Unit2 Arithmetics

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UNIT 2

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Addition/subtraction of signed numbers

si =

ci+1=

13

7+Y

1

0

0

0

1

0

1

1

0

0

1

1

0

11

0

0

1

1

0

1

0

01

0

0

0

0

1

1

1

1

0

0

0

0

1

1

11

Example:

10= = 0

01 1

11 1 0 0

1

1 1 10

Legend for stage i

xi

yi

Carry-inci

Sumsi

Carry-outci+1

X

Z

+ 6 0+xiyi

si

Carry-outci+1

Carry-inci

xiyici

xiyici

xiyici

xiyici xi yi ci =

+ + +

yicixicixiyi+ +

At the ith stage:Input:

ci is the carry-in

Output:

si is the sum

ci+1 carry-out to (i+1)st

state

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Addition logic for a single stage

Full adder(FA)

ci

ci 1+

si

Sum Carry

yixi

ci

yi

xi

ci

yi

x

i

xi

ci

yi

si

ci 1+

Full Adder (FA): Symbol for the complete circuit

for a single stage of addition.

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n-bit adderCascade n full adder (FA) blocks to form a n-bit adder.Carries propagate or ripple through this cascade, n-bit ripple carry adder.

FA c0

y1

x1

s1

FA

c1

y0

x0

s0

FA

cn 1-

yn 1-

xn 1-

cn

sn 1-

Most significant bit(MSB) position

Least significant bit(LSB) position

Carry-in c0 into the LSB position provides a convenient way to

perform subtraction.

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K n-bit adderK n-bit numbers can be added by cascading kn-bit adders.

n-bit c0

yn

xn

sn

cn

y0

xn 1-

s0

ckn

sk 1-( )n

x0

yn 1-

y2n 1-

x2n 1-

ykn 1-

sn 1-

s2n 1-

skn 1-

xkn 1-

addern-bitaddern-bitadder

Each n-bit adder forms a block, so this is cascading of blocks.Carries ripple or propagate through blocks, Blocked Ripple Carry Adder

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n-bit subtractor

FA 1

y1

x1

s1

FAc1

y0

x0

s0

FAcn 1-

yn 1-

xn 1-

cn

sn 1-

Most significant bit(MSB) position

Least significant bit(LSB) position

RecallXYis equivalent to adding 2s complement ofYtoX.

2s complement is equivalent to 1s complement + 1.

XY = X + Y + 1

2s complement of positive and negative numbers is computed similarly.

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n-bit adder/subtractor (contd..)

Add/Subcontrol

n-bit adder

xn 1-

x1

x0

cn

sn 1- s1s0

c0

yn 1-

y1

y0

Add/sub control = 0, addition.

Add/sub control = 1, subtraction.

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Detecting overflows

Overflows can only occur when the sign of the two operands isthe same.

Overflow occurs if the sign of the result is different from the

sign of the operands.

Recall that the MSB represents the sign. xn-1, yn-1, sn-1 represent the sign of operand x, operand yand result s

respectively.

Circuit to detect overflow can be implemented by the

following logic expressions:

111111 nnnnnn syxsyxOverflow

1 nn ccOverflow

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Computing the add timeConsider 0th stage:

x0 y0

c0c1

s0

FA

c1 is available after 2 gate delays.

s1 is available after 1 gate delay.

ci

yi

xi

ci

yi

xi

xi

ci

yi

si

ci 1+

Sum Carry

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Computing the add time (contd..)

x0 y0

s2

FA

x0 y0x0 y0

s1

FAc2

s0

FAc

1

c3

c0

x0 y0

s3

FA

c4

Cascade of 4 Full Adders, or a 4-bit adder

s0 available after 1 gate delays, c1 available after 2 gate delays.

s1

available after 3 gate delays, c2

available after 4 gate delays.

s2 available after 5 gate delays, c3 available after 6 gate delays.

s3 available after 7 gate delays, c4 available after 8 gate delays.

For an n-bit adder,sn-1is available after 2n-1 gate delayscnis available after2ngate delays.

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Fast additionRecall the equations:

iiiiiii

iiii

cycxyxc

cyxs

1

Second equation can be written as:

iiiiii cyxyxc )(1

We can write:

iiiiii

iiii

yxPandyxGwhere

cPGc

1

Giis called generate function andPiis called propagate functionGi and Pi are computed only from xiand yi and not ci, thus they canbe computed in one gate delay after Xand Yare applied to the

inputs of an n-bit adder.

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Carry lookaheadci1 Gi Pici

ci Gi1 Pi1ci1

ci1 Gi Pi(Gi1 Pi1ci1)

continuing

ci1 Gi Pi(Gi1 Pi1(Gi2 Pi2ci2 ))

until

ci1 Gi PiGi1 PiPi1Gi2 . . PiPi1. .P1G0 PiPi1. ..P0c0

All carries can be obtained 3 gate delays afterX, Yandc0are applied.

-One gate delay for Pi and Gi-Two gate delays in the AND-OR circuit for ci+1

All sums can be obtained 1 gate delay after the carries are computed.Independent of n, n-bit addition requires only 4 gate delays.This is called Carry Lookahead adder.

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Carry-lookahead adder

Carry-lookahead logic

B cell B cell B cell B cell

s3

P3

G3

c3

P2

G2

c2

s2

G1

c1

P1

s1

G0

c0

P0

s0

.c

4

x1

y1

x3

y3

x2

y2

x0

y0

Gi

ci

..

.

Pi

si

xi

yi

B cell

4-bit

carry-lookahead

adder

B-cell for a single stage

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Carry lookahead adder (contd..) Performing n-bit addition in 4 gate delays independent ofn

is good only theoretically because of fan-in constraints.

Last AND gate and OR gate require a fan-in of (n+1) for a n-bit adder. For a 4-bit adder (n=4) fan-in of 5 is required. Practical limit for most gates.

In order to add operands longer than 4 bits, we can cascade4-bit Carry-Lookahead adders. Cascade of Carry-Lookaheadadders is called Blocked Carry-Lookahead adder.

ci1 Gi PiGi1 PiPi1Gi2 ..PiPi1..P1G0 PiPi1...P0c0

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4-bit carry-lookahead Adder

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Blocked Carry-Lookahead adder

c4 G3 P3G2 P3P2G1 P3P2P1G0 P3P2P1P0c0

Carry-out from a 4-bit block can be given as:

Rewrite this as:

P0I

P3P2P1P0

G0IG3 P3G2 P3P2G1 P3P2P1G0

Subscript I denotes the blocked carry lookahead and identifies the block.

Cascade 4 4-bit adders, c16can be expressed as:

c16 G3I P3

IG2

I P3

IP2

IG1

IP3

IP2

IP1

0G0

IP3

IP2

IP1

0P0

0c0

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Blocked Carry-Lookahead adder

Carry-lookahead logic

4-bit adder 4-bit adder 4-bit adder 4-bit adder

s15-12

P3

IG3

I

c12

P2

IG2

I

c8

s11-8

G1

I

c4

P1

I

s7-4

G0

I

c0

P0

I

s3-0

c16

x15-12

y15-12

x11-8

y11-8

x7-4

y7-4

x3-0

y3-0

.

Afterxi, yi and c0are applied as inputs:- Gi and Pi for each stage are available after 1 gate delay.- PI is available after 2 and GI after 3 gate delays.- All carries are available after 5 gate delays.- c16 is available after 5 gate delays.- s15 which depends on c12 is available after 8 (5+3)gate delays

(Recall that for a 4-bit carry lookahead adder, the last sum bit isavailable 3 gate delays after all inputs are available)

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Multiplication of unsigned numbers

Product of 2 n-bit numbers is at most a 2n-bit number.

Unsigned multiplication can be viewed as addition of shifted

versions of the multiplicand.

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Multiplication of unsigned

numbers (contd..) We added the partial products at end.

Alternative would be to add the partial products at each stage.

Rules to implement multiplication are:

If the ith bit of the multiplier is 1, shift the multiplicand and add the

shifted multiplicand to the current value of the partial product.

Hand over the partial product to the next stage

Value of the partial product at the start stage is 0.

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Multiplication of unsigned numbers

ith multiplier bit

carry incarry out

jth multiplicand bit

ith multiplier bit

Bit of incoming partial product (PPi)

Bit of outgoing partial product (PP(i+1))

FA

Typical multiplication cell

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Combinatorial array multiplier

Multiplicand

m3 m2