Arithmetic

1
Arithmetic Circuits
2
A Generic Digital Processor
3
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
4
An Intel Microprocessor
Itanium has 6 integer execution units like this
5
Bit-Sliced Design
6
Bit-Sliced Datapath
7
Itanium Integer Datapath
Fetzer, Orton, ISSCC02
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Adders
9
Full-Adder
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The Binary Adder
11
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
B
Å
Delete/kill

A
B
S
C
D and P
Can also derive expressions for
and
based on

o
Note that we will be sometimes using an alternate
definition for

Propagate (P) A
B
12
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
13
Complimentary Static CMOS Full Adder
28 Transistors
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Inversion Property
15
Minimize Critical Path by Reducing Inverting
Stages
Exploit Inversion Property
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A Better Structure The Mirror Adder
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Mirror Adder
Stick Diagram
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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.

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Transmission Gate Full Adder
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Manchester Carry Chain
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Manchester Carry Chain
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Manchester Carry Chain
Stick Diagram
23
Carry-Skip Adder
Also called Carry-bypass
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Carry-Skip Adder (cont.)
tadder tsetup Mtcarry (N/M-1)tbypass
(M-1)tcarry tsum
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Carry Ripple versus Carry Bypass
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Carry-Select Adder
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Carry Select Adder Critical Path
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Linear Carry Select
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Square Root Carry Select
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Adder Delays - Comparison
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LookAhead - Basic Idea
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Look-Ahead Topology
Expanding Lookahead equations
All the way
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Logarithmic Look-Ahead Adder
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Carry Lookahead Trees
Can continue building the tree hierarchically.
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Tree Adders
16-bit radix-2 Kogge-Stone tree
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Tree Adders
16-bit radix-4 Kogge-Stone Tree
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Sparse Trees
16-bit radix-2 sparse tree with sparseness of 2
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Tree Adders
Brent-Kung Tree
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Example Domino Adder
Propagate
Generate
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Example Domino Adder
Propagate
Generate
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Example Domino Sum
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Multipliers
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The Binary Multiplication
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The Binary Multiplication
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The Array Multiplier
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The MxN Array Multiplier Critical Path
Critical Path 1 2
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Carry-Save Multiplier
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Multiplier Floorplan
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Wallace-Tree Multiplier
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Wallace-Tree Multiplier
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Wallace-Tree Multiplier
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Multipliers Summary
53
Shifters
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The Binary Shifter
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The Barrel Shifter
Area Dominated by Wiring
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4x4 barrel shifter
Widthbarrel 2 pm M
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Logarithmic Shifter
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0-7 bit Logarithmic Shifter
A
3
Out3
A
2
Out2
A
1
Out1
A
0
Out0