Chebyshev Filter Design
Defined as Hc(S) Hc(-S) =
μ= measure of allowable deviation in the pass band.
CN(x) = Cos(NCos-1(x)) is the Nth order polynomial.
Let x = Cos θ
CN(x) = Cos(Nθ)
C0(x) = 1
C1(x) = Cosθ =x
C2(x) = Cos2θ = 2 Cos2θ -1 = 2x2-1
C3(x) = Cos3 θ = 4 Cos3 θ -3 Cos θ = 4x3-3x etc..
|4||8x4- 8x2 +1|
Two features of Chebyshev poly are important for the filter design
1.| CN (x)| ≤ 1 for|x| ≤ 1
for 0 ≤ Ω ≤ Ωp
Transfer function lies in the range for 0 ≤ Ω ≤Ωp Whereas the frequency value important for the design of the Butterworth filter was the Ωc the relevant frequency for the Chebyshev filter is the edge of pass band Ωp.
2. |X| >>1|CN,l Increases as the Nth power of x. this indicates that for Ω>>Ωp, the magnitude response decreases as Ω-N, or -6N dB Octane. This is identical to Butterworth filter.
Now the ellipse is defined by major & minor axis.
N = Order of filter.
SP = r Cosθ+j R Sinθ
-1<|H ( jΩ)|2dB ≤ -60 for 0 ≤ Ω ≤ 1404π
|H ( jΩ)|2dB < -60 for Ω ≥ 8268π
Value of μ is determined from the pass band
10 log(1 +μ 2 )-1 > -1dB -1dB = 0.794
μ< [ 100.1-1 ]1/2 = 0.508
Value of N is determined from stop band inequality
C3(5.9) = 804 C4(5.9) = 9416 therefore N = 4 is sufficient.
Since this last inequality is easily satisfied with N=4 the value of μ can be reduced to as small as 0.11, to decrease pass band ripple while satisfying the stop band. The value μ =0.4 provides a margin in both the pass band and stop band. We proceed with the design with μ =0.508 to show the 1dB ripple in the pass band.
Axes of Ellipse: