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

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), Choose -1 Positive Radix Negative Radix Of these, 1 is Pos. Radix 1 is Neg. Radix The following is the Radix-complement:

General FR Number Systems - Properties Largest Representable Integer Smallest Representable Integer p i are Digits of P

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

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:

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

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

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

GFRNS – Complement Recall that a complement of a digit, x i, is: The Complement of a Value, X, is Calculated as: Q X Thus,

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

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

Signed Digit Example Redundancy 19 possible digits For n = 2, range is 199 values and 19 2 = 361 representations implies redundancy Example redundant representation:

Restricting Redundancy

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

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 w i and transfer digit t i+1 where and Step 2: Find final sum s i positional sum p i

Signed Digit Addition Hardware

SD Addition Example Let a = 6 for r = 10

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

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