Introduction

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● Digital systems are concerned with digital signals
● Digital signals can take many forms
● Here we will concentrate on binary signals since these are the most common form of digital signals
    – can be used individually
● perhaps to represent a single binary quantity or the state of a single switch
    – can be used in combination
● to represent more complex quantities

Binary quantities and variables

● A binary quantity is one that can take only 2 states

● A binary arrangement with two switches in series

● A binary arrangement with two switches in parallel


● Three switches in series


● Three switches in parallel


● A series/parallel arrangement


● Representing an unknown network

Logic gates

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● The building blocks used to create digital circuits are logic gates
● There are three elementary logic gates and a range of other simple gates
● Each gate has its own logic symbol which allows complex functions to be represented by a logic diagram
● The function of each gate can be represented by a truth table or using Boolean notation

● The AND gate


● The OR gate


● The NOT gate (or inverter)


● A logic buffer gate


● The NAND gate


● The NOR gate


● The Exclusive OR gate


● The Exclusive NOR gate


● The symbols shown earlier for the various logic gates are the "distinctive shape‟ symbols
● Other symbols are also used such as those described in IEC 617 (shown under "Alternative symbol‟ here)

Boolean algebra

● Boolean constants
– these are "0" (false) and "1" (true)
● Boolean variables
– variables that can only take the values "0" or "1"
● Boolean functions
– each of the logic functions (such as AND, OR and NOT) are represented by symbols as described above
● Boolean theorems
– a set of identities and laws – see text for details


● Boolean identities


● Boolean laws

Combinational logic

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● Digital systems may be divided into two broad categories:
– combinational logic
● where the outputs are determined solely by the current states of the inputs
– sequential logic
● where the outputs are determined not only by the current inputs but also by the sequence of inputs that led to the current state
● In this lecture we will look at combination logic


● Implementing a function from a Boolean expression
Example – see Example 24.1 in the course text
Implement the function X=A+B\overline{C}


● Implementing a function from a Boolean expression
Example – see Example 24.2 in the course text
Implement the function Y=\overline{\overline{A}B+C\overline{D}}




● Generating a Boolean expression from a logic diagram
Example – see Example 24.3 in the course text

– work progressively from the inputs to the output adding logic expressions to the output of each gate in turn



● Implementing a logic function from a description
Example – see Example 24.4 in the course text
The operation of the Exclusive OR gate can be stated as:
“The output should be true if either of its inputs are true, but not if both inputs are true”
This can be rephrased as:
“The output is true if A OR B is true,
AND if A AND B are NOT true.”
We can write this in Boolean notation as X=(A+B)\bullet(\overline{AB})

The logic function X=(A+B)\bullet(\overline{AB}) can then be implemented as before

● Implementing a logic function from a truth table
Example – see Example 24.6 in the course text

Implement the function of the following truth table



The logic function X=\overline{A}\ \overline{B}C+A\overline{B}C+AB\overline{C} can then be implemented as before


– Complex logic diagrams are often made easier to understand by the use of labels, rather than showing complex interconnections – the earlier circuit becomes

Boolean algebraic manipulation

● We can use the various laws of Boolean algebra to change the form of circuits
 – the following diagram shows the implications of the associative laws




● Modifying circuits to use only NAND gates
– All combinational arrangements can be constructed as



– from De Morgan‟s theorem A+B+B=\overline {\overline A\,\overline B\, \overline C}
– therefore

– therefore

– hence any combinational arrangement can be constructed using only NAND gates

Example – see Example 24.7 in the course text.
Implement the following using only NAND gates


This becomes


● Modifying circuits to use only NOR gates
– Similarly, from De Morgan‟s theorem

A\bullet B\bullet C=\overline{\overline{A}+\overline{B}+\overline{C}}
– therefore


– therefore

– hence any combinational arrangement can be constructed using only NOR gates

● Example – see Example 24.8 in the course text
Implement the following using only NOR gates


This becomes

Algebraic simplification

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● All combination circuits can be described in a sum-of-products form
– This consists of a number of terms (minterms or products) that are ORed together
– Examples include

A\overline{B}+\overline{A}B

XYZ+\overline{X}Y\overline{Z}+X\overline{Y}Z

AB\overline{C}D+\overline{AB}CD

– The sum-of-products must not include inversions of a series of term, as in

A\overline{BC}D+\overline{AB}CD

Example – see Example 24.9 in the course text
Implement the following expression:

X=ABC+\overline{A}BC+AC+A\overline{C}


Example (continued)
Alternatively, the expression can be simplified

X=ABC+\overline{A}BC+AC+A\overline{C}
    =BC(A+\overline{A})+A(C+C)
    =BC+A

which can be implemented as

De Morgan's Theorem

Modifying circuits to use only NAND gates

– from De Morgan‟s theorem

A+B+C=\overline{\overline{A}\:\overline{B}\:\overline{C}}

– therefore


Example - From the logic expression draw a circuit using NAND gates only





A+B+C=\overline{\overline{A}\:\overline{B}\:\overline{C}}



● Example – see Example 23.7 in the course text.

● Implement the following using only NAND gates


● This becomes