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Test: Chebyshev Filters - 1


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10 Questions MCQ Test Signals and Systems | Test: Chebyshev Filters - 1

Test: Chebyshev Filters - 1 for Electrical Engineering (EE) 2022 is part of Signals and Systems preparation. The Test: Chebyshev Filters - 1 questions and answers have been prepared according to the Electrical Engineering (EE) exam syllabus.The Test: Chebyshev Filters - 1 MCQs are made for Electrical Engineering (EE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Chebyshev Filters - 1 below.
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Test: Chebyshev Filters - 1 - Question 1

Which of the following defines a chebyshev polynomial of order N, TN(x)? 

Detailed Solution for Test: Chebyshev Filters - 1 - Question 1

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

Test: Chebyshev Filters - 1 - Question 2

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

Detailed Solution for Test: Chebyshev Filters - 1 - Question 2

Explanation: 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 chebyshev polynomial using the following recursive formula
TN(x)= 2xTN-1(x)- TN-2(x), N ≥ 2.

Test: Chebyshev Filters - 1 - Question 3

What is the value of chebyshev polynomial of degree 0?

Detailed Solution for Test: Chebyshev Filters - 1 - Question 3

Explanation: 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.

Test: Chebyshev Filters - 1 - Question 4

 What is the value of chebyshev polynomial of degree 1?

Detailed Solution for Test: Chebyshev Filters - 1 - Question 4

Explanation: 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.

Test: Chebyshev Filters - 1 - Question 5

 What is the value of chebyshev polynomial of degree 3? 

Detailed Solution for Test: Chebyshev Filters - 1 - Question 5

Explanation: 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 3, we obtain
T3(x)=2xT2(x)-T1(x)=2x(2x2-1)-x= 4x3-3x.

Test: Chebyshev Filters - 1 - Question 6

 What is the value of chebyshev polynomial of degree 5? 

Detailed Solution for Test: Chebyshev Filters - 1 - Question 6

Explanation: 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.

Test: Chebyshev Filters - 1 - Question 7

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

Detailed Solution for Test: Chebyshev Filters - 1 - Question 7

Explanation: 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.

Test: Chebyshev Filters - 1 - Question 8

 Chebyshev polynomials of odd orders are:

Detailed Solution for Test: Chebyshev Filters - 1 - Question 8

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

Test: Chebyshev Filters - 1 - Question 9

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

Detailed Solution for Test: Chebyshev Filters - 1 - Question 9

Explanation: 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.

Test: Chebyshev Filters - 1 - Question 10

 TN(-x)=(-1)NTN(x) 

Detailed Solution for Test: Chebyshev Filters - 1 - Question 10

Explanation: 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).

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