Polyphase Interpolators with Reversed Order of Up-Sampling and Down-Sampling

In its simplest form, a rational ratio interpolator performs a cascade of 1-to-P up-sampling, a digital filter at the higher sample rate, and a Q-to-1 down-sampling to raise the input sample rate by the ratio P/Q. If P and Q are relatively prime we can reverse the order of the operations and perform a cascade of Q-to-1 down-sampling, the same digital filter in an altered configuration at the reduced sample rate, and a 1-to-P up-sampling. Contrary to first impression, the input down sampling does not alias the input series. The transformation between the two options is performed by a sequence of elementary resampling steps including the Noble identity and polyphase partitions including the Noble identity and polyphase partitions.