1. Selecting a to Eliminate Carry Chain in SD For no carry, require

2. Selecting a to Eliminate Carry Chain in SD

3. Binary SD Addition Implies no guarantee that s i = w i + t i will not produce a carry Looking at algorithm: Step 1:

4. Unmodified Binary SD Addition Table x i,y i t i+1 wiwi Step 2: Based on calculation of w i and t i+1 Note: redundancy allows choices for w i and t i+1

5. How Useful is Unmodified Table? Works if operands do not contain If operands contain only 0’s and 1’s, no carry generated. Example Why not use this approach to break carry chain for unsigned binary number?

6. Limitations of Table Example (-9) 10 + (29) 10 Does not work if operands contain

7. SD Addition Table Choices Takagi, 1985

8. Modified Binary SD Addition Table x i,y i x i-1,y i-1 - neither isat least one is neither isat least one is -- t i+1 wiwi

9. Repeating Example with Modified Table Example (-9) 10 + (29) 10

10. Two SD Encodings x Encoding 1 x h x l Encoding 2 x h x l 4!=24 possible encodings Only nine are distinct under permutation and logical negation two’s complement

11. Encoding 1 Satisfies simple relation x = x l - x h and 11 has a valid numerical value of 0. SD to two’s complement conversion performed by two’s complement subtraction

12. Encoding 2 Satisfies relation x i = -2x i h + x i l This means that x i l and x i-1 h have the same weight Also simplified addition table possible by regrouping bits

13. Two’s Complement/BSD Conversion Two’s Complement to SD Bits can be encoded directly with MSB negative one BSD to Two’s Complement One algorithm simpler than complete binary adder z i is two’s complement result c 0 = 0 Example -10 10

14. Binary SD Representations Representation of a value with the minimum number of non-zero digits – Important in multiplication and division since each zero eliminates an operation Minimal SD representation of X = 5 X = 5, n = 4, r = 2