A basic single-coefficient allpass section is described here. It is a basic block of the polyphase recursive IIR filter structures described in this book. The commonly used structures of such a filter are shown in Figure A-1:

Figure A-1: Equivalent structures of one-coefficient allpass filter.

Structures (a) and (d) require less space of the integrated circuit than the other two ones due to less number of mathematical operations required. One multiplier, two adders and a small memory is all they need. Structure (d) is very useful for cascading allpass sections. The z ^?N delayers can be shared by successive allpass sections.

The transfer function of this basic building block is given by:

The basic allpass section impulse response:

Figure A-2: Pole-zero plots of the second-order (N=2) allpass filter.

The phase response:

The group delay response:

The step response obtained by convolution in time domain:

Total energy:

Impulse response centre of gravity:

Average time delay:

Figure A-3: Phase shape of the basic allpass filter (N=2) for different coefficient values of ?.

Figure A-4: Phase response of the basic allpass filter ( ?=0.5) for different filter orders of N.

Figure A-5: Group delay of the basic allpass filter (N=2) for different coefficient values of ?.

Figure A-6: Group delay of the basic allpass filter ( ?=0.5) for different filter orders of N.