Lagrange Interpolation Optimality

As derived in §4.2.14, Lagrange fractional-delay filters are
*maximally flat* in the frequency domain at dc. That is,

where is the interpolation error expressed in the frequency domain:

where and are defined in §4.2.2 above. This is the same optimality criterion used for the power response of (recursive)

Figure 4.11 compares Lagrange and optimal Chebyshev fractional-delay
filter frequency responses. Optimality in the *Chebyshev
sense* means minimizing the worst-case
error over a given frequency band (in this case,
). While Chebyshev optimality is often the most desirable
choice, we do not have closed-form formulas for such solutions, so they
must be laboriously pre-calculated, tabulated, and interpolated to
produce variable-delay filtering [361].

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