Chebyshev filter

Using Chebyshev filter design, there are two sub groups,
Type-I Chebyshev Filter
Type-II Chebyshev Filter
The major difference between butterworth and chebyshev filter is that the poles of butterworth filter lie on the circle while the poles of chebyshev filter lie on ellipse.

1. Designing using Chebyshev Approximation 1 Mohammad Akram,Assistant Professor,JIT,Barabanki

2. Chebyshev Filter Ω Ω p Using Chebyshev filter design, there are two sub groups, Type-I Chebyshev Filter Type-II Chebyshev Filter Type-I Chebyshev Filter: These filters are all pole filters . In the pass band, these filters show equiripple behaviour and they have have monotonic characteristics in the stop band. The filter characteristics is shown in fig.1 IH( Ω )I 2 1 Fig.1 Type-I Chebyshev Filter Characteristics 2 Mohammad Akram,Assistant Professor,JIT,Barabanki

3. The magnitude squared frequency response of chebyshev filter is given by: Here = Chebyshev po;ynomial of order N. ε = Ripple parameter in the passband Type-II Chebyshev Filter : These filters contains zeros as well as poles. In the stop band, these filters show equiripple behaviour and they have have monotonic characteristics in the pass band. The major difference between butterworth and chebyshev filter is that the poles of butterworth filter lie on the circle while the poles of chebyshev filter lie on ellipse. 3 Mohammad Akram,Assistant Professor,JIT,Barabanki

4. Chebyshev low pass filter design : Design equation and design steps: Let A p =Attenuation in passband A s =Attenuation in stop band Ω p = Passband edge frequency Ω c =Cut off frequency Ω s = Stopband edge frequency Step I: Calculation of parameter ε : It is given by : If A P is not in dB then ε is calculated using the equation, 4 Mohammad Akram,Assistant Professor,JIT,Barabanki

5. Step II: Calculation of order N of the filter: When stop band attenuation (As) is given in dB then ‘N’ is calculated using the equation: When stop band attenuation (As) is not given in dB then ‘N’ is calculated using the equation: Step III: Calculation of poles: The position of poles of Chebyshev filter lie on ellipse at coordinates x k and y k given by: x k =r cos θ k , k=0,1,2,…….N-1 y k =R sin θ k , k=0,1,2,…….N-1 5 Mohammad Akram,Assistant Professor,JIT,Barabanki

6. Where , k=0,1,2,……N-1 r represents minor axis of ellipse and is given by, R represents major axis of ellipse and is given by, Here the parameter β is given by, Thus the pole positions are denoted by s p , s p = r cos θ k + jR sin θ k 6 Mohammad Akram,Assistant Professor,JIT,Barabanki

7. Step IV: Calculation of system function H(s): The system transfer function H(s) of analog filter is given by, After simplification this equation can be written as, Here b = Constant term in the denominator Now the value of ‘k’ can be calculated as follows for ‘N’ odd for ‘N’ even Step V: Design the digital filter using impulse invariance method or bilinear transformation method. 7 Mohammad Akram,Assistant Professor,JIT,Barabanki

8. Numerical Problem-1: Design a Chebyshev analog filter with maximum passband attenuation of 2.5 dB at Ω p = 20 rad /sec and stop band attaenuation of 30 dB at Ω s = 50 rad /sec. Solution: Given data: Ap =2.5 db, Ω p = 20 rad /sec , As= 30 dB , Ω s = 50 rad /sec. Step-I: Calculation of parameter ε : = 0.882 Step II: Calculation of order N of the filter: After solving, N=0.95 Thus order of the filter, N≈1 8 Mohammad Akram,Assistant Professor,JIT,Barabanki

9. Step III: Calculation of poles: The pole positions are denoted by s p , s p = r cos θ k + jR sin θ k First we will calculate parameter ‘ β ’ = = 2.64 Now we will calculate the values of ‘r’ and ‘R’ = = 22.6 = 30.19 Now we will calculate values of θ k , k=0 to N-1 9 Mohammad Akram,Assistant Professor,JIT,Barabanki

10. = π Now we can write pole positions as s = r cos θ + jR sin θ = 22.6 cos π + j (30.19) sin π = -22.6 Step IV: Calculation of system function H(s): The system transfer function H(s) of analog filter is given by, Here for N=1, we have k= b = 22.6 This is the required transfer function. 10 Mohammad Akram,Assistant Professor,JIT,Barabanki

11. Thanks 11 Mohammad Akram,Assistant Professor,JIT,Barabanki