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| And Gate Table |
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| OR Gate Table |
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| NOT Gate Table |
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| NAND Gate Table |
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| NOT Gate Table |
Friday, January 1, 2016
Boolean Function and Logic gates
Boolean function
Boolean functions is an expression with true binary
variables, the true binary operations (+,.) in unary operator, (NOT)
parenthesis and = sign. For a given value of variables, the functions can be
either 0 or 1. For e.g. F1=x.y.z’ represents F1=1 only if
x=1,y=1 and z=0 otherwise F1=0. A Boolean function may be
transformed from algebraic expression into a logic
diagram compared of AND,OR and NOT gate. The implementation of two functions F1
and F2 can be done using following.
 |
| Logic Diagram |
Digital Logic Gates
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| And Gate |
OR Gate
 |
| NOT gate |
 |
| NAND gate |
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| NOR gate |
Boolean algebra and basic postulates
Chapter
2
Boolean Algebra And Logic Gate
Boolean Algebra like any other mathematical system can be
defined with a set of elements, a set of operator and a no of unproved
postulates. This algebra in two valued system can be used to represent the
properties of bistable circuit.
Two Valued Boolean Algebra
It is defines on a set of two elements B= {0,1} with
rules for the two binary operators as following:
x
|
y
|
x+y
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
0
|
1
|
1
|
1
|
1
|
x
|
y
|
x.y
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
0
|
0
|
1
|
1
|
1
|
Basic
Postulates
These postulates are used to prove various Boolean
theorems and do not need any proof:
POSTULATES:-
1) x+0=
x
2) x.1=x
3) x+x’=1
4) x.x’=0
5) x(y+z)=xy+xz
6) x+y+z=(x+y)(x+z)
7) x+y=y+x
8) x.y=y.x
Basic theorems
The above listed postulates are used to prov some basic
theorems. The theorems are listed as following:-
Theorem 1
a)
x+x= x
(x+x)= (x+x)(x+x’)= x+x.x’= x
b)
x.x= x
x.x= x.x+0= x.x+x.x’= x (x+x’)
= x
Theorem 2
a)
x+1=1
x+1= (x+1).1= (x+1)(x+x’)=
x+x’=1
b)
x.0=0 (by the duality of a)
Theorem 3
(x’)’=x
(INVOLUTION)
Proof:
performing complement of variables two times preserving the same original
value.
Theorem 4
a)
x+xy=x (absorption)
x+xy= x (1+y) = x.1= x
b)
x(x+y)=x
x(x+y) = x.x+x.y= x+xy= x (1+Y)
= x
De Morgan’s Theorem
De Morgan’s Theorem
1)
(x+y)’=(x’.y’)
2)
(x.y)’=(x’+y’)
The validity of the de Morgan’s theorem is proved by
using truth table.
x
|
y
|
x+y
|
(x+y)’
|
x’
|
y’
|
x’+y’
|
0
|
0
|
0
|
1
|
1
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
0
|
0
|
1
|
0
|
1
|
0
|
0
|
1
|
0
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
Both columns are same. Hence
verified
|
x
|
y
|
x.y
|
(x.y)’
|
x’
|
y’
|
x’+y’
|
|
0
|
0
|
0
|
1
|
0
|
1
|
1
|
|
0
|
1
|
0
|
1
|
0
|
0
|
1
|
|
1
|
0
|
0
|
1
|
1
|
1
|
1
|
|
1
|
1
|
1
|
0
|
1
|
0
|
0
|
Both columns are same. Hence
verified
Use of De Morgan’s
theorem
It is used to realize a NORing operation with the use of
only AND gates and NOT gates and similarly NANDing operation by using only OR
gates can also be achieved.
Integrated Circuits
Integrate Circuit(IC)
An integrated circuit is a silicon semi-conductor crystal
called a chip containing electronic components for constructing digital
devices. Various gates are inter-connected inside the chip to form required
circuit. The chip is mounted on a plastic container and connected connections
are welded to external pins.
Advantage of IC’s
1.
Extremely small size and thousands times smaller
than conventional circuit due to fabrication of larger number of element in a
single chip.
2.
Very small weight
3.
Very low cost due to simultaneous production of
hundreds of similar types (Mass production Concept).
4.
More reliable because of elimination of soldered
joints and need for fewer interconnection.
5.
Low power consumptions because of their smaller
size.
6.
Easy replacement as it is in organized and
discrete form.
7.
Increased operating speed because of the absence
of parasitic capacitance.
Linear and Digital IC’s
A linear IC is an analog device characterized by infinite
number of possible operating states of input levels. The best known and common
linear IC is pe-rational amplifier Digital IC process only. On-off signal that
is discrete signal. This IC is made in microprocessor.
BCD code, XS-3 code, ASCII code , EBCDIC code, Gray code, Error detection codes and other codes........
Binary Coded Decimal Code (BCD) codes
A binary code will have some unassigned bit combinations if the number of element in the set is not a multiple power of two. The decimal digit from such a set is the binary code distinguished among 10 elements must contain at least 4 bits and this code is called BCD. So in BCD, 6 out of 16 possible combinations remain unassigned and each decimal digit is treated differently and assigned the corresponding 4 bit code for every digit. A number with K decimal digit will require 4K bits in BCD. For e.g. BCD for 132 is 0001 0011 0010.
Advantages
Even though THE BCD representations requires more numbers of bit then in binary, it is easier for data manipulations and the observations for the user can understand decimal codes rather than binary.
BCD addition
IN BCD addition the normal addition process is followed if the sum is less than 10 (1010)2. But if the um is greater than or equal to 10 (1010)2 we have to add 6(0110) to the sum. The addition of 6 (0110)2 to the binary sum converts it to the correct digit and produces a carry as required. This is because the carry is the most significant bit position of the binary sum and decimal carry differ by 16-10=6
Example
4 (0100) + 5 (0101) = 9 (1001)
4 (0100) + 8 (1000) = 12 (1100) which is greater than 10 so adding 6 (0110) to it we get 10010. (0010)2 is 2
Excess-3 code (XS-3)
Excess-3 code also called as XS-3 code is a non-weighted code used to express decimal numbers. This code is used in some older computer. The excess-3 code for a given number is determined by adding 3 to each decimal digit and replacing the newly formed decimal number by its 4-bit binary equivalent.
Decimal no
|
Binary value
|
XS-3 code
|
0
|
0000
|
0011
|
1
|
0001
|
0100
|
2
|
0010
|
0101
|
3
|
0011
|
0110
|
…….
|
………
|
……..
|
9
|
1001
|
1100
|
ASCII CODES
Alphanumeric codes is the set of elements that includes 10 decimal digits, 26 letter of alphabets in a number of special characters such as $, +, α etc.
The standard alphanumeric binary code is the ASCII (American Standard Code for Information Interchange), which use 7 bits to code 128 characters. For e.g. A may be stored as 1000001 and = may be stored as 0111101.
EBCDIC
It represents (Extended Binary Coded Decimal Interchange Code). It uses 8 bit to represent characters. In this system number has a zone portion of 1111 and characters have 3 types of zone portion as:-
1100 (A-I)
1101 (J-R)
1110 (S-Z)
E.g. A= 11000001 J=11010001 S=11100001 0=11110000
Gray Code
The gray code (also called as reflected binary) is sometimes used for the representation of digital data in place of binary data. The Gray code for each Binary code is tabulated as following:-
Number
|
Binary
|
Gray code
|
0
|
000
|
000
|
1
|
001
|
001
|
2
|
010
|
011
|
3
|
011
|
010
|
4
|
100
|
110
|
5
|
101
|
111
|
6
|
110
|
101
|
7
|
111
|
100
|
Advantages of gray code:-
Advantage of gray code over straight binary number sequence is that only one group of code group changes in going from one number to the next. This can be used in the applications I which normal sequence of binary number produce an error during the transition from one number to the next. If binary numbers are used or changed, for example on going from 0111 to 1000 i.e. 7 to 8 it may produce an intermediate errorless number 10010. If the value of right most bit takes longer to change than other two bits, the gray code eliminates this problem. The typical application of gray code occurs when analog data are represented by continuous change of shaft position (rotor shaft movement).
Error detection codes
During the binary information transmission from source to receiver due to external noise in a system. The binary digits may change from 0 to 1 and vice versa. The error detection codes are binary codes that detect such digital error. However the error cannot be corrected. The particular error information is re-transmitted.
Parity bit:
It is the most common error detection code. The parity bit is an extra bit included with a binary message to make the total number of 1’s either odd or even. A message of three bit and two possible parity bit tabulated as following when p (odd) & p (even) represents the parity bit in odd parity and even parity system respectively.
Message
|
P (odd)
|
P (even)
|
0000
|
1
|
0
|
0001
|
0
|
1
|
0010
|
0
|
1
|
0011
|
1
|
0
|
0100
|
0
|
1
|
0101
|
1
|
0
|
0110
|
1
|
0
|
……
|
…….
|
….
|
1111
|
1
|
0
|
In a particular system one or other parity system is adopted.
Parity Generator and Parity Checker:-
A parity generator is the logical circuit the generates parity bit depending upon the input message and the system (either even or odd parity system). The message is applied to the generator at the sending end then the message including the parity bit is transmitted to the destination. At the receiving end all incoming bit including parity bit are applied to a logic circuit known as parity checker. Parity checker then checks adopted parity (even or odd), an error is detected if the checked parity does not match the adopted parity.
 |
| 3 bit even parity generator |
 |
| 4 bit odd parity checker |
Other Decimal Codes
The BCD is weighted code as it can be found out by representing a no with a specific weight 8421, so BCD can also be called as 8421 code.
Example: - BCD of 9= 1*8+0*4+0*2+1*1= 1001 and 0101 is the BCD of 0*8+1*4+0*2+1*1= 5
Another weighted decimal codes is 2421 code. The numbers like 2, 4, 2, 1in 2421 code are coefficient of each binary digit. For example 1001 is the 2421 code of (1*2+0*4+0*2+1*1)=3. Also, the 2421 code for 6 is found out as: - 6=1*2+1*4+0*2+0*1= 1100
For some of the numbers there may be two 2421 code for example the 2421 code for 6 may be 1100 or 0110.
