Fractional Delays using Lagrange Interpolators
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 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.