5HeadedBloggers: 2012

TOPIC : DIGITAL LOGIC

Assalamualaikum and hi everyone. ^^

Before I moving on further, just a simple introduce about my self.
The names given is Nur Umira Binti Mustafa. Well,you can just call me Nur. ^^
I studying in Technical University Malaysia Melaka (UTEM) , and taking software development as my course.
Alright,lets quick chatting and start with our main topic for today’s discussion. .. ^^

LEARNING OUTCOMES

In order for you to fully understand and take a full control of this topic, you need to dig out a few information about the basic component in this topic.

Lets we analyst our learning outcomes of this discussion

1) understand and apply basic logic gates

2) understand and apply law and rules of Boolean,algebra and DeMorgan’s theorems to Boolean expressions

3) Able to design a combinational logic circuit for given Boolean output expression.

// The learning outcomes seems though isn’t it..?

But don’t worry, neither you are beginner nor expert, I will share all information about this topic with you, to enable you to get more detail information about this topic.

WHAT IS LOGIC GATE ?

So,friends, have you ever wondering what is meant by LOGIC GATE ?

Do you still remember what is BOOLEAN ALGEBRA ?

(if you can answer 2 simple warming up questions above, you are ready to explore this topic ^^ )

Boolean algebra

ALTERNATIVE NAME : Boolean logic

WHAT IS IT ?

: is a logical calculus of truth values.

:It resembles the algebra of real numbers

but with the numeric operations of >

-multiplication xy

-addition x + y

-negation −x replaced by the respective logical operations of conjunction x∧y,

-disjunction x∨y

-negation ¬x.

-The Boolean operations are these and all other operations that can be built from these, such as x∧(y∨z). These turn out to coincide with the set of all operations on the set {0,1} that take only finitely many arguments; there are 2^2^n such operations when there are n arguments.

Basic logic gates

Boolean algebra uses variables and operation to represent the logic circuit. The variables and the function have the only and value, 0 and 1 .
The compliment of the value is shown by a bar over the letter or a single quotes on the right side above the letter.
Example : A’ / B’

Type

Distinctive shape

Rectangular shape

Boolean algebra between A & B

Truth table

AND

A.B

INPUT		OUTPUT
A	B	A AND B
0	0	0
0	1	0
1	0	0
1	1	1

A+B

INPUT		OUTPUT
A	B	A OR B
0	0	0
0	1	1
1	0	1
1	1	1

NOT

INPUT	OUTPUT
A	NOT A
0	1
1	0

NAND

INPUT		OUTPUT
A	B	A NAND B
0	0	1
0	1	1
1	0	1
1	1	0

NOR

INPUT		OUTPUT
A	B	A NOR B
0	0	1
0	1	0
1	0	0
1	1	0

XOR

INPUT		OUTPUT
A	B	A XOR B
0	0	0
0	1	1
1	0	1
1	1	0

XNOR

INPUT		OUTPUT
A	B	A XNOR B
0	0	1
0	1	0
1	0	0
1	1	1

DECORDER :

} Use decoder to build larger components

} The most common type of decoder has an n-bit input and 2ⁿ outputs, where only one output is asserted for each input combination.

} This decoder translates the n-bit input into a signal that corresponds to the binary value of the n-bit input.

} The outputs are shows a 3-bit decoder and the truth table. This decoder is called a 3-to-8 decoder

} since there are 3 inputs and 8 (23) outputs. There is also a logic element called an

} encoder that performs the inverse function of a decoder, taking n inputs and producing

} an n-bit output.

} This decoder is called a 3-to-8 decoder means 3 inputs and 8 (2³) outputs. There is also a logic element called an encoder that performs the inverse function of a decoder, taking n inputs and producing an n-bit output.

Multiplexor (Selector)

} The output is one of the inputs that is selected by a CONTROL

} The left side shows this multiplexor has three inputs: two data values and a selector (or control) value. The selector value determines which of the inputs becomes the output. We can represent the logic function computed by a two-input multiplexor, shown in gate form on the right side as C = (A & S’ ) + (B & S)

} Multiplexors can be created with an arbitrary number of data inputs.

} If there are only two inputs, the selector is a single signal that selects one of the inputs

} if it is true (1) and the other if it is false (0). If there are n data inputs, there will need to be ⎡log₂n⎤ selector inputs.

} In this case, the multiplexor basically consists of 3 parts:

} 1. A decoder that generates n signals, each indicating a different input value

} 2. An array of n AND gates, each combining one of the inputs with a signal from the decoder

} 3. A single large OR gate that incorporates the outputs of the AND gates

} To associate the inputs with selector values, we often label the data inputs numerically

How to deal with COMBINATIONAL CIRCUITS ?

~ For your information , a logic block contains no memory and computes the output given the current inputs.

~ combinational circuits can be defined in three ways.

1) truth table – the truth table shows many possible combination of input values, in tabular from between the input values and the result of a specific Boolean operator or combination on the input variables.

IMPORTANT !

For n inputs,there are 2n of combination

2) Graphical symbol – the layout of connected gates that represent the logic circuit

examples :

3) Boolean equations- Boolean function that consist possible combination of inputs that produce an output signal.

Boolean Equations Forms.

A Boolean algebra is the combination of variables and operators. Typically , it has one or more input and produces an output in the range of 0 or 1. The compliment of a variable is shown by bar or a single quotes on the right side above the letter.
Example : A’ / B’

All Boolean equation can be represent in two forms :

Sum of Products Expansion(SOP )

Definition

- Combination of input values that produce

1s is convert into equivalent variables, AND ed, together then OR ed with other combination variables with same output.

- SOP is easier to derive from the truth

Table.

IMPORTANT !

Sum of products for Boolean function is a Boolean expression constructed by:

-Where each OUTPUT result is a 1

-form product of all variables

-sum each product

Definition 1

Sum-of-products expansion

is the sum of minterms.

Disjunctive normal form

is the same as sum-of-products.

Definition 2

Literal

is a Boolean variable or its complement.

Minterm

of the Boolean variables x₁, x₂, ..., x_n is a

Boolean product y₁, y₂, ..., y_n where y_i=x_i or y_i=x_i

Example 1:

F and G of Table 1 are expressed as sum of minterms.

F(x,y,z) = xyz The sum of one minterm.

G(x,y,z) = xyz + xyz The sum of two minterms.

Example 2 :

Minterm that is 1 for x₁= 0, x₂= 0, x₃= 1, x₄= 1

x₁x₂x₃x₄= 1

Example 3

Find Boolean EQUATION based on TRUTH Table 1.

H = X’Y’ + XYZ’

TABLE 1 :

X	Y	Z	H
0	0	0	1
0	0	1	1
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	0
1	1	0	0
1	1	1	0t

SOLUTION

X	Y	Z	X’Y’	XYZ’	H
0	0	0	1	0	1 X’Y’Z’
0	0	1	1	0	1 X’Y’Z
0	1	0	0	0	0
0	1	1	0	0	0
1	0	0	0	0	0
1	0	1	0	0	0
1	1	0	0	0	0
1	1	1	0	0	0

H = (X’Y’Z’) + (X’Y’Z)

By: Nur Umira

Product-of-sum( POS )

Definition

- Input combination that produce 0 in the sum term ( OR’ed variables ) are AND’ed together.

- Convert input values that produce 0s into equivalent variables, OR’ed the variables, the AND’ed with other OR’ED forms

- Usually use if more 1s produce in output function.

same output.

IMPORTANT !

PRODUCT of SUM for Boolean function is a Boolean expression constructed by:

-Where each OUTPUT result is a 0

-form SUM of all variables

-MULTIPLY each SUM OF VARIABLE

EXAMPLE 4:

Find Boolean EQUATION based on TRUTH Table 2.

H = (A+B+C)(A+B+C’)(A+B’+C)(A’+B+C’)

PRODUCT TERM

(1)A+B+C

(2)A+B+C’

(3)A+B’+C

(4)A’+B+C’

A	B	C	H
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	1
1	0	1	0
1	1	0	1
1	1	1	1

(1)

(2)

(3)

(4)

POS Expressions :

H = (A+B+C)(A+B+C’)(A+B’+C)

-note : this is not simplified versions

SIMPLIFICATION OF BOOLEAN EQUATIONS

Rules of Boolean Algebra

Table 4-1 lists 12 basic rules that are useful in manipulating and simplifying Boolean expressions. Rules 1 through 9 will be viewed in terms of their application to logic gates. Rules 10 through 12 will be derived in terms of the simpler rules and the laws previously discussed.

Table 4-1 Basic rules of Boolean algebra.

Rule 1. A + 0 = A 

A variable ORed with 0 is always equal to the variable. If the input variable A is 1, the output variable X is 1, which is equal to A. If A is 0, the output is 0, which is also equal to A. This rule is illustrated in Fig.(4-6), where the lower input is fixed at 0.

Fig.(4-6)

Rule 2. A + 1 = 1

 A variable ORed with 1 is always equal to 1. A 1 on an input to an OR gate produces a 1 on the output, regardless of the value of the variable on the other input. This rule is illustrated in Fig.(4-7), where the lower input is fixed at 1.

Fig.(4-7)

Rule 3. A . 0 = 0 

A variable ANDed with 0 is always equal to 0. Any time one input to an AND gate is 0, the output is 0, regardless of the value of the variable on the other input. This rule is illustrated in Fig.(4-8), where the lower input is fixed at 0.

Fig.(4-8)

Rule 4. A . 1 = A 

A variable ANDed with 1 is always equal to the variable. If A is 0 the output of the AND gate is 0. If A is 1, the output of the AND gate is 1 because both inputs are now 1s. This rule is shown in Fig.(4-9), where the lower input is fixed at 1.

Fig.(4-9)

Rule 5. A + A = A 

A variable ORed with itself is always equal to the variable. If A is 0, then 0 + 0 = 0; and if A is 1, then 1 + 1 = 1. This is shown in Fig.(4-10), where both inputs are the same variable.