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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 .

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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