# Connexions

You are here: Home » Content » Digital filter design

### Recently Viewed

This feature requires Javascript to be enabled.

# Digital filter design

Module by: Ricardo Vargas. E-mail the author

When designing digital filters for signal processing applications one is often interested in creating objects hRNhRN in order to alter some of the properties of a given vector xRMxRM (where 0<M,N<0<M,N<). Often the properties of xx that we are interested in changing lie in the frequency domain, with X=F(x)X=F(x) being the frequency domain representation of xx given by

x F X = A X e j ω φ X x F X = A X e j ω φ X
(1)

where AXAX and φXφX are the amplitude and phase components of xx, and F(·):RNRF(·):RNR is the Fourier transform operator defined by

F { h } = H ( ω ) n = 0 N - 1 h n e - j ω n ω [ - π , π ] F { h } = H ( ω ) n = 0 N - 1 h n e - j ω n ω [ - π , π ]
(2)

So the idea in filter design is to create filters hh such that the Fourier transform HH of hh posesses desirable amplitude and phase characteristics.

The filtering operator is the convolution operator (**) defined by

( x * h ) ( n ) = m x ( m ) h ( n - m ) ( x * h ) ( n ) = m x ( m ) h ( n - m )
(3)

An important property of the convolution operator is the Convolution Theorem[1] which states that

x * h F X · H = ( A X · A H ) e j ω ( φ X + φ H ) x * h F X · H = ( A X · A H ) e j ω ( φ X + φ H )
(4)

where AX,φXAX,φX and AH,φHAH,φH represent the amplitude and phase components of XX and HH respectively. It can be seen that by filteringxx with hh one can apply a scaling operator to the amplitude of xx and a biasing operator to its phase.

A common use of digital filters is to remove a certain band of frequencies from the frequency spectra of xx. Consider the lowpass filter from Figure 1; note that only the desired amplitude response is shown (not the phase response). Other types of filters include band-pass, high-pass or band-reject filters, depending on the range of frequencies that they alter.

## The notion of approximation in lplp filter design

Once a filter design concept has been selected (such as that from Figure 1), the design problem becomes finding the optimal vector hRnhRn that most closely approximates our desired frequency response concept (we will denote such optimal vector by hh). This approximation problem will heavily depend on the measure by which we evaluate all vectors hRNhRN to choose hh.

In this document we consider the discrete lplp norms defined by

a p = k | a k | p p a R N a p = k | a k | p p a R N
(5)

as measures of optimality, and consider a number of filter design problems based upon this criterion. The work explores the Iterative Reweighted Least Squares (IRLS) approach as a design tool, and provides a number of algorithms based on this method. Finally, this work considers critical theoretical aspects and evaluates the numerical properties of the proposed algorithms in comparison to existing general purpose methods commonly used. It is the belief of the author (as well as the author's advisor) that the IRLS approach offers a more tailored route to the lplp filter design problems considered, and that it contributes an example of a made-for-purpose algorithm best suited to the characteristics of lplp filter design.

## References

1. Ziemer, Rodger E. and Tranter, William H. and Fannin, D. Ronald. (1998). Signals and Systems: Continuous and Discrete. (Fourth). Prentice Hall.

## 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.

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