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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 GatesRead More
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