Arithmetic Circuits II
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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Arithmetic Circuits II
Anselmo Lastra
2 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
3 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Examples• 4-bit signed numbers• 7 + 7• 7 – 7♦ Generates carry but result OK
• -7 -7• 4 + 4♦ Generates no Cout, but overflowed
4 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Overflow Detection• Condition is that either Cn-1 or
Cn is high, but not both
5 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Multiplier• Multiply by doing single-bit
multiplies and shifts• Look at combinational circuit to
do this
6 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Combinational MultiplierAND computes A0 B0
Half adder computes sum. Will need FA for larger multiplier.
7 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Larger Multiplier
8 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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
9 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Contraction• Can implement other functions♦ Like increment, decrement
• By using basic arithmetic circuits♦ Adder
• And removing unused portions• This is called contraction
10 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Example: Incrementing• Very common♦ Next address computation
• Specialize an adder
11 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Eliminate Unneeded Gates
12 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Multiplication by Constant
• What if constant is a power of two?
• What is the circuit?
13 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Multiplication by Pwr of 2
• Just wires
• Division by pwr of 2 similar
14 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Arbitrary Constant
15 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
Contraction Not Always Best
• Sometimes it pays to rethink the function
• Example of decrementer in book
16 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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
17 The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL
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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