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The Digital Logic Level

The Digital Logic Level. Computer Organization Lecture #6 Jahan Zeb. Gates and Boolean Algebra.

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The Digital Logic Level

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  1. The Digital Logic Level Computer Organization Lecture #6 Jahan Zeb

  2. Gates and Boolean Algebra • The symbols and functional behavior for the five basic gates. • Boolean function of n variable is completely described by table with 2n rows, known as truth table

  3. Majority Function (a) Truth table for majority function of three variables. (a) A circuit for (a). • M=ABC+ABC+ABC +ABC

  4. Circuit Equivalence • Construction of (a)NOT, (b)AND, and (c)OR gates using only NAND gates or only NOR gates.

  5. Circuit Equivalence (Cont…) • To reduce complexity designers try to reduce no. of gates • Circuit computing same function as original with fewer gates is preferred • Boolean algebra can be valuable • Using Distributive law: AB + AC = A (B+C)

  6. Boolean Algebra Main Identities • To Circuit designers start with a Boolean function and then apply the laws of Boolean algebra to it • In an attempt to find a simpler but equivalent one • From the final function, circuit can be constructed

  7. Alternate Symbols for NAND, NOR, AND, OR • De Morgan’s law suggests an alternative notation, (a) an OR gate is equivalent to the NAND gate with inverted inputs • (b) NOR gate can be drawn with an AND gate with inverted inputs • By negating both forms we arrive at (c) and (d) respectively

  8. XOR, Conversion to NAND Gate Only • (a) The truth table for the XOR function • (b-d) Three circuits for computing it

  9. Circuit Equivalence (Cont…) • Same physical gate can compute different functions, depending on the convention used (a) Electrical characteristics of a device. (b) Positive logic, (c) Negative logic, AND function OR function

  10. Integrated Circuits An SSI chip containing four gates

  11. Multiplexers • An eight-input • multiplexer circuit

  12. Multiplexers (a) An MSI multiplexer. (b) The same multiplexer wired to compute the majority function.

  13. Decoders • A 3-to-8 decoder • circuit

  14. Comparators • A simple 4-bit • comparator • .

  15. Programmable Logic Arrays • A 12-input, 6-output • programmable logic array • The little squares represent fuses that can be burned out • . • .

  16. Shifters A 1-bit left/right shifter

  17. Half Adder (a) A truth table for 1-bit addition (b) A circuit for a half adder

  18. Full Adder (a) Truth table for a full adder (b) Circuit for a full adder

  19. Arithmetic Logic Units • A 1-bit ALU

  20. Arithmetic Logic Units • Eight 1-bit ALU slices connected to make an 8-bit ALU • The enables and invert signals are not shown for simplicity

  21. Clocks (a) A clock (b) The timing diagram for the clock (c) Generation of an asymmetric clock

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