Takahasi's Results on Tchebycheff and Butterworth Ladder Networks

A Japanese paper published by H. Takahasi in 1951 gives formulas for element values of ladder networks with Tchebycheff characteristics. For a resistance-terminated low-pass ladder with a series inductance as the first reactance, these formulas are given by
L_1= \frac{R_{1} s_{1}}{(k-k^{-1}) - (h - h^{-1})}
K_{r,r+1}=\frac{s_{2r-1}s_{2r+1}}{b_r} (r = 1, 2, \cdots , n-1 )
where
R_1
, is the input resistance and
K_{r,r+1}= L_{r} C_{r+1}
if
r
is odd;
C_{r}L_{r+1}
if
r
is even.
b_r = \xi^{2} - c{2r}\xi \eta + \eta^{2} + s_{2r}^2
s_r = 2 \sin \frac{\pi r}{2n}
c_r = 2 \cos \frac{\pi r}{2n}
\xi = k - k^{-1}
\eta = h - h^{-1}
The positive constants
k
and
h
are related to the zeros and poles of the squared magnitude of the reflection coefficient
|\rho (j \omega )|^{2}
; more specifically, the poles are
\alpha_{2m+1}= k \epsilon^{2m+1} + k^{-1} \epsilon^{-(2m+1)}
for
m = 0, 1, 2, \cdots , 2n - 1
and
\epsilon = e^{\frac{j \pi}{2n}}
, and the zeros are
\beta_{2m+1}= h \epsilon^{2m+1} + h^{-1} \epsilon^{-(2m+1)}
for
m = 0, 1,2, \cdots , 2n -1
. The final reactance can also be related to the output resistance RP so that the elements can be determined by starting from either the first or last element:
L_n = \frac{R_{2} s_{1}}{(k-k^{-1})+(h-h^{-1})}
if
n
is odd
C_n = \frac{ s_{1}}{R_{2}[(k-k^{-1})+(h-h^{-1})]}
if
n
is even The formulas for the Butterworth characteristic are derived from those for the Tehebycheff characteristic by a limiting process. A proof is also furnished by Takahasi. These results, which have been previously unknown to most network theorists, anticipate the work of many authors in the field. In the present paper Takahasi's results are given and his concise proof is expanded so that its potential application to presently unsolved problems may be more easily investigated.