A Symbolic Analysis of Relay and Switching Circuits

35
From the symetry of thi s
of expandi'n~ "vhich v~ill
functlon there is no other Vv8y
reduce the number of elements.
If the r11nctions ere stlbstituted in the other order,
we
~et:
This oomvletes the proof that these functions require
the most elements- To show that each requires (3_2 n - 1 _2)
elements, let the number of elements required be denoted
by s(n). Then from (19) we ~at the differenoe
equation:
s(n) : 2s(n-l) + 2
With s(2) = 4. This is linear, ~v1th constant coefficients,
end may be solved by the usual me thods (5).
The solution is:
n-l
s(n) = 3.2 -2
a s may be ~a s Uy verlf1 ad by subst1 tu tin g in the dlfterence
equation and toundary condition.
Note t"hat the above only apPlies to 8
ssriesparallel
realization. In a later section it Will be
n
shown that the fllnction~Xk and its negative may be
1
reslized with 4:(n-l) elements usin~ 8 more p:eneral
type of circuit.
The fUnction requiring the most
elements usin~
any type of circuit has not as yet
been determined.