Application of Boolean Algebra / Minterm and Maxterm Expansion

Unit 4 1

Unit 4

Application of Boolean Algebra / Minterm and Maxterm Expansion

Unit 4 2

Outline

․Conversion of sentences to Boolean equations ․Truth table-based logic design ․Minterm and maxterm expansions ․Incompletely specified functions ․Binary adders and subtracters ․Speeding up integer additions ․Binary multiplication

Unit 4 3

..6

:

'

'

A

D

CB

Z

closednotiswindowthe

andmpafterisitorclosednotisdoorthe

andonisalarmofpowertheiffringwillalarmTheEx

'' CDABZ

Conversion of Sentences to Boolean Equations

Unit 4 4

By “1's” function

BCA ABCA'

ABAB'BCA' ' ' ' ' '

ABCABCCABCABBCAf

B

C

AF

Truth Table-based Logic Design (1/2)

0111101011011010100101110100101010010000

'ffCBA

ABC

f

ABC

f

Unit 4 5

By “0's” function

1 , 0 1 , 0

0 ,

' )'()( ' '

BCACBA

CBAifisf

BCACBBACBABACBACBACBAf

“0”

By 'f

CBACBACBAf

BCACBACBAf

''

''''''''

Truth Table-based Logic Design (2/2)

0111101011011010100101110100101010010000

'ffCBA

Unit 4 6

',,', : Literals,,,,:Variables

YYXXCBAZYX

'''' BCBAABCF

Example3 variables 7 literals

Minterm & Maxterm Expansions (1/10)

01M' C' B'A mABC111701MC' B'A m' ABC011601M' CB'A mC' AB101501MCB'A m' C' AB001401M' C' BAmBCA'110310MC' BAm' BCA'010210M' CBAmC' BA'100110MCBAm' C' B'A 0000

' MaxtermsMintemsCBANo. Row

77

66

55

44

33

22

11

00

ff

Unit 4 7

Minterm & Maxterm Expansions (2/10)

00maxterm

)2,1,0(),,()'()'()( 210

f

MCBAfMMMCBACBACBAf

1 1termmin)7,6,5,4,3(),,(

'''''76543

f

mCBAformmmmm

ABCABCCABCABBCAf

Unit 4 8

Minterm & Maxterm Expansions (3/10)

)7,6,5,4,3(''''')'('

)2,1,0(''')'()''(

),,(')7,6,5,4,3(),,(

76543

210

7654376543

210210

210

MMMMMMmmmmmmmmmmf

MMMMmmmmmmff

mmmCBAfmCBAf

Unit 4 9

Minterm & Maxterm Expansions (4/10)

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

101111010111101011100011011101010101101001100001011110010110101010010010011100010100101000100000

'

mmmmmmmmmmmmmmmm

ggDCBAExample Another

)15,13,12,9,8,5,1,0(

)15,13,12,9,8,5,1,0('

]')15,13,12,9,8,5,1,0([

)15,13,12,9,8,5,1,0('

)15,13,12,9,8,5,1,0(

)14,11,10,7,6,4,3,2(

M

m

mg

mg

Mor

mg

Unit 4 10

Minterm & Maxterm Expansions (5/10)

0000 0001 0010 0011

f a 'b ' a 'd acd ' a 'b ' c c ' d d ' a 'd b b ' c c ' acd ' b b '

a 'b 'c 'd ' a 'b 'c 'd a 'b 'cd ' a 'b 'cd a 'b 'c 'd a 'b 'cd a 'bc 'd a 'bcd abcd ' ab 'cd '

0101 0111 1110 10100 1 2 3 5 7 1 0 14

4 6 8 9 11 12 13 15

m , , , , , , ,

Find Maxterm Expansion f M , , , , , , ,

Unit 4 11

Minterm & Maxterm Expansions (6/10)

)7,6,4,3,1,0(

)'(')'()'(''

)7,6,4,3,1,0(

)'(c'b')a'(a)b'(bca'

463701

670413

mmmmmmm

bbacbcaaccbaRHS

mmmmmmm

ccabLHS

Unit 4 12

Minterm & Maxterm Expansions (7/10)

7

0 0 1 1 2 2 7 7 0

0 0 1 1 2 2 7 77

0

F ... (a)

If term exists 1

OR......

( )

If term does not exist 1

1 1

i ii

th i

i ii

th i

i

a m a m a m a m a m

i a

F (a M )(a M )(a M ) (a M )

(a M ) b

i a

( M )

7

6

5

4

3

2

1

0

111011101001110010100000

aaaaaaaaFCBA

Unit 4 13

Minterm & Maxterm Expansions (8/10)

77

66

55

44

33

22

11

00

11111011010100010110101011000000

MmMmMmMmMmMmMmMm

FCBA

Unit 4 14

Minterm & Maxterm Expansions (9/10)

)'(

)''(]')([')(

)'()''(

')(]')([')(

ii

iiii

iiii

iiii

Ma

mamaFaFrom

maMa

MaMaFbFrom

Unit 4 15

2 1 2 1

0 0

2 1 2 1

0 0

2 1

1 20

General Form : F ( )

F' ' ( ' )

: 0 i j Ex:(ABCD)(ABCD) 0

So: f f

n n

n n

n

i i i ii i

i i i ii i

i j

i ii

a m a M

a m a M

property m m if

a m

2 1

0

2 1 2 1 2 1 2 1

1 2 0 0 0 0

1 2

f f ( )( )

( terms 0) Ex: f (0,2,3,5,9,11) f (0,3,9,11,13,14

n

n n n n

j jj

i i j j i j i ji j i j

i i i

b m

a m b m a b m m

a b m i jm m

1 2

) f f (0,3,9,11)m

Minterm & Maxterm Expansions (10/10)

Unit 4 16

Incompletely Specified Functions (1/2)

ABC

FN2N1xyz

Unit 4 17

1111011

0101000111100010

caret don' 1001000

2

X

X

FCBA

N

1111011

0101000111100010

caret don' 1001000

2

X

X

FCBA

N

Ⅰ.X , X = 0 , 0

Ⅱ. X , X = 1 , 0

Ⅲ. X , X = 1 , 1

BC' C' B' AABCBC' A' C' B' AF

BCB'' AABCBC' AC' B' A' C' B' AF

ABBC' B' AABC' ABCBC' AC' B' A' C' B' AF

1,6D 5,4,2Mor

6,1d7,3,0mF

Incompletely Specified Functions (2/2)

Unit 4 18

Binary Adders & Subtracters (1/9)

․Half Adder:

011101110000

SumYX

SumHalfAdder

X

Y

Unit 4 19

Binary Adders & Subtracters (2/9)

․Full Adder:

SumFullAdder

XY

Cout

Cin

1111101011011011000101110100101010000000

SumCCYX outin

Unit 4 20

․ The logic equation for the full adder:

Binary Adders & Subtracters (3/9)

XYXCYCXYCXYCXYCCXYXYCYCX

XYCXYCCXYYCXCCYXCYXCYX

YCCYXYCCYXXYCCXYYCXCYXSum

inin

inininininin

ininininout

ininin

inininin

inininin

)'()'()'('''

)'()(')''()''('

''''''　

Unit 4 21

․The logic circuit of full adder:

Binary Adders & Subtracters (4/9)

xyCin

Sum

xy

x

y

Cin

Cin

Cout

Unit 4 22

Binary Adders & Subtracters (5/9)

․ 4-Bit Parallel Adder (Ripple Carry Adder)Adds two 4-bit unsigned binary numbers

4-bit parallel adder

Unit 4 23

Binary Adders & Subtracters (6/9)

․ 4-Bit Parallel Adder (carries)

1 1 0 0 1 1 1 1A3 B3 A2 B2 A1 B1 A0 B0

1 0 1 1 0

C4 C3 C2 C1 C0

0 1 1 0S3 S2 S1 S0

FullAdder

FullAdder

FullAdder

FullAdder

End-around carry for 1’s complement

Parallel Adder Composed of Four Full Adders

Unit 4 24

Binary Adders & Subtracters (7/9)

․Binary Subtracter Using Full Adders:Subtraction of binary numbers is most easily accomplished by adding the complement of the number to be subtracted

FullAdder

FullAdder

FullAdder

FullAdder

A3 B3 A2 B2 A1 B1A4 B4

Binary Subtracter Using Full Adder

S4 S3 S2 S1

C4 C3 C2C1=1C5

B’1B’2B’3B’4

Unit 4 25

Binary Adders & Subtracters (8/9)

․Full Subtracter

xi=0 ，yi=1，bi=1

1111100011001011000101110110101110000000

1 iiiii dbbyx

Truth Table for Binary Full subtracter

)1(01111

100

1

ii

i

i

i

BorrowAfteriColumnBorrowBeforeiColumn

bdyb

x

Unit 4 26

Binary Adders & Subtracters (9/9)

․Parallel Subtracter : Direct subtraction can be accomplished by employing a subtracter.

FullSubtracter

FullSubtracter

FullSubtracter

FullSubtracter

Cell i

xi yi x2 y2 x1 y1xn yn

dn di d2 d1

b1=0bn+1 bn bi+1 bi b3 b2

Unit 4 27

Speeding Up Integer Additions

․Ripple carry adder Simple, regular Long delay

Last carry out needs 2n delay for n-bit adder

․Carry Lookahead Adder (CLA)

․Carry Select Adder

Unit 4 28

Carry Lookahead Adder (1/4)

A3 B3 A2 B2 A1 B1 A0 B0

C4 C3 C2 C1 C0

S3 S2 S1 S0

FullAdder

FullAdder

FullAdder

FullAdder

( )in

out in

Sum A B CC AB A B C

1( )i iC AB A B C

1 1 1 1 1( )i i i i i iC A B A B C

Unit 4 29

Carry Lookahead Adder (2/4)

1 1 1 1 1( )i i i i i iC A B A B C

1 0 0 0C g p C

1 1 1

1 1 1

;;

i i i

i i i

g A Bp A B

generate function

propagate function

2 1 1 1 1 1 0 0 0

1 1 0 1 0 0

( )C g p C g p g p Cg p g p p C

1 1 1i i i iC g p C

Unit 4 30

3 2 2 2 2 2 1 1 0 1 0 0

2 2 1 2 1 0 2 1 0 0

( )C g p C g p g p g p p Cg p g p p g p p p C

Carry Lookahead Adder (3/4)

1 1 2 1 2 3

1 2 1 0 1 2 0 0

n n n n n n n

n n n n

C g p g p p gp p p g p p p C

Delay of 4-bit adder: 2+23 =8 (ripple carry adder)2+3=5 (CLA)

Unit 4 31

Carry Lookahead Adder (4/4)

gn-1

cn

gn-2pn-1 pn-2 gn-3

p1 p0g0 c0

……

Unit 4 32

Carry Select Adder (1/2)

․Two additions are performed in parallel One assumes the carry-in is 0; the other assumes 1

․When the carry-in is finally known, the correct sum is selected (has been precomputed)

s7 s6 s5 s4

s3 s2 s1 s0

c0

a3 b3 a2 b2 a1 b1 a0 b0

a4b4

a4b4a7b7

0

1

c4

Unit 4 33

Carry Select Adder (2/2)

0 0 0

111

Unit 4 34

Binary Multiplication ․Performed in the same way as with decimal numbers

Multiplicand B, multiplier A Partial product Shift one bit left Sum of partial products B

0123

0111

0010

01

01

CCCCBABA

BABAAABB

HA HA

A0

A1

B0

B0B1

B1

C0C1C2C3