Digital Logic Design
ENGG1015
1st Semester, 2010
Dr. Kenneth Wong Dr. Hayden So
Department of Electrical and Electronic Engineering
Lowered Abstraction
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 2
Applications Systems
Digital Logic
Circuits
Electrical Signals
High Level
Low Level
• Computer & Embedded Systems
• Computer Network
• Mobile Network
• Image & Video Processing
• Combinational Logic
• Boolean Algebra
• Basic Circuit Theory
• Voltage, Current
• Power & Energy
Last week
This week
Motivation
How do you build a computer system?
Possible answer: “use electronic circuits”
Partially correct… but way too complicated for human (or even computers) to handle
• Too many things to consider: voltage, current, resistance, loading effect…
The study of digital logic helps to design large digital systems with a easier and mathematically sound abstraction
CPU Control +
Datapath
ALU
+
Digital Logic Design
In a digital system, all signals take on discrete values.
• Also referred as states
Most modern digital systems operate on 2 discrete states
• binary logic system
We represent the two states as
• True and false
• 1 and 0
• High and Low
Remember: They are not binary numbers -- although you can represent binary numbers using logical states
They are not voltage value – although they are usually represented using voltage
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 4
Logic Function
A logic function takes 1 or more logic input to produce 1 single logic output
Sometimes one may “define” logic functions that produce more than 1 output
But note that a multi-output function can always be implemented as a grouping of multiple
single-output functions
• i.e. just a short hand
Mathematically, it a function produce only 1 output
€
y = f (x1, x2,…, xn )
€
(y1, y2,…, ym ) = f (x1, x2,…, xn)
Representing Logic Operations
Each function can be represented equivalently in 3 ways:
• Truth table
• Boolean logic expression
• Schematics
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 6
Truth Table
Boolean Expression Schematics
Truth Tables
Describe how a logic circuit’s output depends on the logic levels present at the inputs.
All the possible combinations of inputs are listed
If the truth table is known, we completely know how the circuit behave!!
3 Basic Logic Functions
Also called a logic gate
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 8
NOT OR
AND
OR Gate
The output of an OR gate is HIGH iff one or more inputs are HIGH
Truth table
0 = LOW 1 = HIGH
Boolean expression
Timing Diagram
time
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X = A + B
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• OR gate can have more than 2 inputs:
Summary of OR operation:
• Produce a result of 1 whenever any input is 1. Otherwise 0.
• The expression x=A+B is read as “x equals A OR B”
Alarm is activated whenever the temperature exceeds a
maximum value VTR or whenever the pressure goes above a certain limit VPR
• Example of the use of an OR gate in an alarm system
More examples
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• Review questions:
• What is the only set of input conditions that will produce a LOW output for any OR gate?
-- Ans: all inputs LOW
• Write the Boolean expression for a six-input OR gate.
-- Ans: X=A+B+C+D+E+F
• If the A input in previous example is permanently kept at the 1 level, what will the resultant output waveform be?
-- Ans: constant HIGH
AND Gate
The output of an AND gate is HIGH only when all inputs are HIGH.
Boolean expression Truth table
0 = LOW 1 = HIGH
Timing Diagram
€
X = AB
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Summary of the AND operation
• The AND operation is performed the same as ordinary multiplication of 1s and 0s.
• An AND gate output will be 1 only for the case when all inputs are 1; for all other cases the output will be 0.
• The expression x=A•B is read as “x equals A AND B.”
Review Questions
What is the only input combination that will produce a HIGH at the output of a five-input AND gate?
• all 5 inputs = 1
What logic level should be applied to the second input of a two-input AND gate if the logic signal at the first input is to be inhibited
(prevented) from reaching the output?
• A LOW input will keep the output LOW
True or false: An AND gate output will always differ from an OR gate output for the same input conditions.
• False
The NOT Operation & Inverter
The output of a NOT gate is always the complement (opposite) of the input.
A NOT gate is sometimes referred as an inverter, especially in circuit designs
Boolean expression Truth table
0 = LOW 1 = HIGH
€
X = A
Other Simple Gates
ALL logic functions, no matter how
complex, can be completely expressed using the 3 basic operations AND, OR, NOT.
However, many systems utilizes more than just the 3 basic logic gates
because it makes the design cleaner and easier to understand (for human).
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 16
NAND Gate
Output 0 iff ALL inputs are 1s
Complement of an AND gate
Note the “bubble” at the output of the
symbol, and the bar over the expression AB. Both of them signifies the
complement nature to the AND gate
€
X = AB
bubble
NOR Gate
Output 1 iff ALL inputs are 0s
Complement of an OR gate
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 18
€
X = A + B
XOR Gate
Exclusive-OR gate
Output 1 iff exactly one input is 1
Similar to an OR gate, except that when both inputs are 1, the output is 0
€
X = A ⊕ B
A B X 0 0 0 0 1 1 1 0 1 1 1 0
3 Representations of Logic Functions
Recall that any complex logic function can be expressed in 3 ways: Truth
Table, Boolean Expression, Schematics
Only Truth Table representation is unique
We can convert representation from one form to the other
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 20
Schematics to Boolean Expression
Example: logic circuit with its Boolean expression
Question: how to interpret A·B+C?
• Is it A·B ORed with C ? Is it A ANDed with B+C ?
Order of precedence for Boolean algebra: AND before OR. Parentheses make the expression clearer, but they are not needed for the case on the preceding slide.
Therefore the case below is different:
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Whenever an INVERTER is present in a logic-circuit diagram, its output expression is simply equal to the input expression with a bar over it.
More examples:
Precedence
Given an Boolean expression
• First, perform all inversions of single terms
• Perform all operations with parentheses
• Perform an AND operation before an OR operation unless parentheses indicate otherwise
• If an expression has a bar over it, perform the operations inside the expression first and then invert the result
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Determining output level from a diagram
Example: Draw the circuit diagram to implement the expression
Example: Draw the circuit diagram that implements the expression
using gates having no more than three inputs.
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X = (A + B)(B + C)
Boolean Expressions to Schematics
When the operation of a circuit is defined by a Boolean expression, we can draw a logic-circuit diagram directly from that expression.
Example: draw the circuit for
Done in two steps
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In conclusion…
AND, OR, NOT are 3 basic logic gates that can implement all logic functions
All logic functions can be represented as (1) truth table (2) schematics (3)
Boolean expressions
The same logic functions can be converted between the 3
representations easily.
Only truth table representation is unique