Arithmetic Circuits II
Anselmo Lastra
The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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
we’ll 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
• Can’t 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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We’ve Covered
• Adders
♦ Ripple carry
♦ Carry lookahead
• Subtracting unsigned numbers
♦ New design for adder-subtractor
• Signed numbers
♦ Signed addition/subtraction
• Multiplication – just basic
• Modified Circuits
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