# Antennas Questions and Answers – Adaptive Array – Chebyshev Polynomials Properties

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This set of Antennas Questions and Answers for Entrance exams focuses on “Adaptive Array – Chebyshev Polynomials Properties”.

1. Which of the following statement is true about the Chebyshev function Tm(x)?
a) It is a continuously increasing function after x=1
b) It is a continuously decreasing function after x=1
c) It is a continuously increasing function after x=0
d) It is a continuously decreasing function after x=0

Explanation: T0(x) = 1
T1(x) = x
T2(x) = 2x2-1
⇨ T3(x) = 4x3-3x
This is the chebyshev polynomial and it increases continuously after x=1

2. How many times the polynomial T5(x) crosses the x-axis between [-1, 1]?
a) 5
b) 4
c) 2
d) 6

Explanation: The polynomial Tm(x) crosses the x axis m times between -1 and 1. Given polynomial is T5(x).
M=5 therefore it crosses the axis 5 times between [-1, 1].

3. Which of the following statements is true?
a) The polynomials are unstable at interval [-1, 1]
b) The polynomials are marginally stable at interval [-1, 1]
c) The polynomial doesn’t oscillate at interval [-1, 1]
d) The polynomials crosses the axis m-1 times at [-1, 1]

Explanation: The polynomials oscillate between -1 and 1 interval. So they are either stable or marginally stable. The polynomial crosses the axis m times in the interval [-1, 1].

4. What is the possible level from the following for the minor lobe when the main beam level is at 50db and SLL at 10 db according to Chebyshev?
a) 40dB
b) 45dB
c) 50dB
d) 80dB

Explanation: The minor lobes are present below the main beam level at a value 1/SLL. SLL is the side lobe level.
Possible level for the minor lobe is 50-10=40dB

5. The condition for the existence of the main lobe according to the Chebyshev is _________
a) |x| > 1
b) |x| < 1
c) |x| = 0
d) 2|x| > 1

Explanation: The condition for the existence of the main lobe according to the Chebyshev is |x| > 1.
The condition for the existence of the minor lobe according to the Chebyshev is |x| < 1.

6. The condition for the existence of the main lobe according to the Chebyshev is _________
a) |x| > 1
b) |x| < 1
c) |x| = 0
d) 2|x| > 1

Explanation: The condition for the existence of the main lobe according to the Chebyshev is |x| > 1.
The condition for the existence of the minor lobe according to the Chebyshev is |x| < 1.

7. Which of the following statements regarding Chebyshev polynomial is true?
a) The polynomial Tm(x) is symmetric for m = even
b) The polynomial Tm(x) is symmetric for m = odd
c) The polynomial Tm(x) is anti-symmetric for m = even
d) The polynomial Tm(x) is symmetric for m = even and odd

Explanation: The polynomial Tm(x) is symmetric for m=even and is anti-symmetric for m=odd.
For m=even, at x=0 it is 1. For m=odd, at x=0 it is 0.

8. Which of the following properties of Chebyshev polynomial is false?
a) The minor lobes have unequal amplitudes
b) The polynomial Tm(x) is symmetric for m = even
c) The polynomial Tm(x) crosses the x axis m times between -1 and 1
d) Minor lobes exists for |x| < 1

Explanation: The minor lobes have equal amplitudes. The polynomial Tm(x) crosses the x axis m times between -1 and 1.
Minor lobes exist for |x| < 1 and major lobes exist for |x| > 1.

9. All the polynomials of the order m pass through the point ____________
a) (1, 1)
b) (0, 0)
c) (0, 1)
d) (-1, 0)

Explanation: All the polynomials of the order m of the chebyshev pass through the point (x, Tm(x))
= (1, 1).
T0(x) = 1
T1(x) = x
T2(x) = 2x2-1
T3(x) = 4x3-3x

10. All the nulls occur at (-1, 1) in the Chebyshev polynomial.
a) True
b) False

Explanation: All the nulls in the Chebyshev polynomial occur in the range -1≤ x≤1.
(-1, 1) here is representing a point. So it is false.

11. As the order of the polynomial increases, the slope becomes steeper.
a) True
b) False

Explanation: The Chebyshev polynomial is given by
T0(x) = 1 m=0
T1(x) = x m=1
T2(x) = 2x2-1 m=2
T3(x) = 4x3-3x m=3
As the order of the polynomial increases, the slope becomes steeper.

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