General Fixed Radix Number Systems

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General Fixed Radix Number Systems

• Nonredundant
• Positive radix, ß
• n digits in digit set
• Vector

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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

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General FR Number Systems - Properties
Largest Representable Integer
Smallest Representable Integer
pi are Digits of P
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General FR Number Systems - Properties
Using the pi Expression and Forming the Radix
Polynomial for P
digit
Define as Q
weight
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General FR Number Systems - Properties
Q is the value represented by the following
n-tuple if all ?i1
For N, the Smallest Representable Value
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General FR Number Systems - Properties
Using Similar Analysis as With the Case of P
digit
Define as Q
weight
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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

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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
2s Complement
(1 more negative number)
System
(1 more positive number)
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GFRNS Complement
Recall that a complement of a digit, xi, is
The Complement of a Value, X, is Calculated as
X
Q
Thus,
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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
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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
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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
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Restricting Redundancy
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Signed-Digit Characteristics

• Positive radix, ß gt 0
• X 0 is unique
• Easy to convert
• Constant Delay for Add/Sub Regardless of Word
  Size

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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 ti1 where
 
positional sum pi
and
Step 2 Find final sum si
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Signed Digit Addition Hardware
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SD Addition Example
Let a 6 for r 10
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SD Addition Example (Continued)
Let X 1634, Y 3366 Using normal addition
produces a carry chain
But by the carry-free algorithm
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Converting Decimal to SD
Let r 10, a 6 Consider the value as xi yi
and use algorithm
Converting from SD to decimal just sum plus and
minus weights 2030 204 1826
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Selecting a to Eliminate Carry Chain in SD
For no carry, require
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Selecting a to Eliminate Carry Chain in SD
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Binary SD Addition
Implies no guarantee that si wi ti will not
produce a carry Looking at algorithm Step 1
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Unmodified Binary SD Addition Table
Step 2 Based on calculation
of wi and ti1
xi,yi
ti1
wi
Note redundancy allows choices for wi and ti1
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How Useful is Unmodified Table?
Works if operands do not contain If operands
contain only 0s and 1s, no carry generated.

Example
Why not use this approach to break carry chain
for unsigned binary number?
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Limitations of Table
Does not work if operands contain
Example (-9)10 (29)10
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SD Addition Table Choices
Takagi, 1985








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Modified Binary SD Addition Table
xi,yi
xi-1,yi-1 - neither is at least one is neither is at least one is - -
ti1
wi
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Repeating Example with Modified Table
Example (-9)10 (29)10
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Two SD Encodings
4!24 possible encodings Only nine are distinct
under permutation and logical negation
twos complement
x Encoding 1 xh xl Encoding 2 xh xl

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Encoding 1
Satisfies simple relation
x xl - xh
and 11 has a valid numerical value of 0.
SD to twos complement conversion performed by
twos complement subtraction
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Encoding 2
Satisfies relation
xi -2xih xil
This means that xil and xi-1h have the same
weight
Also simplified addition table possible by
regrouping bits
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Twos Complement/BSD Conversion
Twos Complement to SD
Bits can be encoded directly with MSB negative one
BSD to Twos Complement
One algorithm simpler than complete binary adder
zi is twos complement result c0 0
Example -1010
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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
 
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Alternate Class of BSD Addition Tables
Motivation Previous tables based on calculation
of where wi and ti require 2-bit encoding
xi,yi
ti1
wi
Note In the discussion ci will be used in place
of ti and ui for wi
see M. Thornton, A Signed Binary Addition
Circuit Based on an Alternative Class of Addition
Tables
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Basic Idea for Alternate SBD Representation

• Restrict

• Add 2bi1 since it is borrowed from i1 column
• Subtract bi since it is borrowed from i-1
  column

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Basic SD Addition Tables
1 0 0 1
1 2 2 1 0
0 1 1 0 1
1 0 0 1 2
1 0 0 1
1 10 10 01 00
0 01 01 00 01
1 00 00 01 10
Both have inherent propagation limitations
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Alternative 1 BSD Addition Table
bi1
1 0 1
1 1 0 1 1 0 0 1 1 1
1 0 0 1 1 1 1 1 0
0 1 0 0 0 0 1 0 0 0
1 1 1 0 0 0 0 1 1
1 1 1 1 0 0 0 0 1 1
1 1 0 0 1 1 0 1 0
bi1 ci1 ui
bi1 ci1 ui
bi0
Table 2 of Thornton paper
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Alternative 2 BSD Addition Table
bi0
1 0 1
1 0 1 0 0 1 1 1 1 0
0 1 1 0 0 0 1 1 1
0 0 1 1 0 0 0 1 1 1
0 0 0 0 0 1 1 0 0
1 1 1 0 1 1 1 1 0 0
1 1 1 1 0 0 1 0 1
bi1 ci1 ui
bi1 ci1 ui
bi1
Table 4 of Thornton paper
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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