Journal on Applied Signal Processing 2004 7 949-963 2004 Hindawi Publishing Corporation Multirate Simulations of String Vibrations Including Nonlinear Fret-String Interactions Using the Functional Transformation Method L. Trautmann Multimedia Communications and Signal Processing University of Erlangen-Nuremberg Cauerstrasse 7 91058 Erlangen Germany Email traut@ Laboratory of Acoustics and Audio Signal Processing Helsinki University of Technology P O. Box 3000 02015 Espoo Finland Email R. Rabenstein Multimedia Communications and Signal Processing University of Erlangen-Nuremberg Cauerstrasse 7 91058 Erlangen Germany Email rabe@ Received 30 June 2003 Revised 14 November 2003 The functional transformation method FTM is a well-established mathematical method for accurate simulations of multidimensional physical systems from various fields of science including optics heat and mass transfer electrical engineering and acoustics. This paper applies the FTM to real-time simulations of transversal vibrating strings. First a physical model of a transversal vibrating lossy and dispersive string is derived. Afterwards this model is solved with the FTM for two cases the ideally linearly vibrating string and the string interacting nonlinearly with the frets. It is shown that accurate and stable simulations can be achieved with the discretization of the continuous solution at audio rate. Both simulations can also be performed with a multirate approach with only minor degradations of the simulation accuracy but with preservation of stability. This saves almost 80 of the computational cost for the simulation of a six-string guitar and therefore it is in the range of the computational cost for digital waveguide simulations. Keywords and phrases multidimensional system vibrating string partial differential equation functional transformation nonlinear multirate approach. 1. INTRODUCTION Digital sound synthesis methods can mainly be .
