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Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Document Description: Butter worth Filter Design for Electronics and Communication Engineering (ECE) 2022 is part of Signals and Systems preparation. The notes and questions for Butter worth Filter Design have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about Butter worth Filter Design covers topics like and Butter worth Filter Design Example, for Electronics and Communication Engineering (ECE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Butter worth Filter Design.

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Butter worth Filter Design

The butterworth LP filter of order N is defined as HB(s) HB(-s) = Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Where s = jΩc

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)OR Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

It has 2N poles

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Ex: for N=3

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) = 1200, 1800, 2400, 3000, 3600, 600

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Poles that are let half plane are belongs to desired system function.

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

For a large Ω, magnitude response decreases as Ω -N, indicating the LP nature of this filter.

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

As Ω → ∞

= -20 N log10Ω

= -20 N dB/ Decade = -6 N dB/Octane

As N increases, the magnitude response approaches that of ideal LP filter.
The value of N is determined by Pass & stop band specifications.

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Ex: Design Butterworth LPF for the following specifications.

Pass band:

-1< H ( jΩ) 2 dB ≤ 0 for 0≤ Ω ≤ 1404π ( W π = 1404π )   

Stop band:

H ( jW)2 dB < -60  for W ≥ 8268π   ( Ωs = 8268π )

If the Ωc is given

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Since Ωc is not given, a guess must be made.

The specifications call for a drop of -59dB, In the frequency range from the edge of the pass band (1404π ) to the edge of stop band (8268π ). The frequency difference is equal to 

log2(8268/1404)=2.56 octaves.

1 oct ---- - 6N dB

2.56 ------ ?

=> 2.56 X - 6N dB = -59 dB’s

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Ω2N > 106 Ωc 2 N

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Ω c <1470.3π

Let Ω c =1470.3π

At this Ω c it should satisfy pass band specifications.

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

= 0.59

This result is below the pass band specifications. Hence N=4 is not sufficient.

Let N=5

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

In the pass band  Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Since N=5

Ω c = 2076π

S1 = -2076π

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

1. Magnitude response is smooth, and decreases monotonically as Ω increases from 0 to ∞

2. the magnitude response is maximally flat about Ω =0, in that all its derivatives up to order N are equal to zero at Ω =0

Ex: Ωc=1, N=1

HB( jΩ)2= (1+ Ω 2)-1

The first derivative

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The second derivative

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

3. The phase response curve approaches  Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) for large Ω , where N is the no. of poles of butterworth circle in the left side of s-plane.

Advantages:

1. easiest to design

2. used because of smoothness of magnitude response .

Disadvantage: 

Relatively large transition range between the pass band and stop band.

Other procedure

When Ω c = 1      Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

If n is even S2N = 1 = e j ( 2k -1)π

The 2N roots will be Sk= Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) k=1,2,….2N

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Therefore: 

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

If N is odd

S2n =1 = e j 2kπ

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

where θ k = Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

choosing this value for n, results in two different selections for Ω. If we wish to satisfy our requirement at Ω1 exactly and do better than our req. at Ω2 , we use 

Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  orButter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)for better req at Ω2

The document Butter worth Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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