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Test: Chebyshev Filters - 1

10 Questions MCQ Test Digital Signal Processing | Test: Chebyshev Filters - 1

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Attempt Test: Chebyshev Filters - 1 | 10 questions in 10 minutes | Mock test for Electrical Engineering (EE) preparation | Free important questions MCQ to study Digital Signal Processing for Electrical Engineering (EE) Exam | Download free PDF with solutions
QUESTION: 1

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

Solution:

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.

QUESTION: 2

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

Solution:

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.

QUESTION: 3

What is the value of chebyshev polynomial of degree 0?

Solution:

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.

QUESTION: 4

What is the value of chebyshev polynomial of degree 1?

Solution:

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.

QUESTION: 5

What is the value of chebyshev polynomial of degree 3?

Solution:

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.

QUESTION: 6

What is the value of chebyshev polynomial of degree 5?

Solution:

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.

QUESTION: 7

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

Solution:

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.

QUESTION: 8

Chebyshev polynomials of odd orders are:

Solution:

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

QUESTION: 9

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

Solution:

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.

QUESTION: 10

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

Solution:

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). Use Code STAYHOME200 and get INR 200 additional OFF Use Coupon Code

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