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2 Combinational Circuit Design

2.4 Circuit Simplification and Minimization

When a Boolean Function is realized as a canonical Normal Form (cDNF or cCNF) normally a large number of gates is necessary with a large number of gate inputs.

A simplified circuit implementation is therefore required. The following considerations indicate how such a reduction in circuit complexity can be achieved.

Seen from the Boolean Algebra viewpoint this shows that the creation of larger areas means nothing else but repeatedly applying the equation

. (2.7)

A Boolean function whose expression contains combined areas of this type does of course not present a canonical normal form anymore, but just a conjunctive (CNF) or disjunctive normal form (DNF).


The following example demonstrates on the K-Map how these combined fields can be formed:

Figure 2.21: Forming combined areas.

In this example the following 1-formations are possible:

Taking into consideration these field combinations, the function will be:

(2-tier) (2.8)

or, introducing the function XOR:

(3-tier). (2.9)

Obviously this example does not allow the formation of any 0-areas, so that only the realization as a cKNF is possible:

. (2.10)


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