05-KarnaughMaps (1)

8/16/2019 05-KarnaughMaps (1)

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Karnaugh Maps for

Simplication

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Boolean algebra helps us simplify expressions

and circuits

Karnaugh Map: A graphical technique forsimplifying a Boolean expression into eitherform: minimal sum of products (MSP) minimal product of sums (MPS)

oal of the simpli!cation" #here are a minimal number of product$sum terms %ach term has a minimal number of literals

&ircuit'ise this leads to a minimal to'le*elimplementation

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A to'*ariable function has four possible minterms"+e can re'arrangethese minterms into a Karnaugh map

,o e can easily see hich minterms containcommon literals Minterms on the left and right sides contain y- and y

respecti*ely Minterms in the top and bottom ros contain x- and x

respecti*ely

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x y minterm

0 0 x’y’

0 1 x’y

1 0 xy’

1 1 xy

Y

0 1

0 x’y’ x’yX

1 xy’ xy

Y0 1

0 x’ y’ x’ yX

1 x y’ x y

Y’ Y

X’ x’ y’ x’ y

X x y’ x y


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.magine a to'*ariable sum of minterms:

x-y- / x-y

Both of these minterms appear in the top roof a Karnaugh map hich

means that they both contain the literal x-

+hat happens if you simplify this expressionusing Boolean algebra0

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x’y’ + x’y = x’(y’ + y) [ Distributive ]= x’ • 1 [ y + y’ = 1 ]= x’ [ x • 1 = x ]

Y

x’y’ x’yX xy’ xy


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Another example expression is x-y / xy Both minterms appear in the right side here y is

uncomplemented

#hus e can reduce x-y / xy to 1ust y

2o about x-y- / x-y / xy0 +e ha*e x-y- / x-y in the top ro corresponding to x-

#here-s also x-y / xy in the right side correspondingto y

#his hole expression can be reduced to x- / y

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Y

x’y’ x’y

X xy’ xy

Y

x’y’ x’y

X xy’ xy


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3or a three'*ariable expression ith inputs

x y 4 the arrangement of minterms is more tric5y:

Another ay to label the K'map (usehiche*er you li5e):

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Y

x’y’z’ x’y’z x’yz x’yz’

X xy’z’ xy’z xyz xyz’

Y

m0 m1 m3 m2

X m4 m5 m! m6

Y

00 01 11 10

0 x’y’z’ x’y’z x’yz x’yz’X

1 xy’z’ xy’z xyz xyz’

Y

00 01 11 10

0 m0 m1 m3 m2X

1 m4 m5 m! m6


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+ith this ordering any group of 6 7 or 8 ad1acent squares onthe map

contains common literals that can be factored out

9Ad1acency includes rapping around the left and right sides:

+e-ll use this property of ad1acent squares to do oursimpli!cations"

!

x’y’z + x’yz= x’z(y’ + y)= x’z • 1= x’z

x’y’z’ + xy’z’ + x’yz’ + xyz’= z’(x’y’ + xy’ + x’y + xy)= z’(y’(x’ + x) + y(x’ + x))= z’(y’+y)

= z’

Y



Y




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+e can !ll in the K'map directly from a truth

table #he output in ro i of the table goes into square

mi of the K'map

;emember that the rightmost columns of the K'

map are 9sitched

"

Y

m0 m1 m3 m2

X m4 m5 m! m6

x y z #(x$y$z)

0 0 0 0

0 0 1 10 1 0 0

0 1 1 0

1 0 0 0

1 0 1 1

1 1 0 1

1 1 1 1

Y

0 1 0 0

X 0 1 1 1


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#he most di=cult step is grouping togetherall the >s in the K'map Ma5e rectangles around groups of one to four

or eight >s

All of the >s in the map should be included in atleast one rectangle

?o not include any of the @s

%ach group corresponds to one product term

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Y

0 1 0 0

X 0 1 1 1


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Ma5e as fe rectangles as possible tominimi4e the number of products in the!nal expression"

Ma5e each rectangle as large as possible

to minimi4e the number of literals in eachterm"

;ectangles can be o*erlapped if that

ma5es them larger"

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et-s consider simplifying f(xy4) xy / y-4

/ x4


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.n our example e can rite f(xy4) into equi*alent ays

.n either case the resulting K'map isshon belo

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Y



#(x$y$z) = x’y’z + xy’z + xyz’ + xyz

Y

m0 m1 m3 m2

X m4 m5 m! m6

#(x$y$z) = m1 + m5 + m6 + m!

Y

0 1 0 0

X 0 1 1 1


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Simplify the sum of minterms m> / mF / mG / mH

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Y

X

Y

m0 m1 m3 m2

X m4 m5 m! m6


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2ere is the !lled in K'map ith all groups

shon #he magenta and green groups o*erlap hich

ma5es each of them as

large as possible

Minterm mH is in a group all by its lonesome

#he !nal MSP here is x-4 / y-4 / xy4-

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Y

0 1 1 0

X 0 1 0 1


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#here may not necessarily be a unique MSP" #he K'map beloyields to

*alid and equi*alent MSPs because there are to possibleays toinclude minterm mI

;emember that o*erlapping groups is possible as shon abo*e

1!

Y

0 1 0 1

X 0 1 1 1

y’z + yz’ + xy y’z + yz’ + xz

Y

0 1 0 1

X 0 1 1 1

Y

0 1 0 1

X 0 1 1 1


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+e can do four'*ariable expressions tooE #he minterms in the third and fourth columns and in the third

and fourth ros are sitched around" Again this ensures that ad1acent squares ha*e common literals

rouping minterms is similar to the three'*ariable casebut: 6 7 8 or >H minterms


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1%

Y

m0 m1 m3 m2

m4 m5 m! m6m12 m13 m15 m14

X

'm" m% m11 m10

Y

(’x’y’z’ (’x’y’z (’x’yz (’x’yz’

(’xy’z’ (’xy’z (’xyz (’xyz’(xy’z’ (xy’z (xyz (xyz’

X

'(x’y’z’ (x’y’z (x’yz (x’yz’


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#he expression is already a sum of

minterms so here-s the K'map:

+e can ma5e the folloing groups resultingin the MSP x-4- / xy-4

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Y1 0 0 1

0 1 0 0

0 1 0 0X

'1 0 0 1

Ym0 m1 m3 m2

m4 m5 m! m6m12 m13 m15 m14

X

'm" m% m11 m10

Y

1 0 0 1

0 1 0 00 1 0 0

X

'1 0 0 1

Y

’x’ y’z’ ’x’y’z ’x’yz ’x’ yz’

’xy’z’ ’xy’z ’xyz ’xyz’xy’z’ xy’z xyz xyz’

X

'x’ y’z’ x’y’z x’yz x’ yz’


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21

= 0 = 1

Ym0 m1 m3 m2

m4 m5 m! m6

m12 m13 m15 m14X

'm" m% m11 m10

Ym16 m1! m1% m"m20 m21 m23 m22m2" m2% m31 m30

X

'm24 m25 m2! m26


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= 0 = 1

1 1

1 1 1

1

1 1

1

1 1

1 11

# = X’ *m(4$6$12$14$20$22$2"$30)+ ’'’Y’ *m(0$1$4$5)+ '’Y’’ *m(0$4$16$20)+ 'XY *m(30$31)+ ’'X’Y m11


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Maxterms are grouped to !nd minimal PoSexpression

x +y+z x+y+z’ x+y’+z’ x+y’+z

x’ +y+z x’+y+z’ x’+y’+z’ x’+y’+z

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00 01 11 10

0

1x

yz


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3(+J67G)

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x +y+z x+y+z’ x+y’+z’ x+y’+z

x’ +y+z x’+y+z’ x’+y’+z’ x’+y’+z

00 01 11 10

0

1

x

yz

0 0 1 0

0 0 1 1

00 01 11 10

0

1

x

yz

&('$X$Y$)= Y (X + )


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&('$X$Y$)= *m(0$1$2$5$"$%$10)

= , -(3$4$6$!$11$12$13$14$15)

00 00

0

0 0 0 0

&('$X$Y$)= ('’ + X’)(Y’ + ’)(X’ +

)

.r$

&('$X$Y$)= X’Y’ + X’’ + '’Y’

'/i/ ne is t/e minim ne


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&('$X$Y$)= , -(0$2$3$4$5$6)

= *m(1$!$"$%$10$11$12$13$14$15)

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1 1 1 1

1 1 1 1

&('$X$Y$)= ' + XY + X’Y’


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3ind a MSP for

f(xy4) Σm(@67G8>7>G) d(xy4) Σm(I>@>F)

#his notation means that input combinations xy4 @>>> >@>@ and>>@>

(corresponding to minterms mI m>@ and m>F) are unused"

2"

Y

1 0 0 1

1 1 x 0

0 x 1 1X

'1 0 0 x


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3ind a MSP for:

f(xy4) Σm(@67G8>7>G) d(xy4) Σm(I>@>F)

2%

Y

1 1

1 1 x

x 1 1 X'

1 x

#($x$y$z)= x’z’ + ’xy’ + xy


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K'maps are an alternati*e to algebra for simplifying expressions

#he result is a MSP$MPS hich leads to a minimal to'le*elcircuit

.t-s easy to handle don-t'care conditions K'maps are really only good for manual simpli!cation of small

expressions"""

#hings to 5eep in mind:

;emember the correct order of minterms$maxterms on the K'map +hen grouping you can rap around all sides of the K'map and

your groups can o*erlap Ma5e as fe rectangles as possible but ma5e each of them as

large as possible" #his leads to feer but simpler product terms #here may be more than one *alid solution

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05-KarnaughMaps (1)