Farrow Structure for Variable Delay FIR Filters

In §K.1.2, we noted that Lagrange interpolation is maximally flat in the frequency domain, while the Farrow filters Eq.(K.3) yielded a maximally flat error in the time domain. This section derives Farrow filters for which implement Lagrange interpolation [515]. Interestingly, the Farrow filters in this case are classic finite difference filters (low-order approximations to true differentiators).

As described in [515], solve the equations

for the FIR transfer functions , each order in general (must invert a constant Vandermonde matrix)

• Each coefficient of any Nth-order FIR interpolating filter can be expressed as an Nth-order polynomial in the fractional delay parameter (see Farrow reference in [272])
• Each polynomial coefficient depends only on filter input samples and not on very convenient to modulate the delay parameter
• When the polynomial above is evaluated using Horner's rule, the resulting filter structure is as shown in the above figure

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Order a Hardcopy of Physical Audio Signal Processing

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written by 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.