Skip to content Skip to navigation

Connexions

You are here: Home » Content » Noble Identities

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of a lens

      Lenses

      A lens is a custom view of Connexions content. You can think of it as a fancy kind of list that will let you see Connexions through the eyes of organizations and people you trust.

      What is in a lens?

      Lens makers point to Connexions materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

      Who can create a lens?

      Any individual Connexions member, a community, or a respected organization.

      What are tags? tag icon

      Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

    • External bookmarks
  • E-mail the author
  • Rate this module (How does the rating system work?)

    Rating system

    Ratings

    Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

    How to rate a module

    Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

    		(0 ratings)

Recently Viewed

This feature requires Javascript to be enabled.

Noble Identities

Module by: Phil Schniter

Summary: Introduction to the topic of noble identities.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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 .

Comments, questions, feedback, criticisms?

Send feedback