CSE 246: Computer Arithmetic Algorithms and Hardware Design

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CSE 246 Computer Arithmetic Algorithms and
Hardware Design
Fall 2006 Lecture 1 Introduction and Numbers

• Instructor
• Prof. Chung-Kuan Cheng

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Agenda

• Administration
• Motivation
• Lecture 1 Numbers

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Administration

• Textbook Computer Arithmetic Algorithms and
  Hardware Designs, Behrooz Parhami, Oxford
• Recommended Art of Computer Programming, Volume
  2, Seminumerical Algorithms (3rd Edition), Donald
  E. Knuth
• Numerical Computing with IEEE Floating Point
  Arithmetic, Michael L. Overton, SIAM
• Computer Arithmetic Algorithms, Israel Koren, A K
  Peters, Natick, Massachusetts
• Digital Arithmetic, Milos D. Ercegovac and Tomas
  Lang, Morgan Kaufmann
• CMOS VLSI Design, Neil H.E. Weste and David
  Harris, Addison Wesley
• Principles and Practices of Interconnection
  Networks, William James Dally and Brian Towles,
  Morgan Kaufmann
• In addition set of papers to read

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Administration

• No classes on the following days
• Tu 10/17 BIBE
• Tu 10/24 EPEP
• Tu 11/7 ICCAD

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Administration

• Grading
• Homework 20
• Midterm 35
• Project
• Report 25
• Presentation 20
• Midterm Thursday, 10/2/06

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Administration

• Potential project samples
• Interconnect and switch modules
• Data path components using FPGAs, nano
  technologies
• Low power logic styles
• Reconfigurable blocks
• Low power data path components
• Low power/reliable coding systems

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Agenda

• Administration
• Motivation
• Lecture 1 Numbers

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Motivation

• Why do we care about arithmetic algorithms and
  hardware design?
• Classic problems well defined
• Advancements will have a huge impact
• Solutions will be widely used
• New paradigms
• Interconnect effects clock, control, bus, signal
• Low power designs
• Wider bit width
• Reliability centric designs
• FPGAs and nano technologies

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Motivation

• Should a new business focus on building market or
  new technology?
• New technology a market will be built around new
  technology

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Topics

• Numbers
• Binary numbers, negative numbers, redundant
  numbers, residual numbers
• Addition/Subtraction
• Prefix adders (zero deficiency)
• Multiplication/Division
• Floating point operations
• Functions (sqrt),log, exp, CORDIC
• Optimization, analysis, fault tolerance

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Other Topics

• Potential focus on the following topics
• Power reduction
• Interconnect
• FPGAs

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Goals/Background

• Why do you want to take this class? What would
  you like to learn?
• Fulfill course requirement
• Hardware
• Software
• Work
• Research
• Curiosity

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Agenda

• Administration
• Motivation
• Lecture 1 Numbers

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Numbers

• Special Symbols
• Symbols used to represent a value
• Roman Numerals
• 1 I 100 C
• 5 V 500 D
• 10 X 1000 M
• 50 L
• For example 2004 MMIV

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Numbers

• Position Symbols
• The value depends on the position of the number
• For example
• 125 100 20 5
• One 100, Two 10s, and Five 1s
• Another example
• 1 hour, 3 minutes
• Positional systems includes radixes
• 2, -2, 2, 2j (imaginary)

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Numbers

• Summation of positional numbers
• Given
• Value is (where y is the base)
• For example
• Consider

4 -2 1
0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 -2 -1 4 5 2 3

• Note that position systems provide a complete
  range of numbers (e.g. 2 to 5)

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Numbers (Radix Conversion)
(xk-1, , x0 . X-1, , x-l)r(XK-1, , X0 . X-1,
, X-L)R

• Repeatedly divide the integer by (R)r
• Repeatedly multiply the fraction by (R)r
• Example
• (201.31)10(13001.123)5

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Numbers

• Avoid Division (Montgomory System)
• Simplify Mod operation
• mod r-1, mod r1

Example 29110 mod 9 291 mod 9 12 mod 9
3 29110 mod 11 2-91mod 11 -6 mod 11 5
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Signed Numbers

• Biased numbers
• Signed Bit
• Complementary representation
• Positive number x (mod p)
• Negative number (M-x) (mod p)
• (Note mod p is added implicitly)
• Ones complement Twos complement

0 0 0 1 0 1
1 0 1 1 -1 -0
0 0 0 1 0 1
1 0 1 1 -2 -1
M2n-1
M2n
Flip each bit
Flip each bit 1

• Twos complement can be used for subtraction

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Signed Numbers

• Twos complement subtraction
• (M-xM-y) mod M M-(xy)
• Twos complement conversion
• Positive number
• To negative

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Signed Numbers

• Twos complement

Proof as follows Which leads to
Example
0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 2 3 -4 -3 -2 -1
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Next time

• Talk about redundant numbers