On Parallelepiped-Shaped Passbands for Multidimensional Nonseparable Mth-Band Low-Pass Filters

Abstract

The “shape” of the desired frequency passband is an important consideration in the design of nonseparable multidimensional ($M$ -D) filters in $M$-D multirate systems. For $M$-D ${bf M}$th-band filters, the passband shape should be chosen such that the ${bf M}$th-band constraint is satisfied. The most commonly used shape of the passband for $M$-D ${bf M}$ th-band low-pass filters is the so-called symmetric parallelepiped (SPD) ${rm SPD}(pi {bf M}^{- {rm T}})$ . In this paper, we consider the more general parallelepiped passband ${rm SPD}(pi {bf L} ^{rm T})$, and derive conditions on $ {bf L} $ such that the ${bf M}$ th-band constraint is satisfied. This result gives some flexibility in designing $M$-D ${bf M}$th-band filters with parallelepiped shapes other than the commonly used case of $ {bf L} = {bf M}^{- 1}$. We present design examples of 2-D ${bf M}$th-band filters to illustrate this flexibility in the choice of $ {bf L} $.