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Switching circuits Composed of switching elements called “gates” that implement logical blocks or switching expressions Positive logic convention (active high): High voltage or H Boolean 1 Low voltage or L Boolean 0 Negative logic convention (active low): Low voltage or L Boolean 1 High voltage or H Boolean 0

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**Switching circuits Logic variables inputs/outputs “signals”**

Signals “asserted” when the voltage level assumes the corresponding “1” value Positive logic asserted by H Negative logic asserted by L Logic variables are written complemented when they are active low Active high signals: a, b, c Active low signals: ā, ē, ū

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Logic gates Logic gates switching functions Gate symbols – two sets

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Logic gates Gate symbols – two sets

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Logic gates The NAND logic function and gate

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Logic gates The NAND gate can be used to implement all 3 elementary operations of switching algebra: AND, OR, NOT

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Logic gates The set {AND, OR, NOT} implements any switching function (by definition): it is functionally complete Therefore, the “NAND” gate can be used to implement any switching function It is functionally complete, or “primitive”

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Logic gates The NOR logic function and gate

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Logic gates The NOR function can be used to implement all 3 elementary operations of switching algebra: AND, OR, NOT It is functionally complete too

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Logic gates The NOR logic function and gate

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**Logic gates and equivalence**

CMOS is “inverting” logic NOR and NAND are easier to implement than OR and AND They are implemented as NOR or NAND followed by an inverter More than one representation is possible for the same switching function Different circuits of logic gates might perform the same switching function Simpler networks are preferable Need to analyze for equivalence and transform

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**Logic gates and equivalence**

Equivalent logic networks

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**Logic gates and equivalence**

Proving the equivalence

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**Digital circuits Analysis Synthesis**

Given a circuit, abstract the Boolean function it is implementing and try to improve the implementation or verify the function From gate diagrams From timing diagrams Synthesis Given a switching function, obtain the corresponding switching network

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Analysis Timing diagram

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Analysis Truth table

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Analysis Switching network

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**Combinational analysis**

... derives truth table

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Signal expressions Multiply out: F = ((X + Y¢) × Z) + (X¢ × Y × Z¢) = (X × Z) + (Y¢ × Z) + (X¢ × Y × Z¢)

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**New circuit, same function**

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**Any number of manipulations can yield equivalent circuits**

e.g. F = ((X + Y’)Z) + X’YZ’ Note: [X’YZ’]Z = 0 (X + Y’)X’YZ’ = 0 (X’YZ’)(X’YZ’) = X’YZ’ So, F = [(X + Y’) + X’YZ’][Z + X’YZ’] =(X + Y’ + X’)(X + Y’ + Y)(X + Y’ + Z’)(Z + X’)(Z + Y)(Z + Z’) =(1)(1)(X + Y’ + Z’)(X’ + Z)(Y + Z)(1) = (X + Y’ + Z’)(X’ + Z)(Y + Z) Circuit:

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**Push bubbles to obtain cancellations**

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**Push bubbles to obtain cancellations**

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Conclude: given circuit ==> many equivalent equations circuit does not determine equation

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**Also, equation does not determine circuit:**

Two-level AND-OR Two-level NAND-NAND Three-level equivalent

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**Combinational analysis**

given circuit, determine function Combinational synthesis given function, determine circuit

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**Prime number detector: F = (1, 2, 3, 5, 7, 11, 13)**

AND-OR design

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Alarm: Derive truth table or expand: A = P + E EX’ (W D G)’ = P + E EX’ (W’ + D’ + G’) = P + E EX’ W’ + E EX’ D’ + E EX’ G’

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**A = P + E EX’ W’ + E EX’ D’ + E EX’ G’**

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**NANDs, NORs have fewer transistors than ANDs, ORs**

AND-OR converts readily to NAND-NAND

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**Complication if some inputs go directly to second stage:**

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OR-AND to NOR-NOR

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**Bubble-pushing produces non-standard gate**

Solution: inverters

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**Bubble-pushing produces non-standard gate**

Solution: inverters

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**Bubble-pushing produces non-standard gate**

Solution: inverters

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Propagation delay

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Propagation delay

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**Synthesis SOP functions -> AND – OR networks**

POS functions -> OR – AND networks Not always possible to design directly Fan-in and out restrictions Most designs are modular and multi-level Modern designs are too complex Design and testing by computers VLSI - CAD

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**Logic simulation Two states only for an ideal logic signal**

Two gates driving the same line in opposite directions Input left not connected or “floating” Third state ‘X’ is added to the set of states Truth tables change

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**Synthesis approaches illustrated to this point:**

Truth table derivation of minterms Ad hoc construction of logic equation Need systematic approach that minimizes hardware Karnaugh maps Quine-McCluskey algorithm

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Boolean Algebra and Combinational Logic

Boolean Algebra and Combinational Logic

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