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Digital Integrated Circuits A Design Perspective. Jan M. Rabaey Anantha Chandrakasan Borivoje Nikolic. Arithmetic Circuits. January, 2003. A Generic Digital Processor. Building Blocks for Digital Architectures. Arithmetic unit. Bit-sliced datapath.

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Digital Integrated Circuits A Design Perspective

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Digital integrated circuits a design perspective

Digital Integrated CircuitsA Design Perspective

Jan M. Rabaey

Anantha Chandrakasan

Borivoje Nikolic

Arithmetic Circuits

January, 2003

A generic digital processor

A Generic Digital Processor

Digital integrated circuits a design perspective

Building Blocks for Digital Architectures

Arithmetic unit

Bit-sliced datapath

(adder, multiplier, shifter, comparator, etc.)



- RAM, ROM, Buffers, Shift registers


- Finite state machine (PLA, random logic.)

- Counters


- Switches

- Arbiters

- Bus

An intel microprocessor

An Intel Microprocessor

Itanium has 6 integer execution units like this

Bit sliced design

Bit-Sliced Design

Bit sliced datapath

Bit-Sliced Datapath

Itanium integer datapath

Itanium Integer Datapath

Fetzer, Orton, ISSCC’02



Full adder


The binary adder

The Binary Adder

Express sum and carry as a function of p g d

Express Sum and Carry as a function of P, G, D

Define 3 new variable which ONLY depend on A, B

Generate (G) = AB

Propagate (P) = A



Delete =





D and P

Can also derive expressions for


based on


Note that we will be sometimes using an alternate definition for


Propagate (P) = A


The ripple carry adder

The Ripple-Carry Adder

Worst case delay linear with the number of bits

td = O(N)

tadder = (N-1)tcarry + tsum

Goal: Make the fastest possible carry path circuit

Complimentary static cmos full adder

Complimentary Static CMOS Full Adder

28 Transistors

Inversion property

Inversion Property

Minimize critical path by reducing inverting stages

Minimize Critical Path by Reducing Inverting Stages

Exploit Inversion Property

A better structure the mirror adder

A Better Structure: The Mirror Adder

Mirror adder

Mirror Adder

Stick Diagram

The mirror adder

The Mirror Adder

  • The NMOS and PMOS chains are completely symmetrical. A maximum of two series transistors can be observed in the carry-generation circuitry.

  • When laying out the cell, the most critical issue is the minimization of the capacitance at node Co. The reduction of the diffusion capacitances is particularly important.

  • The capacitance at node Co is composed of four diffusion capacitances, two internal gate capacitances, and six gate capacitances in the connecting adder cell .

  • The transistors connected to Ci are placed closest to the output.

  • Only the transistors in the carry stage have to be optimized for optimal speed. All transistors in the sum stage can be minimal size.

Transmission gate full adder

Transmission Gate Full Adder

Manchester carry chain

Manchester Carry Chain

Manchester carry chain1

Manchester Carry Chain

Manchester carry chain2

Manchester Carry Chain

Stick Diagram

Carry bypass adder

Carry-Bypass Adder

Also called Carry-Skip

Carry bypass adder cont

Carry-Bypass Adder (cont.)

tadder = tsetup + Mtcarry + (N/M-1)tbypass + (M-1)tcarry + tsum

Carry ripple versus carry bypass

Carry Ripple versus Carry Bypass

Carry select adder

Carry-Select Adder

Carry select adder critical path

Carry Select Adder: Critical Path

Linear carry select

Linear Carry Select

Square root carry select

Square Root Carry Select

Adder delays comparison

Adder Delays - Comparison

Lookahead basic idea

LookAhead - Basic Idea

Look ahead topology

Look-Ahead: Topology

Expanding Lookahead equations:

All the way:

Logarithmic look ahead adder

Logarithmic Look-Ahead Adder

Carry lookahead trees

Carry Lookahead Trees

Can continue building the tree hierarchically.

Tree adders

Tree Adders

16-bit radix-2 Kogge-Stone tree

Tree adders1

Tree Adders

16-bit radix-4 Kogge-Stone Tree

Sparse trees

Sparse Trees

16-bit radix-2 sparse tree with sparseness of 2

Tree adders2

Tree Adders

Brent-Kung Tree

Example domino adder

Example: Domino Adder



Example domino adder1

Example: Domino Adder



Example domino sum

Example: Domino Sum



The binary multiplication

The Binary Multiplication

The binary multiplication1

The Binary Multiplication

The array multiplier

The Array Multiplier

The mxn array multiplier critical path

The MxN Array Multiplier— Critical Path

Critical Path 1 & 2

Carry save multiplier

Carry-Save Multiplier

Multiplier floorplan

Multiplier Floorplan

Wallace tree multiplier

Wallace-Tree Multiplier

Wallace tree multiplier1

Wallace-Tree Multiplier

Wallace tree multiplier2

Wallace-Tree Multiplier

Multipliers summary

Multipliers —Summary



The binary shifter

The Binary Shifter

The barrel shifter

The Barrel Shifter

Area Dominated by Wiring

4x4 barrel shifter

4x4 barrel shifter

Widthbarrel ~ 2 pm M

Logarithmic shifter

Logarithmic Shifter

0 7 bit logarithmic shifter

0-7 bit Logarithmic Shifter













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