250+ TOP MCQs on Chebyshev Filters – 1 & Answers

Digital Signal Processing Multiple Choice Questions on “Chebyshev Filters-1”.

1. Which of the following defines a chebyshev polynomial of order N, T_N(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
Answer: C
Clarification: 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^-1x), |x|≤1
cosh(Ncosh^-1x), |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)
Answer: D
Clarification: 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 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
Answer: A
Clarification: 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.

4. What is the value of chebyshev polynomial of degree 1?
A. 1
B. x
C. -1
D. -x
Answer: B
Clarification: 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.

5. What is the value of chebyshev polynomial of degree 3?
A. 3x³+4x
B. 3x³-4x
C. 4x³+3x
D. 4x³-3x
Answer: D
Clarification: 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.

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
Answer: C
Clarification: 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.

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
Answer: A
Clarification: 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.

8. Chebyshev polynomials of odd orders are _____________
A. Even functions
B. Odd functions
C. Exponential functions
D. Logarithmic functions
Answer: B
Clarification: 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
Answer: D
Clarification: 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.

10. T_N(-x)=(-1)^NT_N(x).
A. True
B. False
Answer: A
Clarification: 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).