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Differentiator Filter Bank

Since, in the time domain, a Taylor series expansion of $ x(n-\Delta)$ about time $ n$ gives

&=& x(n) -\Delta\, x^\prime(n)
+ \frac{\Delta^2}{2!} x^{\prime\prime}(n)
+ \cdots
+ \frac{(-\Delta)^k}{k!}x^{(k)}(n)
+ \cdots \nonumber \\
X(z)\left[1 - \Delta\, D(z) + \frac{\Delta^2}{2!} D^2(z) + \cdots
+ \frac{(-\Delta)^k}{k!}D^k(z) + \cdots \right]

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

$\displaystyle C_m(z) \eqsp \frac{(-1)^m}{m!}D^m(z), \protect$ (5.12)

in the limit, as the number of terms $ M$ goes to infinity. In other terms, the coefficient $ C_m(z)$ of $ \Delta^m$ in the polynomial expansion Eq.$ \,$ (4.10) must become proportional to the $ m$ th-order differentiator as the polynomial order increases. For any finite $ N$ , we expect $ C_m(z)$ to be close to some scaling of the $ m$ th-order differentiator. Choosing $ C_m(z)$ as in Eq.$ \,$ (4.12) for finite $ N$ gives a truncated Taylor series approximation of the ideal delay operator in the time domain [185, p. 1748]. Such an approximation is ``maximally smooth'' in the time domain, in the sense that the first $ N$ derivatives of the interpolation error are zero at $ x(n)$ .5.7 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 ( $ \Delta\leftarrow t$ ).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University