# Connexions

You are here: Home » Content » DSPA » Filter Design for Multirate Systems

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Janko Calic. E-mail the author

# Filter Design for Multirate Systems

Module by: Douglas L. Jones. E-mail the author

The filter design techniques learned earlier can be applied to the design of filters in multirate systems, with a few twists.

## Example 1

Design a factor-of-LL interpolator for use in a CD player, we might wish that the out-of-band error be below the least significant bit, or 96dB down, and < 0.05 % < 0.05 % error in the passband, so these specifications could be used for optimal L L filter design.

In a CD player, the sampling rate is 44.1kHz, corresponding to a Nyquist frequency of 22.05kHz, but the sampled signal is bandlimited to 20kHz. This leaves a small transition band, from 20kHz to 24.1kHz. However, note that in any case where the signal spectrum is zero over some band, this introduces other zero bands in the scaled, replicated spectrum (Figure 1).

So we need only control the filter response in the stopbands over the frequency regions with nonzero energy. (Figure 2) The extra "don't care" bands allow a given set of specifications to be satisfied with a shorter-length filter.

## Direct polyphase filter design

Note that in an integer-factor interpolator, each set of output samples x 1 Ln+p x 1 L n p , p=01L1 p 0 1 L 1 , is created by a different polyphase filter g p n g p n , which has no interaction with the other polyphase filters except in that they each interpolate the same signal. We can thus treat the design of each polyphase filter independently, as an NL N L -length filter design problem. (Figure 3)

Each g p n g p n produces samples x 1 Ln+p= x 0 n+pL x 1 L n p x 0 n p L , where n+pL n p L is not an integer. That is, g p n g p n is to produce an output signal (at a T 0 T 0 rate) that is x 0 n x 0 n time-advanced by a non-integer advance pL p L .

The desired response of this polyphase filter is thus H D p ω=ejωpL H D p ω ω p L for |ω|π ω , an all-pass filter with a linear, non-integer, phase. Each polyphase filter can be designed independently to approximate this response according to any of the design criteria developed so far.

### Exercise 1

What should the polyphase filter for p=0 p 0 be?

#### Solution

A delta function: h 0 n=δ n h 0 n δ n

### Example 2: Least-squares Polyphase Filter Design

• Deterministic x(n): Minimize n=|xn x d n|2 n x n x d n 2 Given xn=xn*hn x n x n h n and x d n=xn* h d n x d n x n h d n . Using Parseval's theorem, this becomes
minn=|xn x d n|2=min12πππ|XωHωXω H d ω|2dω=min12πππ|Hω H d ω||Xω|2dω n x n x d n 2 1 2 ω X ω H ω X ω H d ω 2 1 2 ω H ω H d ω X ω 2
(1)
This is simply weighted least squares design, with |Xω|2 X ω 2 as the weighting function.
• stochastic X(ω):
minE|xn x d n|2=E|xn*hn h d n|2=min12πππ| H d ωHω|2 S x x ωdω x n x d n 2 x n h n h d n 2 1 2 ω H d ω H ω 2 S x x ω
(2)
S x x ω S x x ω is the power spectral density of xx. S x x ω=DTFT r x x k S x x ω DTFT r x x k r x x k=Exk+lxl* r x x k x k l x l Again, a weighted least squares filter design problem.

#### Problem 1

Is it feasible to use IIR polyphase filters?

##### Solution

The recursive feedback of previous outputs means that portions of each IIR polyphase filter must be computed for every input sample; this usually makes IIR filters more expensive than FIR implementations.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

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

##### What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

##### What are tags?

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

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

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

##### What is in a lens?

Lens makers point to 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 member, a community, or a respected organization.

##### What are tags?

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