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Unformatted text preview: straightforward conversion to very nice gating networks which are
more desirable from most implementation points of view. In their purest, nicest form they
go into twolevel networks, which are networks for which the longest path through which
the signal must pass from input to output is two gates.
Conversion Between Canonical Forms
The complement of a function expressed as the sumofminterms equals the sumofminterms missing from the original function. This is because the original function is
expressed by those minterms that make the function equal to 1, while its complement is a
1 for those minterms for which the function is a 0. For example, the function
F (A, B, C) = (1,4, 5,6, 7) = nij +m4 +m5+m6 +m7
has a complement that can be expressed asf
F (A, B, C) = (0, 2, 3) = m0 +m2 +m3
Now, if we take the complement of F, by De Morgan's theorem we obtain F back in a
different form:
F = m0 +m2 +m3
= m0 • m2 • m3
= M0M2M3
= II (0, 2, 3)
The last conversion follows from the definition of minterms and maxterms as shown in
Figure 6.16. From the figure, it is clear that the following relation holds true:
That is, the maxterm with subscript/ is a complement of the minterm with the same
subscript/, and viceversa.
The last example has demonstrated the conversion between a function expressed in sumofminterms and its equivalent in productofmaxterms. A similar argument will show
that the conversion between the productofmaxterms and the sumofminterms is similar.
We now state a general conversion procedure: "To convert from one canonical form to another, interchange the symbol and list those
numbers missing from the original form."
For example, the function
F (x, y, z) = TI (0, 2, 4, 5)
is expressed in the productofmaxterms form. Its conversion to sumofminterms is:
F(x,y,z) = I(l,3,6,7)
Note that in order to find the missing terms, one must realize that the total number of
minterms or maxterms is always 2n, where n is the number of binary variables in the
function.
LOGIC GATES
All operations within a co...
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 Spring '14

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