The Noble identities (illustrated in Figure 1 and Figure 2) 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 Figure 1:

For the left side of the diagram, we have Yz=HzL V 1 z Y z H z L V 1 z where V 1 z=XzL V 1 z X z L Yz=HzLXzL Y z H z L X z L while for the right side, Yz= V 2 zL Y z V 2 z L where V 2 z=HzXz V 2 z H z X z Yz=HzLXzL 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 Figure 2:

For the left side of the preceding diagram, we have V 1 z=1M∑k=0M-1Xⅇ-ⅈ2πMkz1M V 1 z 1 M k 0 M 1 X 2 M k z 1 M

while for the right side, where V 2 z=XzHzM V 2 z X z H z M Thus we have established the Noble identity for decimation. Note that the impulse response of HzL H z L is the LL-upsampled impulse response of Hz H z .