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Datapath Components Prof. Usagi Recap: Digital circuits only have 0s and 1s 1 0 2 Recap: Converting from decimal to binary 2 321 2 160 1 2 80 0 2 40 0 2 20 0 2 10 0 2 5 0 2 2

  1. Datapath Components Prof. Usagi

  2. Recap: Digital circuits only have 0s and 1s… 1 0 2

  3. Recap: Converting from decimal to binary 2 321 2 160 …… 1 2 80 …… 0 2 40 …… 0 2 20 …… 0 2 10 …… 0 2 5 …… 0 2 2 …… 1 1 …… 0 321 = 0b 101000001 3

  4. Recap: 2-variable K-map example A’ A A 0 1 Input Output B A B 0 0 1 B’ B’ 0 1 1 0 1 1 1 0 1 1 1 0 B 1 1 0 A’ F(A, B) = A’ + B’ 4

  5. Recap: 3-variable K-map • Reduce to 2-variable K-map — 1 dimension will represent two variables • Adjacent points should differ by only 1 bit • So we only change one variable in the neighboring column • 00, 01, 11, 10 — such numbering scheme is so-called Gray–code A’B’ A’B AB AB’ Input (A, B) Output 0,0 0,1 1,1 1,0 A B C C C’ 0 0 0 1 0 1 1 1 1 0 0 1 1 C’ 0 1 0 1 0 1 1 1 1 1 1 0 0 C 1 0 0 1 A’ 1 0 1 0 1 1 0 1 F(A, B, C) = A’ + C’ 1 1 1 0 5

  6. Recap: 4-variable K-map • Reduce to 2-variable K-map — both dimensions will represent two variables • Adjacent points should differ by only 1 bit • So we only change one variable in the neighboring column • Use Gray-coding — 00, 01, 11, 10 A’B’ A’B AB AB’ 00 01 11 10 A’B’C’ C’D’ 00 1 0 0 0 F(A, B, C) = A’B’C’+B’CD’ C’D 01 1 0 0 0 CD 11 0 0 0 0 CD’ 10 1 0 0 1 B’CD’ 6

  7. Recap: K-Map with “Don’t Care”s You can treat “X” as either 0 or 1 — depending on which is more advantageous A’B’ A’B AB AB’ (A, B) 0,0 0,1 1,1 1,0 C 0 1 C’ C’ 0 1 X 1 1 AC’ C 1 1 1 0 0 A’C A’B’ A’C If we treat the “X” as 0? If we treat the “X” as 1? F(A,B,C)=A’B’+A’C+AC’ F(A,B,C) = C’ + A’C 7

  8. BCD+1 — Binary coded decimal + 1 Input Output • 0x0 — 1 I8 I4 I2 I1 O8 O4 O2 O1 0 0 0 0 0 0 0 1 • 0x1 — 2 0 0 0 1 0 0 1 0 • 0x2 — 3 I8 O8 0 0 1 0 0 0 1 1 Comparator • 0x3 — 4 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 1 I4 O4 • 0x4 — 5 0 1 0 1 0 1 1 0 Input Output • 0x5 — 6 0 1 1 0 0 1 1 1 I2 O2 0 1 1 1 1 0 0 0 • 0x6 — 7 1 0 0 0 1 0 0 1 I1 O1 • 0x7 — 8 1 0 0 1 0 0 0 0 • 0x8 — 9 1 0 1 0 X X X X 1 0 1 1 X X X X • 0x9 — 0 1 1 0 0 X X X X • 0xA — 0xF — Don’t care 1 1 0 1 X X X X 1 1 1 0 X X X X 8 1 1 1 1 X X X X

  9. K-maps Input Output O8 O4 I8’I4’ I8’I4 I8I4 I8I4’ I8’I4’ I8’I4 I8I4 I8I4’ I8 I4 I2 I1 O8 O4 O2 O1 00 01 11 10 00 01 11 10 0 0 0 0 0 0 0 1 00 0 0 X 1 00 0 1 X 0 I2’I1’ I2’I1’ 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 01 0 0 X 0 01 0 1 X 0 I2’I1 I2’I1 0 0 1 1 0 1 0 0 I2I1 11 0 1 X X I2I1 11 1 0 X X 0 1 0 0 0 1 0 1 10 0 0 X X 10 0 1 X X I2I1’ I2I1’ 0 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 O2 O1 I8’I4’ I8’I4 I8I4 I8I4’ I8’I4’ I8’I4 I8I4 I8I4’ 0 1 1 1 1 0 0 0 1 0 0 0 1 0 0 1 00 01 11 10 00 01 11 10 1 0 0 1 0 0 0 0 I2’I1’ 00 0 0 X 0 I2’I1’ 00 1 1 X 1 1 0 1 0 X X X X 01 1 1 X 0 01 0 0 X 0 I2’I1 I2’I1 1 0 1 1 X X X X 1 1 0 0 X X X X 11 0 0 X X 11 0 0 X X I2I1 I2I1 1 1 0 1 X X X X 10 1 1 X X 10 1 1 X X I2I1’ I2I1’ 1 1 1 0 X X X X 9 1 1 1 1 X X X X

  10. Outline • Revisiting the binary number system • Adders • Multiplexer 10

  11. What do we want from a number system? • Obvious representation of 0, 1, 2, ...... • Represent positive/negative/integer/floating points • Efficient usage of number space • Equal coverage of positive and negative numbers • Easy hardware design • Minimize the hardware cost/reuse the same hardware as much as possible • Easy to distinguish positive numbers • Easy to negation 11

  12. Representing a positive number • Assume that we have 4 bits Decimal Binary Decimal Binary 0 0000 4 0100 1 0001 5 0101 2 0010 6 0110 3 0011 7 0111 • Example binary arithmetic 3 + 2 = 5 3 + 3 = 6 1 carry 1 1 0 0 1 1 0 0 1 1 + 0 0 1 0 + 0 0 1 1 0 0 1 0 1 1 1 0 12

  13. Poll close in The first proposal Decimal Binary Decimal Binary • How many of the following goals can “ simply 0 0000 -0 1000 using the most significant bit as the 1 0001 -1 1001 2 0010 -2 1010 signed bit ” to represent a negative number 3 0011 -3 1011 fulfill in the number system? 4 0100 -4 1100 ① Obvious representation of 0, 1, 2, ...... 5 0101 -5 1101 ② Efficient usage of number space 6 0110 -6 1110 7 0111 -7 1111 ③ Equal coverage of positive and negative numbers ④ Easy hardware design A. 0 B. 1 C. 2 D. 3 E. 4 13

  14. The first proposal Decimal Binary Decimal Binary • How many of the following goals can “ simply 0 0000 -0 1000 using the most significant bit as the 1 0001 -1 1001 2 0010 -2 1010 signed bit ” to represent a negative number 3 0011 -3 1011 fulfill in the number system? 4 0100 -4 1100 ① Obvious representation of 0, 1, 2, ...... 5 0101 -5 1101 ② Efficient usage of number space 6 0110 -6 1110 7 0111 -7 1111 ③ Equal coverage of positive and negative numbers ④ Easy hardware design A. 0 B. 1 C. 2 D. 3 E. 4 14

  15. Can this work? • 3 + 2 = 5 1 0 0 1 1 + 0 0 1 0 0 1 0 1 1 • 3 + (-2) = 1 0 0 1 1 + 1 0 1 0 1 = -5 (Not 1) 1 0 1 Doesn’t work well and you need a separate procedure to deal with negative numbers! 15

  16. The first proposal Decimal Binary Decimal Binary • How many of the following goals can “simply 0 0000 -0 1000 using the most significant bit as the signed 1 0001 -1 1001 2 0010 -2 1010 bit” to represent a negative number fulfill in 3 0011 -3 1011 the number system? 4 0100 -4 1100 ① Obvious representation of 0, 1, 2, ...... 5 0101 -5 1101 ② Efficient usage of number space 6 0110 -6 1110 7 0111 -7 1111 ③ Equal coverage of positive and negative numbers ④ Easy hardware design A. 0 B. 1 C. 2 D. 3 E. 4 16

  17. Poll close in The second proposal — 1’s complement Decimal Binary Decimal Binary • How many of the following goals can “1’s 0 0000 -0 1111 complement — flip/not every bit in the 1 0001 -1 1110 corresponding positive number ” to represent 2 0010 -2 1101 3 0011 -3 1100 a negative number fulfill in the number system? 4 0100 -4 1011 ① Obvious representation of 0, 1, 2, ...... 5 0101 -5 1010 ② Efficient usage of number space 6 0110 -6 1001 ③ Equal coverage of positive and negative 7 0111 -7 1000 numbers ④ Easy hardware design A. 0 B. 1 C. 2 D. 3 E. 4 17

  18. The second proposal — 1’s complement Decimal Binary Decimal Binary • How many of the following goals can “1’s 0 0000 -0 1111 complement — flip/not every bit in the 1 0001 -1 1110 corresponding positive number ” to represent 2 0010 -2 1101 3 0011 -3 1100 a negative number fulfill in the number system? 4 0100 -4 1011 ① Obvious representation of 0, 1, 2, ...... 5 0101 -5 1010 ② Efficient usage of number space 6 0110 -6 1001 ③ Equal coverage of positive and negative 7 0111 -7 1000 numbers ④ Easy hardware design A. 0 B. 1 C. 2 D. 3 E. 4 18

  19. Second proposal: 1’s complement • 3 + 2 = 5 1 0 0 1 1 + 0 0 1 0 0 1 0 1 • 3 + (-2) = 1 1 1 1 1 0 0 1 1 overflow + 1 1 0 1 = 0 (Still not 1) 0 0 0 0 Still does not work, but seems closer... 19

  20. The second proposal — 1’s complement Decimal Binary Decimal Binary • How many of the following goals can “1’s 0 0000 -0 1111 complement — flip/not every bit in the 1 0001 -1 1110 corresponding positive number ” to represent 2 0010 -2 1101 3 0011 -3 1100 a negative number fulfill in the number system? 4 0100 -4 1011 ① Obvious representation of 0, 1, 2, ...... 5 0101 -5 1010 ② Efficient usage of number space 6 0110 -6 1001 ③ Equal coverage of positive and negative 7 0111 -7 1000 numbers ④ Easy hardware design A. 0 B. 1 C. 2 D. 3 E. 4 20

  21. Poll close in The third proposal — 2’s complement Decimal Binary Decimal Binary • How many of the following goals can “2’s 0 0000 -1 1111 complement — take the 1’s complement of 1 0001 -2 1110 corresponding positive number and then 2 0010 -3 1101 3 0011 -4 1100 +1 ” to represent a negative number fulfill in the 4 0100 -5 1011 number system? 5 0101 -6 1010 ① Obvious representation of 0, 1, 2, ...... 6 0110 -7 1001 ② Efficient usage of number space 7 0111 -8 1000 ③ Equal coverage of positive and negative numbers ④ Easy hardware design A. 0 B. 1 C. 2 D. 3 E. 4 21

  22. The third proposal — 2’s complement Decimal Binary Decimal Binary • How many of the following goals can “2’s 0 0000 -1 1111 complement — take the 1’s complement of 1 0001 -2 1110 corresponding positive number and then 2 0010 -3 1101 3 0011 -4 1100 +1 ” to represent a negative number fulfill in the 4 0100 -5 1011 number system? 5 0101 -6 1010 ① Obvious representation of 0, 1, 2, ...... 6 0110 -7 1001 ② Efficient usage of number space 7 0111 -8 1000 Does not waste 1111 anymore ③ Equal coverage of positive and negative numbers ④ Easy hardware design A. 0 B. 1 C. 2 D. 3 E. 4 22

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