Comments on “Multi-dimensional filter banks and wavelets-A system theoretic perspective”

In [S. Basu, Multi-dimensional filter banks and wavelets-a system theoretic perspective, J. Franklin Inst. 335B(8) (1998) 1367-1409], the problems of constructing and parameterizing multi-dimensional (MD) filter banks (FB) and wavelets were studied extensively, especially in the situation of two-dimensional (2D) and quincunx-downsampled systems. Finding the other filter complementary to a given 2D filter for quincunx sampling and linear phase perfect reconstruction (LPPR) system is based on the fact that the construction is possible if and only if the polyphase components of the given filter do not share any common zero. The construction is elaborated in another book chapter [S. Basu, Multidimensional signals, circuits and systems, in: K. Galkowski, J. Wood (Eds.), On the Structure of Linear Phase Perfect Reconstruction Quincunx Filter Banks, Taylor & Francis, London, 2001, pp. 193-208 (Chapter 10)] by the same author. This approach further relies on the fact that the resultant of the polyphase components is monomial. In this comment, we show that both above mentioned facts are not necessarily true for the quincunx FB as well as for general MD FB.