Digital Design Binary Codes

Digital Design – © 2020 Mouli Sankaran Email: mouli.sankaran@yahoo.com 2
Session 2: Focus
 Binary codes and their classifications
 Weighted codes
◦ Binary Coded Decimal (BCD)
 Various BCD formats
 Unsigned BCD arithmetic
 Packed and unpacked BCD
 Non-weighted codes
◦ Excess-3
◦ Gray code
◦ Optical Encoder Example using Gray Code

Binary Codes

What are Binary Codes and
Why are they needed?
 Digital systems represent and manipulate not only
binary numbers, but also
◦ Many other discrete elements of information.
◦ Example: Digital speech signal, Character symbols (a, b, etc.)
 Any discrete element of information that is distinct
among a group of quantities can be represented
◦ With binary code, i.e., a pattern of 0’s and 1’s
 The digital data is represented, stored and transmitted
as group of binary bits
 A set of eight elements requires a three‐bit code and a
set of 16 elements requires a four‐bit code

Binary Codes
 Both letters, numbers, symbols are represented by
binary codes
◦ Various fonts used in word processors are examples of
binary codes
 Binary codes are used in computer applications and
digital data communication
 Binary codes ease implementation of digital circuits to
process encoded digital data

Classification of Binary Codes
 Weighted Codes
 Non-Weighted Codes
 Binary Coded Decimal Code
 Alphanumeric Codes
 Error Detecting Codes – Not covered in this course
 Error Correcting Codes – Not covered in this course

Weighted Codes
Binary Coded Decimal (BCD)

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.
 In these codes each decimal digit is represented by a group of
four bits
Courtesy: Tutorial Point

Binary Coded Decimal (BCD)
 Only the first 9
combinations of four bits
are valid
 The remaining 6
combinations are unused in
BCD
 Easier to represent decimal
digits in BCD which is
straightforward
 Example: (12.75)10
 BCD (8421):
 (0001 0010.0111 0101)8421
 (0001 0010.1101 1011)2421

Other BCDs

Unsigned BCD (8421) Addition
 Invalid (out of range) BCD values are corrected
by adding a value of 6 (0110) to it
 The reason for adding six is that six binary
combinations are unused in BCD (8421)
sequence
 Note that if just BCD is mentioned, it means 8421
Valid BCD
Invalid BCD

N-bit Unsigned BCD Addition
 It follows the same procedure with carry propagating
from right to left
Carry
BCD

Packed and Unpacked BCD
 When a BCD number is stored in a register inside
a processor, when each BCD number occupies one
register
◦ It is called Unpacked BCD
◦ Only lower four bits will be having the values, which is
not efficient in terms of register usage
 In packed BCD, based on the size of register, two
or more BCD numbers are stored in a register or
memory
◦ In a 8 bit register, two BCD numbers are stored
◦ In a 16 bit register, four BCD number are stored
◦ An 8 bit register having 57BCD reads 0101 0111

Non-weighted Codes
Excess-3 Code

Non-weighted Codes: Excess-3
 No positional weights are assigned in non-weighted codes
◦ Examples: Excess-3 code and Gray code
Weighted Non-weighted
Courtesy: Tutorial Point

Excess-3 Codes
 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 by adding 3 (0011) to each code word in 8421.
 This is a self complementing code
 A self complementing code for Base-10 is a code for which the
complement of a given encoded digit d[0-9] will result in 9-d, its
9's complement
◦ It helps in building digital circuits for subtraction

Excess-3 Code: Example
 Find Excess-3 code for the decimal value 597:
 The addition of ‘3’ to each digit yields three new
digits/numbers ‘8’, ‘12’ and ‘10’
 The corresponding four-bit binary equivalents are
1000, 1100 and 1010 respectively
 The excess-3 code for 597 is therefore given by: 1000
1100 1010
 59710 = 1000 1100 1010XS-3

Self complementing Code: Excess-3
 Self complementing means that
the 9’s complement of the
Excess-3 code can be got, by
just complementing the bits.
 For example:
 9’s complement of 95 is 4 (99-
95)
◦ Excess-3 of 95 : 1100 1000
◦ Complement bits: 0011 0111
◦ In Excess-3, it is : 04
◦ Which is 9’s complement of 95

Non-weighted Codes:
Gray Codes

Constructing Gray Codes
Value
Gray code Gray code Gray code Gray code

Gray Codes

Gray Codes
 It is non-weighted code and it is not arithmetic code.
◦ i.e., Gray code cannot be used for arithmetic operations
 There are no specific weights assigned to the bit
positions in Gray codes
 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
 The gray code is a cyclic code.
◦ Circular shifts of each codeword gives another word that
belongs to the code

Use of Gray Codes
 It is a non-weighted code which belongs to a class of
codes called minimum change codes
 Here, two adjacent code numbers differs from each
other by only one bit
 Gray code is popularly used in the shaft position
encoders.
◦ A shaft position encoder produces a code word which
represents the angular position of the shaft
 It is also used in the transmission of digital signals
 The Gray code is used for labelling the axes of
Karnaugh maps

Optical Encoder
Coded Disc
Connected to a
Rotating Shaft

Session 2: Summary
 Binary codes and their classifications
 Weighted codes
◦ Binary Coded Decimal (BCD)
 Various BCD formats
 Unsigned BCD arithmetic
 Packed and unpacked BCD
 Non-weighted codes
◦ Excess-3
◦ Gray code
◦ Optical Encoder Example using Gray Code

References
Ref 1 Ref 2

Digital Design Binary Codes

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Digital Design Binary Codes

Similar to Digital Design Binary Codes (20)

More from International Institute of Information Technology - Bangalore

More from International Institute of Information Technology - Bangalore (10)

Recently uploaded

Recently uploaded (20)

Digital Design Binary Codes