passive network with transcendental poles or zeros

Apr 14, 2011 20 Replies

Just to add, I think they are called "Bessel" filters because of Storch.

H(s) =3D e^(-s) =3D 1/(cosh s + sinh s) =3D (1/sinh s)/( cosh s/sinh s + 1 )

cosh s =3D 1 + s^2/2! + s^4/4! + ... sinh s =3D s + s^3/3! + s^5/5! + ...

Continued fraction expansion of cosh s/sinh s yields:

cosh s/sinh s =3D 1/s + (3/s)||((5/s)||((7/s)||((....

truncating at n=3D3, for example, yields the approximation

cosh s/sinh s ~=3D~ (6s^2 + 15)/(s^3 + 15s)

Adding the numerator and denominator and inserting into the orginal, yields:

H(s) =3D e^(-s) ~=3D~ B_3(s) =3D K/((s^3 + 6s^2 + 15s + 15)

That is the denominator for a bessel LP of order 3. It works the same way for other orders. The polynomials Storch found via this procedure were apparently Bessel type.

Kind of ingenious. I just copied this out of Humpherys. It's in Su, too, Weinburg too, and probably most of them.

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