General Fixed Radix Number Systems

1. General Fixed Radix Number Systems • Nonredundant • Positive radix, ß • n digits in digit set • Vector:

2. General Fixed Radix Number Systems • For Given Radix and n, how many number systems? • ANSWER: Number equal to all possible permutations of n choose –1 (or +1),  Positive Radix Negative Radix Choose -1 • Of these, 1 is Pos. Radix 1 is Neg. Radix • The following is the Radix-complement:

3. General FR Number Systems - Properties Largest Representable Integer Smallest Representable Integer pi are Digits of P

4. General FR Number Systems - Properties Using the pi Expression and Forming the Radix Polynomial for P digit Define as Q weight

5. General FR Number Systems - Properties Q is the value represented by the following n-tuple if all i=1 For N, the Smallest Representable Value:

6. General FR Number Systems - Properties Using Similar Analysis as With the Case of P: digit Define as Q weight

7. General FR Number Systems –Symmetry Summarizing: Where • In General These Bounds are Asymmetric • Measure of Asymmetry is: • Therefore, Q is a Measure of Asymmetry for Generalized Fixed Radix Number Systems

8. GFRNS – Asymmetry Examples Consider the Negative Radix System: Asymmetric Range:n even   times as many negative as positive valuesn odd   times as many positive as negative values 2’s Complement: (1 more negative number) System: (1 more positive number)

9. GFRNS – Complement Recall that a complement of a digit, xi, is: The Complement of a Value, X, is Calculated as: X Q Thus,

10. Signed-Digit Number Systems • Fixed radix (positional) • Allows each digit to carry a sign example This signed digit (SD) is a new definition of the digit complement

11. Signed-Digit Example for a total of 19 possible digits If n = 2 199 values, however there are 192 = 361 representations possible which implies this is a redundant number system

12. Signed-Digit Example - Redundancy 19 possible digits For n = 2, range is 199 values and 192 = 361 representations implies redundancy Redundancy Index,  =  +  + 1 – r for digit set is [- ,  ] Here,  = 9 + 9 + 1 – 10 = 9, but if  = 0 &  =9, then  = 0. Example redundant representation:

13. Restricting Redundancy

14. Signed-Digit Characteristics • Positive radix, ß > 0 • X = 0 is unique • Easy to convert • Constant Delay for Add/Sub Regardless of Word Size

15. Breaking the Carry Chain Using SD Can make sum only a function of two digit positions Carry-Free Addition Algorithm Step 1: Find interim sum wi and transfer digit ti+1where positional sum pi and Step 2: Find final sum si

16. Signed Digit Addition Hardware

17. SD Addition Example Let a = 6 for r = 10

18. SD Addition Example (Continued) Let X = 1634, Y = 3366 Using normal addition produces a carry chain But by the carry-free algorithm

19. Converting Decimal to SD Let r = 10, a = 6 Consider the value as xi + yiand use algorithm Converting from SD to decimal – just sum plus and minus weights 2030 – 204 = 1826

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

21. Selecting a to Eliminate Carry Chain in SD

22. Binary SD Addition Implies no guarantee that si= wi + ti will not produce a carry Looking at algorithm: Step 1:

23. Unmodified Binary SD Addition Table Step 2: Based on calculation of wi and ti+1 Note: redundancy allows choices for wi and ti+1

24. 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?

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

26. SD Addition Table Choices Takagi, 1985

27. Modified Binary SD Addition Table

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

29. Two SD Encodings 4!=24 possible encodings Only nine are distinct under permutation and logical negation two’s complement

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

31. Encoding 2 Satisfies relation xi = -2xih + xil This means that xiland xi-1hhave the same weight Also simplified addition table possible by regrouping bits

32. 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 zi is two’s complement result c0 = 0 Example -1010

33. 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 X = 5, n = 4, r = 2 Minimal SD representation of X = 5

34. Alternative restricts these values to {0,1} or {1,0} Alternate Class of BSD Addition Tables* Motivation: Previous tables based on calculation of where wi and tirequire 2-bit encoding Note: In the discussion ci will be used in place of tiand ui for wi *see M. Thornton, A Signed Binary Addition Circuit Based on an Alternative Class of Addition Tables

35. Basic Idea for Alternate SBD Representation • Restrict • Add 2bi+1 since it is “borrowed” from i+1 column • Subtract bi since it is “borrowed” from i-1 column

36. Basic SD Addition Tables Both have inherent propagation limitations

37. Alternative 1 BSD Addition Table* bi=1 bi=0 * Table 2 of Thornton paper

38. Alternative 2 BSD Addition Table* bi=0 bi=1 * Table 4 of Thornton paper

40. Encoding Scheme For Even Parity* • Single bit error coverage over digit pairs • Choose • Then each successive pair of signed binary digits can • be grouped with even parity *see M. Thornton, Signed Binary Addition Circuitry with Inherent Even Parity Outputs