CHAPTER 2 (b) : AND CODES

Transcription

1 DKT 122 / 3 DIGITAL SYSTEMS 1 CHAPTER 2 (b) : NUMBER SYSTEMS OPERATION AND CODES m.rizal@unimap.edu.my sitizarina@unimap.edu.my

2 DECIMAL VALUE OF SIGNED NUMBERS SIGN-MAGNITUDE: Decimal values of +ve & -ve no are determined summing the weights in all the mag. bit positions where there are 1 s and ignoring i those positions where there are zeros. The sign is determined by examination Of the sign bit

3 DECIMAL VALUE OF SIGNED NUMBERS Sign Magnitude - Determine the decimal value of this signed binary number expressed in sign-magnitude summing the weight where there is 1s = 21 The sign bit is 1; dec no is -21

4 DECIMAL VALUE OF SIGNED NUMBERS 1 s COMPLEMENT: Decimal values of +ve no are determined by summing the weights in all bit positions where there are 1s and ignoring those positions where there are zeros. Decimal number of ve no are determined by assigning a ve value to the weight of sign bit, summing all the weights where there are 1s and adding 1 to the results

5 DECIMAL VALUE OF SIGNED NUMBERS 1 s Complement form (example: +ve value) - Determine the decimal value of this signed binary number expressed in 1 s compliment summing the weight where there is 1s g g = +23

6 DECIMAL VALUE OF SIGNED NUMBERS 1 s Complement form (example: -ve value) - Determine the decimal value of this signed binary number expressed in 1 s compliment summing the weight where there is 1s = - 24 Adding 1 to the result, the final decimal no is = -23

7 DECIMAL VALUE OF SIGNED NUMBERS 2 S COMPLEMENT: Decimal values of +ve and ve no are determined by summing the weights in all by summing the weight in all positions where there are 1s and ignoring those positions where there are zeros The weight of the sign bit in a ve no is given g g g a ve value

8 DECIMAL VALUE OF SIGNED NUMBERS 2 s Complement form (example: +ve value) - Determine the decimal value of this signed binary number expressed in 1 s compliment summing the weight where there is 1s = +86

9 DECIMAL VALUE OF SIGNED NUMBERS 2 s Complement form (example: -ve value) - Determine the decimal value of this signed binary number expressed in 1 s compliment summing the weight where there is 1s = -86

10 THIS WEEK: Arithmetic operation with signed numbers Hexadecimal Numbers Octal Numbers Binary Coded Decimal (BCD) Digital Codes

11 ARITHMETIC OPERATION WITH SIGNED NOs Addition: Conditions: 1. Both number +ve 2. +ve number with magnitude larger than ve no 3. -ve number with magnitude larger than +ve no 4. Both no ve

12 ARITHMETIC OPERATION WITH SIGNED NOs Substraction: Remember: The sign of a +ve or ve binary number is changed by taking its 2 s complement To substract two signed numbers, take the 2 s complement of the subtrahend and add. Discard any final carry bit

13 ARITHMETIC OPERATION WITH SIGNED NOs Multiplication: 2 methods: Direct addition lengthy 2. Partial product most common Partial Product: Step 1 determine if the sign of the multiplicand are the same or different. That s will determines the end result If th i th th d t - If the sign are the same, the product = +ve - If the sign are different, the product = -ve

14 Step 2 Change any number to uncomplemented form. Usually from 2 s complemented form to true number Step 3 Do a partial product multiplication. Use only the mag. bits. Ignore sign bit Step 4 Add each successive partial product to get the final product. Step 5 If sign bit is ve, take the 2 s complement of the product. Else just leave it as the final result. Don t forget to add the sign bit

15 ARITHMETIC OPERATION WITH SIGNED NOs Division: Step 1 determine if the sign of the divident and divisor are the same or different. That s will determines the sign of the quotient - If the sign are the same, the quotient t = +ve - If the sign are different, the quotient = -ve Step 2 Substract the divisor from the dividend using 2 s compliment addition to get the 1 s partial remainder and add 1 to quotient - If the partial remainder is +ve, go to step 3. If ve, division i i is completed.

16 Step 3 Substract the divisor from the partial remainder and add 1 to quotient. If result = +ve, repeat for the next partial remainder. If result = 0 or ve, division is complete.

17 Binary Octal Hex Dec A B C D D E F

18 Significant Digits Binary: Most significant digit Least significant digit Hexadecimal: 1D63A7A Most significant digit Least significant digit

19 Hexadecimal Number System Base 16 system Uses digits 0-9 & letters A,B,C,D,E,F Groups of four bits represent each base 16 digit

20 Hexadecimal to Decimal Conversion Convert 3B4F 16 to its decimal equivalent: Hex Digits Positional Values Products 3 B 4 F x x x x ,183 10

21 Decimal to Hexadecimal Conversion Convert to its hexadecimal equivalent: 830 / 16 = 51 R14 = E in Hex 51 / 16 = 3 R3 3 / 16 = 0 R3 33E 16

22 Number Conversion Binary to Hexadecimal Conversion (vice versa) 1. Grouping the binary position in 4-bit groups, starting from the least significant position.

23 Binary to Hexadecimal Conversion The easiest method for converting binary to hexadecimal is to use a substitution code Each hex number converts to 4 binary digits

24 Number Conversion Example: Convert the following binary numbers to their hexadecimal equivalent (vice versa). a) b) 1F.C 16 Answer: a) b)

25 Substitution Code Convert to hex using the 4-bit substitution code : 5 6 A E 6 A AE6A 16

26 Binary Octal Hex Dec N U M B B E R R S A Y S T B C D T E M E F M S

27 Octal Number System Also known as the Base 8 System Uses digits 0-7 Readily converts to binary Groups of three (binary) digits can be used to represent each octal digit Also uses multiplication and division algorithms for conversion to and from base 10

28 Octal to Decimal Conversion Convert to its decimal equivalent: Octal Digits Positional Values Products x x x

29 Decimal to Octal Conversion Convert to its octal equivalent: 427 / 8 = 53 R3 Divide by 8; R is LSD 53 / 8 = 6 R5 Divide Q by 8; R is next digit 6 / 8 = 0 R6 Repeat until Q =

30 Number Conversion Binary to Octal Conversion (vice versa) 1. Grouping the binary position in groups of three starting at the least significant position.

31 Octal to Binary Conversion Each octal number converts to 3 binary digits To convert to binary, just substitute code:

32 Number Conversion Example: Convert the following binary numbers to their octal equivalent (vice versa). a) b) c) Answer: a) b) c)

33 Substitution Code Substitution code can also be used to convert binary to octal by using 3-bit groupings:

34 Digital Codes BCD (Binary Coded Decimal) Code 1. Represent each of the 10 decimal digits (0~9) as a 4-bit binary code. Example: Convert 15 to BCD C BCD Convert 10 to binary and BCD.

35 Digital Codes ASCII (American Standard Code for Information Interchange) g) Code 1. Used to translate from the keyboard characters to computer language g

36 Digital Codes The Gray Code Decim Binary Gray Only 1 bit changes al Code Can t be used in arithmetic circuits Binary to Gray Code and vice versa