Digital Logic

 Logic Gate

• Boolean algebra

 -to represent  logic circuits
 - the variable and funtion have only one value ,0 and 1.
 -the complement of a variable  is shown by a bar over the letter such as A' 

• Basic Logic Gate

                                                                       Example1

Example2

                                              

Combination Circuits

A logic block no memory and compute the output given the current input
        
~can be define in 3 ways:

1.  Truth table
2. Graphical symbols
3. Boolean equation

Example :Truth table

 Example : Graphical symbol

Example : Boolean equation

Boolean function that consist possible combination of inputs that produce and ouput signal.

Boolean equation can be represent in 2 forms:
Sum-Of_Product(SOP) :

• Combination of input value that produce 1.
• SOP is easier to derive from truth table

Product-Of-Sum(POS) :

• Input combination is 0 in sum term
• usually use if more than 1 output function

Simplify of  Boolean equation

There are 2 ways to simplify boolean equation:

         1. Law Of Boolean Algebra-rule to simplify Boolean Expression

         2.Karnaugh Map-a grid-like representation of a truth table

                               Example of Karnaugh Map:

                                                        

LAW  OF BOOLEAN ALGEBRA

• Boolean expression can be simplified or manipulated.Table below shows basic laws of Boolean Algebra to help manipulating logic equations.

Example:

Absorbtion Law derivation

A(A+B)=AA+BB        ~> A.A=A
             =A+AB          ~>A(1+B)=A(1)
             =A








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.  


 At first, it might seem that the Karnaugh Map is just another way of presenting the information in a truth table.  In one way that's true.  However, any time you have the opportunity to use another way of looking at a problem advantages can accrue to you.  In the case of the Karnaugh Map the advantage is that the Karnaugh Map is designed to present the information in a way that allows easy grouping of terms that can be combined.


        Let's start by looking at the Karnaugh Map we've already encountered.  Look at two entries side by side.  We'll start by focussing on the ones shown below in gray.
        Let's examine the map again.

• The term on the left in the gray area of the map corresponds to:
• 
• The term on the right in the gray area of the map corresponds to:
• 
• These two terms can be combined to give
• 

        The beauty of the Karnaugh Map is that it has been cleverly designed so that any two adjacent cells in the map differ by a change in one variable.  It's always a change of one variable any time you cross a horizontal or vertical cell boundaries.  

        Notice that the order of terms isn't random.  Look across the top boundary of the Karnaugh Map.  Terms go 00, 01, 11, 10.  If you think binary well, you might have ordered terms in order 00, 01, 10, 11.  That's the sequence of binary numbers for 0,1,2,3.  However, in a Karnaugh Map terms are not arranged in numerical sequence!  That's done deliberately to ensure that crossing each horizontal or vertical cell boundary will reflect a change of only one variable.  In the numerical sequence, the middle two terms, 01, and 10 differ by two variables!  Anyhow, when only one variable changes that means that you can eliminate that variable, as in the example above for the terms in the gray area.

        Let's check the claim made on above.  Click on the buttons to shade groups of terms and to find out what the reduced term is.
        The Karnaugh Map is a visual technique that allows you to generate groupings of terms that can be combined with a simple visual inspection.  The technique you use is simply to examine the Karnaugh Map for any groups of  ones that occur.  Grouping ones into the largest groups possible and ensuring that all ones in the table have been included are the first step in using a Karnaugh Map.

        In the next section we will examine how you can generate groups using Karnaugh Maps.  First, however, we will look at some of the kinds of groups that occur in Truth Tables, and how they appear in Karnaugh Maps.

        Click on these buttons to show some groupings.  There's one surprise, but it really is correct.  In each case, be sure that you understand the term that the group represents.


        There is a small surprise in one grouping above.  The lower left and the lower right 1s actually form a group.  They differ only in having B and its' inverse.  Consequently they can be combined.  You will have to imagine that the right end and the left end are connected.

        So far we have focussed on K-maps for three variables.  Karnaugh Maps are useful for more than three variables, and we'll look at how to extend ideas to four variables here.  Shown below is a K-map for four variables.

Universal Gate

Gate that can be use to implement any gates like AND,OR,and NOT or any combination of these basic gates are called universal gate,NAND and NOR are such example.

NAND GATE

A NAND gate (Negated AND or NOT AND) is a logic gate which produce an output that is false only if all its outputs are true.

Truth table and Graphical symbol for NAND gates

NOR GATE

A NOR gate (Negated OR or NOT OR) is a logic gate which produce a HIGH output(1) result if both the input to  the gate are OW(0);a LOW output (0) result if one or both input is HIGH(1).

Truth table and Graphical symbol for NOR gates

 Arithmetic Logic Unit (ALU)
Definition:

• Represents the fundamental building block of the central processing unit (CPU) of a computer.
•  An ALU is a digital circuit used to perform arithmetic and logic operations.
•  It is the computational capacity of the ALU that determines the power of a computer system’s CPU. 
•  The ALU is divided into two units: the arithmetic unit (AU) and the logic unit (LU).

Characteristics of the ALU: 

 1.The ALU is responsible for performing all logical and arithmetic operations.

2.Some of the arithmetic operations are as follows: addition, subtraction, multiplication and division.3. Some of the logical operations are as follows: comparison between numbers, letter and or special characters.

4.The ALU is also responsible for the following conditions: Equal-to conditions, Less-than condition and greater than condition.