This set of Digital Signal Processing quiz focuses on “Chebyshev Filters – 2”.

1. What is the value of |T_{N}(±1)|?

a) 0

b) -1

c) 1

d) None of the mentioned

View Answer

Explanation: We know that a chebyshev polynomial of degree N is defined as

T

_{N}(x) = cos(Ncos

^{-1}x), |x|≤1

cosh(Ncosh

^{-1}x), |x|>1

Thus |T

_{N}(±1)|=1.

2. The chebyshev polynomial is oscillatory in the range |x|<∞.

a) True

b) False

View Answer

Explanation: The chebyshev polynomial is oscillatory in the range |x|≤1 and monotonic outside it.

3. If N_{B} and N_{C} are the orders of the Butterworth and Chebyshev filters respectively to meet the same frequency specifications, then which of the following relation is true?

a) N_{C}=N_{B}

b) N_{C}<N_{B}

c) N_{C}>N_{B}

d) Cannot be determined

View Answer

Explanation: The equi-ripple property of the chebyshev filter yields a narrower transition band compared with that obtained when the magnitude response is monotone. As a consequence of this, the order of a chebyshev filter needed to achieve the given frequency domain specifications is usually lower than that of a Butterworth filter.

4. The chebyshev-I filter is equi-ripple in pass band and monotonic in the stop band.

a) True

b) False

View Answer

Explanation: There are two types of chebyshev filters. The Chebyshev-I filter is equi-ripple in the pass band and monotonic in the stop band and the chebyshev-II filter is quite opposite.

5. What is the equation for magnitude frequency response |H(jΩ)| of a low pass chebyshev-I filter?

a) \(\frac{1}{\sqrt{1-ϵ T_N^2 (\frac{Ω}{Ω_P})}}\)

b) \(\frac{1}{\sqrt{1+ϵ T_N^2 (\frac{Ω}{Ω_P})}}\)

c) \(\frac{1}{\sqrt{1-ϵ^2 T_N^2 (\frac{Ω}{Ω_P})}}\)

d) \(\frac{1}{\sqrt{1+ϵ^2 T_N^2 (\frac{Ω}{Ω_P})}}\)

View Answer

Explanation: The magnitude frequency response of a low pass chebyshev-I filter is given by

|H(jΩ)|=\(\frac{1}{\sqrt{1+ϵ^2 T_N^2(\frac{Ω}{Ω_P})}}\)

where ϵ is a parameter of the filter related to the ripple in the pass band and T

_{N}(x) is the N

^{th}order chebyshev polynomial.

6. What is the number of minima’s present in the pass band of magnitude frequency response of a low pass chebyshev-I filter of order 4?

a) 1

b) 2

c) 3

d) 4

View Answer

Explanation: In the magnitude frequency response of a low pass chebyshev-I filter, the pass band has 2 maxima and 2 minima(order 4=2 maxima+2 minima).

7. What is the number of maxima present in the pass band of magnitude frequency response of a low pass chebyshev-I filter of order 5?

a) 1

b) 2

c) 3

d) 4

View Answer

Explanation: In the magnitude frequency response of a low pass chebyshev-I filter, the pass band has 3 maxima and 2 minima(order 5=3 maxima+2 minima).

8. The sum of number of maxima and minima in the pass band equals the order of the filter.

a) True

b) False

View Answer

Explanation: In the pass band of the frequency response of the low pass chebyshev-I filter, the sum of number of maxima and minima is equal to the order of the filter.

9. Which of the following is the characteristic equation of a Chebyshev filter?

a) 1+ϵ^{2}T_{N}^{2}(s/j)=0

b) 1-ϵ^{2}T_{N}^{2}(s/j)=0

c) 1+ϵ T_{N}^{2}(s/j)=0

d) None of the mentioned

View Answer

Explanation: We know that for a chebyshev filter, we have

|H(jΩ)|=\(\frac{1}{\sqrt{1+ϵ^2 T_N^2(\frac{Ω}{Ω_P})}}\)

=>|H(jΩ)|

^{2}=\(\frac{1}{\sqrt{1+ϵ^2 T_N^2(\frac{Ω}{Ω_P})}}\)

By replacing jΩ by s and consequently Ω by s/j in the above equation, we get

=>|H

_{N}(s)|

^{2}=\(\frac{1}{1+ϵ^2 T_N^2 (s/j)}\)

The poles of the above equation is given by the equation 1+ϵ

^{2}T

_{N}

^{2}(s/j)=0 which is called as the characteristic equation.

10. The poles of H_{N}(s).H_{N}(-s) are found to lie on ___________

a) Circle

b) Parabola

c) Hyperbola

d) Ellipse

View Answer

Explanation: The poles of H

_{N}(s).H

_{N}(-s) is given by the characteristic equation 1+ϵ

^{2}T

_{N}

^{2}(s/j)=0.

The roots of the above characteristic equation lies on ellipse, thus the poles of H

_{N}(s).H

_{N}(-s) are found to lie on ellipse.

11. If the discrimination factor ‘d’ and the selectivity factor ‘k’ of a chebyshev I filter are 0.077 and 0.769 respectively, then what is the order of the filter?

a) 2

b) 5

c) 4

d) 3

View Answer

Explanation: We know that the order of a chebyshev-I filter is given by the equation,

N=cosh

^{-1}(1/d)/cosh

^{-1}(1/k)=4.3

Rounding off to the next large integer, we get N=5.

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