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Noble Identities

Module by: Phil Schniter

Summary: Introduction to the topic of noble identities.

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:

Figure 1
Figure 1 (m10432fig1.png)

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:

Figure 2
Figure 2 (m10432fig2.png)

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

Yz=Hz V 1 z=Hz1M∑k=0M-1Xⅇ-ⅈ2πMkz1M Y z H z V 1 z H z 1 M k 0 M 1 X 2 M k z 1 M (1)
while for the right side,
Yz=1M∑k=0M-1 V z ⅇ-ⅈ2πMkz1M Y z 1 M k 0 M 1 V z 2 M k z 1 M (2)
where V 2 z=XzHzM V 2 z X z H z M
Yz=1M∑k=0M-1Xⅇ-ⅈ2πMkz1MHⅇ-ⅈ2πMkMzMM=Hz1M∑k=0M-1Xⅇ-ⅈ2πMkz1M 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 (3)
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 .

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