Simplification Using K -maps- Overviewedutalks.org/downloads/digtaltheory.pdf · Simplification...
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Simplification Using K-maps- Overview SUHAIL T A LECTURER IN ECE AL- AMEEN ENGG. COLLEGE, SHAORANUR www.edutalks.org
Transcript of Simplification Using K -maps- Overviewedutalks.org/downloads/digtaltheory.pdf · Simplification...
Simplification Using K-maps- Overview
SUHAIL T A
LECTURER IN ECE
AL- AMEEN ENGG. COLLEGE,
SHAORANUR
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Simplification Using K-maps- Overview Karnaugh-map (K-map) is an abstract form of Venn diagram, organised
as a matrix of squares, where
adjacent squares always differ by just one literal (so that the unifying theorem may apply: a + a' = 1)
Note: Literal - A variable in its complemented or uncomplemented form K- map is based on the Theorem:
A + A' = 1
In a K-map, each cell containing a ‘1’ corresponds to a minterm of a given function F.
Each group of adjacent cells containing ‘1’ (group must have size in powersof twos: 1, 2, 4, 8, …) then corresponds to a simpler product term of F. Grouping 2 adjacent squares eliminates 1 variable grouping 4 squares eliminates 2 variables grouping 8 squares eliminates 3 variables, and so on. In general,
grouping 2n squares eliminates n variables.www.edutalks.org
Simplification Using K-maps
Group as many squares as possible.The larger the group is, the fewer the number of literals in the
resulting product term.
Select as few groups as possible to cover all the squares (minterms) of the function.The fewer the groups, the fewer the number of product terms in
the minimized function.
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Simplification Using K-maps Possible valid groupings of a 4-variable K-map include:
1
11
1
1
1
1
1
P
1
11
1 1
111
P
1
11
1
P
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Simplification Using K-maps Groups of minterms must be
(1) rectangular, and (2) have size in powers of 2’s.
Otherwise they are invalid groups. Some examples of invalid groups:
1
11
1 1
111
1
1
O
1
1
1
1
1
1
Owww.edutalks.org
Simplest SOP Expressions
To find the simplest possible sum of products (SOP) expression from a K-map, you need to obtain:minimum number of literals per product term; andminimum number of product terms
This is achieved in K-map using bigger groupings of minterms where possible; and no redundant groupings
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Simplest SOP Expressions Use bigger groupings where possible.
11 1
111
O
11 1
111 P
No redundant groups:
1
1
1
11
1
P1
1
1
1
1
11
1
O1
1
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Incompletely Specified Functions In practice, there are applications where the function is not specified
for certain combinations of the input variables. For example, in the 4-
bit BCD code for the decimal digits, the outputs are unspecified for the
input combinations 1010-1111.
Functions that have unspecified outputs for some input combinations
are called incompletely specified functions.
The unspecified minterms of a function are called the don’t-care
conditions, or simply the don’t-cares, and are denoted as x’s.
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Incompletely Specified Functions
If N1 output does not generate all possible combination of A,B,C, the output of N2(F) has ‘don’t care’ values.
Truth Table with Don’t Cares
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Incompletely Specified Functions Minterm expansion for incompletely specified function
∑∑ += )6,1()7,3,0( dmF
∏ ∏= )6,1()5,4,2( DMF
Don’t Cares
Maxterm expansion for incompletely specified function
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Don’t-care Conditions
In certain problems, some outputs are not specified.
These outputs can be either ‘1’ or ‘0’.
They are called don’t-care conditions, denoted by X (or
sometimes, d).
Don’t-care conditions can be used to help simplify
Boolean expression further in K-maps.
They could be chosen to be either ‘1’ or ‘0’, depending on
which gives the simpler expression. www.edutalks.org
Implementation with AND/OR/NOT & NAND gates
Digital circuits are more frequently constructed with
NAND/NOR gates than with AND/OR/NOT gates due to ease of
fabrication. For example, in gate arrays, only NAND (or NOR)
gates are used.
The conversion process from an expression/schematic with
AND,OR, and NOT gates to one with only NAND or NOR gates
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Implementation with AND/OR/NOT & NAND gates
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Implementation with AND/OR/NOT & NAND gates
Example
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Implementation with AND/OR/NOT & NAND gates
Implementation
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Example 2: Truth Table
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Karnaugh Map
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Simplification Using K-Maps
The complement of a function is represented in the map
by the squares not marked by 1’s (they usually are
marked by 0’s)
The complement of f ’ gives f in POSwww.edutalks.org
Simplification Using K-Maps
Example
Simplify the following Boolean function in the POS form:
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