DESIGN OF BUTTERWORTH AND CHEBYSHEV FILTER Prepared by V.Thamizharasan Assistant professor Department of ECE Erode Sengunthar Engineering College

1. STEPS FOR DESIGN OF BUTTERWORTH AND CHEBYSHEV FILTER
V.THAMIZHARASAN,AP/ECE,ERODE SENGUNTHAR ENGINEERING COLLEGE
Step Butterworth filter
using impulse
invariant
transformation
Butterworth filter using
bilinear transformation
Chebyshev filter using
impulse invariant
transformation
Chebyshev filter using
bilinear transformation
Relation
between
analog and
digital
Frequency
𝜔 = Ω𝑇
𝜔 𝑝 = Ω 𝑝 𝑇
𝜔𝑠 = Ω 𝑠 𝑇
Ω =
2
𝑇
𝑡𝑎𝑛(
𝜔
2
)
Ωp =
2
𝑇
𝑡𝑎𝑛(
𝜔 𝑝
2
)
Ωs =
2
𝑇
𝑡𝑎𝑛(
𝜔𝑠
2
)
𝜔 = Ω𝑇
𝜔 𝑝 = Ω 𝑝 𝑇
𝜔𝑠 = Ω 𝑠 𝑇
Ω =
2
𝑇
𝑡𝑎𝑛(
𝜔
2
)
Ωp =
2
𝑇
𝑡𝑎𝑛(
𝜔 𝑝
2
)
Ωs =
2
𝑇
𝑡𝑎𝑛(
𝜔𝑠
2
)
Find order
of the filter 𝑙𝑜𝑔√100.1𝛼 𝑆−1
10
0.1𝛼 𝑝−1
𝑙𝑜𝑔(
Ω 𝑆
Ω 𝑃
)
≤ 𝑁 Or
log(
𝜆
𝜖
)
𝑙𝑜𝑔(
Ω 𝑆
Ω 𝑃
)
≤ 𝑁
𝑐𝑜𝑠ℎ−1√100.1𝛼 𝑆−1
10
0.1𝛼 𝑝−1
𝑐𝑜𝑠ℎ−1(
Ω 𝑆
Ω 𝑃
)
≤ 𝑁 Or
cosh−1
(
𝜆
𝜖
)
𝑐𝑜𝑠ℎ−1(
Ω 𝑆
Ω 𝑃
)
≤ 𝑁
Find the
poles
Write the transfer function H(S) from the table
𝑆 𝑘 = a cos 𝛷 𝑘 + j b sin 𝛷k
𝛷 𝑘 =
𝜋
2
+
(2𝑘−1) 𝜋
2𝑁
, k=1,2,….N
μ = 𝜀−1
+ √1 + 𝜀−2
𝑎 = Ω 𝑝 [
𝜇
1
𝑁 − 𝜇−
1
𝑁
2
] 𝑏 = Ω 𝑝 [
𝜇
1
𝑁 + 𝜇−
1
𝑁
2
]
Find the
analog
transfer
function
Find Ha(S)=H(S)S is replace by S/Ωc
Ωc =
Ωp
(ε)
1
N
𝐻( 𝑆) = 𝑁𝑟/𝐷𝑟
Dr(S-S1)(S-S2)……(S-SN)
Nrput S=0 in Dr of H(s) and multiply by
1 If N=odd
1/√1 + 𝜀2  If N=EVEN
Find digital
filter
transfer
function
Apply partial fraction
then find H(Z) using
Find H(Z)= Ha(S)S is
replace by
𝑆 =
2
𝑇
[
1−𝑍−1
1+𝑍−1
]
Apply partial fraction to above
H(S) then find H(Z) using
Find H(Z)= H(S)S is
replace by
𝑆 =
2
𝑇
[
1−𝑍−1
1+𝑍−1
]

2. STEPS FOR DESIGN OF BUTTERWORTH AND CHEBYSHEV FILTER
V.THAMIZHARASAN,AP/ECE,ERODE SENGUNTHAR ENGINEERING COLLEGE
𝐻( 𝑍)
= ∑
𝐶 𝑘
1 − 𝑒 𝑃 𝑘 𝑇 𝑍−1
𝑁
𝑘=1
𝐻( 𝑍) = ∑
𝐶 𝑘
1 − 𝑒 𝑃 𝑘 𝑇 𝑍−1
𝑁
𝑘=1
Transfer function for different order of N using Butterworth filter
Order (N) Denominator of H(S)
1 (S+1)
2 (S2+√2 S+1)
3 (S+1)( S2+S+1)
4 (S2+0.765 S+1) (S2+1.847S+1)
5 (S+1)( S2+0.618 S+1) (S2+1.618S+1)
6 ( S2+1.93185S+1) (S2+√2S+1) (S2+0.518S+1)
7 (S+1)( S2+1.809S+1) (S2+1.247S+1) (S2+0.44S+1)
Frequency transmission in Analog Domain
Transformation (or)
conversion
Equation
LPF to LPF LPF to HPF LPF to BPF LPF to BSF
S S / ΩC S ΩC/S 𝑠 →
𝑠2+Ω𝑙Ω 𝑢
𝑠(Ω𝑙−Ω 𝑢)
,
Ω 𝑟 = min{| 𝐴|, | 𝐵|},
𝐴 =
−Ω1
2
+ Ω𝑙Ω 𝑢
Ω1(Ω𝑙−Ω 𝑢)
𝑠 →
𝑠 (Ω𝑙−Ω 𝑢)
𝑠2+Ω𝑙Ω 𝑢
,
Ω 𝑟 = min{| 𝐴|, | 𝐵|},
𝐴 =
Ω1(Ω𝑙−Ω 𝑢)
−Ω1
2
+Ω𝑙Ω 𝑢

3. STEPS FOR DESIGN OF BUTTERWORTH AND CHEBYSHEV FILTER
V.THAMIZHARASAN,AP/ECE,ERODE SENGUNTHAR ENGINEERING COLLEGE
𝐵 =
Ω2
2
− Ω𝑙Ω 𝑢
Ω2(Ω 𝑢−Ω𝑙)
𝐵 =
Ω2(Ω 𝑢−Ω𝑙)
Ω2
2
− Ω𝑙Ω 𝑢