CSE 246: Computer Arithmetic Algorithms and Hardware Design

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CSE 246 Computer Arithmetic Algorithms and
Hardware Design
Winter 2005 Lecture 1 Numbers

• Instructor
• Prof. Chung-Kuan Cheng

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Agenda

• Administrative
• Motivation
• Lecture 1 Numbers

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Administrative

• Textbook Computer Arithmetic Algorithms, Israel
  Koren, 2nd Edition, Published by AK Peters
  Natick, Massachusetts
• Recommended Art of Computer Programming, Volume
  2, Seminumerical Algorithms (3rd Edition), Donald
  E. Knuth
• In addition set of papers to read

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Administrative

• Grading
• Homework 20
• Midterm 35
• Project
• Report 25
• Presentation 20
• Midterm Tuesday, Feb. 8th
• Homework 1
• Due 1/18/05

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Administrative

• Potential project samples
• Design interconnect and switch modules
• Use FPGAs, nano technologies for add/sub
• Design reconfigurable blocks
• Design Low power adder, multiplier
• Invent Low power/reliable number systems

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Agenda

• Administrative
• 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 paradigm
• Interconnect dominated clock, control, bus,
  signal
• Power driven
• Reliability centric
• FPGAs

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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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Motivation

• What if we had a 10GHz chip that was
• 2 cm x 2 cm?
• It takes 2 clock cycles (time of flight) to get
  from one end of the chip to the other
• How would the clock be distributed?
• What if the electrical input is 1 Volt/100 Watts?
  How do we get 100 Amps through the chip?

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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

• Administrative
• 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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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