Skip to content Skip to navigation

Connexions

You are here: Home » Content » Interpolation

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Interpolation

Module by: Phil Schniter. E-mail the author

User rating (How does the rating system work?)
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)

Summary: Introduction of interpolation and it's application.

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.

Interpolation

Interpolation is the process of upsampling and filtering a signal to increase its effective sampling rate. To be more specific, say that xm x m is an (unaliased) TT-sampled version of x c t x c t and vn v n is an LL-upsampled version version of xm x m . If we filter vn v n with an ideal πLL-bandwidth lowpass filter (with DC gain LL) to obtain yn y n , then yn y n will be a TL TL -sampled version of x c t x c t . This process is illustrated in Figure 1.

Figure 1
Figure 1 (m10444fig1.png)

We justify our claims about interpolation using frequency-domain arguments. From the sampling theorem, we know that TT- sampling x c t x c t to create xn x n yields

Xⅇⅈω=1T∑k X c ⅈω−2πkT X ω 1 T k k X c ω 2 k T (1)
After upsampling by factor LL, Equation 1 implies Vⅇⅈω=1T∑k X c ⅈωL−2πkT=1T∑k X c ⅈω−2πLkTL V ω 1 T k k X c ω L 2 k T 1 T k k X c ω 2 L k T L Lowpass filtering with cutoff πL L and gain LL yields Yⅇⅈω=LT∑kL∈ℤ X c ⅈω−2πLkTL=LT∑l X c ⅈω−2πlTL Y ω L T kL k L X c ω 2 L k T L L T l l X c ω 2 l T L since the spectral copies with indices other than k=lL klL (for l∈ℤ l) are removed. Clearly, this process yields a TL TL-shaped version of x c t x c t . Figure 2 illustrates these frequency-domain arguments for L=2L2.

Figure 2
Figure 2 (m10444fig2.png)

Content actions

Give Feedback:

E-mail the module author | Rate 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)

Download:

Module PDF

Add module to:

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 (?)

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