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

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 NB and NC 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) NC=NB

b) NC<NB

c) NC>NB

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?

View Answer

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

where ϵ is a parameter of the filter related to the ripple in the pass band and TN(x) is the Nth 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?

d) None of the mentioned

View Answer

10. The poles of HN(s).HN(-s) are found to lie on:

a) Circle

b) Parabola

c) Hyperbola

d) Ellipse

View Answer

Explanation: The poles of HN(s).HN(-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 HN(s).HN(-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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