1. Chapter 7 Arithmetic Operations and Circuits 1

2. 7-4 Hexadecimal Arithmetic 4 binary bits represent a single hexadecimal digit Addition –Add the digits in decimal –If sum is less than 16, convert to hexadecimal –Is sum is more than 16, subtract 16, convert to hexadecimal and carry 1 to the next-more- significant column 23

3. Example 7-12

4. Hexadecimal Arithmetic Subtraction –When you borrow, the borrower increases by 16 –See example 7-15 24

5. 25 Example 7-15

6. 7-5 BCD Arithmetic Group 4 binary digits to get combinations for 10 decimal digits Range of valid numbers 0000 to 1001 Addition –Add as regular binary numbers –If sum is greater than 9 or if carry out generated: Add 6 (0110) saving any carry out 26

7. 7-6 Arithmetic Circuits Only two inputs are of concern in the LSB column. More significant columns must include the carry-in from the previous column as a third input. 27

8. Arithmetic Circuits The addition of the third input (C in ) is shown in the truth table below. 27

9. Arithmetic Circuits Half-Adder –No carry in (LSB column) –The  0 output is HIGH when A or B, but not both, is high. Exclusive-OR function –C out is high when A and B are high. AND function 28

10. Arithmetic Circuits The half-adder can also be implemented using NOR gates and one AND gate. –The NOR output is Ex-OR. –The AND output is the carry. 28

11. Arithmetic Circuits Full-Adder –Provides for a carry input –The  1 output is high when the 3-bit input is odd. Even parity generator –C out is high when any two inputs are high. 3 AND gates and an OR 29

12. Arithmetic Circuits Full-adder sum from an even-parity generator 32

13. Arithmetic Circuits Full-adder carry out function 33

14. Arithmetic Circuits Logic diagram of a complete full-adder 34

15. Arithmetic Circuits Block diagrams of a half-adder (HA) and a full adder (FA). 35

16. Arithmetic Circuits Block diagram of a 4-bit binary adder 36

17. 7-7 Four-Bit Full-Adder ICs Four full-adders in a single package Will add two 4-bit binary words plus one carry input bit. 37

18. Four-Bit Full-Adder ICs Functional diagram of the 7483 Note that some manufacturers label inputs A 0 B 0 to A 1 B 3 The carry-out is internally connected to the carry-in of the next full-adder. 38

19. Four-Bit Full-Adder ICs Logic diagram for the 7483. 39

20. Four-Bit Full-Adder ICs Logic symbol for the 7483 39

21. Four-Bit Full-Adder ICs Fast-look-ahead carry –Evaluates 4 low-order inputs –High-order bits added at same time –Eliminates waiting for propagation ripple 40

22. 7-9 System Design Application Two’s-Complement Adder/Subtractor Circuit 41

23. System Design Application BCD Adder Circuit 42

24. 7-10 Arithmetic/Logic Units The ALU is a multipurpose device Available in LSI package 74181 (TTL) 74HC181 (CMOS) Mode Control input –Arithmetic (M = L) –Logic (M = H) 44

25. Arithmetic/Logic Units Function Select - selects specific function to be performed 45

26. Summary The binary arithmetic functions of addition, subtraction, multiplication, and division can be performed bit-by-bit using several of the same rules of regular base 10 arithmetic. The two’s-complement representation of binary numbers is commonly used by computer systems for representing positive and negative numbers. 48

27. Summary Two’s-complement arithmetic simplifies the process of subtraction of binary numbers. Hexadecimal addition and subtraction is often required for determining computer memory space and locations. When performing BCD addition a correction must be made for sums greater than 9 or when a carry to the next more significant digit occurs. 49

28. Summary Binary adders can be built using simple combinational logic circuits. A half-adder is required for addition of the least significant bits A full-adder is required for addition of the more significant bits. 50

29. Summary Multibit full-adder ICs are commonly used for binary addition and two’s-complement arithmetic. Arithmetic/logic units are multipurpose ICs capable of providing several different arithmetic and logic functions. The logic circuits for adders can be described in VHDL using integer arithmetic. 51

30. Summary The Quartus II software provides 7400-series macrofunctions and a Library of Parameterized Modules (LPMs) to ease in the design of complex digital systems. Conditional assignments can be made using the IF-THEN-ELSE VHDL statements. 51