Boundedness of Multidimensional Filters Over a Prescribed Frequency Domain

Signal processing often requires the magnitude of a filter simultaneously upper and lower bounded on some specified frequency domains. In this paper, on the basis of the structure of the span of a multivariate matrix polynomial, a sufficient condition is derived for the boundedness of a multidimensional multiple-input multiple-output (MIMO) infinite-impulse-response (IIR) filter over a cuboid frequency domain. This condition is expressed through a linear matrix inequality (LMI) and can be proved to be less conservative than a condition derived from an upper bound of matrix structured singular value. Moreover, using parameter dependent LMIs and the linear fractional transformation structure of the filter, two necessary and sufficient conditions are also obtained for this boundedness verification problem, which are again expressed by LMIs. Furthermore, LMI based conditions are derived for filter output matrix and direct transmission matrix, which are applicable to filter design. Numerical examples are included to illustrate the efficiency and characteristics of the derived theoretical results.