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.

• Reprint

• Matlab code:

  flatdelay.m

  mfaps.m

  mfaps2.m

  mfaps3.m

  nrst.m

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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