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Which of the following defines a chebyshev polynomial of order N, T_{N}(x)?
Explanation: In order to understand the frequencydomain 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^{1}x), x≤1
cosh(Ncosh^{1}x), x>1.
What is the formula for chebyshev polynomial T_{N}(x) in recursive form?
Explanation: We know that a chebyshev polynomial of degree N is defined as
T_{N}(x) = cos(Ncos^{1}x), x≤1
cosh(Ncosh^{1}x), x>1
From the above formula, it is possible to generate chebyshev polynomial using the following recursive formula
T_{N}(x)= 2xT_{N1}(x) T_{N2}(x), N ≥ 2.
What is the value of chebyshev polynomial of degree 0?
Explanation: We know that a chebyshev polynomial of degree N is defined as
T_{N}(x) = cos(Ncos^{1}x), x≤1
cosh(Ncosh^{1}x), x>1
For a degree 0 chebyshev filter, the polynomial is obtained as
T_{0}(x)=cos(0)=1.
What is the value of chebyshev polynomial of degree 1?
Explanation: We know that a chebyshev polynomial of degree N is defined as
T_{N}(x) = cos(Ncos^{1}x), x≤1
cosh(Ncosh^{1}x), x>1
For a degree 1 chebyshev filter, the polynomial is obtained as
T_{0}(x)=cos(cos^{1}x)=x.
What is the value of chebyshev polynomial of degree 3?
Explanation: We know that a chebyshev polynomial of degree N is defined as
T_{N}(x) = cos(Ncos^{1}x), x≤1 cosh(Ncosh^{1}x), x>1
And the recursive formula for the chebyshev polynomial of order N is given as
T_{N}(x)= 2xT_{N1}(x) T_{N2}(x)
Thus for a chebyshev filter of order 3, we obtain
T_{3}(x)=2xT_{2}(x)T_{1}(x)=2x(2x^{2}1)x= 4x^{3}3x.
What is the value of chebyshev polynomial of degree 5?
Explanation: We know that a chebyshev polynomial of degree N is defined as
T_{N}(x) = cos(Ncos^{1}x), x≤1
cosh(Ncosh^{1}x), x>1
And the recursive formula for the chebyshev polynomial of order N is given as
T_{N}(x)= 2xT_{N1}(x) T_{N2}(x)
Thus for a chebyshev filter of order 5, we obtain
T_{5}(x)=2xT_{4}(x)T_{3}(x)=2x(8x^{4}8x^{2}+1)( 4x^{3}3x )= 16x^{5}20x^{3}+5x.
For x≤1, T_{N}(x)≤1, and it oscillates between 1 and +1 a number of times proportional to N.
Explanation: 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^{1}x), x≤1.
Explanation: Chebyshev polynomials of odd orders are odd functions because they contain only odd powers of x.
Explanation: We know that a chebyshev polynomial of degree N is defined as
T_{N}(x) = cos(Ncos^{1}x), x≤1
cosh(Ncosh^{1}x), x>1
For x=0, we have T_{N}(0)=cos(Ncos^{1}0)=cos(N.π/2)=±1 for N even.
Explanation: We know that a chebyshev polynomial of degree N is defined as
T_{N}(x) = cos(Ncos^{1}x), x≤1
cosh(Ncosh^{1}x), x>1
=> T_{N}(x)= cos(Ncos^{1}(x))= cos(N(πcos^{1}x))= cos(NπNcos^{1}x)= (1)^{N} cos(Ncos^{1}x)= (1)^{N}T_{N}(x)
Thus we get, T_{N}(x)=(1)NT_{N}(x).
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