Arithmetic Circuits II

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Arithmetic Circuits II

• Anselmo Lastra

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Overflow

• Two cases of overflow for addition of signed
  numbers
• Two large positive numbers overflow into sign bit
• Not enough room for result
• Two large negative numbers added
• Same not enough bits
• Carry out can be OK

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Examples

• 4-bit signed numbers
• 7 7
• 7 7
• Generates carry but result OK
• -7 -7
• 4 4
• Generates no Cout, but overflowed

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Overflow Detection

• Condition is that either Cn-1 or Cn is high, but
  not both

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Multiplier

• Multiply by doing single-bit multiplies and
  shifts
• Look at combinational circuit to do this

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Combinational Multiplier
AND computes A0 B0
Half adder computes sum. Will need FA for larger
multiplier.
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Larger Multiplier
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Sequential Multiply

• Imagine doing over time rather than in parallel
• Bitwise multiply
• Shift
• Add
• If we have time later in semester well look at
  fancier multipliers

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Contraction

• Can implement other functions
• Like increment, decrement
• By using basic arithmetic circuits
• Adder
• And removing unused portions
• This is called contraction

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Example Incrementing

• Very common
• Next address computation
• Specialize an adder

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Eliminate Unneeded Gates
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Multiplication by Constant

• What if constant is a power of two?
• What is the circuit?

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Multiplication by Pwr of 2

• Just wires
• Division by pwr of 2 similar

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Arbitrary Constant
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Contraction Not Always Best

• Sometimes it pays to rethink the function
• Example of decrementer in book

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Sign Extension

• Changing size of number common in instruction
  sets
• 16-bit immediate to register, for example
• Cant just add zeros (zero fill)
• Would turn 4-bit -5 (1011) to 8-bit 11
  (00001011)
• Sign extend (fill with left digit)
• 1011 to 11111011

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Weve Covered

• Adders
• Ripple carry
• Carry lookahead
• Subtracting unsigned numbers
• New design for adder-subtractor
• Signed numbers
• Signed addition/subtraction
• Multiplication just basic
• Modified Circuits