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

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This set of Antennas Multiple Choice Questions & Answers (MCQs) focuses on “Adaptive Array – Chebyshev Polynomials Fundamentals”.

1. Which of the following is holds true for the Chebyshev polynomial?
a) Tm(-x) = (-1)m Tm(x)
b) Tm(-x) = (-1)m Tm-1(1/x)
c) Tm(x) = (-1)m Tm(1/x)
d) Tm(-x) = (-1)m Tm-1(x)

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
There fore Tm(-x)=(-1)m Tm (x) holds true.

2. The recurrence relation for the Chebyshev polynomial is __________
a) Tm(z) = 2zTm-1(z) – Tm-2(z)
b) Tm (z) = Tm-1(z) – 2zTm-2(z)
c) Tm(z) = 2zTm(z) – Tm-2(z)
d) Tm(z) = 2zTm(z) – Tm-1(z)

Explanation:The Chebyshev polynomial of any order m can be derived from the recursive formula. This is one of the main features of the Chebyshev polynomial. The recurrence relation for the Chebyshev polynomial is
Tm(z) = 2zTm-1(z) – Tm-2(z).

3. The value of Tm(0) for every odd value of m is _____________
a) 0
b) 1
c) -1
d) m

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
For m=odd; Tm(0) = 0
For m=even; Tm(0) = (-1)m/2

4. What is the value of Tm (0) when m is an even number?
a) 0
b) (-1)m
c) -1
d) (-1)m/2

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
For m=even; Tm(0) = (-1)m/2

5. The value of T0(100) is _____
a) 1
b) 100
c) 10000
d) 200

Explanation:The Chebyshev polynomial is given by
T0(x)=1 m=0
⇨ T0(100)=1.

6. The value of Tm(-1)=_______
a) (-1)m
b) (-1)m-1
c) (-1)m/2
d) (-1)2m

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
ThereforeTm(-1)=(-1)m.

7. The value of Tm(z) in the range between -1 to 1 is ____
a) -1 to 1
b) 1
c) 0
d) 0 to infinity

Explanation: The polynomial oscillates between -1 to 1 with amplitude varying from -1 to 1. Hence this is used to get the constant desired side lobe for any order within the desired range. The array factor of the Chebyshev array is approximated to the Chebyshev polynomial.

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