Digital Fundamental - Binary Codes-Logic Gates

Binary Codes,
This ppt gives a brief explanation about the digital and codes and logic gates that can be widely used. Weighted Codes, Binary codes, ASCII Codes, Parity Codes all can be used widely. A logic gate is an idealized model of computation or physical electronic device implementing a Boolean function, a logical operation performed on one or more binary inputs that produces a single binary output.
Logic gates are the basic building blocks of any digital system. It is an electronic circuit having one or more than one input and only one output.
The relationship between the input and the output is based on a certain logic. Based on this, logic gates are named as AND gate, OR gate, NOT gate etc.
, binary digits also be utilized to make such various innumerable combinations. These may be considered as Binary Codes.
Other than the commonly used 8421 code or BCD code, other Binary Codes like 2421 code, 5211 code, reflective code, sequential code, non-weighted code, excess-3 code and Gray code are also popular.

1. Dr.B.Vidhya Assistant Professor, CT Sri Ramakrishna College of Arts & Science UNIT- I Introduction to Codes

2. Introduction to the Codes The first successful system of electrical communication was the telegraph, which was invented by Samuel F.B. Morse in the year 1832. The telegraph operators used a code of clicks to send the messages. If the key pressed for a short time it is the Morse code called ‘dot’ and if the key pressed for a long-time it is ‘Dash’. A sample Morse code is shown in next slide

3. Introduction to the Codes If it is observed logically, various innumerable combinations of dots and dashes of any kind of words (even sentences) can be written utilizing the above code.

4. Introduction to the Codes In the same way, binary digits also be utilized to make such various innumerable combinations. These may be considered as Binary Codes. Other than the commonly used 8421 code or BCD code, other Binary Codes like 2421 code, 5211 code, reflective code, sequential code, non-weighted code, excess-3 code and Gray code are also popular.

5. W eighted Codes Weighted binary codes are those binary codes which obey the positional weight principle. Each position of the number represents a specific weight. Several systems of the codes are used to express the decimal digits through 9. In these codes each decimal digit is represented by a group of four bits.

6. Non- W eighted Codes Some of the codes will not follow the weights of the sequence binary numbers these are called as non-weighted codes. ASCII code and Grey code are some of the examples where they are coded for some special purpose applications and they do not follow the weighted binary number calculations.

7. Binary Code Binary code , code used in digital computers , based on a binary number system in which there are only two possible states, off and on, usually symbolized by and 1. Whereas in a decimal system, which employs 10 digits, each digit position represents a power of 10 (100, 1,000, etc.), in a binary system each digit position represents a power of 2 (4, 8, 16, etc.). A binary code signal is a series of electrical pulses that represent numbers, characters, and operations to be performed.

8. Binary Code In binary code, each decimal number (0–9) is represented by a set of four binary digits, or bits . The four fundamental arithmetic operations (addition, subtraction, multiplication, and division) can all be reduced to combinations of fundamental Boolean algebraic operations on binary numbers. See the table below for how the decimal numbers from to 10 are represented in binary.)

9. Binary W eights Whenever any binary number appears, its decimal equivalent can be found easily as follows. When there is 1 in a digit position, weight of that position should be added. When there is in a digit position, weight of that position should be disregarded. For example binary number 1100 has a decimal equivalent of 8 + 4 + + = 12.

10. Binary Code Decimal The BCD 8421 code is so called because each of the four bits is given a ‘weighting’ according to its column value in the binary system. The least significant bit (lsb) has the weight or value 1, the next bit, going left, the value 2. The next bit has the value 4, and the most significant bit (msb) the value 8, as shown in Table 1.6.1.

11. Binary Code Decimal So the 8421 BCD code for the decimal number 6 10 is 0110 8421 . For numbers greater than 9 the system is extended by using a second block of 4 bits to represent tens and a third block to represent hundreds etc. 24 10 in 8 bit binary would be 00011000 but in BCD 8421 is 0010 0100. 992 10 in 16 bit binary would be 0000001111100000 2 but in BCD 8421 is 1001 1001 0010.

12. Excess Code -3 As mentioned above, some of the codes will not follow the binary weights, Excess-3 code is an example of it and it is an important 4 bit code. The excess – 3 code of a decimal number is achieved by adding the number 3 to the 8421 code. For example to convert 15 to an excess-3 code, first 3 to be added to each digit as shown below.

13. Excess-3 Code examples Find the excess-3 code of (237.75) 10 The excess-3 code for (237) 10 is obtained by adding 3 to all the digits individually, that is 2, 3 and 7 will become 5, 6 and 10 respectively. These 5, 6 and 10 decimals have to be converted into binary form and the result is 010101101010. The excess-3 code for (.75) 10 is obtained by replacing 7 and 5 with 10 and 8 respectively by adding 3 to each digit. That is, the excess-3 code for (.75) 10 is .10101000. Combining the results of the integral and fractional parts, the excess-3 code for (237.75) 10 is 010101101010.10101000.

14. Excess-3 Code examples Find the decimal number of excess-3 number 110010100011.01110101. The excess-3 code is 110010100011.01110101 By separating 4 bits as group the equivalent excess-3 code is given as 1100 1010 0011.0111 0101. Subtracting 0011 from each four-bit group, we obtain the new number as: 1001 0111 0000.0100 0010. Therefore, the decimal equivalent is (970.42) 10 .

15. Gray Code Definition : Gray Code is the minimum-change code category of coding in which, the two consecutive values changes by only a single bit. More specifically we can say, it is a binary number system where while moving from one step to the next, only a single bit shows variation. It is also termed as reflected binary code or cyclic code . It is an unweighted code, as here like other number systems, no particular weight is provided to various bit positions.

16. Gray Code Basically, binary code is changed to gray equivalent in order to lessen the switching operations. As only a single bit is changed at a particular time duration this leads to a reduction in switching from one bit to another. Let us have a look at the tabular representation, showing gray value for different binary values: Decimal Value Binary Code Gray Code 0000 0000 1 0001 0001 2 0010 0011 3 0011 0010 4 0100 0110 5 0101 0111 6 0110 0101 7 0111 0100

17. Gray Code 8 1000 1100 9 1001 1101 A 1010 1111 B 1011 1110 C 1100 1010 D 1101 1011 E 1110 1001 F 1111 1000

18. Gray Code Consider decimal value 7 and 8 , to understand the switching of bits. We know in binary, 7 is written as 0111 while 8 is written as 1000. Thus we can conclude that in a binary system, all 4 bits are getting changed simultaneously. Hence we can say multiple bits are changing at the same time. In gray code, 7 is written as 0100, as against 8 is written as 1100.

19. Introduction to Parity Code In digital electronics system or digital communication system, the information is transmitted from one system to another system or from one circuit to another in a binary form. While transmitting the information, an error may occur due to the presence of noise. An error is said to occur, if a signal corresponding to may change to 1 or vice versa. It is not always possible to avoid those errors, but it is possible to detect and correct the error. First, it is important to detect an error. It is done by adding an extra bit to the transmitted data. There are different codes like parity code, block parity, hamming code , etc to detect and correct the errors. In this section, you will learn in detail about the parity code.

20. Parity Codes The parity code is used for the purpose of detecting errors during the transmission of binary information. The parity code is a bit that is included with the binary data to be transmitted. The inclusion of a parity bit will make the number of 1’s either odd or even. Based on the number of 1’s in the transmitted data, the parity code is of two types. Even parity code Odd parity code

21. Parity Codes In even parity, the added parity bit will make the total number of 1’s an even number. If the added parity bit make the total number of 1’s as odd number, such parity code is said to be odd parity code. Let us consider the 4-bit message(1011) to be transmitted. Adding 1 to the message will make the total number of 1’s in the message to be an even number. Hence it is called as even parity.

22. Parity Codes For the same message, adding with the transmitted message will make the total number of 1’s to be an odd number. Hence it is called as odd parity. It is shown in the example below.

23. Parity Codes The following table shows the some of the 4-bit messages to be transmitted along with the parity bits. The bits in red color are the parity bits. 4-bit message Messag e with Odd parity Messag e with Even Parity 0000 0000 1 0000 0001 0001 0001 1 0010 0010 0010 1 0011 0011 1 0011 0100 0100 0100 1 0101 0101 1 0101 0110 0110 1 0110 0111 0111 0111 1

24. Parity Codes On the receiver side, if the received data is other than the sent data, then it is an error. If the sent date is even parity code and the received data is odd parity, then there is an error. So, both even and odd parity codes are used only for the detection of error and not for the correction in the transmitted data. Even parity is commonly used and it has almost become a convention.

25. ASCII Codes ASCII , abbreviation of American Standard Code For Information Interchange . A standard data-transmission code that is used by smaller and less-powerful computers to represent both textual data (letters, numbers, and punctuation marks) and noninput-device commands (control characters). Like other coding systems, it converts information into standardized digital formats that allow computers to communicate with each other and to efficiently process and store data.

26. ASCII Codes Th e standar d ASCI I cod e use s seven-digit consisting of various sequences of 0’s and 1’s. binary numbers ; i.e., numbers The code can represent 128 different characters, since there are 128 different possible combinations of seven 0’s and 1’s. The binary sequence 1010000, for example, represents an uppercase “P,” while the sequence 1110000 represents a lowercase “p.”

27. ASCII Codes Digital computers use a binary code that is arranged in groups of eight rather than of seven digits, or bits. Each such eight-digit group is called a byte . Because digital computers use eight-bit bytes, the ASCII code is commonly embedded in an eight-bit field consisting of the seven information bits and a parity bit that is used for error-checking purposes or to represent special symbols. The use of an eight-bit system increased the number of characters the code could represent to 256. The eight-bit system, which is known as the extended ASCII code, was introduced in 1981 by the International Business Machines Corporation (IBM) for use with its first model of personal computer . This extended ASCII code soon became the industry-wide standard for personal computers.

28. Introduction to Logic G a tes A logic gate is an idealized model of computation or physical electronic device implementing a Boolean function , a logical operation performed on one or more binary inputs that produces a single binary output. Logic gates are the basic building blocks of any digital system. It is an electronic circuit having one or more than one input and only one output. The relationship between the input and the output is based on a certain logic. Based on this, logic gates are named as AND gate, OR gate, NOT gate etc.

29. Classific a tion of Logic g a tes

30. AND G A TE An AND gate is a logic gate having two or more inputs and a single output. An AND gate operates on logical multiplication rules. In this gate, if either of the inputs is low (0), then the output is also low. If all of the inputs are high (1), then the output will also be high. An AND gate can have any number of inputs, although 2 input and 3 input AND gates are the most common.

31. AND G A TE - Symbol & Time T able A circuit which performs an AND operation is shown in figure. It has n input (n >= 2) and one output .

32. OR G A TE An OR gate is a digital logic gate with two or more inputs and one output that performs logical disjunction . The output of an OR gate is true when one or more of its inputs are true . If all of an OR gate's inputs are false , then the output of the OR gate is false . A circuit which performs an OR operation is shown in figure. It has n input (n >= 2) and one output.

33. OR G A TE - Symbol & Truth T able

34. NOT G a te A logical inverter , sometimes called a NOT gate to differentiate it from other types of electronic inverter devices, has only one input. It reverses the logic state. If the input is 1, then the output is 0. If the input is 0, then the output is 1. NOT gate is also known as Inverter. It has one input A and one output Y.

35. NOT G a te- Truth table & Symbol

36. NAND G a te The NAND gate operates as an AND gate followed by a NOT gate. It acts in the manner of the logical operation "and" followed by negation. The output is "false" if both inputs are "true." Otherwise, the output is "true." A NOT-AND operation is known as NAND operation. It has n input (n >= 2) and one output.

37. NAND G A TE- Truth T able & Symbols

38. NOR G a te The NOR gate is a combination OR gate followed by an inverter. Its output is "true" if both inputs are "false." Otherwise, the output is "false." A NOT-OR operation is known as NOR operation. It has n input (n >= 2) and one output.

39. NOR G a te-Symbols & truth table

40. EX-OR G A TE The Exclusive-OR gate or XOR gate is achieved by combining standard logic gates together. X O R gat e i s used extensi v ely i n er r o r detectio n ci r cuits , computationa l logic comparators and arithmetic logic circuits. The Exclusive OR gate gives an output only if its two inputs are dissimilar, namely if one of them is high (one) and the other is low (zero).

41. EX- OR G a te-Symbols & truth table

42. EX-NOR G a te-Symbols & truth table Exclusive-NOR gate is formed by combining the Exclusive-OR gate (XOR gate) and the NOT gate. On analysing the truth table of Exclusive-NOR gate, we understand that the output is similar to the standard NOR gate except that the output of the Ex-NOR gate is high when both inputs are high. The output of EX-NOR gate is 1 when both the inputs are 1 (high) and if both the inputs are 0 (low). If either one of the input is 1 (high) the output will be 0 (low). In other words, both the inputs should be in the same logic level for the output to be 1.

43. EX-NOR G a te-Symbols & truth table