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Computer Organization and Design Transistors & Logic - II

Computer Organization and Design Transistors & Logic - II. Montek Singh Wed, Oct 16, 2013 Lecture 10. Today ’ s Topics. Synthesis using standard gates Truth tables Universal gates: NAND and NOR Gates with more than 2 inputs Sum-of-Products DeMorgan ’ s Law.

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Computer Organization and Design Transistors & Logic - II

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  1. Computer Organization and DesignTransistors & Logic - II Montek Singh Wed, Oct 16, 2013 Lecture 10

  2. Today’s Topics • Synthesis using standard gates • Truth tables • Universal gates: NAND and NOR • Gates with more than 2 inputs • Sum-of-Products • DeMorgan’s Law

  3. Now We’re Ready to Design Stuff! A Truth Table Y B C If C is 1 thencopy B to Y,otherwise copyA to Y • We need to start somewhere • usually it’s the functional specification If you are like most engineers you’d rather see a table, or formula than parse a logic puzzle. The fact is, any combinational function can be expressed as a table. These “truth tables” are a concise description of the combinational system’s function. Conversely, any computation performed by a combinational system can expressed as a truth table.

  4. We Can Make Most Gates Out of Others XOR B>A A B Y A A y Y B B • How many different gates do we really need?

  5. One Will Do! = = = = = = • NANDs and NORs are universal • one can make any circuit out of just NANDs, or just NORs! • Ah! But what if we want more than 2-inputs?

  6. Gate Trees A 0 0 1 1 B 0 1 0 1 C 0 1 1 0 Suppose we have some 2-input XOR gates: (same idea holds for AND and OR gates) tpd = 1 (latency) And we want an N-input XOR: output = 1 iff number of 1s in input is ODD (“ODD PARITY”) N tpd (latency)= O( ___ ) -- WORST CASE. Can we compute N-input XOR faster?

  7. Gate Trees 22 21 log2N 2 log N N-input TREE has O( ______ ) levels... Signal propagation takes O( _______ ) gate delays. log N

  8. Design Approach: Sum-of-Products Truth Table Three steps: • Write functional spec as a truth table • Write down a Boolean expression for every ‘1’ in the output • Wire up the gates! • This approach will always give us logic expressions in a particular form: • SUM-OF-PRODUCTS (“SOP”) • “SUM” actually means OR • “PRODUCT” actually means AND

  9. Straightforward Synthesis A B C A B C Y A B C A B C • We can implement SUM-OF-PRODUCTS… • …with just three levels of logic: • INVERTERS/AND/OR

  10. Notations • Symbols and Boolean operators:

  11. DeMorgan’s Laws • Change ANDs into ORs and vice-versa

  12. C Useful Gate Structures A C Y B A Y B C  A Y B DeMorgan’s Laws AB=A+B AB=A+B C  A Y B • NAND-NAND • NOR-NOR “Pushing Bubbles” C A Y B C A Y B

  13. An Interesting 3-Input Gate: Multiplexer Truth Table A Y B C If C is 1 thencopy B to Y,otherwise copyA to Y B C Y A • Based on C, select the A or B input to be copied to the output Y. 2-input Multiplexer A 0 B 1 Gate symbol C “schematic” diagram

  14. Multiplexer (MUX) Shortcuts A0 B0 A1 B1 Y0 0 1S A B C D S 0 1 2 3 0 1S Y Y1 I0 I1 I2 I3 A0-3 B0-3 S A2 B2 A3 B3 Y0-3 0 1S Y2 Y 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1S 0 1S Y3 0 1S 0 1S S0 S1 S A 4-bit wide 2-input Mux A 4-input Mux(implemented as a tree)

  15. Let’s do some practice examples On whiteboard

  16. Next Class • Arithmetic circuits

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