Multidimensional Multirate Filters and Filter Banks Derived from One-Dimensional Filters

Abstract

We present a method by which every multidimensional (MD) filter with an arbitrary parallelepiped-shaped passband support can be designed and implemented efficiently. We show that all such filters can be designed starting from an appropriate one-dimensional prototype filter and performing a simple transformation. With D denoting the number of dimensions, we hence reduce the complexity of design as well as implementation of the MD filter from 0{N�) to 0(N). Furthermore, by using the polyphase technique, we can obtain an implementation with complexity of only 2N in the two-dimensional special case. With our method, the Nyquist constraint and zero-phase requirement can be satisfied easily. In the HR case, stability of the designed filters is also easily achieved. Even though the designed filters are in general nonseparable, these filters have separable polyphase components. One special application of this method is in MD multirate signal processing, where filters with parallelepiped-shaped passbands are used in decimation, interpolation, and filter banks. Some generalizations and other applications of this approach, including MD uniform DFT quadrature mirror filter banks which achieve perfect reconstruction, are studied. Several design examples are also given.