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Variable Filter Parametrizations

In practical applications of Lagrange Fractional-Delay Filtering (LFDF), it is typically necessary to compute the FIR interpolation coefficients $ h_\Delta(n)$ as a function of the desired delay $ \Delta$, 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 $ \Delta(t)$.

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 $ h_\Delta(n)$ is tabulated as a function of delay $ \Delta = N/2+\eta$, $ \eta\in[-1/2,1/2)$, $ n=0,1,2,\ldots,N$, where $ N$ is the interpolation-filter order. For each $ n$, $ \Delta$ may be sampled sufficiently densely so that linear interpolation will give a sufficiently accurate ``interpolated table look-up'' of $ h_\Delta(n)$ for each $ n$ and (continuous) $ \Delta$. 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 $ h_\Delta(n)$ as a polynomial in the desired delay $ \Delta$:

$\displaystyle h_\Delta(n) \isdefs \sum_{m=0}^M c_n(m)\Delta^m, \quad n=0,1,2,\ldots,N \protect$ (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

$\displaystyle h_\Delta(n)$ $\displaystyle \isdef$ $\displaystyle \sum_{m=0}^M c_n(m)\Delta^m, \quad n=0,1,2,\ldots,N$  
$\displaystyle \Longleftrightarrow \quad H_\Delta(z)$ $\displaystyle \isdef$ $\displaystyle \sum_{n=0}^N h_\Delta(n)z^{-n}$  
  $\displaystyle =$ $\displaystyle \sum_{n=0}^N \left[\sum_{m=0}^M c_n(m)\Delta^m\right]z^{-n}$  
  $\displaystyle =$ $\displaystyle \sum_{m=0}^M \left[\sum_{n=0}^N c_n(m) z^{-n}\right]\Delta^m$  
  $\displaystyle \isdef$ $\displaystyle \sum_{m=0}^M C_m(z) \Delta^m \protect$ (5.10)

Since $ H_\Delta(z)$ is an $ N$th-order FIR filter, at least one of the $ C_m(z)$ must be $ N$th order, so that we need $ M\ge N$. A typical choice is $ M=N$.

Such a parametrization of a variable filter as a polynomial in fixed filters $ C_m(z)$ 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].

Figure 4.19: Farrow structure for implementing parametrized filters as a fixed-filter polynomial in the varying parameter.
\includegraphics[width=\twidth]{eps/farrow}

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

Farrow Structure Coefficients

Beginning with a restatement of Eq.$ \,$(4.9),

$\displaystyle h_\Delta(n) \isdefs \sum_{m=0}^M c_n(m)\Delta^m, \quad n=0,1,2,\ldots,N, $

we can express each FIR coefficient $ h_\Delta(n)$ as a vector expression:

$\displaystyle h_\Delta(n) \eqsp \underbrace{% \left[\begin{array}{ccccc} 1 & \... ...y}{c} C_n(0) \\ [2pt] C_n(1) \\ [2pt] \vdots \\ [2pt] C_n(M)\end{array}\right] $

Making a row-vector out of the FIR coefficients gives

$\displaystyle \underbrace{\left[\begin{array}{cccc}h_\Delta(0)\!&\!h_\Delta(1)\... ...\vdots \\ C_0(M) & C_1(M) & \cdots & C_N(M) \end{array}\right]}_{\mathbf{C}} $

or

$\displaystyle \underline{h}_\Delta \eqsp \underline{V}_\Delta^T \mathbf{C}. \protect$

We may now choose a set of parameter values $ {\underline{\Delta}}^T=[\Delta_0,\Delta_1,\ldots,\Delta_L]$ over which an optimum approximation is desired, yielding the matrix equation

$\displaystyle \mathbf{H}_{\underline{\Delta}}\eqsp \mathbf{V}_{\underline{\Delta}}\mathbf{C}, \protect$ (5.11)

where

$\displaystyle \mathbf{H}_{\underline{\Delta}}\isdefs \left[\begin{array}{c} \un... ...elta_0}^T \\ [2pt] \vdots \\ [2pt] \underline{h}_{\Delta_L}^T\end{array}\right]$   and$\displaystyle \qquad \mathbf{V}_{\underline{\Delta}}\isdefs \left[\begin{array}... ...ta_0}^T \\ [2pt] \vdots \\ [2pt] \underline{V}_{\Delta_L}^T\end{array}\right]. $

Equation (4.11) may be solved for the polynomial-coefficient matrix $ \mathbf{C}$ by usual least-squares methods. For example, in the unweighted case, with $ L\ge M$, we have

$\displaystyle \zbox {\mathbf{C}\eqsp \left(\mathbf{V}_{\underline{\Delta}}^T\ma... ...ight)^{-1} \mathbf{V}_{\underline{\Delta}}^T \mathbf{H}_{\underline{\Delta}}.} $

Note that this formulation is valid for finding the Farrow coefficients of any $ N$th-order variable FIR filter parametrized by a single variable $ \Delta$. Lagrange interpolation is a special case corresponding to a particular choice of $ \mathbf{H}_{\underline{\Delta}}$.

Differentiator Filter Bank

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

\begin{eqnarray*} x(n-\Delta) &=& x(n) -\Delta\, x^\prime(n) + \frac{\Delta^2... ...D^2(z) + \cdots + \frac{(-\Delta)^k}{k!}D^k(z) + \cdots \right] \end{eqnarray*}

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 [184, 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)$.5.6 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 ( $ \Delta\leftarrow t$).

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Recent Developments in Lagrange Interpolation
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Proof of Maximum Flatness at DC