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**Chebyshev Filter Design**

Defined as H_{c}(S) H_{c}(-S) =

Î¼= measure of allowable deviation in the pass band.

C_{N}(x) = Cos(NCos^{-1}(x)) is the Nth order polynomial.

Let x = Cos Î¸

C_{N}(x) = Cos(NÎ¸)

C_{0}(x) = 1

C_{1}(x) = CosÎ¸ =x

C_{2}(x) = Cos2Î¸ = 2 Cos^{2}Î¸ -1 = 2x^{2}-1

C_{3}(x) = Cos^{3} Î¸ = 4 Cos^{3} Î¸ -3 Cos Î¸ = 4x^{3}-3x etc..

N | C_{N}(x) |

0 | 1 |

1 | x |

2 | 2x^{2}-1 |

3 | 4x^{3}-3x |

4 | 8x^{4}- 8x^{2} +1 |

Two features of Chebyshev poly are important for the filter design

1.| C_{N} (x)| â‰¤ 1 for|x| â‰¤ 1

for 0 â‰¤ Î© â‰¤ Î©p

Transfer function lies in the range for 0 â‰¤ Î© â‰¤Î©p Whereas the frequency value important for the design of the Butterworth filter was the Î©c the relevant frequency for the Chebyshev filter is the edge of pass band Î©p.

2. |X| >>1|C_{N},l Increases as the Nth power of x. this indicates that for Î©>>Î©p, the magnitude response decreases as Î©^{-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 CosÎ¸+j R SinÎ¸

Ex:

Pass band:

-1<|H ( jÎ©)|^{2}dB â‰¤ -60 for 0 â‰¤ Î© â‰¤ 1404Ï€

Stop band:

|H ( jÎ©)|^{2}dB < -60 for Î© â‰¥ 8268Ï€

Value of Î¼ is determined from the pass band

10 log(1 +Î¼ ^{2} )^{-1 }> -1dB -1dB = 0.794

Î¼< [ 10^{0.1}-1 ]1/2 = 0.508

Î¼= 0.508

Value of N is determined from stop band inequality

Evaluating

C_{3}(5.9) = 804 C_{4}(5.9) = 9416 therefore N = 4 is sufficient.

Since this last inequality is easily satisfied with N=4 the value of Î¼ can be reduced to as small as 0.11, to decrease pass band ripple while satisfying the stop band. The value Î¼ =0.4 provides a margin in both the pass band and stop band. We proceed with the design with Î¼ =0.508 to show the 1dB ripple in the pass band.

Axes of Ellipse:

- Chebyshev filter poles are closer to the jÎ© 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 Î© 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 Î¼ will provide some margin for keeping the ripples within the pass band specification. However, too small a value for Î¼ 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.

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