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Abdullah Said Alkalbani alklbany@hotmail University of Buraimi

ECEN2102 Digital Logic Design Lecture 9 Combinational Circuits Binary Adders and Subtractors. Abdullah Said Alkalbani alklbany@hotmail.com University of Buraimi. Overview. Addition and subtraction of binary data is fundamental Need to determine hardware implementation

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Abdullah Said Alkalbani alklbany@hotmail University of Buraimi

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  1. ECEN2102 Digital Logic Design Lecture 9 Combinational Circuits Binary Adders and Subtractors Abdullah Said Alkalbani alklbany@hotmail.com University of Buraimi

  2. Overview • Addition and subtraction of binary data is fundamental • Need to determine hardware implementation • Represent inputs and outputs • Inputs: single bit values, carry in • Outputs: Sum, Carry • Hardware features • Create a single-bit adder and chain together • Same hardware can be used for addition and subtraction with minor changes • Dealing with overflow • What happens if numbers are too big?

  3. A B S C 0 0 0 1 A 0 S 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 B 0 C 1 Dec Binary 1 1 +1 +1 2 10 Half Adder • Add two binary numbers • A0 , B0 -> single bit inputs • S0 -> single bit sum • C1 -> carry out

  4. A3 A2 A1 A0 B3 B2 B1 B0 0 1 0 1 0 1 1 1 A B Ci+1 Ci 0 1 0 1 0 1 1 1 1 1 1 Ai +Bi A B Si 1 1 0 0 Multiple-bit Addition • Consider single-bit adder for each bit position. Each bit position creates a sum and carry

  5. AiBi Ci Ai Bi Si Ci+1 00 01 11 10 Ci 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 Si Full Adder • Full adder includes carry in Ci • Notice interesting pattern in Karnaugh map.

  6. Ci Ai Bi Si Ci+1 Si = !Ci & !Ai & Bi # !Ci & Ai & !Bi # Ci & !Ai & !Bi # Ci & Ai & Bi 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 Full Adder • Full adder includes carry in Ci • Alternative to XOR implementation

  7. Si = !Ci & !Ai & Bi # !Ci & Ai & !Bi # Ci & !Ai & !Bi # Ci & Ai & Bi Si = !Ci & (!Ai & Bi # Ai & !Bi) # Ci & (!Ai & !Bi # Ai & Bi) Si = !Ci & (Ai $ Bi) # Ci & !(Ai $ Bi) Si = Ci $ (Ai $ Bi) Full Adder • Reduce and/or representations into XORs

  8. AiBi Ci Ai Bi Si Ci+1 00 01 11 10 Ci 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 Ci+1 Full Adder • Now consider implementation of carry out • Two outputs per full adder bit (Ci+1, Si) Note: 3 inputs

  9. AiBi Ci Ai Bi Si Ci+1 00 01 11 10 Ci 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 Ci+1 Ci+1 = Ai & Bi # Ci & Bi # Ci & Ai Full Adder • Now consider implementation of carry out • Minimize circuit for carry out - Ci+1

  10. Full Adder Ci+1 = Ai & Bi # Ci !Ai & Bi # Ci & Ai & !Bi Ci+1 = Ai & Bi # Ci & (!Ai & Bi # Ai & !Bi) Ci+1 = Ai & Bi # Ci & (Ai $ Bi) Recall: Si = Ci $ (Ai $ Bi) Ci+1 = Ai & Bi # Ci & (Ai $ Bi)

  11. Half-adder Half-adder Full Adder • Full adder made of several half adders Si = Ci $ (Ai $ Bi) Ci+1 = Ai & Bi # Ci & (Ai $ Bi)

  12. Full Adder • Hardware repetition simplifies hardware design A full adder can be made from two half adders (plus an OR gate).

  13. Block Diagram Full Adder • Putting it all together • Single-bit full adder • Common piece of computer hardware

  14. C 1 1 1 0 A 0 1 0 1 B0 1 1 1 S 1 1 0 0 4-Bit Adder • Chain single-bit adders together. • What does this do to delay?

  15. 110 = 0116 = 00000001 -110 = FF16 = 11111111 12810 = 8016 = 10000000 -12810 = 8016 = 10000000 Negative Numbers – 2’s Complement. • Subtracting a number is the same as: • Perform 2’s complement • Perform addition • If we can augment adder with 2’s complement hardware?

  16. +1 Add A to B’ (one’s complement) plus 1 4-bit Subtractor: E = 1 That is, add A to two’s complement of B D = A - B

  17. Adder- Subtractor Circuit

  18. AN-1 BN-1 Overflow? CN-1 Assumes an N-bit adder, with bit N-1 the MSB Overflow in two’s complement addition • Definition: When two values of the same signs are added: • Result won’t fit in the number of bits provided • Result has the opposite sign.

  19. Addition cases and overflow 00 0010 0011 -------- 0101 01 0011 0110 -------- 1001 11 1110 1101 -------- 1011 10 1101 1010 -------- 0111 00 0010 1100 -------- 1110 11 1110 0100 -------- 0010 -3 -6 7 2 -4 -2 -2 4 2 2 3 5 3 6 -7 -2 -3 -5 OFL OFL

  20. Summary • Addition and subtraction are fundamental to computer systems • Key – create a single bit adder/subtractor • Chain the single-bit hardware together to create bigger designs • The approach is call ripple-carry addition • Can be slow for large designs • Overflow is an important issue for computers • Processors often have hardware to detect overflow • Next time: encoders/decoder.

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