Number Representation, Addition
Tom Kelliher, CS26
Nov. 26, 1996
Properties of number systems:
 Positional.
 Weighted.
Example: decimal
Binary weights:
 Integer.
 Fraction.
 Binary point.
 Bit numbering.
Unsigned value of an binary integer:
Representation error: splitting the gap.
Msb is the sign bit:
 Has no weight.
 If 0, the number is ``positive.''
 If 1, the number is ``negative.''
 Value:
 Problems:
 Multiple representations of 0.
 Hard to build good adders.
 Decimaltobinary conversions. Vice versa.
 Symmetric range.
Msb is the sign bit:
 Has negative weight: .
 If 0, the number is ``positive.''
 If 1, the number is ``negative.''
 Value:
 Problems:
 Multiple representations of 0.
 Wraparound carrys.
 Conversions: decimal, binary, negative, positive.
 Symmetric range.
Msb is the sign bit:
 Has negative weight: .
 If 0, the number is ``positive.''
 If 1, the number is ``negative.''
 Value:
 Problems:
 Asymmetric range.
 Conversions: decimal, binary, negative, positive.
Truth table:
Truth table:
Delay model for a full adder:
 Two gate delays from any input to any output.
 How would you tie full adders together to form an nbit adder?
 What would it's delay be?
 For eight bit adders, what is an example input that would cause a
full precision carry?
 Why bother with speeding up addition?
 Can we precompute the carries?
 Carry generate: .
 Carry propagate: .
 Some carry equations:



 A bit of recursion:
 Base case:
 Recursive step:
 Why this isn't feasible.
 Design a 4bit full carry lookahead adder.
Block diagram:
Block generate, propagate.
 What is the fanin?
 What is the delay model from inputs to outputs?
 Design a 4Group carry lookahead unit.
Block diagram:
Use of block generates, propagates.
 What is the fanin?
 What is the delay model from inputs to outputs?
Cascaded and full carry lookahead.
Cascaded and full carry lookahead.
Thomas P. Kelliher
Mon Nov 25 18:14:25 EST 1996
Tom Kelliher