Representing Data

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Representing Data

• ECE-331, Digital Design
• Dr. Ron Hayne
• Electrical and Computer Engineering

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Binary Number System

• Binary
• Positional number system
• Two symbols, B 0, 1
• Easily implemented using switches
• Easy to implement in electronic circuitry
• Algebra invented by Boole allows easy
  manipulation of symbols

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General Positional Notation

• Consider a Positional Number System Using an
  Arbitrary Radix, r

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Counting Number Representation

• Then, for Counting Numbers,
• e.g., Decimal

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Counting Number Representation

• e.g., Binary (radix 2)

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Representation of Fractions
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Decimal Fraction Example

• Let p 3

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Binary Fraction Example

• Let p 3

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Radix Conversion

• Desire to convert (N.F)r1 --gt (N.F)r2
• R N.F

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Conversion of N

• Number Part of R in r1 can be Represented by a
  Polynomial in r2
• If N is Divided by the Second Radix

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Conversion of N

• Repeated Division Yields All Coefficients

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Counting Number Example
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Counting Number Example
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Conversion of F

• Fraction Part of R in r1 can be Represented by a
  Polynomial in r2
• If F is Multiplied by the Second Radix

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Conversion of F

• Repeated Multiplication Yields All Coefficients

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Fraction Conversion Example
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Another Example

• 23.187510 ?2

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Conversion Considerations

• Purpose of Binary Representation is to Put
  Numbers in a Form which can be Manipulated by
  Digital Logic
• Register Size Limits Resolution
• Not All Numbers in One Radix Can Be Accurately
  Represented in Another Radix

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Binary Arithmetic

• Addition

Input Input Input Output Output
Augend Addend Carry-In Carry-Out Sum
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1
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Example

• 7 14 21

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Negative Numbers

• Signed Magnitude
• Leftmost bit represents sign
• 1 negative
• 0 positive
• Rightmost bits represent magnitude

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Signed Magnitude Example

• Let q 2
• 2 zeroes
• Shifting
• Does not mult/div by radix,
• Does accountfor sign

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Negative Numbers

• Reduced Radix
• 1s complement, 9s complement, etc.
• For 1s complement, alternative to subtraction is
  to complement each bit

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Reduced Radix Example

• Let q 2
• 2 zeroes

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Negative Numbers

• Radix Complement
• 2s complement, 10s complement, etc.
• For 2s complement, alternative to subtraction is
  to take the ones complement and add 1

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Radix Complement Example

• Let q 2
• 1 zero !!!

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Advantages of Radix Comp.

• One Zero
• 2s Complement of Negative Number Yields
  Magnitude
• Multiplication and Division by Radix Yield Scaled
  Results with Correct Sign if Sign Replicated in
  MSB on Divide

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2s Complement

• Forming the 2s Complement is an Operation and
  Can be Done with no Reference to a Negative
  Number
• 2s Complement Notation is an Agreement to use
  the 2s Complement Representation of a Positive
  Number to Represent the Associated Negative Number

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Examples

• 4-bit 2s complement arithmetic
• 3 - 6 - 3
• - 4 - 5 ?

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Binary, Octal, Hexadecimal

• It Is Easy to Convert Binary Numbers to Other,
  More Convenient Representations by Grouping Bits
  Together and Then Converting by Inspection
• Octal 0, 1, ... , 7
• 3-bit groups
• Hexadecimal 0, ... , 9, A, ... , F
• 4-bit groups

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Binary to Octal Example
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Binary to Hex Example
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Weighted Codes

• Arbitrary Weight Assigned to Each Position
• Binary Coded Decimal (BCD)
• 8, 4, 2, 1
• e.g., 1001 8 _ _ 1 9
• Not all codewords used, e.g., (1101)BCD

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Non-Weighted Codes

• Excess 3
• Used for decimal arithmetic
• Representation is 3 more than binary value
• Self-Complementing

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Non-Weighted Codes

• Cyclical
• Sequential codewords differ by only one bit and
  wrap
• Particularly useful for electro-mechanical
  interfaces where all bits must be mechanically
  aligned to generate code
• May have minor inaccuracy but no large jumps
• Reflected Grey Code is most common example
• e.g., Shaft-encoders

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Reflected Grey Code

• Benefits
• Easy to construct to any resolution (number of
  bits)
• Cyclic
• Unique
• Construct by prepending each codeword with zero
  and mirroring codes, complementing MSB of
  mirrored codewords

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3-bit Reflected Grey Code

• Angle Codeword
• 0-44 0 0 0
• 45-89 0 0 1
• 90-134 0 1 1
• 135-179 0 1 0
• 180-224 1 1 0
• 225-269 1 1 1
• 270-314 1 0 1
• 315-359 1 0 0

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End of Lecture
The End