The most commonly used closed-form methods for

delay-line
interpolation may be summarized by the following table:

In the first column, we have linear and first-order

allpass
interpolation, as discussed above in sections §

4.1.1 and
§

4.1.2, respectively. These are the least-expensive methods
computationally, and they find wide use, especially in audio
applications with large

sampling rates. While linear and first-order

allpass interpolation cost the same, only

linear interpolation offers
``random access mode''. That is, since the interpolated

signal is a
(finite)

linear combination of known samples, the signal can be
evaluated at any arbitrary time within the range of known samples --
the interpolation can ``jump around'' as desired. On the other hand,
allpass interpolation has no gain error, so it may preferred inside a

feedback loop to provide slowly varying

fractional delay filtering.
In the second column --

th-order interpolation -- we list
Lagrange for the FIR case (top) and Thiran for the IIR case (bottom),
and these are introduced in §

4.2 and §

4.3,
respectively. Lagrange and

Thiran interpolators, properly
implemented, enjoy the following advantages:

- Gain bounded by 1 at all frequencies
- Coefficients known in closed form as a function of desired delay
- Maximally flat at low frequencies:
- -
*Lagrange:* maximally flat *frequency response* at dc
- -
*Thiran:* maximally flat *group delay* at dc

In the high-order FIR case, one should also consider ``windowed sinc''
interpolation (introduced in §

4.4) as an alternative to

Lagrange interpolation. In fact, as discussed in §

4.2.16,
Lagrange interpolation is a special case of

windowed-sinc
interpolation in which a scaled binomial window is used. By choosing
different windows, optimalities other than ``maximally flat at dc''
can be achieved.
In the most general

th-order case, the interpolation-

filter impulse
response may be designed to achieve any optimality objective, such as
Chebyshev optimality (Fig.

4.11). That is, design a

digital filter
(FIR or IIR) that approximates

optimally in some sense, with coefficients tabulated over a range of

samples (and interpolated on lookup). Tabulated

filter-designs of this nature, while generally giving the best
interpolation quality, are not included in the above table because the
filter-coefficients are not known in closed form.
FIR interpolators have the advantage that they can be used in ``random
access'' mode. IIR interpolators, on the other hand, require a
sequential stream of input samples and produce a sequential stream of
interpolated signal samples (typically implementing a

fractional
delay). In IIR fractional-delay filters, the fractional delay must
change slowly relative to the IIR duration.
Finally, we note in the last column of the above table that if ``order
is no object'' (

), then the ideal bandlimited-interpolator

impulse-response is simply a sampled

sinc function, as discussed in
§

4.4.

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