Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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Chebyshev Filter Design

Defined as Hc(S) Hc(-S) = Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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..

NCN(x)
01
1x
22x2-1
34x3-3x
48x4- 8x2 +1

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Two features of Chebyshev poly are important for the filter design

1.| CN (x)| ≤ 1    for|x| ≤ 1

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) for  0 ≤ Ω ≤ Ωp

Transfer function lies in the range  Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) 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.

Define  Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) N = Order of filter.

SP = r Cosθ+j R Sinθ

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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< [ 100.1-1 ]1/2 = 0.508

μ= 0.508

Value of N is determined from stop band inequality

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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 Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)Chebyshev Filter Design Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

  • 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.
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