introduction
DESCRIPTION
Sanjeev Patel 4xTRANSCRIPT
Introduction
• Digital Computer = H/W + S/W• Digital implies that the information in the computer
is represented by variables that take a limited number of discrete values.
• the decimal digits 0, 1, 2,….,9, provide 10 discrete values, but digital computers function more reliably if only two states are used.
• because of the physical restriction of components, and because human logic tends to be binary(true/false, yes/no),digital component are further constrained to take only two values and are said to be binary.
Bit = binary digit : 0/1Program(S/W)A sequence of instructionS/W = Program + Data» The data that are manipulated by the program constitute the data baseApplication S/W = DB, word processor, Spread SheetSystem S/W = OS, Firmware, Compiler, Device Driver
Digital Components• Computer Hardware(H/W)• CPU • Memory ROM and RAM• I/O Device Interface: 8251 SIO, • Input Device: Keyboard, Mouse,
Scanner• Output Device: Printer, Plotter,
Display• Storage Device(I/O): FDD, HDD,
AND Gate
OR Gate
Boolean Algebra
• Boolean Algebra• Deals with binary variable (A, B, x, y: T/F or
1/0) +• logic operation (AND, OR, NOT…)• Boolean Function: variable + operation• F(x, y, z) = x + y’z• George Boole• Born: 2 Nov 1815 in Lincoln,• Died: 8 Dec 1864 in Ballintemple,• County Cork, Ireland
Boolean Algebra
• Boolean Function: variable + operationF(x, y, z) = x + y’z
• Truth Table: Relationship between a function and variable
• Purpose of Boolean Algebra• To facilitate the analysis and design of digital circuit• Boolean function = Algebraic form = convenient tool• Truth table (relationship between binary variables) Algebraic
form• Logic diagram (input-output relationship :) Algebraic form• Find simpler circuits for the same function : by using Boolean
algebra rules
De Morgan’s law
Boolean Algebra Rule
Karnaugh Map
• Karnaugh Map(K-Map)• Map method for simplifying Boolean expressions• Minterm / Maxterm• Minterm : n variables product ( x=1, x’=0)• Maxterm : n variables sum (x=0, x’=1)
2 variables example
Map
F = x + y’z
Adjacent SquareNumber of square = 2n (2, 4, 8, ….)The squares at the extreme ends of the same horizontal row are to be considered adjacent
The same applies to the top and bottom squares of a column
The four corner squares of a map must be considered to be adjacent
Groups of combined adjacent squares may share one or more squares with one or more group
Half Adder
Logic Diagram
Truth Table
A half adder adds two one-bit binary numbers A and B. It has two outputs, S and C.
The simplest half-adder design, pictured on the right, incorporates an XOR gate for S and an AND gate for C. Half adders cannot be used compositely, given their incapacity for a carry-in bit.
Full AdderA full adder adds binary numbers and accounts for values carried in as well as out. A one-bit full adder adds three one-bit numbers, often written as A, B, and Cin; A and B are the operands, and Cin is a bit carried in.
A full adder can be constructed from two half adders by connecting A and B to the input of one half adder, connecting the sum from that to an input to the second adder, connecting Cin to the other input and OR the two carry outputs
Logic Diagram
Truth Table
SR Flip-Flop
Graphic Symbol
Truth Table
A SR flip-flop has three inputs, S (for set), R (for reset) and C (for clock). It has an output Q.
The undefined condition makes the SR flip-flop difficult to manage and therefore it is seldom used in practice.
D Flip-Flop
Graphic Symbol
Truth Table
The D flip-flop is a slight modification of the SR flip-flop by inserting an inverter between S and R and assigning the symbol D to the single input.
If D=1, the output goes to the state 1, and if D=0, the output of the flip flop goes to the 0 state.
JK Flip-Flop
Graphic Symbol
Truth Table
Inputs J and K behave like inputs S and R. When inputs J and K are both equal to 1, a clock transition switches the output of the flip-flop to their complement state.
T Flip-Flop
Truth Table
Graphic Symbol
The T flip-flop is obtained from a JK flip-flop when inputs J and K are connected to provide a single input designated by T.
The flip-flop thus has only two conditions.
Excitation Tables
During the design of circuits, we need a table that lists the required input combinations for a given change of state. Such table is called a flip flop excitation table.
Sequential Circuits• A sequential circuit
is an interconnection of flip-flops and gates.
Example of a Sequential Circuit
Ax
Bx
Ax+Bx
A’x
x’
A+B (A+B).x
A=Ax+Bx, B=A’x
y=(A+B).x
State Table