Redundant Number Systems and Online Arithmetic Sridhar Rajagopal

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 • Redundant (0. 34578, 0. 35578, …. r=10) – >r possible digits. • Limit carry propagation • Totally Parallel Addition/Subtraction

Contd. . • Radix r can have q values – r+2 <= q <= 2 r-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)

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}

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

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

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

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

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, backconversion

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