Codes: Binary Codes Weighted codes Binary Coded Decimal (BCD) Non-weighted codes Excess – 3 code, Gray code, Alphanumeric codes – ASCII Code, EBCDIC, ISCII Code, Error detectionCodes.pdf

DLCA Bharati Ingale (bharatijingale@gmail.com)
Objective
 Binary Codes
 Weighted codes
 Binary Coded Decimal (BCD)
 Non-weighted codes
 Excess – 3 code,
 Gray code,
 Alphanumeric codes – ASCII Code, EBCDIC, ISCII Code,
 Error detection

Binary Codes
 Computers and digital circuits processes information in the binary
format.
 Each character is assigned 7 or 8 bit binary code to indicate its
character which may be numeric, alphabet or special symbol.

Binary Codes
 Numbers, letters or words are represented by a specific group of
symbols is called a code.
 Here the number, letter or word is said to be encoded.
 Example - Binary number 1000001 represents
 65(decimal) in straight binary code,
 alphabet A in ASCII code and
 41(decimal) in BCD code.

Binary Codes
 The digital data is represented, stored and transmitted as group of
binary bits.
 This group is also called as binary code.
 The binary code is represented by the number as well as
alphanumeric letter.

Advantages of Binary Code
 Binary codes are suitable for both
 the computer applications.
 the digital communications.
 Binary codes make the analysis and designing of digital circuits
easier if binary codes are used.
 Since only 0 & 1 are being used, implementation becomes
easy.

Classification of binary codes
Codes
 Weighted Codes
 Non weighted Codes
 Reflective
 Sequential
 Alphanumeric
 Error Detection & Correcting Codes

Classification of binary codes
Codes
Weighted
Codes
BCD
Codes
8421 2421
Non weighted
Codes
Gray
code
Excess 3
code
Alphanumeric
Codes
ASCII
Code
EBDCIC (Extended
binary coded
decimal
interchange code)
Error Detecting/
Correcting Codes
Parity
Bit
Hamming
Bit

Weighted 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 0
through 9.
 Eg: 8421 & 2421
 In these codes each decimal digit is represented by a group of four bits.

Weighted Codes
2+4+2+1 2+4+2+1

Binary Coded Decimal
(BCD) code
 In this code each decimal digit is represented by a 4-bit binary number.
 BCD is a way to express each of the decimal digits with a binary code.
 In the BCD, with four bits we can represent sixteen numbers (0000 to 1111).
 But in BCD code only first ten of these are used (0000 to 1001).
 The remaining six code combinations i.e. 1010 to 1111 are invalid in BCD.

(BCD) code
Conversion of decimal numbers to BCD
 To express any decimal number in BCD, simply
express each decimal digit with its equivalent 4-bit
BCD code.
 (98)10 = (1001 1000) BCD

Smallest and Largest Digit in BCD
 The smallest digit in BCD is 0000 i.e 0 and
 The largest digit in BCD is 1001 i.e 9
Codes

Binary to BCD Conversion
Steps
Step 1 -- Convert the binary number to decimal.
Step 2 -- Convert decimal number to BCD.
Codes

Binary to BCD Conversion
Example − convert (11101)2 to BCD.
Step 1 − Convert to Decimal
Binary Number − 111012
111012
= ( 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 )10
= (16 + 8 + 4 + 0 + 1)10
= 2910
Codes

BCD code
Binary Number − 111012 = Decimal Number − 2910
Step 2 − Convert to BCD
i. Decimal Number − 2910
ii. Calculate BCD Equivalent.: Convert each digit into
groups of four binary digits equivalent.
(11101)2 = (0010 1001)BCD

BCD to Binary Conversion
Steps
Step 1 -- Convert the BCD number to decimal.
Step 2 -- Convert decimal to binary.
BCD Code

BCD to Binary Conversion
Example − convert (00101001)BCD to Binary.
Step 1 - Convert the BCD number to decimal
BCD Number − (01111001)BCD = Decimal 79
BCD Number − (00101001)BCD = Decimal Number − 2910
Step 2 - Convert to Decimal to Binary
Decimal Number − 7810
Calculating Binary Equivalent −10011102
BCD Code

(BCD) code
 Packed BCD number
 Representation of Decimal Number beyond 9 is called Packed
BCD
Eg: (156)10 = (0001 0101 0110)packed BCD

(BCD) code
 Comparision between binary and BCD
Decimal Binary BCD
(10)10 (1010)2 (0001 0000) BCD
(12)10 (1100)2 (0001 0010) BCD
(14)10 (1110)2 (0001 0100) BCD

1 1 1
1 0 1 1
0
+ 1 0 1 0
1 1 1 0 1
(1100)Binary
0001 0010

(BCD) code
 Advantages of BCD Codes
 It is very similar to decimal system.
 We need to remember binary equivalent of decimal digits 0 to 9
only.
 Disadvantages of BCD Codes
 BCD needs more number of bits than binary to represent the
decimal number. So BCD is less efficient than binary.

Application: 7-segment display
One popular type of decimal display is the 7 segment
display used in LED and LCD numerical displays, where
any decimal digit is made up of 7 segments arranged as
a figure.
BCD Code
These displays therefore require 7 inputs, one to each of the LEDs a to g.
Therefore the 4 bit output in BCD must be converted to supply the correct 7
bit pattern of outputs to drive the display.

BCD Code

Non-Weighted Codes
 In this type of binary codes, the positional weights are not
assigned.
 The examples of non-weighted codes are
 Excess-3 code
 Gray code

16.1 Excess-3 code
 The Excess-3 code is also called as XS-3 code.
 It is non-weighted code used to express decimal numbers.
 The Excess-3 code words are derived from the 8421 BCD
code words adding (0011)2 or (3)10 to each code word in 8421.
 The excess-3 codes are obtained as follows −

(BCD) code
Properties of excess – 3 codes
 Each codeword is equivalent to its decimal value plus 3 in
binary.
 The codewords for 0–4 are complements of the codewords
5–9.

Examples
12 to excess -3 = 1+3 = 4 2+3 = 5
0 0 0 1 0 0 1 0
0 0 1 1 0 0 1 1
0 1 0 0 0 1 0 1
29 to excess -3 = 2+3 = 5 9+3 = 12
1
+ 3
4
0 1 0 0
2
+ 3
5
0 1 0 1
2
+ 3
0 5
0 1 0 1
9
+ 3
1 2
1 1 0 0
Convert the following numbers to Excess – 3 code

Examples
8
+ 3
1 1
1 0 1 1
7
+ 3
1 0
1 0 1 0
Convert the following numbers to Excess – code
1. 87
2. 159
Solution: 1. 87

Examples
5
+ 3
8
1 0 0 0
9
+ 3
1 1
1 0 1 1
1
+ 3
4
0 1 0 0
Convert the following numbers to Excess – code
1. 87
2. 159
Solution: 2. 159

Convert Excess-3 code into BCD and decimal number.
1. Group 4-bit for each group, and
2. Subtract 0011 from given number to get the BCD code
3. Find the decimal equivalent.
Excess-3 code

Convert Excess-3 code 1001001 into BCD and decimal number.
Excess-3 code
1. Group 4-bit for each group,
and
2. Subtract 0011 from given
number to get the BCD code
3. Find the decimal equivalent.
0 1 0 0 1 0 0 1
- 0 0 1 1 0 0 1 1
0 0 0 1 0 1 1 0
1 6
Binary Coded Decimal number is 0001
0110 and decimal number will be 16.

Gray Code
 It is the non-weighted code
 It is not arithmetic codes.
 That means there are no specific weights assigned to the bit position.
 It has a very special feature that, only one bit will change each
time the decimal number is incremented..
 As only one bit changes at a time, the gray code is called as a unit
distance code.

Gray Code
 The gray code is a cyclic code.
 Also called as the reflected binary code
 The reflected binary code or Gray code is an ordering of the binary
numeral system such that two successive values differ in only one bit
(binary digit).
 Gray code cannot be used for arithmetic operation.

 Gray code also known as reflected binary code,
because the first (n/2) values compare with those of
the last (n/2) values, but in reverse order.
Gray code

Gray code exhibits the
reflective property i.e
the 2 LSB for decimal
4 to 7 are the mirror
images of those for
decimal 3 to 0
Gray code
Mirror

Decima
l
G(4) G(3) G(2) G(1) G(4) G(3) G(2) G(1)
0 0 0 0 0 x 0 0 0
1 0 0 0 1 x 0 0 1
2 0 0 1 1 x 0 1 1
3 0 0 1 0 x 0 1 0
4 0 1 1 0 x 1 1 0
5 0 1 1 1 x 1 1 1
6 0 1 0 1 x 1 0 1
7 0 1 0 0 x 1 0 0
8 1 1 0 0 x 1 0 0
9 1 1 0 1 x 1 0 1
10 1 1 1 1 x 1 1 1
11 1 1 1 0 x 1 1 0
12 1 0 1 0 x 0 1 0
13 1 0 1 1 x 0 1 1
14 1 0 0 1 x 0 0 1
15 1 0 0 0 x 0 0 0
4-bit Binary to Gray Truth table
Mirror
the 3 LSB for
decimal 8 to
15 are the
mirror images
of those for
decimal 7 to 0

Binary to Gray Code conversion (Addition)
Steps
Step 1 – Record the MSB as it is
Step 2 – Add MSB to the next bit, record the sum and neglect the
carry
Step 2 – Repeat the process
.
Gray code
b3 b2 b1 b0
1 0 1 1
+1 +0 +1
1 1 1 0
g3 g2 g1 g0

Binary to Gray Code conversion (XOR operation)
Steps
Step 2 – Add MSB to the next bit, record the sum and neglect the
carry
.
Gray code
A B Y
0 0 0
0 1 1
1 0 1
1 1 0

Binary to gray code conversion (Generalized)
Gray Code
G3=
G2=
G1=

Gray to Binary Code conversion (Addition)
Steps
Step 2 – Add MSB to the next bit of Gray code, record the sum
and neglect the carry
.
Gray code
g3 g2 g1 g0
1 1 1 0
1 0 1 1
b3 b2 b1 b0
+ + +

Gray to Binary Code conversion (XOR Operation)
Steps
Step 2 – Add MSB to the next bit of Gray code, record the sum
and neglect the carry
.
Gray code
g3 g2 g1 g0
1 1 1 0
1 0 1 1
b3 b2 b1 b0
+ + +

Gray –to-Binary
Conversion

Gray –to-Binary
Conversion
B4=G4
B3=G4 ⊕G3
B2= G2⊕(G4⊕G3)
B1=G1⊕(G2⊕(G4⊕G3))

Gray Code to Binary
conversion
 Rules of Mod-2 Addition

Binary Code to Gray
conversion
 (46)10=(?)GRAY

We need to digitize English text.
We need to represent:
– 26 uppercase,
– 26 lowercase letters,
– 10 numerals,
– 20 punctuation characters,
– 10 useful arithmetic characters,
– 3 other characters (new line, tab, and backspace)
ASCII code
Totally 95 symbols for English

Assigning Symbols
• To represent 95 distinct symbols, we need 7 bits
– 6 bits gives only 26 = 64 symbols
– 7 bits give 27 = 128 symbols
• 128 symbols is ample for the 95 different characters needed for
English characters
• Some additional characters must also be represented

ASCII
 ASCII is an acronym for American Standard Code for Information
Interchange.
 This is standard set of characters understood by all computers,
 Mostly consists of
i. letters ,
ii. numbers and
iii. few basic symbols.
 It is a code that uses numbers to represent characters.

ASCII
 ASCII is a widely used 7-bit (27) code
 Each letter is assigned a number between 0 and 127.
 There are 128 standard ASCII codes, each of which can be
represented by a 7-digit binary number: 0000000 through 1111111.
 Eg: the character A is assigned the decimal number 65, while a is
assigned decimal 97 as shown below in the ASCII table.

ASCII
Advantages of a “standard”:
 It is relatively easy to exchange information between different programs,
different operating systems, and different computers.
 Computer parts built by different manufacturers can be connected
 Programs can create data and store it so that other programs can
process it later, and so forth

EBCDIC
 EBCDIC, in full extended binary-coded decimal interchange
code.
 It was developed and used by IBM.
 EBCDIC) is an 8-bit binary code for numeric and alphanumeric
characters.
 It can represent a total of 256 (28) different characters.

EBCDIC
 It is a coding representation in which symbols, letters and
numbers are presented in binary language.
 Provide a larger set of codes thanASCII.
 Less common thanASCII
 Rarely used today
 It is use by IBM mainframes today

Error detection and
correction

What is an error?
 An error is an action which is inaccurate or incorrect.
 An error is a deviation from accuracy or correctness.
 The condition of having incorrect or false knowledge
 Error is a condition when the output information does not match with
the input information.

Need of error detection and
error correction
 When transmission of digital signals takes place between two
system, the signal get contaminated due to the addition of “Noise”
to it.
 The noise can introduce an error in the binary bits travelling from one
system to the other.

Need of error detection and
error correction
 That means a 0 may change to 1 or 1 may change to 0.
 These error can become a serious threat to the accuracy of the digital
system.
 Therefore it is necessary to detect and correct the errors.

Types of Errors
 Single-bit errors
 Burst errors

How to detect and correct
error?
 One or more bits are added to data for error detection /correction.
 These extra bits are called Parity bits.
 The data bits along with the parity bits form a code word.
 Error detection methods:
 Parity checking
 Checksum error detection
 Cyclic redundancy code (CRC)

Parity bit
 It is the simplest technique for detecting and correcting errors.
 The MSB of an 8-bits word is used as the parity bit and the remaining 7
bits are used as data or message bits.
 The parity of 8-bits transmitted word can be either even parity or odd
parity.

Parity bit
 Even parity -- Even parity means the number of 1's in the given word
including the parity bit should be even (2,4,6,....).
 Odd parity -- Odd parity means the number of 1's in the given word
including the parity bit should be odd (1,3,5,....).

 For even parity, this bit is set to 1 or 0 such that the no. of "1 bits"
in the entire word is even.
 For odd parity, this bit is set to 1 or 0 such that the no. of "1 bits"
in the entire word is odd.
Use of Parity Bit
 The parity bit can be set to 0 and 1 depending on the type of the
parity required.

How Does Error Detection
Take Place?
 Parity checking at the receiver can detect the presence of an error if
the parity of the receiver signal is different from the expected parity.

Hamming Code
 Hamming code is error detection and correction (single bit error) method.
 It was developed by R.W. Hamming for error correction.
 In this coding method, the source encodes the message by inserting
redundant bits within the message.
 These redundant bits are extra bits that are generated and inserted at
specific positions in the message .
 When the destination receives this message, it performs recalculations
to detect errors and find the bit position that has error.

Hamming Code contd…
 This code uses multiple parity bits and we have to place these parity
bits in the positions of powers of 2.
 The parity bits are inserted at each 2n bit where n =1,2,3,….. Thus P1
is at 20= 1, P2 is at 21=2, P4 is as 22 and P8 is as 23
Parity Bit Bits to be checked
P1 1,3,5,7
P2 2,3,6,7
P3 4,5,6,7

Hamming Bit example
 Ex. A bit word 1 0 1 1 is to be transmitted. Construct the even parity
seven bit Hamming code for this data.
 Step1:
1 0 1 1
D7 D6 D5 P4 D3 P2 P1
1 0 1 1 1
 Step 2: Deciding Parity P1,
P1 =1,3,5,7
=1 1 1 (Apply even parity on data)
= 1
D7 D6 D5 P4 D3 P2 P1

Hamming Bit example contd..
P2 =2,3,6,7
= 0
1 0 1 1 1
D7 D6 D5 P4 D3 P2 P1
1 0 1 1 0 1
D7 D6 D5 P4 D3 P2 P1

Hamming Bit example contd..
P4 =4,5,6,7
= 0
1 0 1 1 0 1
D7 D6 D5 P4 D3 P2 P1
1 0 1 0 1 0 1
D7 D6 D5 P4 D3 P2 P1

 Ex. 2: Encode the data bits 0 1 0 1 in seven bits even parity Hamming
code.
 Solution
0 1 0 1 1 0 1
D7 D6 D5 P4 D3 P2 P1

Receiver side decoding
Received Code:
 Solution:
 Step 1: Check the bits of Parity P1,
P1 =1,3,5,7
=1011 (Apply even parity on data)
= 1 Odd parity
Therefore error exist here, put P1 = 1 in the 1’s position of error word.
 Error word E = =
1 0 1 1 0 1 1
D7 D6 D5 P4 D3 P2 P1
P4 P2 P1 P4 P2 1

 Step 2: Check the bits of ParityP2,
P2 =2,3,6,7
= 0 Means no Error
 Step 3: Check the bits of Parity P4,
P4 =4,5,6,7
= 1 Error
1 0 1
P4 0 1

 E =
=(5) 10
1 0 1
1 0 1 1 0 1 1
Received Code:
D7 D6 D5 P4 D3 P2 P1
 Step 4: Correct the error
 Invert the incorrect bit to obtain the correct code word
 Correct Code word:
Incorrect Bit
P4 D3 P2 P1
1 0 0 1 0 1 1
D7 D6 D5

 A seven bit Hamming code is received as 1 1 1 0 1 0 1. What is the correct
code? Assume the parity to be even.
P4 D3 P2 P1
D7 D6 D5
1 0 1 0 1 0 1
E= 1 1 0 = 6

Limitation of Hamming Code
 The main limitation of Hamming code is that it can detect
and correct only one error

Codes

Codes: Binary Codes Weighted codes Binary Coded Decimal (BCD) Non-weighted codes Excess – 3 code, Gray code, Alphanumeric codes – ASCII Code, EBCDIC, ISCII Code, Error detectionCodes.pdf

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