On Parallelepiped-Shaped Passbands for Multidimensional Nonseparable ${\bf M}$ th-Band Low-Pass Filters

The ldquoshaperdquo 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 Mth-band filters, the passband shape should be chosen such that the Mth-band constraint is satisfied. The most commonly used shape of the passband for M-D Mth-band low-pass filters is the so-called symmetric parallelepiped (SPD) SPD(piM^-T) . In this paper, we consider the more general parallelepiped passband SPD (piL^T), and derive conditions on L such that the Mth-band constraint is satisfied. This result gives some flexibility in designing M-D Mth-band filters with parallelepiped shapes other than the commonly used case of L = M^-1. We present design examples of 2-D Mth-band filters to illustrate this flexibility in the choice of L.