unit 5 karnaugh map - national chiao tung universitymoblie123.cm.nctu.edu.tw/101 logical design/unit...

Department of Communication Engineering, NCTU 1

Unit 5 Karnaugh Map


5.1 Minimum Form of SwitchingFunctions


Logic Design Unit 5 Karnaugh Map Sau-Hsuan Wu

Switching functions can generally be simplified by usingBoolean algebraic techniques

Two problems arise when algebraic procedures are used The procedures are difficult to apply in a systematic way Difficult to tell when a minimum solution is arrived

Remedies systematic way for circuit simplifications Karnaugh map method (for small number of variables) Quine-McCluskey method (for large combinational network)



Karnaugh map techniques lead directly to minimum costtwo-level circuits which are categorized into two classes Sum-of-products: a group of AND gates feed a OR gate Product-of-sums: a group of OR gates feed a AND gate

A minimum sum-of-products (SOP) expression for afunction is defined as a sum of products which has A minimum number of terms A minimum number of literals of all those expressions

which have the same minimum number of terms

The min SOP corresponds directly to a circuit that has A minimum number of gates A minimum number of gate inputs



Given a midterm expansion, the min SOP form can oftenbe obtained by the following procedures Combine terms by repeatedly applying XY'+XY=X Eliminate redundant terms by using the consensus theorem

or other theorems

E.g.



A minimum product-of-sums (POS) expression is definedas a POS which has A minimum number of factors A minimum number of literals of all those expressions

which have the same minimum number of factors

Given a maxterm expansion, the min POS can often beobtained by combining terms using (X+Y)(X+Y') = X


5.2 Two- and Three-VariableKarnaugh Maps



The Karnaugh map of a function specifies the value of thefunction for every combination of values of theindependent variables

A two-variable Karnaugh map



Fig. 5-1 shows the truth table for a function F and thecorresponding Karnaugh map

Each entry of a Karnaugh map is a minterm We can read the minterms from the map just like we can

read them from the truth table



Fig. 3 shows a 3-variable truth table and thecorresponding Karnaugh map



Fig. 5-3 shows the location of the minterms on a three-variable map(a) binary notation (b) decimal notation



A map can be obtained by placing 1’s in the squareswhich correspond to the minterms of F, or by placing 0’sin the squares which correspond to the maxterms of F

Fig. 5-4 shows the plot of

( , , ) (1,3,5) (0, 2, 4,6,7)F a b c m M



Fig. 5-5 illustrates how product terms can be plotted onKarnaugh maps E.g. b’c is 1 when b=1 or c=0, so 1’s are entered in the two

squares in the bc=10 row



It is unnecessary to expand a function to its minterm formbefore plotting it on a map.

E.g.



Terms which differ in only one variable in adjacentsquares on the map can be combined using the theoremXY’+XY=X

E.g.



The Karnaugh map can also illustrate the basic theoremsof Boolean algebra

Fig. 5-8 illustrates the consensus theoremxy+x’z+yz=xy+x’z



If a function has two or more min SOP forms, all of theseforms can be determined from a map

E.g.


5.3 Four-Variable Karnaugh Maps



The location of minterms on a 4-variable map



ExampleThe Karnaugh map of f(a,b,c,d) = acd+a‘b+d’



The simplification of a 4-variable expression on a map Minterms are combined in groups of two, four, or eight



Don’t care minterms are indicated by X’s X’s are only used in they will simplify the resulting

expression



A minimum product of sums (POS) for f’can bedetermined by looping the 0’s on a map of f

The complement of the min SOP for f’is then the minPOS for f

f '=y'z+wxz'+w'xyf =(y+z')(w'+x'+z)(w+x'+y')


5.4 Determination of MinimumExpressions Using Essential Prime

Implicants



A single 1 or any group of 1’s which can be combinedtogether on a map of the function F represents a productterm which is called an implicant of F

A product implicant is called a prime implicant if itcannot be combined with other term to eliminate avariable



E.g. In Fig. 5-15, a‘b’c, a‘cd‘及 ac‘are prime implicants.On the other hand, ab'c', abc', a'b'c'd‘are not



A SOP expression containing a term which is not a primeimplicant cannot be minimum

Because all of the prime implicants of a function aregenerally not needed in forming the minimum SOP, asystematic procedure for selecting prime implicants isneeded

If a minterm is covered by only one prime implicant, thatprime implicant is said to be essential, and it must beincluded in the minimum SOP



If the given minterm and all of the 1’s adjacent to it arecovered by a single term, then that term is an essentialprime implicant



Flow chart fordetermining aminimum SOP usinga Karnaugh Map



If a don’t care minterm is present Do not check it to see if it is covered by one or more

implicants When checking a 1 for adjacent 1’s, treat the adjacent don’t

cares as if they were 1’s


5.5 Five-Variable Karnaugh Maps



A five-variable map can be constructed in 3 dimensionsby placing one four-variable map on top of a second one Terms in the bottom layer are numbered 0 through 15 Terms in the top layer are numbered 16 through 31 Terms in the top or bottom layer combine just like terms on

a four-variable map Two terms in the same square of the top and bottom layers

can be combined Each term can be adjacent to exactly five other terms, four

in the same layer and one in the same square of the otherlayer



When checking for adjacencies, each term should bechecked against the five possible adjacent squares

E.g.

m0 P1 , m8 P3 1 2 3 4p p p p

AB CF or

B CDACD A BCEA B D ABE



E.g. m16 P1 , m3 P2 , m8 P3 , m14 P4

51 2 3 4 pp p p p

C D EF or

AC EB C D B C E A C D A BCD ABDE



Apply consensus theorem in Karnaugh maps Add redundant terms by using the consensus theorem Eliminate terms again using the consensus theorem

E.g. F=ABCD+B'CDE+A'B'+BCE'

F=A'B'+BCE'+ACDE



Two alternative forms for five-variable Karnaugh Maps

Two maps side-by-side One is the mirror image of another

unit 5 karnaugh map - national chiao tung universitymoblie123.cm.nctu.edu.tw/101 logical design/unit...

Documents