This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Chebyshev Filters-1”.

1. Which of the following defines a chebyshev polynomial of order N, T_{N}(x)?

a) cos(Ncos^{-1}x) for all x

b) cosh(Ncosh^{-1}x) for all x

c) cos(Ncos^{-1}x), |x|≤1

cosh(Ncosh^{-1}x), |x|>1

d) None of the mentioned

View Answer

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

T

_{N}(x) = cos(Ncos

^{-1}x), |x|≤1

cosh(Ncosh

^{-1}x), |x|>1

2. What is the formula for chebyshev polynomial T_{N}(x) in recursive form?

a) 2T_{N-1}(x)- T_{N-2}(x)

b) 2T_{N-1}(x)+ T_{N-2}(x)

c) 2xT_{N-1}(x)+ T_{N-2}(x)

d) 2xT_{N-1}(x)- T_{N-2}(x)

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

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

T

_{N}(x)= 2xT

_{N-1}(x)- T

_{N-2}(x), N ≥ 2.

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

a) 1

b) 0

c) -1

d) 2

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

For a degree 0 chebyshev filter, the polynomial is obtained as

T

_{0}(x)=cos(0)=1

a) 1

b) x

c) -1

d) -x

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

For a degree 1 chebyshev filter, the polynomial is obtained as

T

_{0}(x)=cos(cos

^{-1}x)=x

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

a) 3x^{3}+4x

b) 3x^{3}-4x

c) 4x^{3}+3x

d) 4x^{3}-3x

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

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

T

_{3}(x)=2xT

_{2}(x)-T

_{1}(x)=2x(2x

^{2}-1)-x= 4x

^{3}-3x.

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

a) 16x^{5}+20x^{3}-5x

b) 16x^{5}+20x^{3}+5x

c) 16x^{5}-20x^{3}+5x

d) 16x^{5}-20x^{3}-5x

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

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

T

_{5}(x)=2xT

_{4}(x)-T

_{3}(x)=2x(8x

^{4}-8x

^{2}+1)-( 4x

^{3}-3x )= 16x

^{5}-20x

^{3}+5x.

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

a) True

b) False

View Answer

Explanation: 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

^{-1}x), |x|≤1

a) Even functions

b) Odd functions

c) Exponential functions

d) Logarithmic functions

View Answer

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

9. What is the value of T_{N}(0) for even degree N?

a) -1

b) +1

c) 0

d) ±1

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

For x=0, we have T

_{N}(0)=cos(Ncos

^{-1}0)=cos(N.π/2)=±1 for N even.

10. T_{N}(-x)=(-1)^{N}T_{N}(x)

a) True

b) False

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

=> T

_{N}(-x)= cos(Ncos

^{-1}(-x))= cos(N(π-cos

^{-1}x))= cos(Nπ-Ncos

^{-1}x)= (-1)

^{N}cos(Ncos

^{-1}x)= (-1)

^{N}T

_{N}(x)

Thus we get, T

_{N}(-x)=(-1)NT

_{N}(x).

**Sanfoundry Global Education & Learning Series – Digital Signal Processing.**

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