Digital Electronics Codes

1. 1
Digital Electronics
Binary Coded Decimal (BCD)
Prof. Prasenjit Kumar Das
Dept. of CST/CSIT, UEM Kolkata
Email : prasenjit.das@uem.edu.in

2. 2
BCD, ASCII, EBCDIC, Gray codes
and their conversions

3. 3
Binary Coded Decimal - BCD
In computing and electronic systems, binary-coded decimal (BCD) is a class of
binary encodings of decimal numbers where each digit is represented by a fixed
number of bits
BCD was commonly used for displaying alpha-numeric in the past but in
modern-day BCD is still used with real-time clocks or RTC chips to keep track of
wall-clock time and it's becoming more common for embedded microprocessors to
include an RTC.
It's very common for RTCs to store the time in BCD format

4. 4
BCD Vs Binary
 Both binary-coded decimal (BCD) and binary numbers are used in
many digital applications.
 Both have their advantages and disadvantages.
 BCD is commonly used when decimal numbers must be represented in
hardware, as each 4-bit BCD number maps directly to a decimal
number.
 Binary is more efficient for arithmetic, memory storage, and
transmitting information, but is less human-readable.

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Binary Coded Decimal - BCD
 Each single decimal digit represented in binary Coded to 4-bit binary
number
 Radix/Base for decimal number system = 10 So, 10 values possible in
each digit: 0 to 9
 In binary, Radix/Base is = 2 So, only 2 values possible in each digit: 0
and 1

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For example,
 143 is a decimal number 1, 4 and 3 are the separate digits
 Each of the digits 1, 4 and 3, needs to be converted (coded) to binary
separately
 Each digit will be represented as 4 digit binary number

7. 7
 BCD uses ‘positional weights’
 Each decimal digit, in the 4-bit representation, is expressed
in terms of the positional weights
 The ‘positional weights’ used in BCD are:
8-4-2-1

8. 8
 Each bit – binary digit can contain either
0 or 1
 2 values possible in each bit –
expressed in power of 2
 8 – 4 – 2 – 1 is better
understood as
23
– 22
– 21
– 20
 Here, 20 represents the LSB and 23 represents the MSB
for the 4-bit binary code

9. 9
Note
 We use 4 bits in BCD, so 24
= 16 values can be represented
 Out of 16, we use only 10 decimal digits, 0 to 9
 Rest of the values 10 to 15 are invalid,
as they are decimal numbers, not digits
 Note: BCD is the same as 4-bit Binary representation for decimal
digits 0 to 9

10. 10
Representing Decimal Digit 0
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
0 0 0 0 0 0 0 0 0
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

11. 11
Representing Decimal Digit 1
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
1 0 0 0 1 0 0 0 1
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

12. 12
Representing Decimal Digit 2
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
2 0 0 1 0 0 0 1 0
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

13. 13
Representing Decimal Digit 3
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
3 0 0 1 1 0 0 1 1
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

14. 14
Representing Decimal Digit 4
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
4 0 1 0 0 0 1 0 0
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

15. 15
Representing Decimal Digit 5
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
5 0 1 0 1 0 1 0 1
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

16. 16
Representing Decimal Digit 6
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
6 0 1 1 0 0 1 1 0
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

17. 17
Representing Decimal Digit 7
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
7 0 1 1 1 0 1 1 1
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

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Representing Decimal Digit 8
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
8 1 0 0 0 1 0 0 0
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

19. 19
Representing Decimal Digit 9
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
9 1 0 0 1 1 0 0 1
Note: the Decimal Digit is expressed in terms of positional weights,
place 1 in case of the positional weights that are required,
for all others place 0

20. 20
Summary Chart – BCD for Decimal
Digits
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
0 0 0 0 0 0 0 0 0
1 0 0 0 1 0 0 0 1
2 0 0 1 0 0 0 1 0
3 0 0 1 1 0 0 1 1
4 0 1 0 0 0 1 0 0
5 0 1 0 1 0 1 0 1
6 0 1 1 0 0 1 1 0
7 0 1 1 1 0 1 1 1
8 1 0 0 0 1 0 0 0
9 1 0 0 1 1 0 0 1

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2 types of Conversions
Decimal Number
BCD
BCD
Decimal Number

22. 22
Decimal Number to BCD Example 1
Q) Find BCD for Decimal Number 13
●(13)10 is a decimal number, not digit
●2 decimal digits in (13)10, 1 and 3
●First, we need to convert each digit to corresponding BCD value
●Then we place the BCD values together in proper order to get the
final BCD representation

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Finding individual BCD values
●BCD for Decimal Digit 1 = 0001
●
●BCD for Decimal Digit 3 = 0011
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
1 0 0 0 1 0 0 0 1
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
3 0 0 1 1 0 0 1 1

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Final BCD of Decimal Number 13
●Therefore, BCD of (13)10 is:
●0001 0011
BCD of 1 BCD of 3

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Decimal Number to BCD Example 2
●(174)10 is a decimal number, not digit
●3 decimal digits in (174)10 - 1, 7 and 4
●First, we need to convert each digit to
corresponding BCD value
●Then we place the BCD values together in
proper order to get the final BCD
representation

26. 26
Finding individual BCD values
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
1 0 0 0 1 0 0 0 1
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
7 0 1 1 1 0 1 1 1
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
4 0 1 0 0 0 1 0 0
BCD for Decimal Digit 1 = 0001
BCD for Decimal Digit 7 = 0111
BCD for Decimal Digit 4 = 0100

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Final BCD of Decimal Number 174
●Therefore, BCD of (174)10 is:
●0001 0111 0100
BCD of 1 BCD of 7 BCD of 4

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BCD to Decimal Number
Conversion
●The BCD code for each decimal digit is of 4-
bit length
●So for any given BCD code,
●Starting from the right,
find the 4-bit groups in the given BCD
●Using the conversion table, determine the
decimal equivalent for each BCD code
●Place all decimal digits together to find
the final decimal number

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BCD to Decimal Number Example
1
q. Find Decimal Number for BCD 00010101
●Starting with the LSB, make group of 4-bits
●0001 0101
●
●So, we find 2 groups, means 2 decimal digits
●Look up the corresponding decimal digit for
each BCD code in the conversion table

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Finding the corresponding
decimal digits
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
1 0 0 0 1 0 0 0 1
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
5 0 1 0 1 0 1 0 1
So, decimal digit for BCD 0001 is 1
So, decimal digit for BCD 0101 is 5

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Final Decimal Number
●Therefore, Decimal Number for BCD
00010101 = (15)10
●
0001 0101
(1 5)10

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BCD to Decimal Number Example 2
●In many cases, the BCD code may not be
easily seperated in groups of 4-bits
●
q. Find Decimal Number for BCD 100001
●Here, the given BCD is of length 6 !

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Solution
●In this case, the grouping will be:
10 0001
●Clearly, the first group is not a valid BCD code
●To make it valid, add required number of 0s at the
left of the number to change it to 4-bits
●So, the groups now become:
0010 0001
●2 zeroes (highlighted in red) are added before the
MSB to make make 4-bit BCD

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Finding the corresponding
decimal digits
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
2 0 0 1 0 0 0 1 0
Decimal Digit Binary Coded Decimal
8 4 2 1 Final
1 0 0 0 1 0 0 0 1
So, decimal digit for BCD 0001 is 2
So, decimal digit for BCD 0001 is 1

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Final Decimal Number
●Therefore, Decimal Number for BCD
00100001 = (21)10
●
0010 0001
(2 1)10

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●1 3 2 6 – convert to BCD
0001 0011 0010 0110
●11101 – convert to Decimal
0001 1101

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BCD, ASCII, EBCDIC, Gray codes
and their conversions

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ASCII Code
 American Standard Code for Information
Interchange – ASCII
 127 different characters represented
 127 = 27 – 1, so ASCII is a 7-bit code
 Codes are numbered from 0 to 126

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40. 40
Some important ASCII codes
Characters ASCII Code
0 to 9 48 to 57
(A to Z) 65 to 90
(a to z) 97 to 122
For alphabets,
Think of index starting wit
A 0 65
B 1 65+1
C 2 65+2
D 3 65+3
.........
J 9 65+9=74
a 0 97
b 1 98=97+1
c 2 99=97+2
d 3 100
...............
j 9 97+9=106

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Binary to ASCII conversion
●Starting from the right (LSB),
Group every 7 bits together
●For each group of 7 bits,
Determine the corresponding decimal
value
●Use ASCII table to find the ASCII code
for each group

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Binary to ASCII conversion
Example 1
q. Convert 10010110110001 to ASCII
●First, we make 7-bit groups, starting with
the LSB
●1001011 0110001
●Next, we convert each 7-bit binary to
decimal number

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In decimal,
●1001011
6 5 4 3 2 1 0
●= 1x20 + 1x21 + 0x22 + 1x23 + 0x24 + 0x25 + 1x26
●= 1 + 2 + 8 + 64 = 75
●0110001
6 5 4 3 2 1 0
●= 1x20 + 0x21 + 0x22 + 0x23 + 1x24 + 1x25 + 0x26
●= 1 + 16 + 32 = 49
●

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●Finally, we find the corresponding
ASCII values
75 in ASCII = K
49 in ASCII = 1
●So, 10010110110001 in ASCII is K1

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ASCII to Binary Conversion
Example 1
q. Convert Mp4 in ASCII to Binary
●First, find the corresponding decimal
numbers for each ASCII character
●M p 4
77 112 52

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●Next, convert each decimal number to binary
●77 in binary:
2|77 – 1
2|38 – 0
2|19 – 1
2| 9 – 1
2| 4 – 0
2| 2 – 0
1
●112 in binary:
2|112 – 0
2|56 – 0
2|28 – 0
2|14 – 0
2| 7 – 1
2| 3 – 1
1
●52 in binary:
2|52 – 0
2|26 – 0
2|13 – 1
2| 6 – 0
2| 3 – 1
1

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●Next, each of the binary numbers will placed in order
of their corresponding decimal numbers
●So,
77 112
52
1001101 1110000
0110100
●Since we are converting ASCII to binary, all binary
representations have to be of 7-bits
●Here, binary for 52 is of 6 bits, so we add 0 at LSB

48. 48
●So, each binary has to be converted to
7-bit form by adding 0s to the LSB as
required
●Finally, the Binary form of ASCII Mp4
is =
●
(1001101 1110000 0110100)2

49. 49
BCD, ASCII, EBCDIC, Gray codes
and their conversions

50. 50
EBCDIC Code
●Extended Binary Coded Decimal Interchange
Code
●8-bit representation in binary form
●Mainly used in IBM Mainframe machines
●Extension of Binary Coded Decimal form
●Makes use of fixed, 4-bit ‘Zone’ bits
with
4-bit binary representation of digits etc.

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EBCDIC Code equivalent for
Digits 0-9
Character EBCDIC Code
Zone Digit
0 1111 0000
1 1111 0001
2 1111 0010
3 1111 0011
4 1111 0100
5 1111 0101
6 1111 0110
7 1111 0111
8 1111 1000
9 1111 1001
●4-bit Zone bits for digits is 1111
●The 4-bit binary representation is same as BCD codes

52. 52
EBCDIC Code equivalent for
Alphabets
Character EBCDIC
Zone Digits
A 1100 0001
B 1100 0010
C 1100 0011
D 1100 0100
E 1100 0101
F 1100 0110
G 1100 0111
H 1100 1000
I 1100 1001
●The 26 alphabets are divided in 3 groups
A to I (9),
J to R (9), and
S to Z (8)
●Each group has separate 4-bit Zone bits:
A to I: 1100
●For each group, the digit value goes from 1 to 9, and
●corresponding 4-bit binary is used

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EBCDIC Code equivalent for
Alphabets
Character EBCDIC
Zone Digit
J 1101 0001
K 1101 0010
L 1101 0011
M 1101 0100
N 1101 0101
O 1101 0110
P 1101 0111
Q 1101 1000
R 1101 1001
●For this group, the 4-bit Zone bits are
J to R: 1101
●The digit value goes from 1 to 9, and so the same corr

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EBCDIC Code equivalent for
Alphabets
Character EBCDIC
Zone Digit
S 1110 0010
T 1110 0011
U 1110 0100
V 1110 0101
W 1110 0110
X 1110 0111
Y 1110 1000
Z 1110 1001
●For this group, the 4-bit Zone bits are
S to Z: 1110
●The digit value goes from 2 to 9, and so the same
● corresponding 4-bit binary is used

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Decimal Number to EBCDIC Example
1
q. Convert 425 to EBCDIC
●First, determine the zone bits – here it is
1111 (for digits)
●Next, find the BCD of each decimal digit:
●4 5 2
0100 0010 0101

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●To the BCDs, append the Zone bits at LSB
4 2
5
11110100 11110010
11110101
●Bring the 8-bit codes together in order to get
the final EBCDIC representation:
●11110100 11110010 11110101

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EBCDIC to Character Example 1
q. Convert the following EBCDIC to
Decimal:
1111100111111000
●First, make groups of 8-bits each
starting from the right
11111001 11111000

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●The first 4-bits of each group
represents Zone bits,use it to identify
the type of data
●Here Zone bits = 1111, so we have digits
●For each group, starting with the right,
extract the 4 digit bits and find their
corresponding decimal value
●11111001 11111000
1001 1000

59. 59
In decimal,
1001
3 2 1 0
= 1x20 +1x23 = 9
1000
3 2 1 0
= 1x23 = 8
So, placing the decimal digits in order, the
converted value of given EBCDIC code = 98

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BCD, ASCII, EBCDIC, Gray codes
and their conversions

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Gray Code
●Named after scientist Frank Gray
●Called ‘Reflected Binary code’
●Expressions for sequential numbers differ
only by 1 bit
●So, binary expressions are converted to
Gray Code to minimize switching operations
●Also called ‘Unit Distance Code’, ‘Cyclic
Code’

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Binary to Gray Code conversion
●Take the MSB – this is the MSB for Gray
Code
●Add MSB to next bit of Binary, keep the
sum and ignore carry (XOR) – the sum is
the 2nd bit after MSB in Gray Code
●Keep shifting bit by bit towards right,
and repeat the above two steps to obtain
the complete Gray Code

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Generalized Rule
●If a binary number b3b2b1b0 is to be
converted to Gray Code g3g2g1g0,
●g3 = b3,
●g2=b3⊕b2
●g1=b2⊕b1
●g0=b1⊕b0

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Example of Binary to Gray Code
convertion
q. Convert 1101 to Gray Code
Step 1:
●MSB of 1101 is 1
●So, MSB of Gray
Code is set as 1
Binary: 1 1 0 1
Gray Code:1

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Example of Binary to Gray Code
convertion
Step 2:
●Add MSB (1) to ne
●Set the sum (0) a
●Ignore the carry
Binary: 1 1 0 1
+
Gray Code: 1 0

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Example of Binary to Gray Code
convertion
Step 3:
●Add succeeding bi
●Set the sum (1) a
●Ignore the carry
Binary: 1 1 0 1
+
Gray Code: 1 0 1

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Example of Binary to Gray Code
conversion
Step 4:
●Add succeeding bi
●Set the sum (1) a
●Ignore the carry
Binary: 1 1 0 1
+
Gray Code: 1 0 1 1

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Gray Code to Binary conversion
●Take the MSB – this is the MSB for Gray
Code
●Add MSB to next bit of Gray Code, keep
the sum and ignore carry (XOR) – the sum
is the 2nd bit after MSB
●Keep shifting bit by bit towards right,
and repeat the above two steps to obtain
the complete Gray Code

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Generalized Rule
●If a Gray Code g3g2g1g0 is to be
converted to binary number b3b2b1b0,
●b3 = g3,
●b2=b3⊕g2
●b1=b2⊕g1
●b0=b1⊕g0

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Example of Gray Code to Binary
conversion
q. Convert 1011 to Binary
Step 1:
●MSB of 1011 is 1
●So, MSB of Binary
Gray Code: 1 0 1 1
Binary: 1

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Example of Gray Code to Binary
conversion
Step 2:
●XOR the MSB (1) t
●Set the result (1
Gray Code: 1 0 1 1
⊕
Binary: 1 1

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Example of Binary to Gray Code
convertion
Step 3:
●XOR the succeesin
●Set the result (0
Gray Code: 1 0 1 1
⊕
Binary: 1 1 0

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Example of Binary to Gray Code
conversion
Step 4:
●XOR the succeesin
●Set the result (1
Gray Code: 1 0 1 1
⊕
Binary: 1 1 0 1

74. 74
Thank You