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Lagrange Frequency Response Magnitude Bound

The amplitude response of fractional delay filters based on Lagrange
interpolation is observed to be bounded by 1 when the desired delay
lies within half a sample of the midpoint of the coefficient
span [502, p. 92], as was the case in all preceeding examples
above. Moreover, *even*-order interpolators are observed to have
this boundedness property over a *two*-sample range centered on the
coefficient-span midpoint [502, §3.3.6]. These assertions are
easily proved for orders 1 and 2. For higher orders, a general proof
appears not to be known, and the conjecture is based on numerical
examples. Unfortunately, it has been observed that the gain of some
odd-order Lagrange interpolators do exceed 1 at some frequencies when
used outside of their central one-sample range [502, §3.3.6].

**Next Section:**

Even-Order Lagrange Interpolation Summary

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Avoiding Discontinuities When Changing Delay