Ircam - Centre Georges-Pompidou

Fractional Delay Lines using Lagrange Interpolators

Ph. Depalle, S. Tassart
ICMC 96, Hongkong,


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 fixed by the sampling rate which becomes 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), such as the Farrow structure, are a high computation cost and a lack of control over the delay which can only vary in a narrow range of values. Furthermore, there is no explicit method for shrinking or enlarging the fractional delay line.

We propose a new implementation for fractional delay lines based on the formal power series expansion of the exact z-transform. We have developed different fast and modular algorithms for fractional delay lines which make them 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 size of the fractional delay line may be maintained within reasonable limits. We have written a real-time implementation in a MAX-FTS environment. Different examples will illustrate its time-varying properties and its numerical stability.

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Thu Feb 20 16:44:40 1997
Tassart Stéphan

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