Design of doubly-complementary IIR digital filters using a single complex allpass filter, with multirate applications

It is shown that a large class of real-coefficient doubly-complementary IIR transfer function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence corresponds to the output of one of the transfer functions
G(z)
(for example, lowpass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter
H(z)
(for example, highpass). The resulting implementation is structurally lossless, and hence the implementations of
G(z)
and
H(z)
have very low passband sensitivity. Numerical design examples are included, and a typical numerical example shows that the new implementation with 4 bits per multiplier is considerably better than a direct form implementation with 9 bits per multiplier. Multirate filter bank applications (quadrature mirror filtering) are outlined.