CSc 28 Data representation. CSc 28 Fall

Transcription

1 CSc 28 Data representation 1

2 Binary numbers Binary number is simply a number comprised of only 0's and 1's. Computers use binary numbers because it's easy for them to communicate using electrical current -- 0 is off, 1 is on. You can express any base 10 (our number system -- where each digit is between 0 and 9) with binary numbers. The need to translate decimal number to binary and binary to decimal. There are many ways in representing decimal number in binary numbers. 2

3 How does the binary system work? It is just like any other system In decimal system the base is 10 We have 10 possible values 0-9 In binary system the base is 2 We have only two possible values 0 or 1. The same as in any base, 0 digit has non contribution, where 1 s has contribution which depends at there position. 3

4 Example = = 3* * * = =1*2 4 +1*2 2 +1*2 0 4

5 Examples: decimal -- binary Find the binary representation of Find the decimal value represented by the following binary representations:

6 Examples: Representing fractional numbers Find the binary representations of: = = 0.11 Using only 8 binary digits find the binary representations of:

7 Number representation Representing whole numbers Representing fractional numbers 7

8 Integer Representations Unsigned notation Signed magnitude notion Excess notation Two s complement notation 8

9 Unsigned Representation Represents positive integers. Unsigned representation of 157 position Bit pattern contribution Addition is simple: =

10 Advantages: Advantages and disadvantages of unsigned notation One representation of zero Simple addition Disadvantages: Negative numbers can not be represented. The need of different notation to represent negative numbers 10

11 Representation of negative numbers Is a representation of negative numbers possible? Unfortunately: you can not just stick a negative sign in front of a binary number. (it does not work like that) There are three methods used to represent negative numbers. Signed magnitude notation Excess notation Two s complement notation 11

12 Signed Magnitude Representation Unsigned: - and + are the same. In signed magnitude the left-most bit represents the sign of the integer. 0 for positive numbers. 1 for negative numbers. The remaining bits represent to magnitude of the numbers. 12

13 Example Suppose is a signed magnitude representation. The sign bit is 1, then the number represented is negative position Bit pattern contribution The magnitude is with a value = 29 Then the number represented by is 29 13

14 Exercise has in signed magnitude notation. Find the signed magnitude of 37 10? 2. Using the signed magnitude notation find the 8-bit binary representation of the decimal value and Find the signed magnitude of 63 using 8-bit binary sequence? 14

15 Disadvantage of Signed Magnitude Addition and subtractions are difficult. Signs and magnitude, both have to carry out the required operation. They are two representations of = = To test if a number is 0 or not, the CPU will need to see whether it is or

16 Signed-Summary In signed magnitude notation, The most significant bit is used to represent the sign. 1 represents negative numbers 0 represents positive numbers. The unsigned value of the remaining bits represent The magnitude. Advantages: Represents positive and negative numbers Disadvantages: two representations of zero, Arithmetic operations are difficult. 16

17 Excess Notation In excess notation: The value represented is the unsigned value with a fixed value subtracted from it. For n - bit binary sequences the value subtracted fixed value is 2 (n-1) Most significant bit: 0 for negative numbers 1 for positive numbers 17

18 Excess Notation with n bits represent 2 n-1 is the decimal value in unsigned notation. Decimal value In unsigned notation - 2 n-1 = Decimal value In excess notation Therefore, in excess notation: will represent 0. 18

19 Example (1) - excess to decimal Find the decimal number represented by in excess notation. Unsigned value = = = Excess value: excess value = = =

20 Example (2) - decimal to excess Represent the decimal value 24 in 8-bit excess notation. We first add, 2 8-1, the fixed value = = 152 Then, find the unsigned value of = (unsigned notation) = (excess notation) 20

21 Example (3) Represent the decimal value -24 in 8-bit excess notation. We first add, 2 8-1, the fixed value = = 104 then, find the unsigned value of = (unsigned notation) = (excess notation) 21

22 Example (4) Unsigned = = The value represented in unsigned notation is 21 Sign Magnitude The sign bit is 1, so the sign is negative The magnitude is the unsigned value = 5 10 So the value represented in signed magnitude is Excess notation As an unsigned binary integer = subtracting = 2 4 = 16, we get = 5 10 So the value represented in excess notation is

23 Advantages of Excess Notation It can represent positive and negative integers. There is only one representation for 0. It is easy to compare two numbers. When comparing the bits can be treated as unsigned integers. Excess notation is not normally used to represent integers. It is mainly used in floating point representation for representing fractions (later floating point rep.) 23

24 Your turn 1. Find is an 8-bit binary sequence. Find the decimal value it represents if it was in unsigned and signed magnitude Suppose this representation is excess notation, find the decimal value it represents? 2. Using 8-bit binary sequence notation, find the unsigned, signed magnitude and excess notation of the decimal value 11 10? 24

25 Excess notation - Summary In excess notation, the value represented is the unsigned value with a fixed value subtracted from it For n-bit binary sequences the value subtracted is 2 (n-1) Most significant bit: 0 for negative numbers 1 positive numbers Advantages: Only one representation of zero Easy for comparison 25

26 Two s Complement Notation The most used representation for integers. All positive numbers begin with 0. All negative numbers begin with 1. One representation of zero 0 is represented as 0000 using 4-bit binary sequence 26

27 Two s Complement Notation with 4-bits Binary pattern Value in 2 s complement

28 Properties of Two s Complement Notation Positive numbers begin with 0 Negative numbers begin with 1 Only one representation of 0, i.e Relationship between +n and n

29 Advantages of Two s Complement Notation It is easy to add two numbers Subtraction can be easily performed. Multiplication is just a repeated addition. Division is just a repeated subtraction Two s complement is widely used in ALU (Arithmetic Logic Unit) 29

30 Evaluating numbers in two s complement notation Sign bit = 0, the number is positive. The value is determined in the usual way Sign bit = 1, the number is negative Three methods can be used: Method 1 decimal value of (n-1) bits, then subtract 2 n-1 Method 2 Method 3-2 n-1 is the contribution of the sign bit. Binary rep. of the corresponding positive number. Let V be its decimal value. - V is the required value. 30

31 Example in Two s Complement The most significant bit is 1, indicating a negative number Method = +5 ( = = 5-16 = -11) Method = -11 Method 3 Corresponding + number is = = 11 the result is then

32 Two s complement-summary In two s complement the most significant for an n-bit number has a contribution of 2 (n-1) One representation of zero All arithmetic operations can be performed by using addition and inversion. The most significant bit: 0 for positive and 1 for negative. Three methods can the decimal value of a negative number: Method 1 decimal value of (n-1) bits, then subtract 2 n-1 Method 2 Method 3-2 n-1 is the contribution of the sign bit. Binary rep. of the corresponding positive number. Let V be its decimal value. - V is the required value. 32

33 Exercise Determine the decimal value represented by in each of the following four systems. 1. Unsigned notation? 2. Signed magnitude notation? 3. Excess notation? 4. Two s complements? 33

34 Fraction Representation To represent fraction we need other representations: Fixed point representation Floating point representation 34

35 Fixed-Point Representation old position New position Bit pattern Contribution = Radix-point 35

36 Limitation of Fixed-Point Representation To represent large numbers or very small numbers we need a very long sequences of bits. This is because we have to give bits to both the integer part and the fraction part. 36

37 Floating Point Representation In decimal notation we can get around this problem using scientific notation or floating point notation Number Scientific notation Floating-point notation 1,245,000,000,

38 Floating Point could be represented as Sign Mantissa Base * * * A calculator might display 159 E14 Exponent 38

39 Floating point format M B E Sign mantissa or base exponent significand Sign Exponent Mantissa 39

40 Floating Point Representation format Sign Exponent Mantissa The exponent is biased by a fixed value b, called the bias The mantissa should be normalised, e.g. if the real mantissa is of the form 1.f then the normalised mantissa should be f, where f is a binary sequence 40

41 IEEE 745 Single Precision The number will occupy 32 bits The first bit represents the sign of the number; 1= negative 0= positive The next 8 bits will specify the exponent stored in biased 127 form The remaining 23 bits will carry the mantissa normalised to be between 1 and 2. i.e. 1<= mantissa < 2 41

42 Representation in IEEE 754 single precision Sign bit: 0 for positive and, 1 for negative numbers 8 biased exponent by bit normalised mantissa Sign Exponent Mantissa 42

43 Basic Conversion Converting a decimal number to a floating point number Steps: 1. Take the integer part of the number and generate the binary equivalent. 2. Take the fractional part and generate a binary fraction 3. Then place the two parts together and normalise 43

44 IEEE Example 1 Convert 6.75 to 32 bit IEEE format. 1. The Mantissa. The Integer first. 6 / 2 = 3 r 0 = / 2 = 1 r 1 1 / 2 = 0 r 1 2. Fraction next..75 * 2 = * 2 = 1.0 = put the two parts together Now normalise *

45 IEEE Example 1 Convert 6.75 to 32 bit IEEE format. 1. The Mantissa. The Integer first. 6 / 2 = 3 r 0 = / 2 = 1 r 1 1 / 2 = 0 r 1 2. Fraction next..75 * 2 = * 2 = 1.0 = put the two parts together Now normalise *

46 IEEE Example 1 Convert 6.75 to 32 bit IEEE format. 1. The Mantissa. The Integer first. 6 / 2 = 3 r 0 3 / 2 = 1 r 1 1 / 2 = 0 r 1 2. Fraction next..75 * 2 = * 2 = 1.0 = = Put the two parts together = Now normalise *

47 IEEE Biased 127 Exponent To generate a biased 127 exponent Take the value of the signed exponent and add 127 Example 2 16 then = value for the exponent would be 143 = So it is simply now an unsigned value 47

48 Possible Representations of an Exponent Binary Sign Magnitude 2's Complement Biased 127 Exponent {reserved} {reserved} 48

49 Why Biased? The smallest exponent Only one exponent zero The highest exponent is To increase the exponent by one simply add 1 to the present pattern. 49

50 Back to the example Our original example revisited * 2 2 Exponent is =129 or in binary NOTE: Mantissa always ends up with a value of 1 before the Dot. This is a waste of storage therefore it is implied but not actually stored is stored as in 32 bit floating point IEEE representation: sign(1) exponent(8) mantissa(23) 50

51 Representation in IEEE 754 single precision Sign bit: 0 for positive and, 1 for negative numbers 8 biased exponent by bit normalised mantissa Sign Exponent Mantissa 51

52 Example (2) Which number does the following IEEE single precision notation represent? The sign bit is 1, hence it is a negative number. The exponent is = It is biased by 127, hence the real exponent is = 1. The mantissa: It is normalised, hence the true mantissa is 1.01 = Finally, the number represented is: x 2 1 =

53 Single Precision Format The exponent is formatted using excess notation, with an implied base of 2 Example: Exponent: Representation: = 8 The stored values 0 and 255 of the exponent are used to indicate special values, the exponential range is restricted to to The number 0.0 is defined by a mantissa of 0 together with the special exponential value 0 The standard allows also values +/- (represented as mantissa +/-0 and exponent 255 Allows various other special conditions 53

54 In comparison The smallest and largest possible 32-bit integers in two s complement are only and

55 Range of numbers Normalized (positive range; negative is symmetric) smallest largest (1+0) = ( ) ( ) Positive underflow Positive overflow

56 Representation in IEEE 754 double precision format It uses 64 bits 1 bit sign 11 bit biased exponent 52 bit mantissa Sign Exponent Mantissa 56

57 IEEE 754 double precision Biased = bit exponent with an excess of 1023 For example: If the exponent is -1 Add 1023 to it = 1022 Find the binary representation of 1022 which is The exponent field will now hold Represent -1 with an excess of

58 IEEE 754 Encoding Single Precision Double Precision Represented Object Exponent Fraction Exponent Fraction non-zero 0 non-zero 1~254 anything 1~2046 anything +/- denormalized number +/- floating-point numbers /- infinity 255 non-zero 2047 non-zero NaN (Not a Number) 58

59 Floating Point Representation format (summary) Sign Exponent Mantissa Sign bit 0 for positive numbers 1 for negative numbers The exponent is biased by a fixed value B, called the Bias The mantissa should be normalised, e.g. if the real mantissa is of the form 1.M then the normalised mantissa should be M where M is a binary sequence 59

60 Character representation- ASCII ASCII (American Standard Code for Information Interchange) It is the scheme used to represent characters Each character is represented using 7-bit binary code If 8-bits are used, the first bit is always set to 0 Character representation in ASCII 60

61 ASCII example Symbol decimal Binary : ; < = > ? A B C

62 Character strings How to represent character strings? A collection of adjacent words (bit-string units) can store a sequence of letters 'H' 'e' 'l' 'l' o' ' ' 'W' 'o' 'r' 'l' 'd' '\0' Notation: enclose strings in double quotes "Hello world" Representation convention: null character defines end of string Null is sometimes written as '\0' Its binary representation is the number 0 62

63 Layered View of Representation Text string Information Information Sequence of characters Character Data Information Data Information Bit string Data Data 63

64 Working With A Layered View of Representation Represent SI at the two layers shown on the previous slide Representation schemes: Top layer - Character string to character sequence: Write each letter separately, enclosed in quotes. End string with \0 Bottom layer - Character to bit-string: Represent a character using the binary equivalent according to the ASCII table provided 64

65 Solution SI S I \ The colors are intended to help you read it; computers don t care that all the bits run together 65

66 Your turn Use the ASCII table to write the ASCII code for the following: CIS110 6 = 2*3 (1 space before and after the =) 66

67 Unicode - representation ASCII code can represent only 128 = 2 7 characters. It only represents the English Alphabet plus some control characters. Unicode is designed to represent the worldwide interchange. It uses 16 bits and can represents 32,768 characters. For compatibility, the first 128 Unicode are the same as the one of the ASCII. 67

68 Color representation Colours can represented using a sequence of bits. 256 colors how many bits? Hint for calculating To figure out how many bits are needed to represent a range of values, figure out the smallest power of 2 that is equal to or bigger than the size of the range. That is, find x for 2 x => bit color how many possible colors can be represented? Hints 16 million possible colors (why 16 millions?) 68

69 24-bits -- the True colour 24-bit color is often referred to as the true color. Any real-life shade, detected by the naked eye, will be among the 16 million possible colors. 69

70 Example: 2-bit per pixel = (white) 4=2 2 choices (off, off)=white 01 (off, on)=light grey 0 1 = (light grey) 10 (on, off)=dark grey = (dark grey) 11 (on, on)=black 1 0 = (black)

71 Image representation An image can be divided into many tiny squares, called pixels Each pixel has a particular colour The quality of the picture depends on two factors: the density of pixels The length of the word representing colours The resolution of an image is the density of pixels The higher the resolution the more information the image contains 71

72 Bitmap Images Each individual pixel (pi(x)cture element) in a graphic stored as a binary number Pixel: A small area with associated coordinate location Example: each point below is represented by a 4-bit code corresponding to 1 of 16 shades of gray 72

73 Representing Sound Graphically X axis: time Y axis: pressure A: amplitude (volume) : wavelength (inverse of frequency = 1/ ) 73

74 Sampling Sampling is a method used to digitise sound waves. A sample is the measurement of the amplitude at a point in time. The quality of the sound depends on: The sampling rate, the faster the better The size of the word used to represent a sample. 74

75 Digitizing Sound Capture amplitude at these points Lose all variation between data points Zoomed Low Frequency Signal 75

76 Summary Integer representation Unsigned, Signed, Excess notation, and Two s complement. Fraction representation Floating point (IEEE 754 format ) Single and double precision Character representation Colour representation Sound representation 76

77 Exercise 1. Represent +0.8 in the following floating-point representation: 1-bit sign 4-bit exponent 6-bit normalised mantissa (significand). 2. Convert the value represented back to decimal. 3. Calculate the relative error of the representation. 77