Digital Logic

DIGITAL LOGIC

Boolean algebra uses variables and operators to represent logic circuits. The variables and function  have only one value, 0 ad 1. The complement of a variable is shown by a bra over the letter such as Ā or apostrophe (A’). Table 1 summarizes logic gates as the symbol of the functions in  Boolean expressions.

Table 1:




























A logic block contains no memory and computes the output  given the current inputs.

A Logic block can be defined in three ways:
1. Truth table – The Truth table shows many possible combinations of input values, in tabular form between the input values and the result of a specific Boolean operator or combinations on the input variables.
2. Graphical symbols – the layout of connected gates the represent the logic circuit.

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

Boolean Equation Forms
A Boolean algebra is the combinations of variables and operators. Typically, it has one or more inputs and produces an output in the range of 0 or 1. The complement of a variable is shown by a bar over the letter.

All Boolean equation can be represented in two forms:
*Sum –of – products (SOP) – REFER example 1
---Combination of input values that produce 1s is convert into equivalent variables, ANDed together then ORed with other combination variables with the same output.
---SOP  is easier to derive from truth table.

*Product-of-sums (POS) – refer Example 2
---Input combinations that produce 0 in sum terms (ORed variables) are ANDed together.
---Convert input values that produce 0s into equivalent variables, ORed the variables, then ANDed with other ORed forms.
---Usually use if more 1s produce in output function.

Simplification of Boolean Equation
There are two ways to simplify Boolean equation.
1. Laws Of Boolean Algebra – rules to simplify Boolean expression
2. Karnaugh Map – A grid-like representation of a truth table.

Example 1
F = AB’ + ABC’
The truth table:

A	B	C	F
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	0
1	0	0	1
1	0	1	1
1	1	0	1
1	1	1	0

AB’C’    AB’C    ABC’

SOP expression:
F = AB’C’+ AB’C + ABC’

Example 2
F = (A + B + C)(A’+ B + C)(A’ + B’ + C)
The truth table:

A	B	C	F
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	0
1	1	0	0
1	1	1	1

POS expression :
F = (A + B + C)(A’+ B + C)(A’ + B’ + C)

Laws of Boolean Algebra
Boolean expressions can be simplified or manipulated. Table 2 shows basic laws of Boolean Algebra to help manipulating logic equations.

Table 2 : Basic Laws of Boolean Algebra

Absorption Law derivation:
A(A+B) = A(1+B)                  1 + B = 1
              = A (1)                      A.1 = A
              = A

Absorption Law derivation:
A(A + B) = AA + AB                        A.A = A
                = A + AB                A(1 + B) = A(1)
                = A

Absorption Law derivation:
A + AB’ = (A + AB) + A’B               A = A.A
               = (AA + AB) + A’B                        (AA + AB) + A’B = AA + (AB + A’B)
               = AA + AB +A’B
               = (A+A’)(A+B)                   (A+A’)(A+B) = AA + AB + A’B
               = 1.(A+B)                            A + A’ = 1
               = (A + B)

·         Distributive Law reverse derivation:
(A+B).(A+C) = AA + AC + AB + BC            AA = A
                       = A + AC + AB + BC               A (1+C) = A(1) = A
                       = A + AB + BC             A (1 + B) = A(1) = A
                       = A(1 + B) + BC
                       = A.1 + BC
                       = A + BC


DE MORGAN’S LAW

There is a trick to remember the laws . An aid to memory is ,

If you break the line , you change the sign .

A’B’ = A’+B’

A’+B’ = A’. B’

The logic gates that represent de Morgan’s Law is ,

Karnaugh Map

A Karnaugh Map is a grid-like representation of a truth table.  It is really just another way of presenting a truth table, but the mode of presentation gives more insight.  A Karnaugh map has zero and one entries at different positions.  Each position in a grid corresponds to a truth table entry.  Here’s an example taken from the voting circuit presented in the lesson on Minterms.  The truth table is shown first.  The Karnaugh Map for this truth table is shown after the truth table.
A Karnaugh map provides a pictorial method of grouping together expressions with common factors and therefore eliminating unwanted variables. The Karnaugh map can also be described as a special arrangement of a truth table

The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem.

Universal Gates
Gates that can be used to implement any gates like AND, OR and NOT or any combination of these basic gates are called universal gates. NAND and NOR are such examples.

NAND Gate

NOR Gate