### Lagrange Interpolation Optimality

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

*Butterworth filters*in classical analog filter design [343,449]. It can also be formulated in terms of ``Pade approximation'' [373,374]. To summarize, the basic idea of maximally flat filter design is to match exactly as many leading terms as possible in the Taylor series expansion of the desired frequency response. Equivalently, we zero the maximum number of leading terms in the Taylor expansion of the frequency-response error.

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 [358].

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Explicit Lagrange Coefficient Formulas

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Fractional Delay Filters