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) 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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.