MMSE design of interpolation and downsampling FIR filters in the context of periodic nonuniform sampling

The generalized sampling theorem states that any analog signal whose spectrum is limited to 1/T can be exactly recovered from N sequences of samples taken at a rate 2/NT and all having a different sampling phase. When N=2, the exact interpolation formula can be derived quite easily. The ideal interpolation filters have infinite impulse responses (IIRs). This paper addresses first the question of recovering from the two initial sequences any other sequence taken at the same rate 1/T and with a different sampling phase. The design problem is dealt with for finite length filters, and the criterion is the minimization of the mean squared interpolation error. Next, the problem of computing from the two initial sequences a third one at a lower rate is addressed. FIR decimation filters are also designed for an MMSE criterion. These problems are illustrated for two typical covariance functions