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Lecture 21: Combinatorial Circuits II

Lecture 21: Combinatorial Circuits II. Discrete Mathematical Structures: Theory and Applications. Learning Objectives. Learn about Boolean expressions Become aware of the basic properties of Boolean algebra Explore the application of Boolean algebra in the design of electronic circuits

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Lecture 21: Combinatorial Circuits II

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  1. Lecture 21: Combinatorial Circuits II Discrete Mathematical Structures: Theory and Applications

  2. Learning Objectives • Learn about Boolean expressions • Become aware of the basic properties of Boolean algebra • Explore the application of Boolean algebra in the design of electronic circuits • Learn the application of Boolean algebra in switching circuits Discrete Mathematical Structures: Theory and Applications

  3. Discrete Mathematical Structures: Theory and Applications

  4. Discrete Mathematical Structures: Theory and Applications

  5. Discrete Mathematical Structures: Theory and Applications

  6. Discrete Mathematical Structures: Theory and Applications

  7. Discrete Mathematical Structures: Theory and Applications

  8. Discrete Mathematical Structures: Theory and Applications

  9. Logical Gates and Combinatorial Circuits • The diagram in Figure 12.32 represents a circuit with more than one output. Discrete Mathematical Structures: Theory and Applications

  10. Discrete Mathematical Structures: Theory and Applications

  11. Discrete Mathematical Structures: Theory and Applications

  12. Discrete Mathematical Structures: Theory and Applications

  13. Logical Gates and Combinatorial Circuits • A NOT gate can be implemented using a NAND gate (see Figure 12.36(a)). • An AND gate can be implemented using NAND gates (see Figure 12.36(b)). • An OR gate can be implemented using NAND gates (see Figure12.36(c)). Discrete Mathematical Structures: Theory and Applications

  14. Logical Gates and Combinatorial Circuits • Any circuit which is designed by using NOT, AND, and OR gates can also be designed using only NAND gates. • Any circuit which is designed by using NOT, AND, and OR gates can also be designed using only NOR gates. Discrete Mathematical Structures: Theory and Applications

  15. Discrete Mathematical Structures: Theory and Applications

  16. Discrete Mathematical Structures: Theory and Applications

  17. Logical Gates and Combinatorial Circuits • The Karnaugh map, or K-map for short, can be used to minimize a sum-of-product Boolean expression. Discrete Mathematical Structures: Theory and Applications

  18. Discrete Mathematical Structures: Theory and Applications

  19. Discrete Mathematical Structures: Theory and Applications

  20. Discrete Mathematical Structures: Theory and Applications

  21. Discrete Mathematical Structures: Theory and Applications

  22. Logical Gates and Combinatorial Circuits • 1s should be circled in the largest group of a power of 2 (1,2,4,8,… etc.) to which they belong. • There are six steps to be followed when deciding how to circle blocks of 1s. Discrete Mathematical Structures: Theory and Applications

  23. Logical Gates and Combinatorial Circuits • First mark the 1s that cannot be paired with any other 1. Put a circle around them. • Next, from the remaining 1s, find the 1s that can be combined into two square blocks, i.e., 1 x 2 or 2 x 1 blocks, and in only one way. • Next, from the remaining 1s, find the 1s that can be combined into four square blocks, i.e., 2 x 2, 1 x 4, or 4 x 1 blocks, and in only one way. • Next, from the remaining 1s, find the 1s that can be combined into eight square blocks, i.e., 2 x 4 or 4 x 2 blocks, and in only one way. • Next, from the remaining 1s, find the 1s that can be combined into 16 square blocks, i.e., a 4 x 4 block. (Note that this could happen only for Boolean expressions involving four variables.) • Finally, look at the remaining 1s, i.e., the 1s that have not been grouped with any other 1. Find the largest blocks that include them. Discrete Mathematical Structures: Theory and Applications

  24. Discrete Mathematical Structures: Theory and Applications

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