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Even-Order Lagrange Interpolation Summary

We may summarize some characteristics of even-order Lagrange fractional delay filtering as follows:
  • Two-sample bounded-by-1 delay-range instead of only one-sample
  • No gain zero at half the sampling rate for the middle delay
  • No phase-delay discontinuity when crossing the middle delay
  • Optimal (central) delay range is centered about an integer
To stay within the central, one-sample delay range for even-order Lagrange interpolators, the delay range should be

$\displaystyle \Delta\in\left(\frac{N}{2}-\frac{1}{2},\frac{N}{2}+\frac{1}{2}\right),

where $ N$ is the order of the interpolator.
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Lagrange Frequency Response Magnitude Bound