# Connexions

You are here: Home » Content » Practical Filters

### Recently Viewed

This feature requires Javascript to be enabled.

# Practical Filters

Module by: Richard Baraniuk. E-mail the author

Summary: Describes how real-world filters behave.

An ideal filter simply removes all unwanted frequencies, preserving the remaining frequencies exactly. This would resemble some sort of a finite rectangle function in frequency. However, a simple, finite rectangle function in frequency is an infinite sinc function in time. This is a problem. A sinc function is an infinite length signal in both the positive and negative directions, making it impossible to create in the real-world. This leads us to as what would happen if we just made this sinc function causal by "chopping it off" somewhere. What we find when we do this is that the frequency domain representation is no longer a perfect rectangle: it now does not fall off immediately and shows some wiggling where it was flat before.

## The Bands

An ideal filter has two types of bands: the stop band defines the region of frequencies that are eliminated by the filter, while the pass band defines the region of frequencies that the filter allows through. Practical filters add one more, the transition band. This is the area where the filter is moving between the stop band and pass band.

## Filter Design Specifications

In our look at filter design specifications, we will use the example of a lowpass filter. The extension to the other kinds of filters should be fairly straightforward. Figure 2 shows the parameters for a lowpass filter in the frequency domain.

From this image, the passband is the region from ω p ω p to ω p ω p , the transition bands are the regions from ω s ω s to ω p ω p and from ω p ω p to ω s ω s while the stop band is the region less than ω s ω s or greater than ω s ω s .

In the figure above, e p e p and e s e s represent the acceptable tolerance (or error) around the desired level that the passband and stopband respectively may vary. The behavior within the transition band is not specified, allowing anything to occur there, as long as the width is within specifications.

### Example 1

This example will look at a moving average system.

• It is lowpass
• It has linear phase with jumps of π radians when the sinc function changes sign
• The duration of the filter is inversely proportional to its bandwidth
• This filter is finite impulse response (FIR)
• It cannot be built with passive R, L, C circuits
• We do not have independent control over all four design specifications

We are going to design a moving average filter with the following design specs: ω p =100π ω p 100 , e s =0.1 e s 0.1 , e p =0.1 e p 0.1

With this specification, we are allowing ω s ω s to be a dependant variable (since we need one). We can now find the equation for this moving average system.

We begin with

|Hiω|=1.1sinωT2ωT2 H ω 1.1 ω T 2 ω T 2
(1)
We will now solve for TT with
|Hi100π|=1.1sin50πT50πT=0.9 H 100 1.1 50 T 50 T 0.9
(2)
For these specs, T0.007 T 0.007 . This means that |Hiω| H ω does not stay below e s =0.1 e s 0.1 until ω s 771π ω s 771 .

It is very clear from this representation that the transition band is huge ( 671π 671 ). This is a very bad filter, especially when you consider that it cannot be implemented with passive circuitry. Fortunately better filters (e.g. Butterworth, Chebyshev and Elliptical) do exist.

## Beyond Lowpass Filter Design

In the discussion of the different filters (Butterworth, Chebyshev and Elliptical) is common to see explanations based on lowpass filters. This explanation is very nice when first learning about them, because it is sufficient to understand the fundamentals of each of them. It is acceptable, because there exist fairly straightforward techniques to convert these lowpass filters into highpass, bandpass or bandstop filters. These techniques are the lowpass to highpass transformation, lowpass to bandpass transformation and lowpass to bandstop transformation.

## 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