### Variable Filter Parametrizations

In practical applications of Lagrange Fractional-Delay Filtering
(LFDF), it is typically necessary to compute the FIR interpolation
coefficients
as a function of the desired delay
, which is usually time varying. Thus, LFDF is a special case
of FIR *variable filtering* in which the FIR coefficients must be
time-varying functions of a single delay parameter .

#### Table Look-Up

A general approach to variable filtering is to tabulate the filter coefficients as a function of the desired variables. In the case of fractional delay filters, the impulse response is tabulated as a function of delay , , , where is the interpolation-filter order. For each , may be sampled sufficiently densely so that linear interpolation will give a sufficiently accurate ``interpolated table look-up'' of for each and (continuous) . This method is commonly used in closely related problem of sampling-rate conversion [462].

#### Polynomials in the Delay

A more parametric approach is to formulate each filter coefficient
as a *polynomial* in the desired delay :

Taking the

*z*transform of this expression leads to the interesting and useful

*Farrow structure*for variable FIR filters [134].

#### Farrow Structure

Taking the *z* transform of Eq.(4.9) yields

Since is an th-order FIR filter, at least one of the must be th order, so that we need . A typical choice is .

Such a parametrization of a variable filter as a polynomial in
*fixed* filters is called a *Farrow structure*
[134,502]. When the polynomial Eq.(4.10) is
evaluated using *Horner's rule*,^{5.5} the efficient structure of
Fig.4.19 is obtained. Derivations of Farrow-structure
coefficients for Lagrange fractional-delay filtering are introduced in
[502, §3.3.7].

As we will see in the next section, Lagrange interpolation can be implemented exactly by the Farrow structure when . For , approximations that do not satisfy the exact interpolation property can be computed [148].

#### Farrow Structure Coefficients

Beginning with a restatement of Eq.(4.9),

where

#### 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 (
).

**Next Section:**

Recent Developments in Lagrange Interpolation

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Proof of Maximum Flatness at DC