let3-Number Systems

# Computer Arithmetic: Algorithms and Hardware Designs

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CSE 246: Computer Arithmetic  Algorithms and Hardware  Design Numbers: RNS, DBNS, Montgomory Prof Chung-Kuan Cheng Lecture 3

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How to compare two RNS numbers We can approximate the magnitude of a RNS number by the following formula M p p p RNS x x x M x k k k k ) | ... | | ( ) | ... | | ( 0 2 1 0 2 1 - - - - = - = = 1 0 1 ) ( k i MOD i i i P x α j j P M Π = i P i i P M ] ) [( 1 - = where
An Example Suppose, x = (6|3|0) RNS (7|5|3) y = (3|0|1) RNS (7|5|3) Then we have x/105 = [6(1/7) + 3(1/5) + 0(2/3)] mod 1 ≈ 0.457 y/105 = [3(1/7) + 0(1/5) + 1(2/3)] mod 1 ≈ 0.095 Clearly, x ( 48 ) is greater than y ( 10 ).

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Double Base Number System  (DBNS) DBNS is a new kind of number system, where there are two bases, p and q . Any number x is represented by the equation Also, this number system could be redundant, e.g. 54 = 2 0 3 0 +2 1 3 0 +2 1 3 1 +2 0 3 1 +2 0 3 2 +2 2 3 2 = 2 1 3 3 < < = j i ij j i ij p d q p q p d x , 0 ; ,
Double Base Number System  (DBNS) We can represent DBNS numbers in a two-dimensional table. For example we can express 54 by this tabular representation. 1 2 4 1 x x 3 x 9 x x For each entry in the table, we multiply the corresponding row-value and column-value. Then we add up all such entries to get the value of the number represented by the table.

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Double Base Number System  (DBNS) DBMS can be of practical use too in some scenarios. In binary number representation, each bit has approximately 0.5 probability of being 1. But in DBNS, the number of bits that are logic 1 in the tabular representation could be much less. Effectively, we can reduce the number of 0 1 and 1 0 transitions, thus saving power.
Double Base Number System  (DBNS) A greedy approach to minimize the number of TRUE bits in the tabular representation of any integer : GREEDY (x) { if (x > 0) then do{ find the largest 2-integer w such that w ≤ x; write(w); x = x-w; greedy(x); } }

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Double Base Number System  (DBNS) It can be shown that expected number of bits that are ‘turned on’ in a DBNS representation of integer is O[lg x/(lg lg x)], which is significantly lower than the corresponding number in the positional binary system, O(lg x). As an example, consider the integer 2
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let3-Number Systems - CSE 246 Computer Arithmetic...

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