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






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),


![$\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]
$](http://www.dsprelated.com/josimages_new/pasp/img1090.png)
![$\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}}
$](http://www.dsprelated.com/josimages_new/pasp/img1091.png)
![$ {\underline{\Delta}}^T=[\Delta_0,\Delta_1,\ldots,\Delta_L]$](http://www.dsprelated.com/josimages_new/pasp/img1093.png)
where
![$\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]$](http://www.dsprelated.com/josimages_new/pasp/img1095.png)
![$\displaystyle \qquad
\mathbf{V}_{\underline{\Delta}}\isdefs \left[\begin{array}...
...ta_0}^T \\ [2pt] \vdots \\ [2pt] \underline{V}_{\Delta_L}^T\end{array}\right].
$](http://www.dsprelated.com/josimages_new/pasp/img1096.png)






Differentiator Filter Bank
Since, in the time domain, a Taylor series expansion of
about time
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*}](http://www.dsprelated.com/josimages_new/pasp/img1101.png)
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













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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Proof of Maximum Flatness at DC