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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.
The chebyshev polynomial is oscillatory in the range x<∞.
Explanation: The chebyshev polynomial is oscillatory in the range x≤1 and monotonic outside it.
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?
Explanation: The equiripple 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.
The chebyshevI filter is equiripple in pass band and monotonic in the stop band.
Explanation: There are two types of chebyshev filters. The ChebyshevI filter is equiripple in the pass band and monotonic in the stop band and the chebyshevII filter is quite opposite.
What is the equation for magnitude frequency response H(jΩ) of a low pass chebyshevI filter?
Explanation: The magnitude frequency response of a low pass chebyshevI 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.
What is the number of minima’s present in the pass band of magnitude frequency response of a low pass chebyshevI filter of order 4?
Explanation: In the magnitude frequency response of a low pass chebyshevI filter, the pass band has 2 maxima and 2 minima(order 4=2 maxima+2 minima).
What is the number of maxima present in the pass band of magnitude frequency response of a low pass chebyshevI filter of order 5?
Explanation: In the magnitude frequency response of a low pass chebyshevI filter, the pass band has 3 maxima and 2 minima(order 5=3 maxima+2 minima).
The sum of number of maxima and minima in the pass band equals the order of the filter.
Explanation: In the pass band of the frequency response of the low pass chebyshevI filter, the sum of number of maxima and minima is equal to the order of the filter.
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.
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?
Explanation: We know that the order of a chebyshevI 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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