Butterworth Filter Design | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

Butter worth Filter Design

The butterworth LP filter of order N is defined as H_B(s) H_B(-s) =

Where s = jΩ_c

OR

It has 2N poles

Ex: for N=3

 = 120⁰, 180⁰, 240⁰, 300⁰, 360⁰, 60⁰

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

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

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.

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   ( Ω_s = 8268π )

If the Ω_c is given

Since Ω_c 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

Ω_s ^2N > 10⁶ Ω_c ^{2 N}

Ω c <1470.3π

Let Ω c =1470.3π

At this Ω c it should satisfy pass band specifications.

= 0.59

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

Let N=5

In the pass band 

Since N=5

Ω c = 2076π

S1 = -2076π

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

H_B( jΩ)²= (1+ Ω ²)-1

The first derivative

The second derivative

3. The phase response curve approaches   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     

If n is even S^2N = 1 = e ^{j ( 2k -1)}π

The 2N roots will be Sk=  k=1,2,….2N

Therefore: 

If N is odd

S²n =1 = e ^{j 2kπ}

where θ _{k =} 

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

  orfor better req at Ω2