Chebyshev Filter Design | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

Chebyshev Filter Design

Defined as H_c(S) H_c(-S) =

μ= measure of allowable deviation in the pass band.

C_N(x) = Cos(NCos^-1(x)) is the Nth order polynomial.

Let x = Cos θ

C_N(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..

Two features of Chebyshev poly are important for the filter design

1.| C_N (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|C_N,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.

Define 

 N = Order of filter.

S_P = r Cosθ+j R Sinθ

Ex: 

Pass band:

-1<|H ( jΩ)|²dB ≤ -60   for 0 ≤ Ω ≤ 1404π

Stop band:

|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< [ 10^0.1-1 ]1/2 = 0.508

μ= 0.508

Value of N is determined from stop band inequality

Evaluating 

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:

• Chebyshev filter poles are closer to the jΩ axis, therefore filter response exhibits a  ripple in the pass band. There is a peak in the pass band for each pole in the filter, located approximately at the ordinate value of the pole.
• Exhibits a smaller transition region to reach the desired attenuation in the stop band, when compared to Butterworth filter.
• Phase response is similar.
• Because of proximity of Chebyshev filter poles to j Ω axis, small errors in their locations, caused by numerical round off in the computations, can results in significant changes in the magnitude response. Choosing the smaller value of μ will provide some margin for keeping the ripples within the pass band specification. However, too small a value for  μ may require an increase in the filter order.
• It is reasonable to expect that if relevant zeros were included in the system function, a lower order filter can be found to satisfy the specification. These relevant zeros could serve to achieve additional attenuation in the stop band. The elliptic filter does exactly this.