Butter worth Filter Design
The butterworth LP filter of order N is defined as HB(s) HB(-s) =
Where s = jΩc
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.
-1< H ( jΩ) 2 dB ≤ 0 for 0≤ Ω ≤ 1404π ( W π = 1404π )
H ( jW)2 dB < -60 for W ≥ 8268π ( Ωs = 8268π )
If the Ωc is given
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
1 oct ---- - 6N dB
2.56 ------ ?
=> 2.56 X - 6N dB = -59 dB’s
Ωs 2N > 106 Ωc 2 N
Ω c <1470.3π
Let Ω c =1470.3π
At this Ω c it should satisfy pass band specifications.
This result is below the pass band specifications. Hence N=4 is not sufficient.
In the pass band
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
HB( 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.
1. easiest to design
2. used because of smoothness of magnitude response .
Relatively large transition range between the pass band and stop band.
When Ω c = 1
If n is even S2N = 1 = e j ( 2k -1)π
The 2N roots will be Sk= k=1,2,….2N
If N is odd
S2n =1 = e j 2kπ
where θ k =
choosing this value for n, results in two different selections for Ωc . 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