1. Full Adder Display

2. Topics A 1 bit adder with LED display Ripple Adder Signed/Unsigned Subtraction Hardware Implementation of 4-bit adder

3. Implementation of a Full Adder (carry-in)

4. Verilog Implementation Use switches to input binary numbers—x, y, and z. z is the carry-in. Display the output on the LED. Press a button to determine which bit will be displayed. s represents the sum bit. c represents the carry-out bit. A mux is used to determine Whether s or c should be displayed.

5. Multiplexing 7-Segment Displays (Last Week) Get values for an[3:0] from btn[3:0] so that only one LED is displayed. If s[1:0]=00, then x[3:0]. If s[1:0]=01, then x[7:4]. If s[1:0]=10, then x[11:8]. If s[1:0]=11, then x[15:12]. Use Quad 4-to-1 mux

6. Explanation of the Code If btn[0] is pushed, t[0] is 0. If btn[1] is pusehd, t[0] is 1. So we can use t[0] as a selector bit for the MUX. t[ ] =s[] If the output of the MUX is a 0, a 0 Is displayed. If the output of the MUX is a 1, a 1 is displayed.

7. Implementation of a Full Adder (carry-in)

8. Four-Bit Adder C 4 is calculated last because it takes C 0 8 gates to reach C 4. Each FA uses 2 XOR, 2 AND and 1 OR gate. A four-bit adder uses 8 XOR, 8 AND and 4 OR gate.

9. Alternative Naming Convention for the Full Adder

10. Hardware Simplification 2 gate delays for C 3 !

11. Four-bit adder with Carry Lookahead Ripple adder uses 8 XOR, 8 AND and 4 OR gate. Lookahead implementation: 8 XOR, (4+6) AND, 1 2-input OR, 2 3-input OR.

12. Advantages C1, C2 and C3 do not have to wait for C1 and C2 to progate. C3 is propagated at the same time as C1 and C2.

13. carry_lookahead.v

14. four_bit_adder_carry_lookahead.v

15. four_adder_carry_lookahead_top.v

16. Topics Calculations Examples – Signed Binary Number – Unsigned Binary Number Hardware Implementation Overflow Condition

17. Unsigned Number Decimalb1b0 000 101 210 311 (2-bit example)

18. Unsigned Addition 1+2= Decimalb1b0 000 101 210 311 Decimalb1b1 b0b0 101 +210 311

19. Unsigned Addition 1+3= Decimalb1b0 000 101 210 311 Decimalb1b1 b0b0 11 101 +311 4100 (Carry Out) (Indicates Overflow) Overflow can be an issue in unsigned addition.

20. Unsigned Subtraction (1) 1-2= Decimalb1b0 000 101 210 311 Decimalb1b1 b0b0 101 +-210 11 00 01 (1’s complement) (2’s complement)

21. Unsigned Subtraction (2) 2-1= Decimalb1b0 000 101 210 311 Decimalb1b0 1 210 +11 3101 Discarded

22. Summary for Unsigned Addition/Subtraction Overflow can be an issue in unsigned addition Unsigned Subtraction (M-N) – If M≥N, and end carry will be produced. The end carry is discarded. – If M<N, Take the 2’s complement of the sum Place a negative sign in front

23. Signed Binary Numbers 4-bit binary number – 1 bit is used as a signed bit – -8 to +7 – 2’s complement

24. Signed Addition (70+80) b7b7 b6b6 b5b5 b4b4 b3b3 b2b2 b1b1 b0b0 01 7001000110 8001010000 10010110 70=2 1 +2 2 +2 6 =2+4+64 80=2 4 +2 6 =16+64 10010110→01101001 →01101010 2 1 +2 3 +2 5 +2 6 =2+8+32+64=106 10010110↔-106 (Indicates a negative number) 010010110 010010110↔ 2 1 +2 2 +2 4 +2 7 =2+4+16+128=150 Conclusion: There is a problem of overflow Fix: Use the end carry as the sign bit, and let b7 be the extra bit.

25. Signed Subtraction (70-80) b7b7 b6b6 b5b5 b4b4 b3b3 b2b2 b1b1 b0b0 7001000110 -8010110000 11110110 70=2 1 +2 2 +2 6 =2+4+64 80=2 4 +2 6 =16+64=01010000→10101111→10110000 11110110→00001001 →00001010 2 1 +2 3 =10 11110110↔-10 (Indicates a negative number) (No Problem)

26. Signed Subtraction (-70-80) b7b7 b6b6 b5b5 b4b4 b3b3 b2b2 b1b1 b0b0 1011 -7010111010 -8010110000 01101010 70=2 1 +2 2 +2 6 =2+4+64 80=2 4 +2 6 =16+64 (Indicates a positive number! A negative number expected.) 101101010 →010010101 → 010010110 010010110 ↔2 1 +2 2 +2 4 +2 7 =2+4+16+128=150 101101010 ↔-150 Conclusion: There is a problem of overflow Fix: Use the end carry as the sign bit, and let b7 be the extra bit.

27. Observations Given the similarity between addition and subtraction, same hardware can be used. Overflow is an issue that needs to be addressed in the hardware implementation A signed number is not processed any different from an unsigned number. The programmer must interpret the results of addition and subtraction appropriately.

28. Four-Bit Adder-Subtractor

29. The Mode Input (1)

30. The Mode Input (2)

31. M=0 (Addition) 0 B3B3 B2B2 B1B1 B0B0

32. M=1 (Subtraction) 1 2’s complement is generated of B is generated!

33. Unsigned Addition When two unsigned numbers are added, an overflow is detected from the end carry.

34. Detect Overflow in Signed Addition Observe 1.The cary into the sign bit 2.The carry out of the sign bit If they are not equal, they indicate an overflow.

35. FPGA Demo: 12+15

36. FPGA: 15-12

37. FPGA: 12-15