### 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 :

 (5.9)

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
 (5.10)

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),

we can express each FIR coefficient as a vector expression:

Making a row-vector out of the FIR coefficients gives

or

We may now choose a set of parameter values over which an optimum approximation is desired, yielding the matrix equation

 (5.11)

where

and

Equation (4.11) may be solved for the polynomial-coefficient matrix by usual least-squares methods. For example, in the unweighted case, with , we have

Note that this formulation is valid for finding the Farrow coefficients of any th-order variable FIR filter parametrized by a single variable . Lagrange interpolation is a special case corresponding to a particular choice of .

#### 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

 (5.12)

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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Recent Developments in Lagrange Interpolation
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