Arithmetic Building Blocks - people.ee.duke.edupeople.ee.duke.edu/~krish/teaching/Lectures/arithmetic_circuits_1... · Inversion Property AB S Ci FA Co AB S Ci FA Co Inverting all

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ECE 261 Krish Chakrabarty 1

Arithmetic Building Blocks

• Datapath elements• Adder design

– Static adder– Dynamic adder

• Multiplier design– Array multipliers

• Shifters, Parity circuits


A Generic Digital Processor

MEMORY

DATAPATH

CONTROL

Inpu

t-O

utpu

t

2


Building Blocks for Digital Architectures• Arithmetic unit- Bit-sliced datapath (adder, multiplier,

shifter, comparator, etc.)• Memory- RAM, ROM, Buffers, Shift registers• Control- Finite state machine (PLA, random logic.)- Counters• Interconnect- Switches- Arbiters- Bus


Bit-Sliced Design

Bit 3

Bit 2

Bit 1

Bit 0

Control

Tile identical processing elements

Dat

a-in

Dat

a-ou

t

Reg

iste

r

Add

er

Shift

er

Mul

tiplie

r

Signals

Data Control

Metal 2(control)

Metal 1(data)

3


Single-Bit Addition

Half Adder Full Adder

0111100110100000SCoutBA

1111101011011011000101110100101010000000SCoutCBA

A B

S

Cout

A B

C

S

Cout

out

S A BC A B

= ⊕= out ( , , )

S A B CC MAJ A B C

= ⊕ ⊕=


Full-AdderA B

Cout

SumCin

Fulladder

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The Binary Adder

A B

Cout

Sum

Cin Fulladder

Sum = A ⊕ B ⊕ C= ABCi + ABCi + ABCi + ABCi

Co = AB + BCi + ACi


Sum and Carry as a functions of P, G

Define 3 new variable which ONLY depend on A, BGenerate (G) = AB

Propagate (P) = A+B

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The Ripple-Carry Adder

A0 B0

S0

Co,0Ci,0

A1 B1

S1

Co,1

A2 B2

S2

Co,2

A3 B3

S3

Co,3

(= Ci,1)FA FA FA FA

Wo rs t cas e d elay lin ear with th e n u mber o f b itstd = O(N)

Goal: Make the fas tes t poss ible carry path circuit

td = (N-1)tcarry + tsum


Complimentary Static CMOS Full Adder

VDD

VDD

VDD

VDD

A B

Ci

S

Co

X

B

A

Ci A

BBA

Ci

A B Ci

Ci

B

A

Ci

A

B

BA

28 Transistors

Note:1) S = ABCi + Co(A + B + Ci)2) Placement of Ci3) Two inverter stages for

each Co

O(N) delay

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Inversion Property

A B

S

CoCi FA

A B

S

CoCi FA

Inverting all inputs results in inverted outputs


Minimize Critical Path by Reducing Inverting Stages

Co,0Ci,0 Co,1 Co,2 Co,3FA’ FA FA’ FA

Odd CellEven Cell

Exploit Inversion Property

Need two different types of cells, FA’: no inverter in carry path

A0A1B0

A2B1 B2B3A3

S0 S1 S2 S3

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A better structure: the Mirror Adder

VDD

Ci

A

BBA

B

A

A BKill

Generate"1"-Propagate

"0"-Propagate

VDD

Ci

A B Ci

Ci

B

A

Ci

A

BBA

VDD

SCo

24 transistors


The Mirror Adder• Symmetrical NMOS and PMOS chains

– identical rising and falling transitions if the NMOS and PMOS devices are properly sized.

– Maximum of two series transistors in the carry-generation circuitry.

• Critical issue: minimization of the capacitance at Co. – Reduction of the diffusion capacitances important. – The capacitance at Co composed of four diffusion capacitances, two

internal gate capacitances, and six gate capacitances in the connecting adder cell .

• Transistors connected to Ci placed closest to output.• Only the transistors in carry stage have to be optimized for

speed. All transistors in the sum stage can be minimal size.

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NP-CMOS Adder

VDD

φ

φ

Ci0

A0 B0 B0

φ

A0

VDD

φ

B1

φ

A1

VDD

φ

φ

A1 B1

Ci1

Ci2

Ci0

Ci0

B0

A0B0

S0

A0

VDD

φ

φ

VDD

φ

VDD

φ

φ

B1 Ci1

B1

φ

A1A1

VDDφ S1

Ci1

Carry Path

17 transistors,ignoring extra inverters for inputsand outputs


Manchester Carry Chain

P0

Ci,0

P1

G0

P2

G1

P3

G2

P4

G3 G4

VDD

∅

∅

Co,4

• Only nMOS transmission gates used. Why?• Delay of long series of pass gates: add buffers

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Carry-Bypass Adder

FA FA FA FA

P0 G1 P0 G1 P2 G2 P3 G3

Co,3Co,2Co,1Co,0Ci,0

FA FA FA FA

P0 G1 P0 G1 P2 G2 P3 G3

Co,2Co,1Co,0Ci,0

Co,3

BP=P oP1P2P3

Idea: If (P0 and P1 and P2 and P3 = 1)then Co3 = C0, else “kill” or “generate”.


Manchester-Carry Implementation

P0

Ci,0

P1

G0

P2

G1

P3

G2

BP

G3

BP

Co,3

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Carry-Bypass Adder (cont.)

S e tu p

C arry

Propagation

S um

S e tu p

C arry

Propagation

S um

Setup

C arry

Propagation

Sum

S e tu p

C arry

Propagation

S um

Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

Ci,0

Design N-bit adder using N/M equal length stagese.g. N = 16, M = 4What is the critical path?

tp = tsetup + Mtcarry + (N/M-1)tbypass + Mtcarry + tsum , i.e. O(N)


Carry Ripple versus Carry Bypass

N

tp

ripple adder

bypass adder

4..8

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Carry-Select Adder

Setup

"0" Carry Propagation

"1" Carry Propagation

Multiplexer

Sum Generation

Co,k-1 Co,k+3

"0"

"1"

P,G

Carry Vector

Generate carry out for both “0” and “1” incoming carries

4-bit block for bitsk, k+1, k+2, k+3


Carry Select Adder: Critical Path

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

S0-3 S4-7 S8-11 S12-15


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Carry-Select Adder: Linear Configuration

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

S0-3 S4-7 S8-11 S12-15


(1)

(5)(5)(5)(6)

(1)

(7) (8)(5)(5)

Are equal-sized blocks best?


Linear Carry Select

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Bit 0-3 Bit 4-7 Bit 8-11 Bit 12-15

S0-3 S4-7 S8-11 S12-15


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Square Root Carry Select

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"

Setup

"0" Carry

"1" Carry

Multiplexer

Sum Generation

"0"

"1"


(1)

(4)(3)(3)(4)

(1)

(5) (6)(6)(5)

Bit 0-1 Bit 2-4 Bit 5-8 Bit 9-13

i.e., O(√N)


Adder Delays - Comparison

0.0 20.0 40.0 60.0N

0.0

10.0

20.0

30.0

40.0

50.0

tp

ripple adder

linear select

square root select

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Carry Look-Ahead - Basic IdeaA0,B0 A1,B1 AN-1,BN-1...

Ci,0 P0 Ci,1 P1Ci,N-1 PN-1

...

Delay “independent” of the number of bitsS0 S1 SN-1


Carry-Lookahead Adders• High fanin for large N• Implement as CLA slices, or use 2nd level lookahead

generator

4

4

4 4 4

4 4 4 4 4 4 4 16-bit CLA based on 4-bitslices and ripple carry

4

4

4 4 4

4 4 4 4 4 4 4

CLA generator

Faster implementation

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Look-Ahead: TopologyVDD

Gnd

G3

G2

G1

G0

P0

P3

P2

P1

Ci,0

Co,3

Arithmetic Building Blocks - people.ee.duke.edupeople.ee.duke.edu/~krish/teaching/Lectures/arithmetic_circuits_1... · Inversion Property AB S Ci FA Co AB S Ci FA Co Inverting all

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