digital logic design - university of hong kongengg1015/fa10/handouts/06-digitallogic.pdf · digital...
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Digital Logic Design
ENGG1015
1st Semester, 2010
Dr. Kenneth Wong Dr. Hayden So
Department of Electrical and Electronic Engineering
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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
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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
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 3
CPU Control + Datapath
ALU
+
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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
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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
1st semester, 2010 Digital Logic - ENGG1015 - K. Wong/H. So 5
€
y = f (x1,x2,…,xn )
€
(y1,y2,…,ym ) = f (x1,x2,…,xn )
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Representing Logic Operations Each function can be represented
equivalently in 3 ways: • Truth table • Boolean logic expression • Schematics
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Truth Table
Boolean Expression Schematics
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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!!
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3 Basic Logic Functions Also called a logic gate
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NOT OR AND
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OR Gate
The output of an OR gate is HIGH iff one or more inputs are HIGH
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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”
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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
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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
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AND Gate
The output of an AND gate is HIGH only when all inputs are HIGH.
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Boolean expression Truth table 0 = LOW 1 = HIGH
Timing Diagram
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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
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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
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Boolean expression
Truth table 0 = LOW 1 = HIGH
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X = A
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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).
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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
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€
X = AB
bubble
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NOR Gate
Output 1 iff ALL inputs are 0s
Complement of an OR gate
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X = A + B
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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
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€
X = A⊕ BA B X 0 0 0 0 1 1 1 0 1 1 1 0
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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
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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:
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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
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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)
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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
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