FixedPoint Negative Numbers

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Fixed-Point Negative Numbers

• Two Common Forms
• Signed-Magnitude Form
• Complement Forms

Signed-Magnitude Numbers

• First Digit is Sign Digit, Remaining n-1 are the
  Magnitude
• Convention (binary)
• 0 is a Positive Sign bit
• 1 is a Negative Sign bit
• Convention (non-binary)
• 0 is a Positive Sign digit
• ? -1 is a Negative Sign digit
• Only 2 ? n-1 Digit Sequences are Utilized

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Signed-Magnitude Example
Largest Representable Value is
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Signed-Magnitude Example (cont)
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Signed-Magnitude Ternary Example
Notice that fractional part is infinite in ?10
but finite in ?3
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Signed-Magnitude Ternary Bounds
Positive Numbers
Negative Numbers
Range
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Signed-Magnitude Comments

• Two Representations for zero, 0 and 0
• Addition of K and K is not zero
• EXAMPLE
• Disadvantage since algorithm requires comparison
  of signs and, if different, comparison of
  magnitudes

10001010.002 00001010.002 10010100.002
-10101010 Yields a Sum of 2010!!!!!
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Complement Representations

• Two Types of Complement Representations
• Radix Complement (binary 2s-complement)
• Diminished-Radix Complement (binary
  1s-complement)
• Positive Values Represented Same Way as Signed
  Magnitude for Both Types
• Negative Value, -Y, Represented as R-Y Where R is
  a Constant
• Obeys the Identity

• Advantage is No Decisions Needed Based on
  Operand Sign Before Operations are Applied

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Complement Representation Example

• X is Positive, Y is Negative, Compute X Y
  Using Complement Representation

• If Y gt X, Then the Answer is R - (Y - X)
• If X gt Y, Then the Answer Should be X - Y
• But X (R - Y) R (X - Y),
• Thus R Must be Discarded!
• Solution is to Choose the Value of R Carefully

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Requirements for Complementation Value, R

• Select R to Simplify (or Eliminate) Correction
  for the X gt Y Case
• Calculation of Complement of Y or (R-Y) Should be
  Simple and Fast
• Definition of Complement for Single Digit, xi
• Definition of Digit Complement for a Word, X

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Complementation Value, R

• Add Word and Complement Together

Answer to Addition
Now Add 1 ulp

• Therefore, we see that

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Radix-Complement Form

• The Radix Complement Form is Defined When

• Using ? k is Convenient Since Storing Result in
  Register of Length n Causes MSD of 1 to be
  Discarded due to Finite Register Length
• Therefore, it is Easy to Compute the Complement
  of X by
• Take the Digit Complement of X
• Add 1ulp to Complement

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Radix-Complement Form (cont)

• No Correction is Needed When We have Positive X
  and Negative Y Such That

• Since R? k

• And ? k is discarded Due to Finite Register Length

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Radix-Complement Example

• Since n m k ? m 0
• Therefore 1 ulp 20 1
• Given X, the radix complement (2s complement) is

• Range of Positive Numbers is 0000,0111
• 2s Complement of Largest, 0111

• In Radix Complement, There is a Single
  Representation of Zero(0000) and Each Positive
  Number has Corresponding NegativeNumber With
  MSB1

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Radix-Complement Example

• In Radix Complement, There is a Single
  Representation of Zero(0000) and Each Positive
  Number has Corresponding NegativeNumber With
  MSB1
• Accounts for 1(zero)7(pos.)7(neg.), But Extra
  Bit Pattern Left
• One Additional Negative Number, 10002-810,
  -810?X?710

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Diminished-Radix Complement

• In Diminished Radix Complement, the
  Complementation Process is Easier Since the
  Addition of 1 ulp is Avoided

• Range of Positive Numbers is 00002,01112010,
  710
• 1s Complement of Largest is 10002 -710
• 1s Complement of Zero is 11112
• Two Representations of Zero!

• In All Cases MSB is Sign Bit

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Comparison of Twos Complement, Ones Complement
and Signed-Magnitude
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Signed-Number Arithmetic

• Signed Magnitude Only Use Magnitude Digits

Carry-out ? Overflow
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Radix-Complement Arithmetic

• Radix Complement In this case 2s Complement

Carry-out Does NOT Mean Overflow
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2s-Complement Overflow

• If X, Y have opposite signs overflow never
  occurs whether carry-out exists or not

No Carry-out
Carry-out

• If X, Y have same sign and result sign differs,
  overflow occurs

No Carry-out,Overflow
Carry-out, Overflow
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1s-Complement Overflow

• Ones complement carry-out indicates a
  correction is needed

• If X gt Y, then answer should be X-Y however
  registercontains X-Y-ulp since 2n is carry-out
  bit, therefore mustcorrect by adding 1 ulp

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Example of 1s-Complement Overflow
NeedCorrectionSinceOverflow
So-called end-around carry
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End-Around Carry Design
Carry-out

• This is end-around carry always add
  carry-out to LSD

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Other Number Systems

• Binary Number Systems are Most Common
• In terms of building fast systems, we should
  consider
• Negative Radix
• Signed Digit
• Log (logarithm)
• Signed Log
• Complex Radix
• Mixed Radix
• Residue Number Systems

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Negative-Radix Fixed-Position Systems
Nega-decimal example
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Nega-Decimal Number System
Largest Positive Value, Xmax
Smallest Value, Xmin
Finite Register Length, n3 digits
Asymmetric System!!! 10 times more positive
than negative values represented
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Nega-Decimal Number System
Finite Register Length, n4 digits
Now more Negative Values than Positive
Nega-decimal System Characteristics

• Arithmetic Operations Same Regardless of Sign of
  Number
• No Signed Digit/Complement Representation Needed
• Sign of X Determined by Position of First
  Non-zero Digit

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Nega-Binary Number System
Negative Radix
Example
How is this Addition Operation Performed?????
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Nega-Binary Number System
wi Values
(5)10
(-3)10
(114-2)10
(000)10
(4416-8)10
(0-8-8)10
(5-32)-10
Carry-out
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Nega-Binary Adder Design

• Individual Adder Cells Produce Two Carry-out
  Bits
• Design a Circuit at Gate Level for a 4-Digit
  Nega-Binary Adder
• Hint Cout Functions Should Look Familiar!