Ircam - Centre Georges-Pompidou

Fractional delays using Lagrange Interpolators

Équipe Analyse-Synthèse
IRCAM, Institut de Recherche et de Coordination Acoustique-Musique
1 place Igor-Stravinsky, 75004 PARIS, FRANCE,


Many studies have been undertaken on the modeling of physical systems by means of waveguide filters. These methods consist mainly in simulating the propagation of acoustic waves with digital delay lines. These models are constrained to have a spatial step determined by the sampling rate which is a serious drawback when a high spatial resolution in the geometry of the model is needed or when the length of the waveguide needs to vary. One can use digital filters for approximating the exact fractional delay, but length variations usually induce audible distortions because of local instabilities or modification of the filter's structure.

Lagrange Interpolation theory leads to FIR filters which approximate fractional delays according to a maximally flat error criterion. Major drawbacks of current implementations of Lagrange Interpolator Filters (LIF) are a high computation cost and a lack of control over the delay which can only vary in a narrow range of values.

We propose a new implementation of LIF based on a formal power series expansion of the exact z-transform. We have developed different fast and modular algorithms for LIF which make the LIF usable for real-time delay-varying applications. Modularity in the structure is a key point here as it enables one to switch between filters of different order while preserving the continuity of the z-transform. Thus the delay may vary over an unlimited range of values. Furthermore, any arbitrary integer part of the fractional delay can be simulated by a classical delay line so that the actual order of the LIF may be maintained within reasonable limits. This paper will focus on the time-varying properties of our implementation and its numerical stability over a wide range of delays.

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Thu Feb 20 16:44:40 1997
Tassart Stéphan

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