Noble Identities 2.11 2002/01/04 2003/01/16 Phil Schniter schniter@ee.eng.ohio-state.edu Adan Galvan jago@rice.edu Erin Maloney emaloney@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu downsampling noble identities upsampling Introduction to the topic of noble identities.

The Noble identities (illustrated in and ) describe when it is possible to reverse the order of upsampling/downsampling and filtering. We prove the Noble identities showing the equivalence of each pair of block diagrams. The Noble identity for interpolation can be depicted as in :

For the left side of the diagram, we have Y z H z L V 1 z where V 1 z X z L Y z H z L X z L while for the right side, Y z V 2 z L where V 2 z H z X z Y z H z L X z L Thus we have established the Noble identity for interpolation. The Noble identity for decimation can be depicted as in :

For the left side of the preceding diagram, we have V 1 z 1 M k 0 M 1 X 2 M k z 1 M Y z H z V 1 z H z 1 M k 0 M 1 X 2 M k z 1 M while for the right side, Y z 1 M k 0 M 1 V z 2 M k z 1 M where V 2 z X z H z M Y z 1 M k 0 M 1 X 2 M k z 1 M H 2 M k M z M M H z 1 M k 0 M 1 X 2 M k z 1 M Thus we have established the Noble identity for decimation. Note that the impulse response of H z L is the L-upsampled impulse response of H z .