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Redundant Number Systems and Online Arithmetic

Redundant Number Systems and Online Arithmetic. Sridhar Rajagopal June 16, 2000. Outline. Introduction to RNS Introduction to Online An example comparison Why are we doing this ?. Redundant Number Systems . Conventional Systems ( 0.34578 r=10) radix r has r possible digits

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Redundant Number Systems and Online Arithmetic

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  1. Redundant Number Systems andOnline Arithmetic Sridhar Rajagopal June 16, 2000

  2. Outline • Introduction to RNS • Introduction to Online • An example comparison • Why are we doing this ?

  3. Redundant Number Systems • Conventional Systems ( 0.34578 r=10) • radix r has r possible digits • Redundant (0.34578,0.35578,…. r=10) • >r possible digits. • Limit carry propagation • Totally Parallel Addition/Subtraction

  4. Contd.. • Radix r can have q values • r+2 <= q <= 2r-1 (Say why) • Addition of bit I depends on I and I+1 only • Transfer Digit ti = -1,0,1 • Interim Sum |wi| <= r-2 (Hence, r >2) • wmax - wmin >= r-1 (Tell why)

  5. Contd.. • For r >4, more than 1 set of digit values • ceil{(r-1)/2} <= a <= r-1 (Say why) • min/max redundancy • {-wmin-1,…0,…,wmax+1}

  6. Conversion To a RNS • Z = +/- xi r-i ; x  {0,..r-1} • 1. Choose ‘a’ - the amt. Of redundancy • 2. If Z <0,interpret each digit as ‘-’ • 3. Wi = xi - r*ti-1 ;ti-1  f(xi) • 4. Form zi = wi + ti

  7. Conversion from a RNS • Sum of 2 numbers,positive and negative • fed to an adder • Serial manner, from the LSB.

  8. Multi-operand Addition • Can be done totally parallel • |ti| > 1 allowed , r large • n <= floor(r/2) (Tell why) • r =10, 5 digits can be added in parallel

  9. Binary Radix Addition • Two-transfer addition; I,I+1,I+2 • can use for r = 2 • decreased redundancy (r+1) • More complicated addition • Greater transfer digit propagation

  10. Example • R = 10, a = 6,(w=5), t = -1,0,1 • 0.76486 + (-0.39471) = 0.37015 (usual) • 1.36514 + 0.40531 = 0.43025 (redund) • Show conversion, addition, back-conversion

  11. Online Arithmetic • Pipelined Bit-Serial Arithmetic • MSDF computations • Successive computations as soon as  inputs available ( = 1-4, typically) • Advantages of MSDF online (Tell)

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