Electronics and Communication Engineering (ECE) Exam > Electronics and Communication Engineering (ECE) Notes > Signals and Systems > Chebyshev Filter Design
Chebyshev Filter Design | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download
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..
| N | CN(x) |
| 0 | 1 |
| 1 | x |
| 2 | 2x2-1 |
| 3 | 4x3-3x |
| 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.
Define 
N = Order of filter.
SP = 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< [ 100.1-1 ]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.
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FAQs on Chebyshev Filter Design - Signals and Systems - Electronics and Communication Engineering (ECE)
| 1. What is a Chebyshev filter and how does it work? |  |
Ans. A Chebyshev filter is a type of electronic filter that is designed to have a specific frequency response. It is characterized by its ability to have a steeper roll-off rate than other filter types. The filter achieves this by allowing for ripple in the passband, which helps to achieve a sharper cutoff. The amount of ripple and the steepness of the roll-off can be controlled by the design parameters of the filter.
| 2. What are the advantages of using a Chebyshev filter? | |
Ans. Chebyshev filters offer several advantages over other filter types. Firstly, they can achieve a steeper roll-off rate, which means they can provide better attenuation of unwanted frequencies. Secondly, Chebyshev filters are more efficient than other filter types, meaning they require fewer components to achieve the desired frequency response. Lastly, they offer a wide range of options for design parameters, allowing for customization to meet specific application requirements.
| 3. How do I design a Chebyshev filter? | |
Ans. Designing a Chebyshev filter involves determining the desired filter specifications, such as the cutoff frequency and the amount of ripple in the passband. These specifications are then used to calculate the required order of the filter and the values of the components needed. There are various software tools and online calculators available that can assist in the design process by providing the necessary equations and calculations.
| 4. What are the applications of Chebyshev filters? | |
Ans. Chebyshev filters find applications in various fields, including telecommunications, audio processing, and signal processing. They are commonly used in situations where a sharp roll-off and precise frequency response are required, such as in high-speed data transmission, audio equalizers, and instrumentation amplifiers. Chebyshev filters are also used in scientific research and industrial applications for filtering and analyzing signals.
| 5. Are there any limitations or trade-offs when using Chebyshev filters? | |
Ans. Like any other filter type, Chebyshev filters have their limitations and trade-offs. One limitation is that the passband ripple can introduce distortion in the filtered signal. Additionally, Chebyshev filters with steeper roll-off rates may exhibit phase distortion, which can affect the time-domain characteristics of the filtered signal. There is also a trade-off between the steepness of the roll-off and the amount of passband ripple, where increasing one may result in a compromise in the other. Designers need to carefully consider these factors and select the appropriate filter parameters based on the specific application requirements.
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