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Lec 02 data representation part 2


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Lec 02 data representation part 2

  1. 1. Data Representation COAL Computer Organization and Assembly Language
  2. 2. Outline <ul><li>Floating Point Numbers Representation </li></ul><ul><li>Carry and Overflow </li></ul><ul><li>Character Storage </li></ul>
  3. 3. Floating-Point Representation <ul><li>The signed magnitude, one’s complement, and two’s complement representation that we have just presented deal with integer values only. </li></ul><ul><li>Without modification, these formats are not useful in scientific or business applications that deal with real number values. </li></ul><ul><li>Floating-point representation solves this problem. </li></ul>
  4. 4. Floating-Point Representation <ul><li>If we are clever programmers, we can perform floating-point calculations using any integer format. </li></ul><ul><li>This is called floating-point emulation , because floating point values aren’t stored as such, we just create programs that make it seem as if floating-point values are being used. </li></ul><ul><li>Most of today’s computers are equipped with specialized hardware that performs floating-point arithmetic with no special programming required. </li></ul>
  5. 5. Floating-Point Representation <ul><li>Floating-point numbers allow an arbitrary number of decimal places to the right of the decimal point. </li></ul><ul><ul><li>For example: 0.5  0.25 = 0.125 </li></ul></ul><ul><li>They are often expressed in scientific notation. </li></ul><ul><ul><li>For example: </li></ul></ul><ul><ul><ul><li>0.125 = 1.25  10 -1 </li></ul></ul></ul><ul><ul><ul><li>5,000,000 = 5.0  10 6 </li></ul></ul></ul>
  6. 6. Floating-Point Representation <ul><li>Computers use a form of scientific notation for floating-point representation </li></ul><ul><li>Numbers written in scientific notation have three components: </li></ul>
  7. 7. <ul><li>Computer representation of a floating-point number consists of three fixed-size fields: </li></ul><ul><li>This is the standard arrangement of these fields. </li></ul>Floating-Point Representation
  8. 8. <ul><li>The one-bit sign field is the sign of the stored value. </li></ul><ul><li>The size of the exponent field, determines the range of values that can be represented. </li></ul><ul><li>The size of the significand determines the precision of the representation. </li></ul>Floating-Point Representation
  9. 9. <ul><li>The IEEE-754 single precision floating point standard uses an 8-bit exponent and a 23-bit significand. </li></ul><ul><li>The IEEE-754 double precision standard uses an 11-bit exponent and a 52-bit significand. </li></ul>Floating-Point Representation For illustrative purposes, we will use a 14-bit model with a 5-bit exponent and an 8-bit significand.
  10. 10. <ul><li>The significand of a floating-point number is always preceded by an implied binary point. </li></ul><ul><li>Thus, the significand always contains a fractional binary value. </li></ul><ul><li>The exponent indicates the power of 2 to which the significand is raised. </li></ul>Floating-Point Representation
  11. 11. <ul><li>Example: </li></ul><ul><ul><li>Express 32 10 in the simplified 14-bit floating-point model. </li></ul></ul><ul><li>We know that 32 is 2 5 . So in (binary) scientific notation 32 = 1.0 x 2 5 = 0.1 x 2 6 . </li></ul><ul><li>Using this information, we put 110 (= 6 10 ) in the exponent field and 1 in the significand as shown. </li></ul>Floating-Point Representation
  12. 12. <ul><li>The illustrations shown at the right are all equivalent representations for 32 using our simplified model. </li></ul><ul><li>Not only do these synonymous representations waste space, but they can also cause confusion. </li></ul>2.5 Floating-Point Representation
  13. 13. <ul><li>Another problem with our system is that we have made no allowances for negative exponents. We have no way to express 0.5 (=2 -1 )! (Notice that there is no sign in the exponent field!) </li></ul>Floating-Point Representation All of these problems can be fixed with no changes to our basic model.
  14. 14. <ul><li>To resolve the problem of synonymous forms, we will establish a rule that the first digit of the significand must be 1. This results in a unique pattern for each floating-point number. </li></ul><ul><ul><li>In the IEEE-754 standard, this 1 is implied meaning that a 1 is assumed after the binary point. </li></ul></ul><ul><ul><li>By using an implied 1, we increase the precision of the representation by a power of two. (Why?) </li></ul></ul>2.5 Floating-Point Representation In our simple instructional model, we will use no implied bits.
  15. 15. <ul><li>To provide for negative exponents, we will use a biased exponent . </li></ul><ul><li>A bias is a number that is approximately midway in the range of values expressible by the exponent. We subtract the bias from the value in the exponent to determine its true value. </li></ul><ul><ul><li>In our case, we have a 5-bit exponent. We will use 16 for our bias. This is called excess-16 representation. </li></ul></ul><ul><li>In our model, exponent values less than 16 are negative, representing fractional numbers. </li></ul>2.5 Floating-Point Representation
  16. 16. <ul><li>Example: </li></ul><ul><ul><li>Express 32 10 in the revised 14-bit floating-point model. </li></ul></ul><ul><li>We know that 32 = 1.0 x 2 5 = 0.1 x 2 6 . </li></ul><ul><li>To use our excess 16 biased exponent, we add 16 to 6, giving 22 10 (=10110 2 ). </li></ul><ul><li>Graphically: </li></ul>Floating-Point Representation
  17. 17. <ul><li>Example: </li></ul><ul><ul><li>Express 0.0625 10 in the revised 14-bit floating-point model. </li></ul></ul><ul><li>We know that 0.0625 is 2 -4 . So in (binary) scientific notation 0.0625 = 1.0 x 2 -4 = 0.1 x 2 -3 . </li></ul><ul><li>To use our excess 16 biased exponent, we add 16 to -3, giving 13 10 (=01101 2 ). </li></ul>Floating-Point Representation
  18. 18. <ul><li>Example: </li></ul><ul><ul><li>Express -26.625 10 in the revised 14-bit floating-point model. </li></ul></ul><ul><li>We find 26.625 10 = 11010.101 2 . Normalizing, we have: 26.625 10 = 0.11010101 x 2 5 . </li></ul><ul><li>To use our excess 16 biased exponent, we add 16 to 5, giving 21 10 (=10101 2 ). We also need a 1 in the sign bit. </li></ul>Floating-Point Representation
  19. 19. <ul><li>The IEEE-754 single precision floating point standard uses bias of 127 over its 8-bit exponent. </li></ul><ul><ul><li>An exponent of 255 indicates a special value. </li></ul></ul><ul><ul><ul><li>If the significand is zero, the value is  infinity. </li></ul></ul></ul><ul><ul><ul><li>If the significand is nonzero, the value is NaN, “not a number,” often used to flag an error condition. </li></ul></ul></ul><ul><li>The double precision standard has a bias of 1023 over its 11-bit exponent. </li></ul><ul><ul><li>The “special” exponent value for a double precision number is 2047, instead of the 255 used by the single precision standard. </li></ul></ul>Floating-Point Representation
  20. 20. <ul><li>Both the 14-bit model that we have presented and the IEEE-754 floating point standard allow two representations for zero. </li></ul><ul><ul><li>Zero is indicated by all zeros in the exponent and the significand, but the sign bit can be either 0 or 1. </li></ul></ul><ul><li>This is why programmers should avoid testing a floating-point value for equality to zero. </li></ul><ul><ul><li>Negative zero does not equal positive zero. </li></ul></ul>Floating-Point Representation
  21. 21. <ul><li>Floating-point addition and subtraction are done using methods analogous to how we perform calculations using pencil and paper. </li></ul><ul><li>The first thing that we do is express both operands in the same exponential power, then add the numbers, preserving the exponent in the sum. </li></ul><ul><li>If the exponent requires adjustment, we do so at the end of the calculation. </li></ul>Floating-Point Representation
  22. 22. <ul><li>Example: </li></ul><ul><ul><li>Find the sum of 12 10 and 1.25 10 using the 14-bit floating-point model. </li></ul></ul><ul><li>We find 12 10 = 0.1100 x 2 4 . And 1.25 10 = 0.101 x 2 1 = 0.000101 x 2 4 . </li></ul>Floating-Point Representation <ul><li>Thus, our sum is 0.110101 x 2 4. </li></ul>
  23. 23. <ul><li>Floating-point multiplication is also carried out in a manner akin to how we perform multiplication using pencil and paper. </li></ul><ul><li>We multiply the two operands and add their exponents. </li></ul><ul><li>If the exponent requires adjustment, we do so at the end of the calculation. </li></ul>Floating-Point Representation
  24. 24. <ul><li>Example: </li></ul><ul><ul><li>Find the product of 12 10 and 1.25 10 using the 14-bit floating-point model. </li></ul></ul><ul><li>We find 12 10 = 0.1100 x 2 4 . And 1.25 10 = 0.101 x 2 1 . </li></ul>Floating-Point Representation <ul><li>Thus, our product is 0.0111100 x 2 5 = 0.1111 x 2 4. </li></ul><ul><li>The normalized product requires an exponent of 20 10 = 10110 2 . </li></ul>
  25. 25. <ul><li>No matter how many bits we use in a floating-point representation, our model must be finite. </li></ul><ul><li>The real number system is, of course, infinite, so our models can give nothing more than an approximation of a real value. </li></ul><ul><li>At some point, every model breaks down, introducing errors into our calculations. </li></ul><ul><li>By using a greater number of bits in our model, we can reduce these errors, but we can never totally eliminate them. </li></ul>Floating-Point Representation
  26. 26. <ul><li>Our job becomes one of reducing error, or at least being aware of the possible magnitude of error in our calculations. </li></ul><ul><li>We must also be aware that errors can compound through repetitive arithmetic operations. </li></ul><ul><li>For example, our 14-bit model cannot exactly represent the decimal value 128.5. In binary, it is 9 bits wide: </li></ul><ul><ul><li>10000000.1 2 = 128.5 10 </li></ul></ul>Floating-Point Representation
  27. 27. Character Storage <ul><li>Character sets </li></ul><ul><ul><li>Standard ASCII: 7-bit character codes (0 – 127) </li></ul></ul><ul><ul><li>Extended ASCII: 8-bit character codes (0 – 255) </li></ul></ul><ul><ul><li>Unicode: 16-bit character codes (0 – 65,535) </li></ul></ul><ul><ul><li>Unicode standard represents a universal character set </li></ul></ul><ul><ul><ul><li>Defines codes for characters used in all major languages </li></ul></ul></ul><ul><ul><ul><li>Used in Windows-XP: each character is encoded as 16 bits </li></ul></ul></ul><ul><ul><li>UTF-8: variable-length encoding used in HTML </li></ul></ul><ul><ul><ul><li>Encodes all Unicode characters </li></ul></ul></ul><ul><ul><ul><li>Uses 1 byte for ASCII, but multiple bytes for other characters </li></ul></ul></ul><ul><li>Null-terminated String </li></ul><ul><ul><li>Array of characters followed by a NULL character </li></ul></ul>
  28. 28. ASCII Codes <ul><li>Examples: </li></ul><ul><ul><li>ASCII code for space character = 20 (hex) = 32 (decimal) </li></ul></ul><ul><ul><li>ASCII code for ‘A' = 41 (hex) = 65 (decimal) </li></ul></ul><ul><ul><li>ASCII code for 'a' = 61 (hex) = 97 (decimal) </li></ul></ul>
  29. 29. Control Characters <ul><li>The first 32 characters of ASCII table are used for control </li></ul><ul><li>Control character codes = 00 to 1F (hex) </li></ul><ul><li>Examples of Control Characters </li></ul><ul><ul><li>Character 0 is the NULL character  used to terminate a string </li></ul></ul><ul><ul><li>Character 9 is the Horizontal Tab (HT) character </li></ul></ul><ul><ul><li>Character 0A (hex) = 10 (decimal) is the Line Feed (LF) </li></ul></ul><ul><ul><li>Character 0D (hex) = 13 (decimal) is the Carriage Return (CR) </li></ul></ul><ul><ul><li>The LF and CR characters are used together </li></ul></ul><ul><ul><ul><li>They advance the cursor to the beginning of next line </li></ul></ul></ul><ul><li>One control character appears at end of ASCII table </li></ul><ul><ul><li>Character 7F (hex) is the Delete (DEL) character </li></ul></ul>
  30. 30. ASCII Groups <ul><li>4 groups of 32 characters </li></ul><ul><li>G1  Codes 0-1F (31) [Non printing Control characters] </li></ul><ul><li>G2  Codes 30-39H [0---9] + punctuation symbols etc </li></ul><ul><ul><li>Numeric digits differs in H.O Nibble from their numeric values </li></ul></ul><ul><ul><li>Subtract 30 to get numeric value </li></ul></ul><ul><li>G3  Codes 41-5A (65-90) [A---Z + 6 codes fro special symbols] </li></ul><ul><li>G4  Codes 61-7A [a—z] +special symbols </li></ul><ul><ul><li>Upper and Lower differs in bit number 5 </li></ul></ul>
  31. 31. ASCII Groups <ul><li>Bits 5 and 6 determines the groups as shown in table below </li></ul>
  32. 32. Parity Bit <ul><li>Data errors can occur during data transmission or storage/retrieval. </li></ul><ul><li>The 8th bit in the ASCII code is used for error checking. </li></ul><ul><li>This bit is usually referred to as the parity bit . </li></ul><ul><li>There are two ways for error checking: </li></ul><ul><ul><li>Even Parity : Where the 8th bit is set such that the total number of 1s in the 8-bit code word is even. </li></ul></ul><ul><ul><li>Odd Parity : The 8th bit is set such that the total number of 1s in the 8-bit code word is odd. </li></ul></ul>
  33. 33. Boolean Operations <ul><li>NOT </li></ul><ul><li>AND </li></ul><ul><li>OR </li></ul><ul><li>Operator Precedence </li></ul><ul><li>Truth Tables </li></ul>
  34. 34. Boolean Algebra <ul><li>Based on symbolic logic , designed by George Boole </li></ul><ul><li>Boolean expressions created from: </li></ul><ul><ul><li>NOT, AND, OR </li></ul></ul>
  35. 35. NOT <ul><li>Inverts (reverses) a boolean value </li></ul><ul><li>Truth table for Boolean NOT operator: </li></ul>Digital gate diagram for NOT:
  36. 36. AND <ul><li>Truth table for Boolean AND operator: </li></ul><ul><li>Forcing ZERO/FALSE result (AND with ZERO) </li></ul><ul><li>If NO CHANGE required, AND with ONE </li></ul>Digital gate diagram for AND:
  37. 37. OR <ul><li>Truth table for Boolean OR operator: </li></ul><ul><li>Forcing ONE (OR with ONE) </li></ul><ul><li>NO CHANGE (OR with ZERO) </li></ul>Digital gate diagram for OR:
  38. 38. XOR <ul><li>If both operands EQUAL, the result is ZERO else ONE </li></ul><ul><li>Closer to English meaning or the word ‘OR’ </li></ul><ul><li>INVERSING OPERAND (XOR with ONE) </li></ul><ul><li>NO CHANGE (XOR with ZERO) </li></ul>
  39. 39. Operator Precedence <ul><li>Examples showing the order of operations: </li></ul>
  40. 40. Truth Tables (1 of 3) <ul><li>A Boolean function has one or more Boolean inputs, and returns a single Boolean output. </li></ul><ul><li>A truth table shows all the inputs and outputs of a Boolean function </li></ul>Example:  X  Y
  41. 41. Truth Tables (2 of 3) <ul><li>Example: X   Y </li></ul>