Chebyshev Filters

CHEBYSHEV FILTERS

• Two types of chebyshev filters (i) Type I chebyshev filters (ii) Type II chebyshev filters Type I chebyshev filters : Filters are all pole filter that Exhibits equiripple behavior in the pass band and monotonic characteristics in the stop band Type II chebyshev filters : Filters that contains both pole and zeros filter and exhibits monotonic behavior in the pass band and equiripple characteristics in the stop band

Characteristics of chebyshev filters ‫׀‬H(jΩ‫׀‬ 1.0

‫׀‬H(jΩ‫׀‬ 1.0

1 1 +

1

ε2

2 1 + ε

1

1

2 1+ λ

1 + λ2

Ωp Ωs Ω

Ωp Ωs

Ω

•

filter can be expressed as N=0,1,2…. 1

The magnitude response of the Nth order

‫׀‬H jΩ ‫׀‬2 =

1

Ω 2 2 + ξ CN   1  Ωp

Where ξ is the parameter of the related to the ripple in the pass band and C N (x) is the Nth order chebyshev polynomial defined as 2 C N (x) = cos(Ncosh-1x), ‫׀‬x ‫<׀‬1 (pass band) C N (x) = cosh(Ncosh-1x), ‫׀‬x ‫>׀‬1 (stop band) 3

•

The chebyshev polynomial is defined as CN (x) =2x CN-1 (x) - CN-2 (x) ,N>1

Where C0 (x)

4

=1and C1 (x) =x

The chebyshev polynomials described by equ 4 and 2 have the following properties 1. CN (x) = -CN (-x) for the N odd CN (x) = CN (-x) for the N even for the N even CN (0) = (-1)N/2 CN (0) = 0 for the N odd for all N CN (1)= 1 CN (-1)= 1 for N even for N odd CN (-1) = -1

2. CN (x) oscillates with equal ripple between ± 1 for ‫׀‬x ‫≤ ׀‬1 3. For all N 0 ≤ ‫׀‬CN (x) ‫ ≤ ׀‬1 for 0 ≤ ‫׀‬x ‫ ≤ ׀‬1 ‫׀‬CN (x) ‫> ׀‬1 for ‫׀‬x ‫≥ ׀‬1 4. CN (x) is monotonically increasing for ‫׀‬x ‫ > ׀‬1 for all N N odd

‫׀‬H(jΩ‫׀‬

N odd 1.0

1.0

1 1+ ξ 2

1 2 1+ ξ

αp αs

α Ωs

1

1

1 + λ2

αp

1 + λ2 Ωp Ωp

Ω

Ωs

Ωp

Ω

• Taking log for equ 1  20 log ‫׀‬H jΩ‫ = ׀‬10 log 1 – 10 log 1 + ξ



2

 Ω   C2N   Ωp  

5

Let αp is the attenuation in positive dB at the pass band frequency Ω p and αs is the attenuation in positive dB at the atop band frequency Ω s at Ω= Ω p in equ 5 αp = 10 log(1+ξ2 ) (CN =1) which gives ξ = (100.1-1)1/2 6 At Ω= Ω s equ 5 can be written Ωs   αs = 10 log  2 2 

 1 + ξ C N   Ωp   

Ω

 s  >1  • =10 log[1+ε2{cosh(N cosh-1(Ωs/Ωp))}2] 7  Ωp  Sub equ 6 for ε in equ 7 , solving for N and rounding it to the next higher integer 100.1αs -1 -1 8 N ≥ cosh 0.1αp −

10

Cosh-1

-1 Ω s Ω p

for butter worth filter method Cosh-1A

N≥ Cosh-1

Equ 9

cosh-1 evaluated

 1 

 

 k



using the identity cosh-1x =ln[ x+ ] 2

x

-1

9

Poles location for chebyshe filter • The poles for type I filter obtaining by setting the denominator of equ 1 equal to zero That is 1 + ξ 2C 2  −js =o 10 N Ω p

Simply the above equ CN -js = ±j/ε  Ωp

= cos(Ncosh-1  -js )  Ωp

11

We define -js cosh-1  Ωp  

= φ − jθ

12

• ±j/ε =cos [Nφ − jNθ ] • =cos (NΦ) cos (jNθ) + j sin (NΦ) sin (jNθ) • = cos (NΦ) cosh (Nθ) + j sin (NΦ) sinh(Nθ) formula cos θ =ej θ+ e-j θ

13

2

cos jθ =ej(j θ)+ e-j (jθ) 2

cos jθ =e- θ + eθ 2

coshθ Equating

the real and imaginary parts of euq 13

cos (NΦ) cosh (Nθ) =0 14 sin (NΦ) sinh (Nθ) =± 1/ ε 15 Since cosh(Nθ)>0 for θ then in order to satisfy equ 14 we have Φ= (2k − 1)∏ k=1,2,3,…….N 16 2N Form equ 15 we solve for θ when sin (NΦ) =±1

• Now we have θ = ±1/Nsin-1(1/ε)

17

Combine equ 17 ,16 and 12we obtain LHP sk=j Ωp cos (Φ-jθ) =j Ωp [cos Φcosh θ +j sin Φ sin h θ] =jΩp [- sin Φ sin h θ + j cos Φ cosh θ ] Equ 18 can be simplified using the identity sinh-1x =ln[ x+ Sinh-1(ε -1)= In ξ −1+1 + ε−2 Or µ=esinh-1(ε-1) =ε-1 + 1 +x2

(

)

18

1

+x2

]