Differentiator Filter Bank

Since, in the time domain, a Taylor series expansion of about time gives

where denotes the transfer function of the ideal differentiator, we see that the th filter in Eq.(4.10) should approach

in the limit, as the number of terms goes to infinity. In other terms, the coefficient of in the polynomial expansion Eq.(4.10) must become proportional to the th-order differentiator as the polynomial order increases. For any finite , we expect to be close to some scaling of the th-order differentiator. Choosing as in Eq.(4.12) for finite gives a truncated Taylor series approximation of the ideal delay operator in the time domain [184, p. 1748]. Such an approximation is ``maximally smooth'' in the time domain, in the sense that the first derivatives of the interpolation error are zero at .^5.6 The approximation error in the time domain can be said to be maximally flat.

Farrow structures such as Fig.4.19 may be used to implement any one-parameter filter variation in terms of several constant filters. The same basic idea of polynomial expansion has been applied also to time-varying filters ( ).

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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.