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**The Digital Logic Level**

Computer Organization Lecture #6 Jahan Zeb

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**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

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

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

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**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)

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**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

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**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

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**XOR, Conversion to NAND Gate Only**

(a) The truth table for the XOR function (b-d) Three circuits for computing it

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**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

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**An SSI chip containing four gates**

Integrated Circuits An SSI chip containing four gates

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Multiplexers An eight-input multiplexer circuit

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**Multiplexers (a) An MSI multiplexer.**

(b) The same multiplexer wired to compute the majority function.

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Decoders A 3-to-8 decoder circuit

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Comparators A simple 4-bit comparator .

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**Programmable Logic Arrays**

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

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**A 1-bit left/right shifter**

Shifters A 1-bit left/right shifter

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**Half Adder (a) A truth table for 1-bit addition**

(b) A circuit for a half adder

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**Full Adder (a) Truth table for a full adder**

(b) Circuit for a full adder

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**Arithmetic Logic Units**

A 1-bit ALU

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**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

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**Clocks (a) A clock (b) The timing diagram for the clock**

(c) Generation of an asymmetric clock

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