Test: Chebyshev Filters - 1

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
T_N(x) = cos(Ncos^-1x), |x|≤1
cosh(Ncosh^-1x), |x|>1.

Explanation: 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.

Explanation: 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.

Explanation: 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.

Explanation: 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 3, we obtain
T₃(x)=2xT₂(x)-T₁(x)=2x(2x²-1)-x= 4x³-3x.

Explanation: 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.

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^-1x), |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^-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.

Explanation: 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).