Approximation Theory for Filter-Networks

This paper presents a general design theory for filter-networks constructed on the basis of theory of the Abelian Integral. An ideal transmission function is defined to be an Anbelian Integral
w(\lambda)
with the following properties: 1)
Re w(\lambda) = u(\lambda) = A_k
in given regions
B_k
, 2)
\{u(\lambda) - \log | \lambda + a_i|\}
is regular at any given point
a_i
, and 3) otherwise
u(\lambda)
is a harmonic function. The application of appropriate analytic transformation techniques to
w(\lambda)
leads to a generalized characteristic function
\phi(\lambda))
. All kinds of realizable transfer functions can be derived by the use of linear transformations of
\{\phi(\lambda)\}^2
with respect to
\lambda^2
. Further, this treatise gives the design methods for filter-networks having one or two pass bands, in which they exhibit the Tchebycheff performance. Besides the method is also established to synthesize a reactance band-pass network with the order N using only
[(N - 1)/2]
coils. Three design examples, such that the band-pass filter comprising only 6 coils for
N = 13
and 4 coils for
N = 10
, and the double-pass-band filters with
N = 8
and 4 coils, are described in quite detail. All of the examples have the Tchebycheff performance in their pass bands.