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Electrical Engineering & Telecommunications

Lecturer:Ted Spooner

Lecture -Introduction to Digital Logic

Elec1111Elec1111

Rm 124A EE email: e.spooner@unsw.edu.au

Book

• Katz and Borriello, “Contemporary Logic Design”, Prentice Hall, 2nd ed, 2005

• alternative:– Wakerly, “Digital Design: principles and

practices”, Prentice Hall, 4th ed, 2005

Typical electronic systems Analogue and digital signals

• Analogue– continuous– physical

• Digital– discrete– “symbolic”

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Noise in analogue signals

• Loose information below noise level• Linear, accurate systems expensive

– minimise use

Noise in digital signals

• Can be regenerated error-free(usually)• Accuracy arbitrary (number of levels / bits)• Exact computations

Two-level logic

• Almost universal• Voltage encoding• • Level values:

– technology dependant

– power supply dependant

Two-level logic II

• Transfer fun. + gap:• – ensures good noise

margin• Notation:

– Highlow (voltage)– Truefalse (Boolean)– 1-0 (binary digit)

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Why use digital?• Good noise reject.• High reliability• Low drift• High accuracy• Predictable• Low power• Ease of design• Design tools

Why not to use– generates noise– D/A interface costly

Disrete digital components

• Quad 2input NOR 74LS02

Digital technologies

• Discrete– CMOS, TTL, ECL

• FPGA• ASIC

– CMOS, GaAs

Combinational / sequential

• Combinational– output is logic

• function of inputs– no memory

• Sequential– output is logic

• function of inputs and state– memory

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Truth tables

• Exhaustive description of output for all possible inputs

• Example– Phone rings (R=1) if power

on (P=1) and incoming call (C=1)

2N combinations forN inputs

Assigning logic variables

• Assign logic variable to precise statement• Example: David’s purchase

– He buys if he wants an item and has cash or if he needs it and has cash or card

– David needs item (N=1)– David wants item (W=1)– David has sufficient cash for purchase (C=1)– David has brought his EFTPOS card (E=1)– David buys the item (B=1)

David’ s purchase truth table

• N: needs• W: wants• C: cash• E: eftpos• B: buys

Primitive gates

• A logic gate:– implements a

combinational logic function

• – 22^N possible functions of N variables

• – All can be described using primitive gates

• Two-input gates– and (2 input)– or– not– xor– nand, nor, xnor– also other types

(16)

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Logical AND operation

YX

Z

AND gate

Z X Y= ⋅Z XY=

Z=1 if both X and Y are 1

0 0 00 1 01 0 01 1 1

X Y Z

AND truth table

Logicalmultiplication

Z = X and Y

Logical OR operation

Z X Y= +

Z=1 if X or Y or both are 1

0 0 00 1 11 0 11 1 1

X Y Z

OR truth table

Logicaladdition

Y

XZ

OR gate

Z = X or Y

NOT operation(negation, complement)

Z X=

0 11 0

X Z

NOT truth table

X Z

NOT gate (inverter)

(also X’, /X or \X)

Z = not X

Also called Inversion, Negation, Complement

NAND function (not AND)

YX

Z

NAND gate

Z=1 if X is 0 or Y is 0

0 0 10 1 11 0 11 1 0

X Y Z

NAND truth table

)'(

.

XYZXYZ

YXZ

XandYZ

==

=

=

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Drawing conventions

XY

X.YX.Y X

YX.Y

Bubbles invert signal

X

Y

X

YX+Y

Inverted input signified by a bubble on the input….eg:

NOR function (not OR)

Z=1 if both X and Y are 0

0 0 10 1 01 0 01 1 0

X Y Z

NOR truth table

Y

XZ

NOR gate

)'( YXZYXZ

XorYZ

+=+=

=

XOR function (exclusive OR)

Z=1 if X has a different value from Y

0 0 00 1 11 0 11 1 0

X Y Z

XOR truth table

Y

XZ

XOR gate

YXZYZ

⊕== xor X

XNOR function (NOT exclusive OR)

Z=1 if X and Y are the same

0 0 00 1 11 0 11 1 0

X Y Z

XNOR truth table

Y

XZ

XNOR gate

)'(

xnor X

YXZYXZ

YZ

⊕=⊕=

=

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Gates with more than 2 inputs

T X Y Z= + +YX

Z

3-input NOR gate

T

BA

Z

4-input AND gate

CD

Z A B C D= ⋅ ⋅ ⋅

Combining primitive gatesExample XOR

ABBABA ).( +=⊕–as truth tables are identical

Combining primitive gates II

• Primitive gates can implement any logic function

• – 2 input nand can!

David’ s purchase logic

• B = N(C+E)+WC

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Necessary logic operators• Any function expressed by NOT, AND & OR

C

CB.

Example

• Draw the logic diagram represented by:

CBAZ .+=

CBAZ .+=

Draw a truth table for the logic:

111011101001110010100000

MXCBA