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CSE 246: Computer Arithmetic Algorithms and Hardware Design

This lecture provides an introduction to arithmetic algorithms and hardware design, discussing the importance and impact of advancements in these fields. It covers topics such as binary numbers, addition/subtraction, multiplication/division, floating point operations, optimization, and fault tolerance.

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CSE 246: Computer Arithmetic Algorithms and Hardware Design

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  1. CSE 246: Computer Arithmetic Algorithms and Hardware Design Fall 2006 Lecture 1: Introduction and Numbers Instructor: Prof. Chung-Kuan Cheng

  2. Agenda • Administration • Motivation • Lecture 1: Numbers

  3. 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

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

  5. Administration • Grading: • Homework – 20% • Midterm – 35% • Project • Report – 25% • Presentation – 20% • Midterm: Thursday, 10/2/06

  6. 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

  7. Agenda • Administration • Motivation • Lecture 1: Numbers

  8. 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

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

  10. 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

  11. Other Topics • Potential focus on the following topics: • Power reduction • Interconnect • FPGAs

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

  13. Agenda • Administration • Motivation • Lecture 1: Numbers

  14. 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

  15. 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)

  16. Numbers • Summation of positional numbers • Given: • Value is: (where y is the base) • For example: • Consider • Note that position systems provide a complete range of numbers (e.g. –2 to 5)

  17. Numbers (Radix Conversion) • Repeatedly divide the integer by (R)r • Repeatedly multiply the fraction by (R)r • Example • (201.31)10=(13001.123)5 (xk-1, …, x0 . X-1, … , x-l)r=(XK-1, …, X0 . X-1, …, X-L)R

  18. Numbers • Avoid Division (Montgomory System) • Simplify Mod operation • mod r-1, mod r+1 Example: 29110 mod 9 = 2+9+1 mod 9 = 12 mod 9 = 3 29110 mod 11 = 2-9+1mod 11 = -6 mod 11 = 5

  19. 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) • One’s complement Two’s complement M=2n-1 M=2n Flip each bit Flip each bit + 1 • Two’s complement can be used for subtraction

  20. Signed Numbers • Two’s complement subtraction: • (M-x+M-y) mod M = M-(x+y) • Two’s complement conversion: • Positive number: • To negative:

  21. Signed Numbers • Two’s complement Proof as follows: Which leads to: Example:

  22. Next time • Talk about redundant numbers

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