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Fundamental Description of Boolean Functions
In order to simplify (minimize) a Boolean function using Karnaugh Diagrams (K-Maps), the combination (grouping) of fields (boxes, squares) is necessary. This process can be applied to the 0-fields or the 1-fields. Furthermore, to characterize these fields the arguments or their complements can be used.
As can be seen, a multitude of combinations can be distinguished. These possibilities will be summarized here.
Describing a Boolean function using 1-fields:
DNF (AND/OR-Form) - SOP |
||
Transformation using the "de Morgan" Theorem results in: | ||
(NAND-Form) |
Describing a Boolean function using the complement of 1-fields:
(OR/AND-Form) | ||
Transformation using the "de Morgan" Theorem results in: | ||
(NOR/OR-Form) |
Describing a Boolean function using 0-fields:
(AND/OR-Form) | ||
Transformation using the "de Morgan" Theorem results in: | ||
(NAND/AND-Form) |
Describing a Boolean function using the complement of 0-fields:
CNF (OR/AND-Form) - POS | ||
Transformation using the "de Morgan" Theorem results in: | ||
(NOR-Form) |
Considering two-level implementations (of circuits), the AND/OR-Form (DNF) and the OR/AND-Form
(KNF) are of special importance, because they represent very straightforward description forms.
Definition:
A multilevel network with an arbitrary number of inputs and outputs is called an AND/OR-network when on any arbitrary signal path between an input and an output AND and OR functions are alternatingly passed.
Example:
Figure 2.29: Typical AND-OR-Network.
Considering practical applications, circuits implemented with NAND or NOR gates, respectively, are more important than AND/OR networks. The main reason for this is a technological one: basic NAND and NOR gates can be manufactured more easily.
Therefore the transformation from AND/OR/NOT-circuits to NAND/NOR-circuits is of very high importance. This transformation follows again the "DeMorgan" Theorem and can therefore be formulated as a simple three-step process:
Transformation Example (two-level networks):
AND/OR → NAND:
(2.11) |
OR/AND → NOR:
(2.12) |
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