Skip to content Skip to navigation

Connexions

You are here: Home » Content » Efficient Multirate Filter Structures

Navigation

Recently Viewed

This feature requires Javascript to be enabled.

Efficient Multirate Filter Structures

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.

Rate-changing appears expensive computationally, since for both decimation and interpolation the lowpass filter is implemented at the higher rate. However, this is not necessary.

Interpolation

For the interpolator, most of the samples in the upsampled signal are zero, and thus require no computation. (Figure 1)

Figure 1
Figure 1 (imag001.png)
For m=L⌊mL⌋+mmodL m L m L m L and p=mmodL p m L ,
x 1 m=∑m= N 1 N 2 h L p mym=∑k= N 1 L N 2 L g p k x 0 ⌊mL⌋−k x 1 m m N 1 N 2 h L p m y m k N 1 L N 2 L g p k x 0 m L k (1)
g p n=hLn+p g p n h L n p Pictorially, this can be represented as in Figure 2.
Figure 2
Figure 2 (imag002.png)
These are called polyphase structures, and the g p n g p n are called polyphase filters.

Computational cost

If hm h m is a length-NN filter:

  • No simplification: N T 1 =LN T 0 computationssec N T 1 L N T 0 computations sec
  • Polyphase structure: LLN1 T 0 o computationssec=N T 0 L L N 1 T 0 o computations sec N T 0 where LL is the number of filters, NL N L is the taps/filter, and 1 T 0 1 T 0 is the rate.
Thus we save a factor of LL by not being dumb.

Note:

For a given precision, NN is proportional to LL, (why?), so the computational cost does increase with the interpolation rate.

Question:

Can similar computational savings be obtained with IIR structures?

Efficient Decimation Structures

We only want every MMth output, so we compute only the outputs of interest. (Figure 3) x 1 m=∑k= N 1 N 2 x 0 Lm−khk x 1 m k N 1 N 2 x 0 L m k h k

Figure 3
Polyphase Decimation Structure
Polyphase Decimation Structure (imag003.png)
The decimation structures are flow-graph reversals of the interpolation structure. Although direct implementation of the full filter for every MMth sample is obvious and straightforward, these polyphase structures give some idea as to how one might evenly partition the computation over MM cycles.

Efficient L/M rate changers

Interpolate by LL and decimate by MM (Figure 4).

Figure 4
Figure 4 (imag004.png)
Combine the lowpass filters (Figure 5).
Figure 5
Figure 5 (imag005.png)
We can couple the lowpass filter either to the interpolator or the decimator to implement it efficiently (Figure 6).
Figure 6
Figure 6 (imag006.png)
Of course we only compute the polyphase filter output selected by the decimator.

Computational Cost

Every T 1 =ML T 0 seconds T 1 M L T 0 seconds , compute one polyphase filter of length NL N L , or NL T 1 =NLML T 0 =NM T 0 multipliessecond N L T 1 N L M L T 0 N M T 0 multiplies second However, note that NN is proportional to max{LM} L M .

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