Butter worth Filter Design | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

Butter worth Filter Design

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

Where s = jΩc

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

It has 2N poles

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

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

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

Ex: for N=3

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

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

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

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

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

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

Ω2N > 106 Ωc 2 N

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

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

Since N=5

Ω c = 2076π

S1 = -2076π

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

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

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

The second derivative

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

3. The phase response curve approaches  Butter worth Filter Design | 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 | Signals and Systems - Electronics and Communication Engineering (ECE)

Butter worth Filter Design | 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 | Signals and Systems - Electronics and Communication Engineering (ECE) k=1,2,….2N

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

Therefore: 

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

If N is odd

S2n =1 = e j 2kπ

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

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

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

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

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

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

The document Butter worth Filter Design | 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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FAQs on Butter worth Filter Design - Signals and Systems - Electronics and Communication Engineering (ECE)

1. What is a Butterworth filter and how does it work?
Ans. A Butterworth filter is a type of electronic filter designed to have a flat frequency response in the passband. It is also referred to as a maximally flat magnitude filter. The filter achieves this by attenuating the signal in the stopband while minimizing the ripple in the passband. The Butterworth filter is based on a polynomial function called the Butterworth polynomial, which determines the shape and characteristics of the filter's response.
2. What are the key advantages of using a Butterworth filter?
Ans. There are several advantages of using a Butterworth filter. Firstly, it provides a flat frequency response in the passband, ensuring minimal distortion of the desired signal. Secondly, it has a maximally flat magnitude response, resulting in a smooth transition between the passband and stopband. Additionally, Butterworth filters are easy to design and implement, making them widely used in various applications. Lastly, they offer a good trade-off between the sharpness of the filter's cutoff and passband ripple.
3. How do you design a Butterworth filter?
Ans. To design a Butterworth filter, you need to specify the filter order, cutoff frequency, and desired response characteristics. The filter order determines the steepness of the filter's roll-off, while the cutoff frequency defines the frequency at which the filter starts attenuating the signal. Once these parameters are determined, you can use mathematical formulas or software tools to calculate the filter coefficients, which are then used to implement the filter.
4. What are the applications of Butterworth filters?
Ans. Butterworth filters find applications in various fields, including audio processing, telecommunications, image processing, and control systems. They are commonly used in audio equalizers, where they help shape the frequency response of the sound system. Butterworth filters are also used in data communication systems to remove unwanted noise and interference from the transmitted signals. In image processing, they can be utilized for image enhancement and noise reduction.
5. Can a Butterworth filter be used for both low-pass and high-pass filtering?
Ans. Yes, a Butterworth filter can be used for both low-pass and high-pass filtering. By adjusting the cutoff frequency, you can determine whether the filter passes low-frequency signals (low-pass filter) or high-frequency signals (high-pass filter). This flexibility makes Butterworth filters versatile and suitable for a wide range of applications, allowing users to tailor the filter's response to their specific needs.
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