Computer Systems

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Computer Systems

• Data Representation andComputer Arithmetic

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Data Representation

• Binary codes used to represent numbers
• Conversion between number systems
• Binary ?? decimal
• Hexadecimal ?? decimal
• Between arbitrary number systems
• Representation of positive/negative numbers
• Floating point representation

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Bit, Byte and Word

• Bit is the smallest quantity that can be handled
  by computer 1 or 0.
• Byte is a group of 8 bits.
• Word is the basic unit of data that can be
  operated by computer e.g., 16 bits.

• Some architectures have 8, 32, or 64-bit words

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• Content of Word
• Bit Pattern
• May represent many things
• Actual meaning of a particular bit pattern is
  given by the programmer
• Computer itself cannot determine the meaning of
  the word
• Classic question Can you tell the meaning of a
  word picked randomly from the memory?

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• Instruction (op-code)
• A single word defines an action that is to be
  performed by CPU

• Numeric Quantity
• Character
• A to Z, a to z, 0 to 9, , -, , etc.
• ASCII Code
• - Representation of a character by 7 bits.
• - The ASCII for W is 101 01112, or 5716

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ASCII Code
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Pixel (Picture Element)

• The smallest unit to construct a picture
• 1bit/pixel for black-and-white pictures, gt1
  bits/pixel for gray scale or color pictures, e.g.
  24bits/pixel.
• A letter is represented by a group of pixels on
  computer screen.

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Positional Notation

• Weight is associated with the location within a
  number.
• The 9 in 95 has weight 10

• A number N in base b is represented by

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Example
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Number Bases

• Decimal (b10) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

• Binary (b2) 0, 1

• Octal (b8) 0, 1, 2, 3, 4, 5, 6, 7

• Hexadecimal (b16)
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

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How many binary bits are needed to represent an
n-digits decimal?

• The value of an m-bit binary is up to
• The value of an n-digit decimal is up to
• so

OR
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How many bits are needed to represent an n-digits
octal?How many bits are needed to represent
an n-digits hexadecimal?
Questions
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Conversion of Integer - Decimal to Binary

• Divide a decimal successively by 2, record the
  remainder, until the result of the division is 0.

67/2 33 Rem.1 33/2 16 Rem.1 16/2
8 Rem.0 8/2 4 Rem.0
4/2 2 Rem.0 2/2 1 Rem.0 1/2 0 Rem.1
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Conversion of Integer - Decimal to Hexadecimal

• Divide a decimal successively by 16, record the
  remainder, until the result is 0

432/16 27 Rem.0 27/16 1 Rem.B 1/16
0 Rem.1
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Conversion of integer - Binary to Decimal
Conversion of integer - Hexadecimal to Decimal
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Conversion of integer - Hexadecimal Binary
Conversion of integer - Octal Binary
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Conversion of Fraction - Decimal to Binary

• The fraction is repeatedly multiplied by 2, the
  integer part is stripped and recorded.

0.687510 2 1.375 1 0.37510 2
0.75 0 0.7510 2 1.5 1 0.510 2
1.0 1 0.010 2 0.0 end
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Conversion of Fraction - Binary Hexadecimal
0.1010110012 0.1010 1100 10002
0.AC816 0.7BC16 0.0111 1011 11002
0.01111011112
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Binary Coded Decimal (BCD)

• A decimal digit is coded into 4 bits ? 1 byte
  can store 2 digits
• Example
• 1942 is encoded as
• 0001 1001 0100 0010 (2 bytes)
• Disadvantages
• Complex arithmetic
• Inefficient use of storage
• Advantage
• Can represent real number

Decimal BCD 0 0000 1 0001 2
0010 3 0011 4 0100 5 0101 6
0110 7 0111 8 1000 9 1001
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Binary Addition
0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0
1 1 1 1 1 Carry 1 0 0 0
1 1 0 1
Binary Subtraction ?

• use 2s complementary arithmetic (later )

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Signed Integer Representations

• Sign-and-Magnitude
• 1s Complement
• 2s Complement

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Sign-and-Magnitude Representation

• Use the most significant bit (MSB) to indicate
  the sign of the number.
• The MSB is 0 for positive, 1 for negative.
• 8 bits represent -12710 to 12710
• Examples 00101100 for 4410
• 10101100 for -4410

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1s Complement

• Just flip the bits!
• The MSB is 0 for positive, 1 for negative.
• 8 bits represent -12710 to 12710
• Examples 00101100 for 4410
• 11010011 for -4410

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2s Complement

• The 2s complement for N is
• Example Using 5 bits (n 5), if N 7, then

0 1 1 0 0 1 1 0 0 1 1) 0 0 1 0 1
12 -7 5

• 2s complement 1s complement 1

7 0 0 1 1 1 1s complement 1 1 0 0 0
1 2s complement 1 1 0 0 1 -7
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2s Complement (con.)

• To form the twos complement of a number, simply
  invert the bits (1 to0, 0 to 1), and add 1 ? 1s
  complement 1
• 1310 0 0 0 0 1 1 0 1
• -1310 1 1 1 1 0 0 1 0 1
• 1 1 1 1 0 0 1 1

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Properties of 2s Complement

• One unique 0
• MSB 0, positive number
• MSB 1, negative number
• The range is
• For 5 bits, the range is -1610 (10000) to 1510
  (01111)
• The 2s complement of the complement of X is X
  itself (Prove )

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2s Complementary Arithmetic

• Subtraction is performed using addition
• A - B A (-B)
• X 910 010012, Y 610 001102
• -X -910 101112, -Y -610 110102

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Binary Adder/Subtractor
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Arithmetic Overflow

• The overflow happens when
• positive positive ? negative
• negative negative ? positive

• If are MSBs, then the overflow bit v is
  expressed as

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Arithmetic Overflow

• In practice,

• Case I A and B are
• an-1 __ bn-1 __
• cn __ cn-1 __
• V _____
• Case II A and B are
• an-1 __ bn-1 __
• cn __ cn-1 __
• V _____

Most significant stage of full adder
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Fixed-point Arithmetic
3.62510 ? 0011.10102 ? 00111010 6.510 ?
0110.10002 ? 01101000 10.12510
10100010 The fraction point is added 10100010
-gt 1010.00102
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Floating Point Numbers

• Scientific Notation
• a is mantissa, r is radix, e is exponent

• is for binary

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IEEE Floating Point Format
S Sign bit, 1 bit E Exponent, 8 bits B Bias, 8
bits, 127100111 1111 F mantissa
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IEEE Floating Point - Example

• Normalization, e.g.,
• Use sign and magnitude representation for signed
  mantissa.
• Use biased exponent, e.g. in excess 127

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IEEE Floating Point - Example
-2345.125 -100100101001.0012 1)
Normalization 2) Negative mantissa, so S 1 3)
Exponent E 11 127 13810 100010102 4)
Mantissa F 00100101001 0010000 1 10001010
001001010010010000
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IEEE Floating Point Format Special Cases

• Zero exponent and fraction all 0s
• Denormalized exponent all 0s,
• fraction non-zero
• for single precision
• Infinity exponents all 1s, fraction all 0s
• sign bit determines infinity or -infinity
• Not a Number (NaN) exponents all 1s,
• fraction non-zero

• A good reference is available at
• http//research.microsoft.com/hollasch/cgindex/
  coding/ieeefloat.html

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Bit Patterns and Logical Operations

• AND
• OR
• NOT
• Exclusive OR

They are bit-wise operations.
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AND Operation

• x AND y is true, if and only if both bits x and
  y are true.
• A 1 1 0 0 1 0 1 1
• B 0 1 1 0 1 1 0 1
• CA AND B 0 1 0 0 1 0 0 1

OR Operation

• x OR y is true, if either bits x or y is true.
• A 1 1 0 0 1 0 1 1
• B 0 1 1 0 1 1 0 1
• CA OR B 1 1 1 0 1 1 1 1

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NOT Operation

• A 1 becomes 0, and a 0 becomes 1.
• A 1 1 0 0 1 0 1 1
• C NOT A 0 0 1 1 0 1 0 0

EOR (Exclusive OR) Operation

• It is true, if and only if just one of inputs is
  true.
• A 1 1 0 0 1 0 1 1
• B 0 1 1 0 1 1 0 1
• C A EOR B 1 0 1 0 0 1 1 0