Introduction to Digital Electronics Lecture 2: Number Systems.
Digital Electronics. Introduction to Number Systems & Codes Digital & Analog systems, Numerical...
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Transcript of Digital Electronics. Introduction to Number Systems & Codes Digital & Analog systems, Numerical...
Digital Electronics
Introduction to Number Systems & Codes
Digital & Analog systems, Numerical representation, Digital number systems, Binary to Decimal conversions, Decimal to Binary conversions, Hexa-decimal
number systems, BCD code, The Gray code, Alphanumeric codes, Parity method for error detection, Octal & Hexa-decimal numbers, Complements.
1St Cycle
Data can be analog or digital
Analog data refers to information that is continuous
Analog data take on continuous values
Analog signals can have an infinite number of values in a range
Digital data refers to information that has discrete states
Digital data take on discrete values
Digital signals can have only a limited number of values
In data communications, we commonly useperiodic analog signals and non-periodic digital signals.
ANALOG AND DIGITAL
ANALOG & DIGITAL SYSTEMS
An Analog system contains devices that manipulate physical quantities that are represented in analog form.
In an analog system, the quantities can vary over a continuous range of values.
For example, the amplitude of the output signal to the speaker in a radio receiver can have any value between zero and its maximum limit.
Other common analog systems are audio amplifiers, magnetic tape recording and playback equipment and a simple light dimmer switch.
More over, a 1 can be encoded as a positive voltage and a 0 as zero voltage.
A digital signal can have more than two levels.
In this case, we can send more than 1 bit for each level.
A Digital system is a combination of devices designed to manipulate logical information or physical quantities that are represented in digital form; that is, the quantities can take on only discrete values.
These devices are most often electronic, but they can also be mechanical, magnetic or pneumatic.
Some of the more familiar digital systems include digital computers, calculator, Digital audio and video equipments and telephone system etc.
Continue………………
Digital Signal
Analog Signal
Advantages of Digital Techniques
Very much easier to design.
Information storage is easy.
Accuracy and precision are easier to maintain throughout the system.
Operation can be programmed.
Less affected by noise.
More circuits can be fabricated on IC chips.
Limitations of Digital Techniques:
The real world is analog
Processing digitized signals takes time.
NUMBER SYSTEMS
Many Number systems are in use in digital technology.
The most common are-DecimalBinary Octal andHexadecimal.
Second Cycle
Analysis & synthesis of digital logic circuitsBasic logic functions
OR operation with OR gates, AND operation with AND gates, NOR operation with NOR gates, Describing logic circuits algebraically, Evaluating logic circuit outputs, Implementing circuits from Boolean expressions, NOR gates & NAND
gates.
Boolean Algebra
Boolean theorems, Demorgan’s Theorems, Universality of NAND gates & NOR gates
DIGITAL LOGIC GATES
Boolean functions are implemented in digital
computer circuits called gates.
A gate is an electronic device that produces a result
based on two or more input values.
– In reality, gates consist of one to six transistors,
but digital designers think of them as a single unit.
– Integrated circuits contain collections of gates
suited to a particular purpose.
Logic gates
Electronic gates require a power supply.
Gate Input’s are driven by voltage having two nominal values.
The Output of a gate provides two nominal values of voltage.
There is always a Time Delay between an input being applied and the output responding.
Point’s Need To Understand
Different Types of Logic gates
Digital systems are said to be constructed by using gates. These gates are --
AND GATE
OR GATE
NOT GATE
NAND GATE
Ex-OR GATE
Ex-NOR GATE
Truth Tables
Used to describe the functional behavior of a Boolean
expression and/or Logic circuit.
Each row in the truth table represents a unique
combination of the input variables.
For n input variables, there are 2n rows.
The output of the logic function is defined for each row.
Each row is assigned a numerical value, with the rows
listed in ascending order.
The order of the input variables defined in the logic
function is important.
AND Gate
The AND gate is an electronic circuit that gives a high output only if all its inputs are high.
A dot(.) is used to show the AND operation.
Truth table
2 Input AND gate
A B Z=A.B
0 0 0
0 1 0
1 0 0
1 1 1
Circuit Symbol
A
B
AND
Z=A.B
Summary of the AND gates
The AND operation is performed the same as ordinary multiplication of 1s and 0s.
An AND gate is a logic circuit that performs the AND operation on the circuit’s input.
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 Z=A.B is read as “Z equals A AND B.”
OR Gate
The OR gate is an electronic circuit that gives a high output. One or more of its inputs are high.
A plus (+) is used to show the OR operation.
Truth table
2 Input OR gate
A B Z=A+B
0 0 0
0 1 1
1 0 1
1 1 1
A
B
OR
Z=A+B
Circuit Symbol
Summary of the OR gates
The OR operation produces a result (output) of 1
whenever any input is a1. Otherwise the output is 0.
An OR gate is a logic circuit that performs an OR
operation on the circuit’s input.
The expression Z=A+B is read as “Z equals A OR
B.”
NOT Gate
The NOT gate is an electronic circuit that produces an inverted version of the input at its output.
More commonly called an Inverter.
Truth tableCircuit Symbol
NOT GATE
A A
0 1
1 0
A A
NAND Gate
The NAND gate which is to NOT - AND gate.
Truth table
Circuit Symbol
A.BAB
NAND
2 Input NAND gate
A B A.B
0 0 1
0 1 1
1 0 1
1 1 0
NOR Gate
The NOR gate which is to NOT - OR gate.
Truth table
2 Input NOR gate
A B A.B
0 0 1
0 1 0
1 0 0
1 1 0
Circuit Symbol
A+BA
B
NOR
EX-OR Gate
The Exclusive –OR gate is a circuit which will give a
high output.An encircled plus sign() is used to show
the EX-OR operation.
Truth table
Circuit Symbol
A BA
B
EX-OR
2 Input EXOR gate
A B A B
0 0 0
0 1 1
1 0 1
1 1 0
EX-NOR Gate
The Exclusive –NOR gate is circuit does the opposite to
the EX-NOR gate. The symbol is an EX-NOR gate with a
small circle on the output. The small circle represents
inversion. Truth table
2 Input EXNOR gate
A B A B
0 0 1
0 1 0
1 0 0
1 1 1
Circuit Symbol
A BA
B
EX-NOR
NAND and NOR are known as universal gates
because they are inexpensive to manufacture and
any Boolean function can be constructed using only
NAND or only NOR gates.
Universal Gates
Boolean Expressions
Boolean expressions are composed of Literals – variables and their complements Logical operations
Examples F = A.B'.C + A'.B.C' + A.B.C + A'.B'.C'
F = (A+B+C').(A'+B'+C).(A+B+C) F = A.B'.C' + A.(B.C' + B'.C)
literals logic operations
Boolean Expressions
Boolean expressions are realized using a
network (or combination) of logic gates. Each logic gate implements one of the logic
operations in the Boolean expression
Each input to a logic gate represents one of
the literals in the Boolean expression
f
A B
logic operationsliterals
Boolean Expressions
Boolean expressions are evaluated by Substituting a 0 or 1 for each literal. Calculating the logical value of the
expression.
A Truth Table specifies the value of the Boolean expression for every combination of the variables in the Boolean expression.
For an n-variable Boolean expression, the truth table has 2n rows (one for each combination).
Boolean Algebra
George Boole developed an algebraic description for processes involving logical thought and reasoning.
Became known as Boolean Algebra Claude Shannon later demonstrated that Boolean
Algebra could be used to describe switching circuits. Switching circuits are circuits built from devices that
switch between two states (e.g. 0 and 1). Switching Algebra is a special case of Boolean
Algebra in which all variables take on just two distinct values
Boolean Algebra is a powerful tool for analyzing and designing logic circuits.
Basic Laws and Theorems
Commutative Law A + B = B + A A.B = B.AAssociative Law A + (B + C) = (A + B) + C A . (B . C) = (A . B) . CDistributive Law A.(B + C) = AB + AC A + (B . C) = (A + B) . (A + C)Null Elements A + 1 = 1 A . 0 = 0Identity A + 0 = A A . 1 = A
A + A = A A . A = AComplement A + A' = 1 A . A' = 0Involution A'' = AAbsorption (Covering) A + AB = A A . (A + B) = ASimplification A + A'B = A + B A . (A' + B) = A . BDeMorgan's Rule (A + B)' = A'.B' (A . B)' = A' + B'Logic Adjacency (Combining) AB + AB' = A (A + B) . (A + B') = AConsensus AB + BC + A'C = AB + A'C (A + B) . (B + C) . (A' + C) = (A + B) . (A' + C)
Idempotence
DeMorgan's Laws
Can be stated as follows: The complement of the product (AND) is the
sum (OR) of the complements. (X.Y)' = X' + Y'
The complement of the sum (OR) is the product (AND) of the complements.
(X + Y)' = X' . Y'
Easily generalized to n variables. Can be proven using a Truth table
Proving DeMorgan's Law
(X . Y)' = X' + Y'
DeMorgan's Theorems
x 1
x 2
x 1
x 2
x 1 x 2 x 1 x 2 + = (a)
x 1 x 2 + x 1 x 2 = (b)
x 1
x 2
x 1
x 2
x 1
x 2
x 1
x 2
Importance of Boolean Algebra
Boolean Algebra is used to simplify Boolean
expressions.
– Through application of the Laws and
Theorems discussed
Simpler expressions lead to simpler circuit
realization, which, generally, reduces cost, area
requirements, and power consumption.
The objective of the digital circuit designer is to
design and realize optimal digital circuits.
Algebraic Simplification
Justification for simplifying Boolean expressions:
– Reduces the cost associated with realizing
the expression using logic gates.
– Reduces the area (i.e. silicon) required to
fabricate the switching function.
– Reduces the power consumption of the
circuit.
In general, there is no easy way to determine when
a Boolean expression has been simplified to a
minimum number of terms or minimum number of
literals.
– No unique solution
Algebraic Simplification
Boolean (or Switching) expressions can be
simplified using the following methods:
1. Multiplying out the expression.
2. Factoring the expression.
3. Combining terms of the expression.
4. Eliminating terms in the expression.
5. Eliminating literals in the expression
6. Adding redundant terms to the expression.
Examples