EEM336 Microprocessors I

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EEM336 Microprocessors I

• Representation of Numbers

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Contents

• Representation of Numbers
• Unsigned, Sign magnitude, Ones complement, Twos
  complement, Biased
• Real Numbers
• Single Precision, Double Precision
• BCD Numbers
• ASCII Numbers
• Shift and Rotate

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Data Representation

• Computers operate on binary values (as a result
  of being built from transistors)
• There are different binary representations for
  integers possible qualifications
• Positive numbers only
• Positive and negative numbers
• Ease of human readability
• Speed of computer operations

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Integer Representation

• There are 5 commonly known integer
  representations
• Unsigned
• Sign magnitude
• Ones complement
• Twos complement
• Biased (not commonly known)

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Integer Representation (continued)

• All have been used at various times for various
  reasons.
• Virtually all modern computers operate based on
  twos complement representation. Because,
• Hardware is faster
• Hardware is simpler (which makes it faster)

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Unsigned Numbers

• The standard binary encoding that you already
  know
• Only positive integer numbers are represented
• Range 0 to 2n - 1 for n bits
• Examples
• 5 is represented as 0000 0101
• 134 is represented as 1000 0110

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Sign Magnitude

• Both positive and negative integers can be
  represented
• The most significant bit (MSB) is used to
  represent the sign
• For positive numbers, MSB is 0 and for negative
  numbers, MSB is 1
• Examples
• 5 is represented in 8-bit as 0000 0101
• -5 is represented in 8-bit as 1000 0101

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Ones Complement

• Positive integers use the same representation as
  unsigned
• Negation is done by taking a bitwise complement
  of the positive representation
• Examples
• 5 is represented in 8-bit as 0000 0101
• -5 is represented in 8-bit as 1111 1010

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Twos Complement

• Positive integers use the same representation as
  unsigned
• Negation is done by taking the ones complements
  and adding 1
• Examples
• 5 is represented in 8-bit as 0000 0101
• -5 is represented in 8-bit as 1111 1011

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Biased Representation

• A bias value is chosen as either 2n-1 or 2n-1-1,
  according to the representation algorithm
• The integer number to be represented is added to
  the bias
• Unsigned representation of the result is the
  biased representation of the number
• Examples
• In 8-bit, if bias is 127, then 5 is 1000 0100
  (132)
• In 8-bit, if bias is 127, then -5 is 0111 1010
  (122)

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Sign Extension in Twos Complement

• Assume that you have a signed number in 8 bits
  and you want to convert it to 16 bits
• This type of conversion from smaller number of
  bits to larger number of bits is called sign
  extension
• In this case, the original number is copied into
  LSBs and the sign bit is copied into the MSBs
• Examples
• 5 0000 0101 in 8 bits ? 0000 0000 0000 0101 in
  16 bits
• -5 1111 1010 in 8 bits ? 1111 1111 1111 1010 in
  16 bits

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Character Representation

• ASCII codes are used to represent the characters
• Characters are generally represented in 8-bits
• Examples
• A is represented as 0100 0001 (65 or 41h)
• B is represented as 0100 0010 (66 or 42h)
• 0 is represented as 0011 0000 (48 or 30h)
• 9 is represented as 0011 1001 (57 or 39h)
• In Assembly, characters should be written into
  single-quotes (i.e. dont use double-quotes)

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Converting from String to Integer

• The 0 character and the number 0 are different
• If you want to convert the string 354 to
  integer, you should follow this algorithm
• Read 3 and translate it to 3 (just subtract
  30h)
• Read 5 and translate it to 5
• integer 3 10 5 35
• Read 4 and translate it to 4
• integer 35 10 4 354

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Representation of Real Numbers

• Real numbers (or floating-point numbers) are
  often encountered in computer programs
• A real number should be converted to binary
  system and encoded into bits
• There are several representations but Intel
  microprocessors use IEEE 754, version 10.0
  standard
• Intel 8086 has no floating point instructions, so
  you have to code them by yourself

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Converting Real Numbers into Binary

• Separate the number into integer and fractional
  parts
• You can easily convert the integer part into
  binary
• You can convert the fractional part by
  multiplying it by 2 continuously and taking the
  integer parts of the results

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Example Convert 64.2 to Binary

• Separate it into 64 and 0.2
• Binary representation of 64 is 100 0000
• 0.2 can be gotten using the algorithm
• 0.2 2 0.4 ? 0 (MSB)
• 0.4 2 0.8 ? 0
• 0.8 2 1.6 ? 1
• 0.6 2 1.2 ? 1
• 0.2 2 0.4 ? 0 (and it repeats itself)
• ? 0.2 0.0011 0011 0011
• Then, 64.2 is 100 0000.0011 0011 0011

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Scientific Notation

• The real numbers should be normalized and written
  in scientific notation
• In scientific notation, a number has three
  components
• Sign
• Mantissa
• Exponent
• In normalized form, the integer part of the
  mantissa should be 1.

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Example Scientific Notation of 64.2

• 64.2 can be written in binary as
• 1.00 0000 0011 0011 0011 26
• Here, 1.00 0000 0011 0011 0011 is called the
  mantissa
• 6 is the exponent
• Sign is plus ()

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IEEE 754 Standard

• According to the IEEE 754 standard,
  floating-point numbers can be represented in
  either 32 bits or 64 bits
• 32 bit version is called single-precision and 64
  bit version is called double-precision

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Single Precision
31
30
23
22
0

• The first bit (bit 31) represents the sign (0 for
  positive and 1 for negative numbers)
• Next 8 bits (23-30) represent the biased exponent
  (bias 127)
• Next 23 bits (0-22) represent the 24 bits of the
  mantissa (24 bits in 23 bits!!! Is it correct?)
• Yes! The first bit of the mantissa, which comes
  from the integer part, is not needed to be coded

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Double Precision
63
62
52
51
0

• The first bit (bit 63) represents the sign (0 for
  positive and 1 for negative numbers)
• Next 11 bits (52-62) represent the biased
  exponent (bias 1023)
• Next 52 bits (0-51) represent the 53 bits of the
  mantissa (53 bits in 52 bits!!! Is it correct?)
• Yes! The first bit of the mantissa, which comes
  from the integer part, is not needed to be coded

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64.2 in Single Precision

• Since 64.2 is positive, sign bit should be 0
• Exponent is 6 add 127 and obtain biased value (6
  127 133 1000 0101)
• Mantissa is 1.00 0000 0011 0011 0011
• Ignore 1 in integer part and take the 23 bits
  from the fractional part 000 0000 0110 0110 0110
  0110
• Combine them 0100 0010 1000 0000 0110 0110 0110
  0110 (or 42806666h)

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64.2 in Double Precision

• Since 64.2 is positive, sign bit should be 0
• Exponent is 6 add 1023 and obtain biased value
  (6 1023 1029 100 0000 0101)
• Mantissa is 1.00 0000 0011 0011 0011
• Ignore 1 in integer part and take the 52 bits
  from the fractional part 0000 0000 1100 1100
  1100 1100 1100 1100 1100 1100 1100 1100 1100
• Combine them 0100 0000 0101 0000 0000 1100 1100
  1100 1100 1100 1100 1100 1100 1100 1100 1100 (or
  40500CCCCCCCCCCCh)

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Exceptions

• Zero is stored as all zeros.
• The number infinity is stored as all ones in the
  exponent and all zeros in the mantissa. The sign
  bit indicates either a positive or a negative
  infinity.
• 8 0 1111 1111 000 0000 0000 0000 0000 0000
• -8 1 1111 1111 000 0000 0000 0000 0000 0000

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Exercises

• Calculate the single and double precision
  representations of 12.3 (ans 4144 CCCDh and 4028
  9999 9999 999Ah)
• Calculate the single and double precision
  representations of -23.4 (ans C1BB 3333h and
  C037 6666 6666 6666h)
• What does the single precision floating point
  number 3F80 0000h represent?

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BCD (Binary Coded Decimal) Data

• In some cases, you might consider representing
  each digits of an integer number in 4 bits
• For example, 1234h might represent the decimal
  value 1234
• These type of numbers are called BCD numbers
• There are some special instructions to be used
  with BCD numbers

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DAA

• DAA Decimal Adjust after Addition
• Corrects the result of addition of two packed BCD
  values on AL register
• Example
• MOV AL, 15 AL 0Fh
• DAA AL 15h

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DAA Example

• Sum two 4-digit BCD numbers stored in BX and DX
  and store the result in CX as a BCD.
• MOV DX, 1234h Load 1234 BCD
• MOV BX, 3099h Load 3099 BCD
• MOV AL, BL Sum low bytes
• ADD AL, DL AL 34h 99h CDh
• DAA CF 1, AL 33h (133)
• MOV CL, AL Store low byte of result
• MOV AL, BH AL 30h (sum high bytes)
• ADC AL, DH AL 43h
• DAA CF 0, AL 43h
• MOV CH, AL Store high byte of result

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DAS

• DAS Decimal Adjust after Subtraction
• Corrects the result of subtraction of two packed
  BCD values on AL register
• Similar to DAA but DAS follows a subtraction

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ASCII Coded Numbers

• You may use ASCII codes of numbers to represent
  them
• For example, you can represent 19 as 3139h since
  ASCII codes of 1 and 9 are 31h (49) and 39h
  (57) respectively
• There are some instructions to be used on ASCII
  coded numbers AAA, AAD, AAM, and AAS.

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AAA

• AAA ASCII Adjust after Addition
• Corrects result in AH and AL after addition of
  two ASCII coded numbers
• Works on AL and changes AL and AH
• 3030h should be added to AX after AAA
• Example
• MOV AX, '01' AX 3031h (ASCII code for 1)
• ADD AL, '9' AL 0Ah
• AAA AH 01h, AL 00h
• ADD AX, 3030h AX 3130 (ASCII code for 10)

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Other ASCII Code Operations

• AAD ASCII Adjust before Division
• AAM ASCII Adjust after Multiplication
• AAS ASCII Adjust after Subtraction

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Shift and Rotate

• Shift and rotate instructions manipulate binary
  numbers at the binary bit level
• Common applications in low-level software used to
  control I/O devices
• Shift/Rotate count may be an immediate value or
  CL
• Examples
• SHL AX, 1
• SHR BX, CL

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Shift

• Position or move numbers to the left or right
  within a register or memory location.
• also perform simple arithmetic as multiplication
  by powers of 2n (left shift) and division by
  powers of 2-n (right shift).
• The microprocessors instruction set contains
  four different shift instructions
• two are logical two are arithmetic shifts

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The Shift Instructions

• logical shifts move 0 in the rightmost bit for a
  logical left shift
• 0 to the leftmost bit position for a logical
  right shift
• arithmetic right shift copies the sign-bit
  through the number
• logical right shift copies a 0 through the number.

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Logical vs. Arithmetic?

• Logical shifts multiply or divide unsigned data
  arithmetic shifts multiply or divide signed data.
  
• a shift left always multiplies by 2 for each bit
  position shifted
• a shift right always divides by 2 for each
  position
• shifting a two places, multiplies or divides by 4

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Rotate
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References

• http//pages.cs.wisc.edu/smoler/x86text/TOC.lectn
  otes.html
• emu8086 Emulators Documentation
• Textbook