Skip to content Skip to navigation

Connexions

You are here: Home » Content » Interpolation, Decimation, and Rate Changing by Integer Fractions

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Interpolation, Decimation, and Rate Changing by Integer Fractions

Module by: Douglas L. Jones. 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)

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: by an integer factor L

Interpolation means increasing the sampling rate, or filling in in-between samples. Equivalent to sampling a bandlimited analog signal LL times faster. For the ideal interpolator,

X 1 ω= X 0 Lωif|ω|<πL0ifπL≤|ω|≤π X 1 ω X 0 L ω ω L 0 L ω (1)
We wish to accomplish this digitally. Consider Equation 2 and Figure 1.
ym= X 0 mLifm=0±L±2L…0otherwise y m X 0 m L m 0 ± L ± 2 L … 0 (2)
Figure 1
Figure 1 (imag002.png)
The DTFT of ym y m is
Yω=∑m=-ω∞ymⅇ-ⅈωm=∑n=-∞∞ x 0 nⅇ-ⅈωLn=∑n=-∞∞xnⅇ-ⅈωLn= X 0 ωL Y ω m ω y m ω m n x 0 n ω L n n x n ω L n X 0 ω L (3)
Since X 0 ω ′ X 0 ω ′ is periodic with a period of 2π 2 , X 0 Lω=Yω X 0 L ω Y ω is periodic with a period of 2πL 2 L (see Figure 2).
Figure 2
Figure 2 (imag003.png)
By inserting zero samples between the samples of x 0 n x 0 n , we obtain a signal with a scaled frequency response that simply replicates X 0 ω ′ X 0 ω ′ LL times over a 2π 2 interval!

Obviously, the desired x 1 m x 1 m can be obtained simply by lowpass filtering ym y m to remove the replicas.

x 1 m=ym* h L m x 1 m y m h L m (4)
Given H L m=1if|ω|<πL0ifπL≤|ω|≤π H L m 1 ω L 0 L ω In practice, a finite-length lowpass filter is designed using any of the methods studied so far (Figure 3).
Figure 3
Interpolator Block Diagram
Interpolator Block Diagram (imag004.png)

Decimation: sampling rate reduction (by an integer factor M)

Let ym= x 0 Lm y m x 0 L m (Figure 4)

Figure 4
Figure 4 (imag005.png)
That is, keep only every LLth sample (Figure 5)
Figure 5
Figure 5 (imag006.png)
In frequency (DTFT):
Yω=∑m=-∞∞ymⅇ-ⅈωm=∑m=-∞∞ x 0 Mmⅇ-ⅈωm=∑n=-∞∞ x 0 n∑k=-∞∞δn−Mkⅇ-ⅈωnM|n=Mm=∑n=-∞∞ x 0 n∑k=-∞∞δn−Mkⅇ-ⅈ ω ′ n| ω ′ =ωM=DTFT x 0 n*DTFT∑δn−Mk Y ω m y m ω m m x 0 M m ω m n M m n x 0 n k δ n M k ω n M ω ′ ω M n x 0 n k δ n M k ω ′ n DTFT x 0 n DTFT δ n M k (5)
Now DTFT∑δn−Mk=2π∑k=0M−1Xkδ ω ′ −2πkM DTFT δ n M k 2 k 0 M 1 X k δ ω ′ 2 k M for |ω|<π ω as shown in homework #1 , where Xk X k is the DFT of one period of the periodic sequence. In this case, Xk=1 X k 1 for k∈01…M−1 k 0 1 … M 1 and DTFT∑δn−Mk=2π∑k=0M−1δ ω ′ −2πkM DTFT δ n M k 2 k 0 M 1 δ ω ′ 2 k M .
DTFT x 0 n*DTFT∑δn−Mk= X 0 ω ′ *2π∑k=0M−1δ ω ′ −2πkM=12π∫-ππ X 0 μ ′ 2π∑k=0M−1δ ω ′ − μ ′ −2πkMd μ ′ =∑k=0M−1 X 0 ω ′ −2πkM DTFT x 0 n DTFT δ n M k X 0 ω ′ 2 k 0 M 1 δ ω ′ 2 k M 1 2 μ ′ X 0 μ ′ 2 k 0 M 1 δ ω ′ μ ′ 2 k M k 0 M 1 X 0 ω ′ 2 k M (6)
so Yω=∑k=0M−1 X 0 ωM−2πkM Y ω k 0 M 1 X 0 ω M 2 k M i.e., we get digital aliasing.(Figure 6)
Figure 6
Figure 6 (imag007.png)
Usually, we prefer not to have aliasing, so the downsampler is preceded by a lowpass filter to remove all frequency components above |ω|<πM ω M (Figure 7).
Figure 7
Figure 7 (imag008.png)

Rate-Changing by a Rational Fraction L/M

This is easily accomplished by interpolating by a factor of LL, then decimating by a factor of MM (Figure 8).

Figure 8
Figure 8 (imag009.png)
The two lowpass filters can be combined into one LP filter with the lower cutoff, Hω=1if|ω|<πmax{LM}0ifπmax{LM}≤|ω|≤π H ω 1 ω L M 0 L M ω Obviously, the computational complexity and simplicity of implementation will depend on LM L M : 2/3 23 will be easier to implement than 1061/1060 10611060!

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