Arithmetic Functions

BIL- 223 Logic Circuit Design

Ege UniversityDepartment of Computer

Engineering

Cell n-1Xn-1

Y n-1

A n-1Bn-1

Cn-1

Xn

Y nCell 1

X1

Y 1

A 1

C1

Cell 0X0

Y 0

B0

C0

X2

Y 2

Iterative Arrays

Example: n = 32 Number of inputs = ? Truth table rows = ? Equations with up to ? input variables Equations with huge number of terms Design impractical!

Iterative array takes advantage of the regularity to make design feasible

Half Adder (1-bit)A B S(um

)C(arry)

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

HalfAdder

A B

SC AB C

BABABAS

B

A

0 1

321

1

S B

A

0 1

32 1

C

A

BSum

Carry

Full AdderCin A B S(um

)Cout

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

FullAdder

A B

SCarry Out(Cout)

Carry In(Cin)

00 01 11 10

0 0 1 0 1

1 1 0 1 0

CinAB

BACin

B)(ACin)BACin(

)BABA(CinAB)BACin(

BACinCinABBACinBACinS

00 01 11 10

0 0 0 1 0

1 0 1 1 1

CinAB

ABCinACinBCout

B)Cin(AAB)BABACin(ABCout

Full AdderBACinS

A

B

Cin

Cout

S

H.A. H.A.

B)Cin(AABCout

Cout

S

HalfAdder

S

C

A

B

HalfAdder

S

C

A

BB

A

Cin

4-bit Ripple Adder using Full Adder

FullAdder

A B

CinCout

S

S0

A0 B0

FullAdder

A B

CinCout

S

S1

A1 B1

FullAdder

A B

CinCout

S

S2

A2 B2

FullAdder

A B

CinCout

S

S3

A3 B3

Carry

A

BS

C

Half Adder

A

B

CinCout

SH.A. H.A.

Full Adder

Half Subtractor(1-bit)A B Difference

(D)Borrow (B)

0 0 0 0

0 1 1 1

1 0 1 0

1 1 0 0

HalfSubtractor

A B

DiffBorrow

BA Borrow

BABABADifference

A

BDiff

Borrow

Full SubtractorBorrow (in)

A B Difference (D)

Borrow (out)

0 0 0 0 0

0 0 1 1 1

0 1 0 1 0

0 1 1 0 0

1 0 0 1 1

1 0 1 0 1

1 1 0 0 0

1 1 1 1 1

FullSubtractor

A B

DifferenceBorrowOut(Bout)

Borrow In(Bin)

BABDifference in

ininout BBBABAB

4-bit Ripple Subtractor using Full Subtractor

A B

BinBout

D

D0

A0 B0

A B

BinBout

D

D1

A1 B1

A B

BinBout

D

D2

A2 B2

FullSubtractor

A B

BinBout

D

D3

A3 B3

BorrowFull

SubtractorFull

SubtractorFull

Subtractor

Adder/Subtractor Design

A – B = A + (-B) Take 2’s complement of B Perform addition of A and 2’s complement of B

FullAdder

A B

CinCout

S

S0

A0

FullAdder

A B

CinCout

S

S1

A1

FullAdder

A B

CinCout

S

S2

A2

FullAdder

A B

CinCout

S

S3

A3

B0B1B2B3

C

Subtract

Overflow/Underflow

01001000 (+72)00111001 (+57)+__________ (+129)

8-bit Signed number addition

10000001 (-127)11111010 (-6)-_____________ (-133)

8-bit Signed number addition

Cn-1 = 1

Cn = 0

Cn-1 = 0

Cn = 1

What is largest positive number represented by 8-bit?

What is smallest negative number represented by 8-bit?

Overflow/Underflow DetectionCn-1 An-1 Bn-1 Sn-1 Cn OF

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

Examine the MSB bit Bottom line:

P: positive; N: negative N + N = N P + P = P P+N or N+P always fall

into the range E.g. -128+P cannot be

smaller than -128 or bigger than 127

Problem lies in N+N = P P+P = N

Discarded

Overflow/Underflow DetectionCn-1 An-1 Bn-1 Sn-1 Cn OF

0 0 0 0 0 0

0 0 1 1 0 0

0 1 0 1 0 0

0 1 1 0 1 1

1 0 0 1 0 1

1 0 1 0 1 0

1 1 0 0 1 0

1 1 1 1 1 0

BACABCOF

Discarded

n1n CCOF

or

Overflow/Underflow Detection

A B

CinCout

S

S0

A0 B0

A B

CinCout

SS1

A1 B1

A B

CinCout

SS2

A2 B2

A B

CinCout

SS3

A3 B3

Carry

Overflow/Underflow

n-bit Adder/Subtractor

Overflow/Underflow

Cn

Cn-1

Design by Contraction Contraction is a technique for simplifying the logic

in a functional block to implement a different function

Contraction of a ripple carry adder to incrementer for n = 3 Set B = 001

Unsigned Integer Multiplier (2-bit)

s

00100111

1011

0001

01

01

ba baba ba

ba ba

ba ba

b b x

a a

carrycarry out p0

a0b0

H.A.

p1

cs

a1b0a0b1

H.A.cs

p2p3

a1b1

Unsigned Integer Multiplier (3-bit)

ba ba ba

ba ba ba

ba ba ba

b b b x

a a a

202122

101112

000102

012

012

p0

a0b0

s

F.A.

p1

a1b0a0b1

0co

s

ci

c

F.A.

p2

s

a2b0a1b1

co

s

ci

c

F.A.

s

co

s

ci

a0b2

00

c

F.A.

p3

a2b1

co

s

ci

c

F.A.co

s

ci

a1b2

0

s

s

s

c

p4

c

F.A.co

s

ci

a2b2

p5

Multiplication by a Constant

Multiplication of B(3:0) by 101

B1B 2B 300 B 0B1B 2B 3

Carry

output

4-bit Adder

Sum

B0

C 0C 1C2C3C4C5C6