Number Representation, Addition

Tom Kelliher, CS26

Nov. 26, 1996

Number Representation

Properties of number systems:

1. Positional.

2. Weighted.

Binary weights:

1. Integer.

2. Fraction.

3. Binary point.

4. Bit numbering.

Unsigned value of an binary integer:

Representation error: splitting the gap.

Sign-and-Magnitude Representation

Msb is the sign bit:

1. Has no weight.

2. If 0, the number is ``positive.''

3. If 1, the number is ``negative.''

4. Value:

   

5. Problems:
   1. Multiple representations of 0.

   2. Hard to build good adders.

6. Decimal-to-binary conversions. Vice versa.

7. Symmetric range.

1's-Complement Representation

Msb is the sign bit:

1. Has negative weight: .

2. If 0, the number is ``positive.''

3. If 1, the number is ``negative.''

4. Value:

   

5. Problems:
   1. Multiple representations of 0.

   2. Wrap-around carrys.

6. Conversions: decimal, binary, negative, positive.

7. Symmetric range.

2's-Complement Representation

Msb is the sign bit:

1. Has negative weight: .

2. If 0, the number is ``positive.''

3. If 1, the number is ``negative.''

4. Value:

   

5. Problems:
   1. Asymmetric range.

6. Conversions: decimal, binary, negative, positive.

Unsigned Addition

Half Adder

Truth table:

Full Adder

Truth table:

Delay model for a full adder:

1. Two gate delays from any input to any output.

Ripple-Carry Addition

1. How would you tie full adders together to form an n-bit adder?

2. What would it's delay be?

3. For eight bit adders, what is an example input that would cause a full precision carry?

Carry Lookahead Addition

1. Why bother with speeding up addition?

2. Can we pre-compute the carries?

3. Carry generate: .

4. Carry propagate: .

5. Some carry equations:
   1. 

   2. 

   3. 

6. A bit of recursion:
   1. Base case:

   2. Recursive step:

7. Why this isn't feasible.

4-Bit Carry Lookahead Adder

1. Design a 4-bit full carry lookahead adder.

   Block diagram:

   

   Block generate, propagate.

2. What is the fan-in?

3. What is the delay model from inputs to outputs?

4-Group Carry Lookahead Unit

1. Design a 4-Group carry lookahead unit.

   Block diagram:

   

   Use of block generates, propagates.

2. What is the fan-in?

3. What is the delay model from inputs to outputs?

16-Bit Carry Lookahead Adders

Cascaded and full carry lookahead.

32-Bit Carry Lookahead Adders

Cascaded and full carry lookahead.