بسم الله الرحمن الرحيم

    

 

 

 

     
 

Binary Signed Digit Numbers

Main Features:

Ø      Radix: r =2

Ø      Digit set: , with  a =1

Ø     

Properties of BSD Numbers:

  1. BSD numbers have positive and negative digits (no need for a rule to represent negative numbers)
  2. No carry propagation for addition, which means addition is fast
  3. BSD numbers are inherently redundant

 

Redundancy of BSD numbers:

The radix of BSD numbers is r = 2; and a =1; let n be the number of digits.

Thus,

*        Range (R):   To

*        Number of representation: R =2n+1 +1 numbers

*        Possible number of combination: P = 3n

*        % redundancy = 

  

Encoding of SD Binary Numbers:

Two codes have been used in practice, which are:

Encoding #2 is the two's complement representation of the signed digit number x and will be used though out this topic.

  

String Property:

 

This property is used to set equivalent representations for the BSD numbers.

 Note that:        (1)

                         (2)

 Example (1):

Express (23)10 in binary using 6 digits; then find 5 different representations in BSD.

 

           

 

Minimal Representations of Binary SD Numbers:

¨       Representation of BSD numbers with a minimal number of nonzero digits

¨       Important for fast multiplication and division algorithms

¨       The canonical recoding algorithm generates minimal SD representations of given binary numbers

 

Example (2):

Represent X=(7)10 and find the minimal representation:

           

  is the minimal representation

 

Addition of Binary SD Numbers

Interim sum and carry in addition algorithm:

 

 

Summary of rules:

 

 When we use this rule there is no guarantee that a new carry will not be generated in the second step of the algorithm.

 

Addition of Binary SD Numbers: Modified Rules

 

Example (3):

 

 Direct summation of the two operands results in which is equivalent to , all represent (3)10

 

Number Systems

    - Conventional Number System

         >Properties

         >Binary, Decimal, Hexadecimal

         >Number Base Conversion

    - Unconventional Number System

         >Roman Number System

         >Signed Digit Number System

         >Binary SD Number System

Fast Addition

    - Half and Full Adders

    - Ripple Carry Adder

    - Carry Look-Ahead Adders

Fast Multiplication

    - Carry Save Adders

    - Additive & Non-additive Multiplier

    - Counters