| Résumé |
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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