I. W. Selesnick,
Lowpass Filters Realizable as Allpass Sums: Design via a New Flat Delay Filter,
*IEEE Trans. on Circuits and Systems Part II*,
46(1):40-50, January 1999.

*Abstract:*
This paper describes a new class of maximally flat lowpass recursive
digital filters. The filters are realizable as a parallel sum of
two allpass filters, a structure for which low-complexity low-noise
implementations exist.
Note that, with the classical Butterworth filter of degree N, which
is retrieved as a special case, it is
not possible to adjust the delay (or phase-linearity). However,
with the more general class of filters described in this paper, the
adjustment of the delay becomes possible, and the trade-off
between the delay and the phase-linearity can be chosen.
The construction of these lowpass
filters depends upon a new maximally flat
delay allpole filter, for which
the degrees of flatness at w=0 and w=pi
are not necessarily equal.
For the coefficients of this flat delay filter,
an explicit solution is introduced, which also
specializes to a previously known result.

Lowpass filters realized as a parallel sum of allpass fitlers have several interesing properties.

- This structure has its origins in classical analog lattice structures.
- Digital filters that are obtained from the classical analog
(Butterworth, Chebyshev, and elliptic) prototypes via the bilinear
transformation, can be realized as allpass sums.
- The complementary filter can be realized without additional filtering.
- Approximately linear phase IIR filters are obtained when
one of the allpass branches is a pure delay.
- Parallel sums of allpass filters have recently arisen in a
variety of applications, one example being orthogonal and
perfect reconstruction filter banks.
That means they are useful for the construction of wavelets.
- For allpass filters, there are low-complexity low-noise implementations.

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