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Characteristics of Prototype Analog Filters 411
The equiripple response of the Chebyshev filters is due to this polynomial
T N (x). Its key properties are (a) for 0 < x < 1, T N (x) oscillates between
−1 and 1, and (b) for 1 Ω c ), |H a (jx)| 2 decreases monotonically to 0.
• At x =Ω r , |H a (jx)| 2 = 1 A 2 .
To determine a causal and stable H a (s), we must find the poles of
H a (s)H a (−s) and select the left half-plane poles for H a (s). The poles of
H a (s)H a (−s) are obtained by finding the roots of
1+ɛ 2 T 2 N
( s
jΩ c
)
The solution of this equation is tedious if not difficult to obtain. It can be
shown that if p k = σ k + jΩ k , k =0,...,N − 1 are the (left half-plane)
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