Important Formulas Digital Electronics - Digital Circuits - Electronics and Communication

 Page 1


IMPORTANT FORMULAS ON DIGITAL ELECTRONICS 
 
Number System and Codes 
 
Fig. 1 
Types of Number System: 
The number can be represented in various ways to show the data and process it on the 
processing devices. 
 
Decimal Number 
System 
Hexadecimal Number 
System 
Octal Number 
System 
Binary Number System 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
A 
B 
C 
D 
E 
F 
0 
1 
2 
3 
4 
5 
6 
7 
10 
11 
12 
13 
14 
15 
16 
17 
0000 
0001 
0010 
0011 
0100 
0101 
0110 
0111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
   Table 1: Counting in different number system 
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1. 
Decimal to other codes conversions:  
To convert decimal number into other system with base ‘r’, divide integer part by r and 
multiply fractional part with r. 
Other codes to Decimal Conversions: 
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A   ,        
2 –1 –2
2 1 0 1 2
A  x r x r  x y r y r = + + + + 
Hexadecimal to Binary:  
Convert each Hexadecimal digit into 4 bits binary. 
Page 2


IMPORTANT FORMULAS ON DIGITAL ELECTRONICS 
 
Number System and Codes 
 
Fig. 1 
Types of Number System: 
The number can be represented in various ways to show the data and process it on the 
processing devices. 
 
Decimal Number 
System 
Hexadecimal Number 
System 
Octal Number 
System 
Binary Number System 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
A 
B 
C 
D 
E 
F 
0 
1 
2 
3 
4 
5 
6 
7 
10 
11 
12 
13 
14 
15 
16 
17 
0000 
0001 
0010 
0011 
0100 
0101 
0110 
0111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
   Table 1: Counting in different number system 
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1. 
Decimal to other codes conversions:  
To convert decimal number into other system with base ‘r’, divide integer part by r and 
multiply fractional part with r. 
Other codes to Decimal Conversions: 
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A   ,        
2 –1 –2
2 1 0 1 2
A  x r x r  x y r y r = + + + + 
Hexadecimal to Binary:  
Convert each Hexadecimal digit into 4 bits binary. 
( )
( )
2
16
1111 0101
1010
5AF
5 A F
? 
Binary to Hexadecimal:  
Grouping of 4 bits into one hex digit. 
( ) ( )
2 16
110101.11 00110101.1100 35.C ?? 
Octal Binary and Binary to Octal:  
Same procedure as discussed above but here group of 3 bits is made. 
Codes: 
Binary coded decimal (BCD): 
• In BCD code each decimal digit is represented with 4 bit binary format. 
 ( )
??
??
??
??
10
93 4
BCD
Eg: 943 1001 0100 0011  
• It is also known as 8421 code. 
• Invalid BCD codes are the codes whose decimal equivalent is more than 9. i.e. Valid 
BCD codes ranges from 0 to 9. 
 Total number of codes possible 
4
2 16 ??  
 Valid BCD codes 10 ? 
 Invalid BCD codes 16 10 6 -? 
 There 1010, 1011, 1100, 1110 and 1111 
 
Excess-3 codes: (BCD + 0011) 
• It can be derived from BCD by adding ‘3’ to each coded number.  
• It is unweighted and self-complementing code.  
 
Gray Code:  
It is also known as minimum change codes or unit distance code or reflected code. 
 
Binary code to Gray code: 
In order to find Gray code from Binary code, XOR Gate is applied between the present 
binary bit and the next binary bit starting from the MSB side(keeping MSB of Binary and 
Gray as same) e.g. 
 
Page 3


IMPORTANT FORMULAS ON DIGITAL ELECTRONICS 
 
Number System and Codes 
 
Fig. 1 
Types of Number System: 
The number can be represented in various ways to show the data and process it on the 
processing devices. 
 
Decimal Number 
System 
Hexadecimal Number 
System 
Octal Number 
System 
Binary Number System 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
A 
B 
C 
D 
E 
F 
0 
1 
2 
3 
4 
5 
6 
7 
10 
11 
12 
13 
14 
15 
16 
17 
0000 
0001 
0010 
0011 
0100 
0101 
0110 
0111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
   Table 1: Counting in different number system 
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1. 
Decimal to other codes conversions:  
To convert decimal number into other system with base ‘r’, divide integer part by r and 
multiply fractional part with r. 
Other codes to Decimal Conversions: 
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A   ,        
2 –1 –2
2 1 0 1 2
A  x r x r  x y r y r = + + + + 
Hexadecimal to Binary:  
Convert each Hexadecimal digit into 4 bits binary. 
( )
( )
2
16
1111 0101
1010
5AF
5 A F
? 
Binary to Hexadecimal:  
Grouping of 4 bits into one hex digit. 
( ) ( )
2 16
110101.11 00110101.1100 35.C ?? 
Octal Binary and Binary to Octal:  
Same procedure as discussed above but here group of 3 bits is made. 
Codes: 
Binary coded decimal (BCD): 
• In BCD code each decimal digit is represented with 4 bit binary format. 
 ( )
??
??
??
??
10
93 4
BCD
Eg: 943 1001 0100 0011  
• It is also known as 8421 code. 
• Invalid BCD codes are the codes whose decimal equivalent is more than 9. i.e. Valid 
BCD codes ranges from 0 to 9. 
 Total number of codes possible 
4
2 16 ??  
 Valid BCD codes 10 ? 
 Invalid BCD codes 16 10 6 -? 
 There 1010, 1011, 1100, 1110 and 1111 
 
Excess-3 codes: (BCD + 0011) 
• It can be derived from BCD by adding ‘3’ to each coded number.  
• It is unweighted and self-complementing code.  
 
Gray Code:  
It is also known as minimum change codes or unit distance code or reflected code. 
 
Binary code to Gray code: 
In order to find Gray code from Binary code, XOR Gate is applied between the present 
binary bit and the next binary bit starting from the MSB side(keeping MSB of Binary and 
Gray as same) e.g. 
 
Fig. 2 
 
Gray code to Binary code: 
In order to find Binary code from Gray code, XOR Gate is applied between the present 
binary bit and the next gray bit starting from the MSB side (keeping MSB of Gray and 
Binary as same) e.g. 
 
 
Fig. 3 
Alpha Numeric code:  
EBCDIC (Extended BCD interchange code) 
It is an 8-bit code. It can represent 128 possible characters.  
• Parity method is most widely used schemes for error detection.  
• Hamming code is most useful error correcting code. 
• BCD code is used in calculators, counters. 
Complements:  
Its base is r then we can have two complements. 
(i) (r – 1)’s complement 
(ii) r’s complement 
To determine(r – 1)’s complement: First write maximum possible number in the given 
system and subtract the given number. 
To determine r’s complements: (r – 1)’s complement + 1 
i.e. First write (r – 1)’s complement and then add 1 to LSB 
Example: 
Q.  Find 7’s and 8’s complement of 2456 
Sol. 7’s Complement 
7777
2456
5321
- 
 8’s Complement 
5321
1
5322
+ 
Q.   Find 2’s complement of 101.110 
Sol. 1’s complements 010.001 
 For 2’s complement add 1 to the LSB 
Page 4


IMPORTANT FORMULAS ON DIGITAL ELECTRONICS 
 
Number System and Codes 
 
Fig. 1 
Types of Number System: 
The number can be represented in various ways to show the data and process it on the 
processing devices. 
 
Decimal Number 
System 
Hexadecimal Number 
System 
Octal Number 
System 
Binary Number System 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
A 
B 
C 
D 
E 
F 
0 
1 
2 
3 
4 
5 
6 
7 
10 
11 
12 
13 
14 
15 
16 
17 
0000 
0001 
0010 
0011 
0100 
0101 
0110 
0111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
   Table 1: Counting in different number system 
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1. 
Decimal to other codes conversions:  
To convert decimal number into other system with base ‘r’, divide integer part by r and 
multiply fractional part with r. 
Other codes to Decimal Conversions: 
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A   ,        
2 –1 –2
2 1 0 1 2
A  x r x r  x y r y r = + + + + 
Hexadecimal to Binary:  
Convert each Hexadecimal digit into 4 bits binary. 
( )
( )
2
16
1111 0101
1010
5AF
5 A F
? 
Binary to Hexadecimal:  
Grouping of 4 bits into one hex digit. 
( ) ( )
2 16
110101.11 00110101.1100 35.C ?? 
Octal Binary and Binary to Octal:  
Same procedure as discussed above but here group of 3 bits is made. 
Codes: 
Binary coded decimal (BCD): 
• In BCD code each decimal digit is represented with 4 bit binary format. 
 ( )
??
??
??
??
10
93 4
BCD
Eg: 943 1001 0100 0011  
• It is also known as 8421 code. 
• Invalid BCD codes are the codes whose decimal equivalent is more than 9. i.e. Valid 
BCD codes ranges from 0 to 9. 
 Total number of codes possible 
4
2 16 ??  
 Valid BCD codes 10 ? 
 Invalid BCD codes 16 10 6 -? 
 There 1010, 1011, 1100, 1110 and 1111 
 
Excess-3 codes: (BCD + 0011) 
• It can be derived from BCD by adding ‘3’ to each coded number.  
• It is unweighted and self-complementing code.  
 
Gray Code:  
It is also known as minimum change codes or unit distance code or reflected code. 
 
Binary code to Gray code: 
In order to find Gray code from Binary code, XOR Gate is applied between the present 
binary bit and the next binary bit starting from the MSB side(keeping MSB of Binary and 
Gray as same) e.g. 
 
Fig. 2 
 
Gray code to Binary code: 
In order to find Binary code from Gray code, XOR Gate is applied between the present 
binary bit and the next gray bit starting from the MSB side (keeping MSB of Gray and 
Binary as same) e.g. 
 
 
Fig. 3 
Alpha Numeric code:  
EBCDIC (Extended BCD interchange code) 
It is an 8-bit code. It can represent 128 possible characters.  
• Parity method is most widely used schemes for error detection.  
• Hamming code is most useful error correcting code. 
• BCD code is used in calculators, counters. 
Complements:  
Its base is r then we can have two complements. 
(i) (r – 1)’s complement 
(ii) r’s complement 
To determine(r – 1)’s complement: First write maximum possible number in the given 
system and subtract the given number. 
To determine r’s complements: (r – 1)’s complement + 1 
i.e. First write (r – 1)’s complement and then add 1 to LSB 
Example: 
Q.  Find 7’s and 8’s complement of 2456 
Sol. 7’s Complement 
7777
2456
5321
- 
 8’s Complement 
5321
1
5322
+ 
Q.   Find 2’s complement of 101.110 
Sol. 1’s complements 010.001 
 For 2’s complement add 1 to the LSB 
 2’s complement 
010.001
1
010.010
+ 
 
 
Data Representation: 
 
   
Fig. 4 
Unsigned Magnitude:  
 Range with n bit 
 
n1
0 to 2 5 101
-
? + ? 
 5 Not possible -? 
 
Signed Magnitude:  
 Range with n bit 
 
sign bit sign bit
with 4 bits with 8 bits
6 1 110 1 0000110 -? 
 
1’s complements:  
 Range with n bit 
 
( ) ( )
n 1 n 1
2 1 to 2 1
--
? - - + - 
 + ? - ? 6 0110 6 1001 
 
2’s complements:  
 With n bit range –2
n-1
 to (2
n-1
 –1) 
 + ? - ? 6 0110 6 1010 
 In any representation +ve numbers are represented similar to +ve number in sign 
magnitude. 
Page 5


IMPORTANT FORMULAS ON DIGITAL ELECTRONICS 
 
Number System and Codes 
 
Fig. 1 
Types of Number System: 
The number can be represented in various ways to show the data and process it on the 
processing devices. 
 
Decimal Number 
System 
Hexadecimal Number 
System 
Octal Number 
System 
Binary Number System 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
A 
B 
C 
D 
E 
F 
0 
1 
2 
3 
4 
5 
6 
7 
10 
11 
12 
13 
14 
15 
16 
17 
0000 
0001 
0010 
0011 
0100 
0101 
0110 
0111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
   Table 1: Counting in different number system 
A number system with base ‘r’, contains ‘r’ different digits and they are from 0 to r –1. 
Decimal to other codes conversions:  
To convert decimal number into other system with base ‘r’, divide integer part by r and 
multiply fractional part with r. 
Other codes to Decimal Conversions: 
( ) ( ) ?
2 1 0 1 2
r 10
x x x .y y A   ,        
2 –1 –2
2 1 0 1 2
A  x r x r  x y r y r = + + + + 
Hexadecimal to Binary:  
Convert each Hexadecimal digit into 4 bits binary. 
( )
( )
2
16
1111 0101
1010
5AF
5 A F
? 
Binary to Hexadecimal:  
Grouping of 4 bits into one hex digit. 
( ) ( )
2 16
110101.11 00110101.1100 35.C ?? 
Octal Binary and Binary to Octal:  
Same procedure as discussed above but here group of 3 bits is made. 
Codes: 
Binary coded decimal (BCD): 
• In BCD code each decimal digit is represented with 4 bit binary format. 
 ( )
??
??
??
??
10
93 4
BCD
Eg: 943 1001 0100 0011  
• It is also known as 8421 code. 
• Invalid BCD codes are the codes whose decimal equivalent is more than 9. i.e. Valid 
BCD codes ranges from 0 to 9. 
 Total number of codes possible 
4
2 16 ??  
 Valid BCD codes 10 ? 
 Invalid BCD codes 16 10 6 -? 
 There 1010, 1011, 1100, 1110 and 1111 
 
Excess-3 codes: (BCD + 0011) 
• It can be derived from BCD by adding ‘3’ to each coded number.  
• It is unweighted and self-complementing code.  
 
Gray Code:  
It is also known as minimum change codes or unit distance code or reflected code. 
 
Binary code to Gray code: 
In order to find Gray code from Binary code, XOR Gate is applied between the present 
binary bit and the next binary bit starting from the MSB side(keeping MSB of Binary and 
Gray as same) e.g. 
 
Fig. 2 
 
Gray code to Binary code: 
In order to find Binary code from Gray code, XOR Gate is applied between the present 
binary bit and the next gray bit starting from the MSB side (keeping MSB of Gray and 
Binary as same) e.g. 
 
 
Fig. 3 
Alpha Numeric code:  
EBCDIC (Extended BCD interchange code) 
It is an 8-bit code. It can represent 128 possible characters.  
• Parity method is most widely used schemes for error detection.  
• Hamming code is most useful error correcting code. 
• BCD code is used in calculators, counters. 
Complements:  
Its base is r then we can have two complements. 
(i) (r – 1)’s complement 
(ii) r’s complement 
To determine(r – 1)’s complement: First write maximum possible number in the given 
system and subtract the given number. 
To determine r’s complements: (r – 1)’s complement + 1 
i.e. First write (r – 1)’s complement and then add 1 to LSB 
Example: 
Q.  Find 7’s and 8’s complement of 2456 
Sol. 7’s Complement 
7777
2456
5321
- 
 8’s Complement 
5321
1
5322
+ 
Q.   Find 2’s complement of 101.110 
Sol. 1’s complements 010.001 
 For 2’s complement add 1 to the LSB 
 2’s complement 
010.001
1
010.010
+ 
 
 
Data Representation: 
 
   
Fig. 4 
Unsigned Magnitude:  
 Range with n bit 
 
n1
0 to 2 5 101
-
? + ? 
 5 Not possible -? 
 
Signed Magnitude:  
 Range with n bit 
 
sign bit sign bit
with 4 bits with 8 bits
6 1 110 1 0000110 -? 
 
1’s complements:  
 Range with n bit 
 
( ) ( )
n 1 n 1
2 1 to 2 1
--
? - - + - 
 + ? - ? 6 0110 6 1001 
 
2’s complements:  
 With n bit range –2
n-1
 to (2
n-1
 –1) 
 + ? - ? 6 0110 6 1010 
 In any representation +ve numbers are represented similar to +ve number in sign 
magnitude. 
Logic Gates