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Carlin's Broodband-Matching Method 219
power transfer function will then be derived, and its derivatives will be
obtained with respect to the piecewise linear fit parameters. An objective
function for the Fletcher-Reeves optimizer (Section 5.4) will be furnished in
another BASIC language program. Finally, utilization of tbe optimal piecewise
linear fit to the required Zq real part to synthesize the matching network will
be described. Actually, this last step has been covered completely in Sections
2.5 and 3.5, so that only the connection between these procedures and Carlin's
method is required.
6.7.1. Piecewise Hilbert Transfonn. Blinchikoff and Zverev (1976, p. 76)
give the well-known Hilbert transform that determines the reactance function
from a given resistance function:
X(w) = If+oo R(y) dy.
'Tr -00 y-w
(6.110)
There is a similar function for the inverse transform. Bode (1945, p. 318) gives
a more useful form for analysis on linear frequency scales:
X(w) = I (OOdRlnly+wldY.
7T J o dy y-w
(6.111)
A restriction on these Hilbert transforms is that the function (impedance in
this case) must have minimum reactance. Restrictions on transfer functions
are similar. Guillemin (1957, p. 301) shows that the Jlbase lag will be least for
any transfer function magnitude if its zeros are restricted to the left-half plane.
The poles are similarly restricted for passive networks. Such functions are thus
called minimum phase; in general, they are associated with ladder (single
signal path) networks that do not contain delay equalizer (bridge) sections.
In this case, it is convenient to presume that the equation Riw) = Re(Zq) in
Figure 6.1 has the piecewise linear form
n
Rq(w)= L: rkak(w),
k=O
where the normalized linear interpolation function is
(6.112)
(6.113)
and a o = I. This linear interpolation function is easily visualized according to
Figure 6.38. The overall form of Rq is illustrated in Figure 6.39. Since this
form of resistance (Rq) will be integrated according to (6.111), it must assume
a zero value, beginning at some finite frequency. Therefore, an arbitrary but