### Avoiding Discontinuities When Changing Delay

We have seen examples (e.g., Figures 4.16 and 4.18) of the general fact that every Lagrange interpolator provides an integer delay at frequency , except when the interpolator gain is zero at . This is true for any interpolator implemented as a real FIR filter, i.e., as a linear combination of signal samples using real coefficients.5.4Therefore, to avoid a relatively large discontinuity in phase delay (at high frequencies) when varying the delay over time, the requested interpolation delay should stay within a half-sample range of some fixed integer, irrespective of interpolation order. This provides that the requested delay stays within the capture zone'' of a single integer at half the sampling rate. Of course, if the delay varies by more than one sample, there is no way to avoid the high-frequency discontinuity in the phase delay using Lagrange interpolation.

Even-order Lagrange interpolators have an integer at the midpoint of their central one-sample range, so they spontaneously offer a one-sample variable delay free of high-frequency discontinuities.

Odd-order Lagrange interpolators, on the other hand, must be shifted by sample in either direction in order to be centered about an integer delay. This can result in stability problems if the interpolator is used in a feedback loop, because the interpolation gain can exceed 1 at some frequency when venturing outside the central one-sample range (see §4.2.11 below).

In summary, discontinuity-free interpolation ranges include

Wider delay ranges, and delay ranges not centered about an integer delay, will include a phase discontinuity in the delay response (as a function of delay) which is largest at frequency , as seen in Figures 4.16 and 4.18.

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