arithmetic circuits ii
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Arithmetic Circuits II. Anselmo Lastra. 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. Examples. 4-bit signed numbers - PowerPoint PPT PresentationTRANSCRIPT
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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 MultiplierAND 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