On the one hand, Thiran introduced in [6] an explicit formulation for the transfer function of the allpass filter which possesses a maximally flat delay characteristic at low frequency. This consists in optimizing an allpass filter in such a way that the derivatives of the phase response up to an order N equal those of the ideal response.
But on the other hand, it appears that the transfer function
is a rational fraction, the numerator and denominator of
which are image mirrored. This is a characteristic of allpass filter
transfer functions. Furthermore, as a Padé approximation, the
continued fraction optimizes its first derivatives. This
proves that is the transfer function of the allpass delay
filter of order N.