Test: Chebyshev Filters - 2

Explanation: We know that a chebyshev polynomial of degree N is defined as
T_N(x) = cos(Ncos^-1x), |x|≤1
cosh(Ncosh^-1x), |x|>1
Thus |T_N(±1)|=1.

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

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.

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.

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.

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).

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).

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.

Explanation: The poles of HN(s).HN(-s) is given by the characteristic equation 1+ϵ²T_N²(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.

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.