<< Chapter < Page Chapter >> Page >

Chebyshev filter properties

The Butterworth filter does not give a sufficiently good approximation across the complete passband in many cases. TheTaylor's series approximation is often not suited to the way specifications are given for filters. An alternate error measure isthe maximum of the absolute value of the difference between the actual filter response and the ideal. This is considered over thetotal passband. This is the Chebyshev error measure and was defined and applied to the FIR filter design problem. For the IIR filter,the Chebyshev error is minimized over the passband and a Taylor's series approximation at ω = is used to determine the stopband performance. This mixture of methods in the IIR case iscalled the Chebyshev filter, and simple design formulas result, just as for the Butterworth filter.

The design of Chebyshev filters is particularly interesting, because the results of a very elegant theory insure thatconstructing a frequency-response function with the proper form of equal ripple in the error will result in a minimum Chebyshev errorwithout explicitly minimizing anything. This allows a straightforward set of design formulas to be derived which can beviewed as a generalization of the Butterworth formulas [link] , [link] .

The form for the magnitude squared of the frequency-response function for the Chebyshev filter is

| F ( j ω ) | 2 = 1 1 + ϵ 2 C N ( ω ) 2

where C N ( ω ) is an Nth-order Chebyshev polynomial and ϵ is a parameter that controls the ripple size. This polynomial in ω has very special characteristics that result in the optimality of the response function [link] .

Chebyshev polynomials

The Chebyshev polynomial is a powerful function in approximation theory. Although the function is a polynomial, it isbest defined and developed in terms of trigonometric functions by [link] , [link] , [link] , [link] .

C N ( ω ) = cos ( N cos - 1 ( ω ) )

where C N ( ω ) is an Nth-order, real-valued function of the real variable ω . The development is made clearer by introducing an intermediate complex variable φ .

C N ( ω ) = cos ( N φ )

where

ω = cos ( φ )

Although this definition of C N ( ω ) may not at first appear to result in a polynomial, the following recursive relation derivedfrom [link] shows that it is a polynomial.

C N + 1 ( ω ) = 2 ω C N ( ω ) - C N - 1 ( ω )

From [link] , it is clear that C 0 = 1 and C 1 = ω , and from [link] , it follows that

C 2 = 2 ω 2 - 1
C 3 = 4 ω 3 - 3 ω
C 4 = 8 ω 4 - 8 ω 2 + 1

etc.

Other relations useful for developing these polynomials are

C N 2 ( ω ) = ( C 2 N ( ω ) + 1 ) / 2
C M N ( ω ) = C M ( C N ( ω ) )

where M and N are coprime.

These are remarkable functions [link] . They oscillate between +1 and -1 for - 1 < ω < 1 and go monotonically to +/- infinity outside that domain. All N of their zeros are real and fall in the domain of - 1 < ω < 1 , i.e., C N is an equal ripple approximation to zero over the range of ω from -1 to +1. In addition, the values for ω where C N reaches its local maxima and minima and is zero are easily calculated from [link] and [link] . For - 1 < ω < 1 , a plot of C N ( ω ) can be made using the concept of Lissajous figures. Example plots for C 0 , C 1 , C 2 , C 3 , and C 4 are shown in [link] .

Figure one is a graph titled Nth order chebyshev polynomials. Its horizontal axis is labeled Frequency, ω, and ranges in value from -2 to 2 in increments of 0.5. The vertical axis is labeled C_N(ω) and ranges in value from -2 to 2 in increments of 0.5. There are four curves in this figure. The first is a diagonal line with constant positive slope that passes through the origin. The second is parabolic in shape with  its vertex as a minimum of the curve at (0, -1). The third starts from the bottom-left of the graph, increases to a peak at (-0.5, 1) and then decreases to a trough at (0.5, -1), where it finally increases to the top-right area of the graph. The fourth begins in the top-left as a decreasing function, and proceeds to make a trough, a peak, and a trough at (-0.5, -1), (0, 1) and (0.5, -1) respectively. The curve then increases and ends in the top-right area.
Chebyshev Polynomials for N = 0, 1, 2, 3, and 4

The filter frequency-response function for N = 5 is given in [link] showing the passband ripple in terms of the parameter ϵ .

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing and digital filter design (draft)' conversation and receive update notifications?

Ask