Formulation of multirate filter design as an approximation problem

We consider the design of filters intended for multirate systems. Applications include sample-rate changers, narrowband filtering and beamforming. Conventional frequency-domain techniques are appropriate for linear time-invariant systems, for which the frequency domain is the eigenvector domain. Multirate systems are not time-invariant, but exhibit block time-invariance, as observed in the literature. As a consequence, a multirate system is appropriately described by a matrix-valued frequency response, just as a linear time invariant system is described by a scalar-valued frequency response. We formulate multirate filter design as approximation of a desired matrix-valued frequency response, after motivating it as a logical extension of classical frequency-domain design of equiripple filters