Design of the Chebyshev I filter A typical valuable response specification is given below (shown for N = 5 and N = 6). The magnitudes at the critical frequencies Ω1 and Ω2 are A and B, respectively. Typically Ω1 is in the pass band or is the edge of the pass band and Ω2 is in the stop band or is the edge of the stop band. In relates of the log-magnitude the analog filter specifications are as below. Note that (20 log A) = K1 dB and (20 log B) = K2 dB. If A and B are less than 1, K1 and K2 are negative.

The magnitude operation of the Nth order Chebyshev I filter is provided by

where ε has to do with pass band ripple and CN(x) is the Nth order Chebyshev cosine polynomial shown as

Chebyshev polynomials are also defined by the recursion formula

C_N ( x) = 2xC_{N -1} (x) - C_{N -2} (x)

with C₀ ( x) =1 and C₁ ( x) = x. Using this recursion formula we get, for N = 2, C₂ ( x) = 2.2xC₁ (x) - C₀ (x) = 2x² - 1.

[Aside If the frequencies are normalized, that is, for a normalized filter, Ω₁ = 1 rad/sec and the value characteristic of the N^th order Chebyshev I filter is provided by

End of Aside]

As a consequence, on the vertical axis (Ω = 0) the magnitude curve starts at |H(j0)| = 1 for odd N and at 

At Ω = Ω₁ we have C_N(1) = cos(N cos ^-11) = cos(0) = 1 for all N. The related magnitude is

This equation is needed to calculate ε from the provided |H(jΩ)|.

The order, N, of the filter is provided by

The symbol || seems that the calculated result is rounded to the next larger integer. For example, if N = 3.2 by the above computation then it is rounded up to 4, and the order of the needed filter is N = 4. In such a type the resulting filter could exceed the specification at both Ω₁ and Ω₂.

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