Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.

Audio Effects Based on Biorthogonal Time-Varying Frequency Warping

Abstract: We illustrate the mathematical background and musical use of a class of audio effects based on frequency warping These effects alter the frequency content of a signal via spectral mapping They can be implemented in dispersive tapped delay lines based on a chain of all-pass filters In a homogeneous line with first-order all-pass sections, the signal formed by the output samples at a given time is related to the input via the Laguerre transform However, most musical signals require a time-varying frequency modification in order to be properly processed Vibrato in musical instruments or voice intonation in the case of vocal sounds may be modeled as small and slow pitch variations Simulation of these effects requires techniques for time-varying pitch and/or brightness modification that are very useful for sound processing The basis for time-varying frequency warping is a time-varying version of the Laguerre transformation The corresponding implementation structure is obtained as a dispersive tapped delay line, where each of the frequency dependent delay element has its own phase response Thus, time-varying warping results in a space-varying, inhomogeneous, propagation structure We show that time-varying frequency warping is associated to an expansion over biorthogonal sets generalizing the discrete Laguerre basis Slow time-varying characteristics lead to slowly varying parameter sequences The corresponding sound transformation does not suffer from discontinuities typical of delay lines based on unit delays

Families of Biorthogonal Rational Functions in a Discrete Variable

Abstract: By using Toscano’s [Boll. Un. Mat. Ital. (3), 4 (1949), pp. 398–409] finite difference representation for generalized hypergeometric functions, two families of biorthogonal rational functions in an integer-valued variable are obtained, one a ${}_3 F_2 $ series and the other a balanced ${}_4 F_3 $ series.

Numerical Algorithms Based on Biorthogonal Wavelets

Abstract: Wavelet bases are used to generate spaces of approximation for the resolution of bidimensional elliptic and parabolic problems. Under some specific hypotheses relating the properties of the wavelets to the order of the involved operators, it is shown that an approximate solution can be built. This approximation is then stable and converges towards the exact solution. It is designed such that fast algorithms involving biorthogonal multi resolution analyses can be used to resolve the corresponding numerical problems. Detailed algorithms are provided as well as the results of numerical tests on partial differential equations defined on the bidimensional torus.

Two-channel multifilter banks and multiwavelets

Abstract: Systematic design methods for multifilters and multiwavelets are presented. Explicit formulae are given for aliasing cancellation in biorthogonal multifilter banks. The relationship with vector transforms over the real or finite fields is described.