Digital Filters Design MCQ [Free PDF] – Objective Question Answer for Digital Filters Design Quiz

In order to understand the frequency-domain behavior of Chebyshev filters, it is of utmost importance to define a Chebyshev polynomial and then its properties. A Chebyshev polynomial of degree N is defined as

T_N(x) = cos(Ncos^-1x), |x|≤1

cosh(Ncosh^-1x), |x|>1.

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

From the above formula, it is possible to generate the Chebyshev polynomial using the following recursive formula

T_N(x)= 2xT_N-1(x)-T_N-2(x), N ≥ 2.

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

For a degree 0 Chebyshev filter, the polynomial is obtained as
T₀(x)=cos(0)=1.

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

For a degree 1 Chebyshev filter, the polynomial is obtained as
T₀(x)=cos(cos^-1x)=x.

We know that a Chebyshev polynomial of degree N is defined as

T_N(x) = cos(Ncos^-1x), |x|≤1; T_N(x) = cosh(Ncosh^-1x), |x|>1

And the recursive formula for the chebyshev polynomial of order N is given as
T_N(x)=2xT_N-1(x)-T_N-2(x)

Thus for a chebyshev filter of order 3, we obtain
T₃(x)=2xT₂(x)-T₁(x)=2x(2x²-1)-x=4x³-3x.

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

And the recursive formula for the Chebyshev polynomial of order N is given as

T_N(x)= 2xT_N-1(x)-T_N-2(x)

Thus for a chebyshev filter of order 5, we obtain

T₅(x)=2xT₄(x)-T₃(x)=2x(8x⁴-8x²+1)-(4x³-3x)=16x⁵-20x³+5x.

For |x|≤1, |T_N(x)|≤1, and it oscillates between -1 and +1 a number of times proportional to N.

The above is evident from the equation,
T_N(x) = cos(Ncos^-1x), |x|≤1.

Chebyshev polynomials of odd orders are odd functions because they contain only the odd powers of x.

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

For x=0, we have T_N(0)=cos(Ncos^-10)=cos(N.π/2)=±1 for N even.

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

=> T_N(-x)= cos(Ncos^-1(-x))=cos(N(π-cos^-1x))=cos(Nπ-Ncos^-1x)=(-1)^N cos(Ncos^-1x)=(-1)^NT_N(x)

Thus we get, T_N(-x)=(-1)^NT_N(x).

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.

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

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.

There are two types of Chebyshev filters. The Chebyshev-I filter is equi-ripple in the passband and monotonic in the stopband and the Chebyshev-II filter is quite opposite.

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 passband and T_N(x) is the N^th order Chebyshev polynomial.

In the magnitude frequency response of a low pass Chebyshev-I filter, the passband has 2 maxima and 2 minima(order 4=2 maxima+2 minimA.

In the magnitude frequency response of a low pass Chebyshev-I filter, the passband has 3 maxima and 2 minima(order 5=3 maxima+2 minimA.

In the passband of the frequency response of the low pass Chebyshev-I filter, the sum of the number of maxima and minima is equal to the order of the filter.

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 ²T_N²(s/j) = 0 which is called as the characteristic equation.

The poles of H_N(s).H_N(-s) is given by the characteristic equation 1+ϵ²T_N²(s/j)=0.

The roots of the above characteristic equation lie on the ellipse, thus the poles of H_N(s).H_N(-s) are found to lie on an ellipse.