# Chebyshev Filter MCQ Quiz – Objective Question with Answer for Chebyshev Filter

1. Which of the following defines a Chebyshev polynomial of order N, TN(x)?

A. cos(Ncos-1x) for all x
B. cosh(Ncosh-1x) for all x
C.cos(Ncos-1x), |x|-1x), |x|>1
D. None of the mentioned

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
TN(x) = cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1.

2. What is the formula for chebyshev polynomial TN(x) in recursive form?

A. 2TN-1(x) – TN-2(x)
B. 2TN-1(x) + TN-2(x)
C. 2xTN-1(x) + TN-2(x)
D. 2xTN-1(x) – TN-2(x)

We know that a chebyshev polynomial of degree N is defined as
TN(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
TN(x)= 2xTN-1(x)-TN-2(x), N ≥ 2.

3. What is the value of the Chebyshev polynomial of degree 0?

A. 1
B. 0
C. -1
D. 2

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

TN(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.

4. What is the value of the Chebyshev polynomial of degree 1?
A. 1
B. x
C. -1
D. -x

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

TN(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.

5. What is the value of Chebyshev polynomial of degree 3?

A. 3x3+4x
B. 3x3-4x
C. 4x3+3x
D. 4x3-3x

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

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

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

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

6. What is the value of Chebyshev polynomial of degree 5?

A. 16x5+20x3-5x
B. 16x5+20x3+5x
C. 16x5-20x3+5x
D. 16x5-20x3-5x

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

TN(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
TN(x)= 2xTN-1(x)-TN-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.

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

A. True
B. False

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

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

8. Chebyshev polynomials of odd orders are _______

A. Even functions
B. Odd functions
C. Exponential functions
D. Logarithmic functions

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

9. What is the value of TN(0) for even degree N?

A. -1
B. +1
C. 0
D. ±1

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

TN(x) = cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1

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

10. TN(-x)=(-1)NTN(x).

A. True
B. False

We know that a chebyshev polynomial of degree N is defined as
TN(x) = cos(Ncos-1x), |x|≤1
cosh(Ncosh-1x), |x|>1

=> TN(-x)= cos(Ncos-1(-x))=cos(N(π-cos-1x))=cos(Nπ-Ncos-1x)=(-1)N cos(Ncos-1x)=(-1)NTN(x)

Thus we get, TN(-x)=(-1)NTN(x).

11. What is the value of |TN(±1)|?

A. 0
B. -1
C. 1
D. None of the mentioned

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

12. The Chebyshev polynomial is oscillatory in the range |x|<∞.

A. True
B. False

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

13. 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

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.

14. The Chebyshev-I filter is equi-ripple in the passband and monotonic in the stopband.
A. True
B. False

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.

15. 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. $$\frac{1}{\sqrt{1+ϵ^2 T_N^2 (\frac{Ω}{Ω_P})}}$$

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 TN(x) is the Nth order Chebyshev polynomial.

16. What is the number of minima present in the passband of the magnitude frequency response of a low pass Chebyshev-I filter of order 4?

A. 1
B. 2
C. 3
D. 4

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.

17. What is the number of maxima present in the passband of the magnitude frequency response of a low pass Chebyshev-I filter of order 5?

A. 1
B. 2
C. 3
D. 4

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.

18. The sum of the number of maxima and minima in the passband equals the order of the filter.
A. True
B. False

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.

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

A. 1+ϵ2TN2(s/j)=0
B. 1-ϵ2TN2(s/j)=0
C. 1+ϵ TN2(s/j)=0
D. None of the mentioned

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)|<sup>2</sup>=$$\frac{1}{\sqrt{1+ϵ^2 T_N^2(\frac{Ω}{Ω_P})}}$$

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

=>|H<sub>N</sub>(s)|<sup>2</sup>=$$\frac{1}{1+ϵ^2 T_N^2 (s/j)}$$

The poles of the above equation is given by the equation 1+ϵ 2TN2(s/j) = 0 which is called as the characteristic equation.

20. The poles of HN(s).HN(-s) are found to lie on ______

A. Circle
B. Parabola
C. Hyperbola
D. Ellipse

The poles of HN(s).HN(-s) is given by the characteristic equation 1+ϵ2TN2(s/j)=0.

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

21. 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

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.

22. The equation for H<sub>eq</sub>(s) is $$\frac{\sum_{K=0}^M b_K s^K}{\sum_{K=0}^N a_K s^K}$$

A. True
B. False

The analog filter in the time domain is governed by the following difference equation,

$$\sum_{K=0}^N a_K y^K (t)=\sum_{K=0}^M b_K x^K (t)$$

Taking Laplace transform on both the sides of the above differential equation with all initial conditions set to zero, we get

$$\sum_{K=0}^N a_K s^K Y(s)=\sum_{K=0}^M b_K s^K X(s)$$

=> H<sub>eq</sub>(s)=Y(s)/X(s)=$$\frac{\sum_{K=0}^M b_K s^K }{\sum_{K=0}^N a_K s^K}$$.

23. What is the first backward difference of y(n)?

A. [y(n)+y(n-1)]/T
B. [y(n)+y(n+1)]/T
C. [y(n)-y(n+1)]/T
D. [y(n)-y(n-1)]/T

A simple approximation to the first-order derivative is given by the first backward difference. The first backward difference is defined by
[y(n)-y(n-1)]/T.

24. Which of the following is the correct relation between ‘s’ and ‘z’?
A. z=1/(1+sT)
B. s=1/(1+zT)
C. z=1/(1-sT)
D. none of the mentioned

We know that s=(1-z-1)/T=> z=1/(1-sT).

25. What is the center of the circle represented by the image of jΩ axis of the s-domain?
A. z=0
B. z=0.5
C. z=1
D. none of the mentioned