Carry-Select Adders

1. Carry-Select Adders • Consider the Following Partitioned Addition: 101 +110 101 +111 110 +001 1  011  0 100  0 111  0 1  100  1 101  1 000  1 1 100 100 111

2. 4-Bit Carry Select X = 1 0 1 1 0 1 1 0 +Y = 0 0 1 0 1 1 0 1 sum 1 1 0 1 0 0 1 1 cout 0 1 sum 1 1 1 0 cout 0 } cin = 0 } cin = 1 • Answer: 1110 0011 • cin 1 for Least Significant Nibble • Must Wait for 4 “Ripples” to Select • Can Divide into Groups of Two

3. Carry-Select Adders

4. 2-Bit Carry Select X = 1 0 1 1 0 1 1 0 +Y = 0 0 1 0 1 1 0 1 sum 1 0 0 1 0 0 1 1 cout 0 1 1 0 sum 1 1 1 0 0 1 cout 0 1 1 } cin = 0 } cin = 1 • Answer: 11 10 00 11 • cin 1 for Least Significant 2 Bits • Must Wait for 2 “Ripples” to Select • Can Divide into Groups of One – Carry Select

5. Carry-Select Adders

6. 1-Bit Carry-Select/Cond. Sum X = 1 0 1 1 0 1 1 0 +Y = 0 0 1 0 1 1 0 1 sum 1 0 0 1 1 0 1 1 cout 0 0 1 0 0 1 0 0 sum 0 1 1 0 0 1 0 cout 1 0 1 1 1 1 1 } cin = 0 } cin = 1 • Answer: 11 10 00 11 • cin 1 for Least Significant 1 Bit • Must Wait for 1 “Ripples” to Select • Can Divide into Groups of One – Conditional Sum

7. Conditional-Sum Adders

8. 1-Bit Conditional-Sum Adder Cell

9. Conditional-Sum Addition

10. Optimizations/Implementation • Carry Select/Conditional Sum Implemented as Adders with MUX Tree (1-bit Cells for Conditional Sum) • Carry Select is Conditional Sum with One Set of Groups - Not Necessarily with Group Size = 1 • CLA/Conditional Sum Approximately Same Speed • CLA Generally has More Efficient Layout • Trees are Hard (irregular) – CLA More Popular • Prefix Generators are Optimal in speed/area Tradeoff

11. Hybrid Adders

12. Hybrid Adders

13. Optimizations in Adders

14. Optimizations in Adders