Computer Organization

1
Computer Organization

• It describes the function and design of the
  various units of digital computers that store and
  process information.
• It also deals with the units of computer that
  receive information from external sources and
  send computed results to external destinations

2
Computer Organization Computer Architecture

• Computer Organization and Computer
  Architecture are the terms used in describing
  digital computer systems.
• Computer architecture deals with the instruction
  sets, address fields, operation codes, addressing
  modes, effective utilization of I/O mechanisms
  and such other aspects which would interest the
  users of computers.
• Computer organization include hardware details
  such as generation of control signals, interface
  between the computer and peripherals, memory
  technologies used, modular design, flow of data,
  control mechanisms and so on.
• In brief, a programmer is generally more
  concerned about the architectural features and a
  designer of computer systems is more concerned
  about the organizational aspects.

3
Computer types

• Digital computer
• It is a fast electronic calculating machine that
  accepts digitized input information, processes it
  according to a list of internally stored
  instructions, and produces the resulting output
  information.
• (1) Personal computer
• It is the most common form of desktop computers.
• Desk top computers have processing and storage
  units, visual display and audio output units, and
  a keyboard that can all be located easily on a
  home or office desk. The storage media include
  hard disks, CD-ROMs and diskettes.
• Portable notebook computers are a compact version
  of the personal computers with all of these
  components packaged into single unit the size of
  a thin briefcase

4
Computer types (contd.,)

• (2) Workstations
• Work stations with high resolution graphics
  input/output capability, although still retaining
  the dimensions of desktop computers, have
  significantly more computational power than
  personal computers.
• Workstations are often used in engineering
  applications, especially for interactive design
  works.
• (3) Enterprise systems or mainframes
• These are used for business data processing in
  medium to large corporations that require much
  more computing power and storage capacity than
  workstations can provide.

5
Computer types (contd.,)

• 4) Servers
• Servers contain sizable database storage units
  and are capable of handling large volumes of
  requests to access the data.
• The Internet and its associated servers have
  become a dominant world wide source of all types
  of information.
• 5) Super Computers
• Super Computers are used for the large scale
  numerical calculations required in applications
  such as weather forecasting, aircraft design and
  simulation.

6
Functional units

• A computer consists of five functionally
  independent main parts
• Input unit
• Memory unit
• Arithmetic and logic unit (ALU)
• Output unit
• Control unit
• The structure of a digital computer is shown
  below

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8
Operation of a computer

• The computer accepts information in the form of
  programs and data through an input unit and
  stores it in the memory.
• Information stored in the memory is fetched,
  under program control, into an arithmetic and
  logic unit, where it is processed.
• Processed information leaves through an output.
• All activities inside the machine are directed by
  the control unit.

9
Input unit

• Computers accept coded information through input
  units.
• The most well known input device is key board.
• Whenever a key is pressed, the corresponding
  letter or digit is automatically translated into
  its corresponding binary code and transmitted
  over a cable to either the memory or the
  processor.

10
Memory unit

• The function of the memory is to store programs
  and data.
• There are two classes of storage, called primary
  and secondary.
• Primary storage is a fast memory that operates at
  electronic speeds.
• Programs must stay in memory while they
  are being executed.
• The memory contains a large number of
  semiconductor storage cells, each capable of
  storing one bit of information.
• Random access memory Memory in which any
  location can be reached in a short and fixed
  amount of time after specifying its address is
  called random access memory (RAM).

11
Memory unit (Contd.,)

• The time required to access one word is called
  the memory access time.
• Cache memory The small, fast, Ram units are
  called caches.
• They are tightly coupled with the
  processor and are often contained on the same
  integrated circuit chip to achieve high
  performance.
• Main memory The largest and slowest unit is
  referred to as the main memory.

12
Memory unit (Contd.,)

• Secondary storage
• Although primary storage is essential, it tends
  to be expensive.
• Thus additional, cheaper, secondary storage is
  used when large amounts of data and many programs
  have to be stored, particularly for information
  that is accessed infrequently.
• A wide selection of secondary storage devices is
  available, including magnetic disks,tapes and
  optical disks (CD-ROMs)

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Arithmetic and logic unit (ALU)

• Most computer operations are executed in in the
  arithmetic and logic unit (ALU) of the processor.
• For example, Suppose two numbers are to be added.
  
• They are brought into the processor, and the
  actual addition is carried out by the ALU.
• The sum may then be stored in the memory or
  retained in the processor for immediate use.
• When operands are brought into the processor,
  they are stored in high-speed storage elements
  called registers.

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Output unit

• The output unit is the counterpart of input unit.
• Its function is to send processed results to the
  outside world.
• The most familiar example of such a device is a
  printer.
• Some units, such as graphic displays, provide an
  input function and an output function. The dual
  role of such units is the reason for using the
  single name I/O unit in many cases.

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Control unit

• The memory, arithmetic and logic, and I/O units
  store and process information and perform input
  and output operations.
• The operations of these units are coordinated by
  control unit.
• The control unit is effectively the nerve center
  that sends control signals to other units and
  senses their states.

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Basic operational concepts

• To perform a given task, an appropriate program
  consisting of a list of instructions is stored in
  the memory.
• Individual instructions are brought from the
  memory into the processor, which execute the
  specified operations.
• Data to be used as operands are also stored
  in the memory

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Basic operational concepts (Contd.,)

• Consider, the two instruction sequence
• Load LOCA, R1
• Add R1, R0
• The first of these instructions transfers the
  contents of memory location LOCA into processor
  register R1, and second instruction adds the
  contents of register R1 as well as those of R0
  and places the sum into R0.
• Note that this destroys the former contents of R1
  as well as R0, where as the original contents of
  memory location LOCA are preserved.

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Connections between the processor and the
memory
Memory
Processor
19
Instruction register (IR)

• The instruction register holds the instruction
  that is currently being executed.
• Its output is available to the control circuits,
  which generate the timing signals that control
  the various processing elements involved in
  executing the instruction

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Program counter (PC)

• The program counter is another specialized
  register.
• It keeps track of the execution of a program.
• It contains the memory address of the next
  instruction to be fetched and executed.
• During the execution of an instruction, the
  contents of of the PC are updated to correspond
  to the address of the next instruction to be
  executed.
• It is customary to say that PC points to the next
  instruction that is to be fetched from the
  memory.

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Memory address register (MAR) Memory data
register (MDR)

• These two registers facilitate communication with
  the memory.
• The MAR holds the address of the location to be
  accessed.
• The MDR contains the data to be written into or
  read out of the addressed location.

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Operating steps for Program execution

• Programs are stored in the memory through the
  input unit.
• Execution of the program starts when the PC is
  set to point to the first instruction of the
  program.
• The contents of the PC are transferred to the MAR
  and a Read control signal is sent to the memory.
• After the time required to access the memory
  elapses, the addressed word (in this case, the
  first instruction of the program) is read out of
  the memory and loaded into the MDR.

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Operating steps for Program execution (Contd.,)

• Next, the contents of the MDR are transferred to
  the IR .
• At this point, the instruction is ready to be
  decoded and executed.
• If the instruction involves an operation to be
  performed by the ALU, it is necessary to obtain
  the required operands.
• If an operand resides in memory ( it could also
  be in a general-purpose register in the
  processor), it has to be fetched by sending its
  address to the MAR and initiating a Read cycle.

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Operating steps for Program execution (Contd.,)

• When the operand has been read from the memory
  into the MDR, it is is transferred from the MDR
  to the ALU.
• After one or more operands are fetched in this
  way, the ALU can perform the desired operation.
• If the result of the operation is to be stored in
  the memory, then the result is sent to the MDR.
• The address of the location where the result is
  to be stored is sent to the MAR, and a write
  cycle is initiated.

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Operating steps for Program execution (Contd.,)

• At some point during the execution of the current
  instruction, the contents of the PC are
  incremented so that the PC points to the next
  instruction to be executed.
• Thus, as soon as the execution of the current
  instruction is completed, a new instruction fetch
  may be started.
• In addition to transferring data between the
  memory and the processor, the computer accepts
  data from input devices and sends data to output
  devices. Thus, some machine instructions with the
  ability to handle I/O transfers are provided.

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Interrupt service routine

• Normal execution of a programs may be preempted
  if some device requires urgent servicing.
• For example, a monitoring device in a
  computer-controlled industrial process may detect
  a dangerous condition.
• In order to deal with the situation
  immediately, the normal execution of the current
  program must be interrupted.
• To do this, the device raises an interrupt
  signal.
• An interrupt is a request from an I/O device for
  service by the processor.
• The processor provides the requested
  service by executing an appropriate
  interrupt-service routine.

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Interrupt service routine (contd.,)

• Because such diversions may alter the internal
  state of the processor, its state must be saved
  in the memory locations before servicing the
  interrupt.
• Normally, the contents of the PC, the general
  registers, and some control information are
  stored in memory.
• When the interrupt service routine is completed,
  the state of the processor is restored so that
  the interrupted program may continue.

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Bus structures

• A group of lines(wires) that serves as a
  connecting path for several devices of a computer
  is called a bus.
• In addition to the lines that carry the data, the
  bus must have lines for address and control
  purposes.
• The simplest way to interconnect functional units
  is to use a single bus, as shown below

processor
input
output
memory
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Bus structures (contd.,)

• All units are connected to this bus. Because the
  bus can be used for only one transfer at a time,
  only two units can actively use the bus at any
  given time.
• Bus control lines are used to arbitrate multiple
  requests for use of the bus.
• The main virtue of the single-bus structure is
  its low cost and its flexibility for attaching
  peripheral devices.
• Systems that contain multiple buses achieve more
  concurrency in operations by allowing two or more
  transfers to be carried out at the same time.This
  leads to better performance but at an increased
  cost.

30
Explain how timing differences are smoothed out
among processors, memories and I/O devices.

• The devices connected to a bus vary widely in
  their speed of operation.
• Electro mechanical devices such as key
  board and printers are relatively slow.
• Others like magnetic or optical disks, are
  considerably faster.
• Memory and processor units operate at
  electronic speeds.
• A common approach to smooth out the timing
  differences is to include buffer registers with
  the devices to hold the information during
  transfers.
• They prevent a high speed processor from being
  locked to a slow I/O device during a sequence of
  data transfers.

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Contd.,

• This allows the processor to switch rapidly from
  one device to another, interweaving its process
  activity with data transfers involving several
  I/O devices.
• Thus, buffer registers smooth out timing
  differences among processors, memories and I/O
  devices.

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Software

• Explain the role of system software in a
  computer.
• System software is responsible for the
  coordination of all activities in a computing
  system.
• System software is a collection of programs that
  are executed as needed to perform functions such
  as
• 1) Receiving and interpreting user
  commands.
• 2) Entering and editing application programs
  and storing them as files in secondary storage
  devices.
• 3) Managing the storage and retrieval of
  files in secondary storage devices.

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Contd.,

• 4) Running standard application programs such
  as word processors, spread sheets, or games, with
  data supplied by the user.
• 5) Controlling I/O units to receive input
  information and produce output results.
• 6) Translating programs from high level
  language to low level language.
• 7) Linking and running user-written application
  programs with existing standard library routines,
  such as numerical computation packages.

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Explain various components of system software.

• Compiler A system software program which
  translates the high-level language program into a
  suitable machine language program.
• Text editor Another important system program
  that all programmers use is a text editor.
• It is used for entering and editing application
  programs.
• The user of this program interactively execute
  commands that allow statements of a source
  program entered at a keyboard to be accumulated
  in a file.

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Contd.,

• Operating system (OS)
• It is a key system software component.
• This is a large program, or actually a collection
  of routines , that is used to control the sharing
  of and interaction among various computer units
  as they execute application programs.
• The OS routines perform the tasks required to
  assign computer resources to individual
  application programs.
• These tasks include assigning memory to program
  and data files, moving data between memory and
  disk units, and handling I/O operations.

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Performance

• Discuss various parameters for improving the
  performance of a computer.
• The most important measure of the performance of
  a computer is how quickly it can execute a
  programs.
• The speed with which a computer executes programs
  is affected by the design of its hardware and its
  machine language instructions.
• Elapsed time The total time required to execute
  a program .
• This elapsed time is a measure of the performance
  of the entire computer system.
• It is affected by the speed of the processor, the
  disk and the printer.

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Contd.,

• Processor time
• Here we have to consider only those periods of
  the elapsed time, during which the processor is
  active.
• The sum of these periods is called processor
  time.
• The processor time depends on the hardware
  involved in the execution of individual machine
  instructions.
• Cache memory
• The processor and a relatively small cache memory
  can be fabricated on a single IC chip.
• The internal speed of performing the basic steps
  of instruction processing on such chips is very
  high and considerably faster than the speed at
  which instructions and data can be fetched from
  the main memory.

38
Contd.,

• A program will be executed faster if the movement
  of instructions and data between the main memory
  and processor is minimized, which is achieved by
  using the cache.

Processor
Main memory
Cache memory
Bus
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Processor clock

• Processor circuits are controlled by a timing
  signal called a clock .
• The clock defines regular time intervals, called
  clock cycles.
• To execute a machine instruction, the processor
  divides the action to be performed into a
  sequence of basic steps, such that each step can
  be completed in one clock cycle.
• The length P of one clock cycle is an important
  parameter that affects processor performance.
• Its inverse is the clock rate, R 1/P , which is
  measured in cycles per second .
• If the clock rate is 500 million cycles per
  second, then the corresponding clock period is 2
  nanoseconds.

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Basic performance equation

• Let T be the processor time required to execute
  a program that has been prepared in some high
  level language.
• Assume that the complete execution of the program
  requires the execution of N machine language
  instructions.
• Suppose that the average number of basic steps
  needed to execute one machine instruction is S.
• If the clock rate is R cycles per second, the
  program execution time is given by
• T N . S
• R
• This is often referred to as the basic
  performance equation.

41
Pipelining and Super scalar operation

• A substantial improvement in performance can be
  achieved by overlapping the execution of
  successive instructions, using a technique called
  pipelining .
• Consider the instruction
• Add R1, R2, R3
• Which adds the contents of registers R1 and R2,
  and places the sum into R3.
• The contents of R1 and R2 are first transferred
  to the inputs of the ALU.
• After the add operation is performed, the sum is
  transferred to R3.
• Processor can read the next instruction from the
  memory while the addition operation is being
  performed.

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Pipelining (contd.,)

• Then, if that instruction also uses the ALU, its
  operands can be transferred to the ALU inputs at
  the same time that the result of Add instruction
  is being transferred to R3.
• Thus, pipelining increases the rate of executing
  instructions significantly and cause the
  effective value of S to approach 1.

43
Super scalar operation

• A higher degree of concurrency can be achieved if
  multiple instruction pipelines are implemented in
  the processor.
• This means that multiple function units are used,
  creating parallel paths through which different
  instructions can be executed in parallel.
• With such an arrangement, it becomes possible to
  start the execution of several instructions in
  every clock cycle.
• This mode of execution is called Super scalar
  operation

44
Clock rate

• There are two possibilities for increasing the
  clock rate, R.
• First, improving the IC technology makes logic
  circuit faster, which reduces the the needed to
  complete a basic step. This allows the clock
  period, P, to be reduced and the clock rate, R,
  to be increased.
• Second, reducing the amount of processing done in
  one basic step also makes it possible to reduce
  the clock period , P.

45
Instruction set CISC and RISC

• The terms CISC and RISC refer to design
  principles and techniques.
• RISC Reduced instruction set computers
• Simple instructions require a small number of
  basic steps to execute.
• For a processor that has only simple
  instructions, a large number of instructions may
  be needed to perform a given programming task.
• This could lead to a large value of N and a
  small value for S.
• It is much easier to implement efficient
  pipelining in processors with simple instruction
  sets.

46
Instruction set CISC and RISC (contd.,)

• CISC Complex instruction set computers.
• Complex instructions involve a large number of
  steps.
• If individual instructions perform more complex
  operations, fewer instructions will be needed,
  leading to a lower value of N and a larger value
  of S.
• Complex instructions combined with pipelining
  would achieve good performance.

47
Compiler

• A compiler translates a high-level language
  program into a sequence of machine instructions
• To reduce N, we need to have a suitable machine
  instruction set and a compiler that makes good
  use of it.
• An optimizing compiler takes advantage of various
  features of the target processor to reduce the
  product N.S .
• The compiler may rearrange program instructions
  to achieve better performance.

48
Performance measurement

• SPEC rating.
• A non profit organization called System
  Performance Evaluation Corporation (SPEC)
  selects and publishes representative application
  programs for different application domains.
• The SPEC rating is computed as follows
• SPEC rating Running time on the
  reference computer
• Running time on
  the computer under test
• Thus SPEC rating of 50 means that the computer
  under test is 50 times faster than the reference
  computer for this particular benchmark.

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Contd.,

• The test is repeated for all the programs in the
  SPEC suite, and the geometric mean of the results
  is computed.
• Let SPECi be the rating for program i in the
  suite.
• The overall SPEC rating for the computer is
  given by
• SPEC rating ? ( SPECi )
• where n is the number of programs in the
  suite.

n
i1
50
Multiprocessors

• Large computer systems may contain a number of
  processor units, in which case they are called
  multiprocessor systems.
• These systems either execute a number of
  different application tasks in parallel, or they
  execute subtasks of a single large task in
  parallel.
• All processors usually have access to all of the
  memory in such systems, and the term
  shared-memory multiprocessor systems is often
  used to make this clear.

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Multiprocessors (contd.,)

• The high performance of these systems comes with
  much increased complexity and cost.
• In addition to multiple processors and
  memory units, cost is increased because of the
  need for more complex interconnection networks.

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Multi-computers

• In contrast to multiprocessor systems, it is
  possible to use an interconnected group of
  complete computers to achieve high total
  computational power.
• The computers normally have access only to their
  own memory units.
• When the tasks they are executing need to
  communicate data, they do so by exchanging
  messages over a communication network.
• This property distinguishes them from
  shared-memory multiprocessors, leading to the
  name message-passing multi- computers.

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Data Representation

• 3-1 Data Types
• Binary information is stored in memory or
  processor registers
• Registers contain either data or control
  information
• Data are numbers and other binary-coded
  information
• Control information is a bit or a group of bits
  used to specify the sequence of command signals
• Data types found in the registers of digital
  computers
• Numbers used in arithmetic computations
• Letters of the alphabet used in data processing
• Other discrete symbols used for specific purpose
• ?? Number ? Letter ?? ??, ?) gray code, error
  detection code,
• Number Systems
• Base or Radix r system uses distinct symbols
  for r digits
• Most common number system Decimal, Binary,
  Octal, Hexadecimal
• Positional-value(weight) System r2 r 1r0.r-1
  r-2 r-3
• Multiply each digit by an integer power of r and
  then form he sum of all weighted digits

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• Decimal System/Base-10 System
• Composed of 10 symbols or numerals(0, 1, 2, 3, 4,
  5, 6, 7, 8, 9, 0)
• Binary System/Base-2 System
• Composed of 10 symbols or numerals(0, 1)
• Bit Binary digit
• Hexadecimal System/Base-16 System Tab. 3-2
• Composed of 16 symbols or numerals(0, 1, 2, 3, 4,
  5, 6, 7, 8, 9, A, B, C, D, E, F)
• Binary-to-Decimal Conversions
• 1011.1012 (1 x 23) (0 x
  22) (1 x 21) (1 x 2o) (1 x 2-1) (0 x 2-2)
  (1 x 2-3)
• 810 0 210 110
  0.510 0 0.12510
• 11.62510
• Decimal-to-Binary Conversions
• Repeated division(See p. 69, Fig. 3-1)
• 37 / 2 18 remainder 1 (binary number
  will end with 1) LSB
• 18 / 2 9
  remainder 0
• 9 / 2 4 remainder 1
• 4 / 2 2
  remainder 0
• 2 / 2 1
  remainder 0
• 1 / 2 0
  remainder 1 (binary number will start with 1)
  MSB

??? ?? 0.375 x 2 0.750 integer
0 MSB 0.750 x 2 1.500 integer 1
. 0.500 x 2 1.000 integer 1 LSB
Read the result downward .37510 .0112
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• Hex-to-Decimal Conversion
• 2AF16 (2 x 162) (10 x 161)
  (15 x 16o)
• 51210 16010
  1510
• 68710
• Decimal-to-Hex Conversion
• 42310 / 16 26 remainder 7
  (Hex number will end with 7) LSB
• 2610 / 16 1 remainder
  10
• 110 / 16 0 remainder
  1 (Hex number will start with 1) MSB
• Read the result upward to give an answer
  of 42310 1A716
• Hex-to-Binary Conversion
• 9F216 9 F
  2
• 1001 1111 0010
• 1001111100102

Table 3-2 Hex Binary Decimal 0 0000
0 1 0001 1 2 0010 2 3 0011 3 4
0100 4 5 0101 5 6 0110
6 7 0111 7 8 1000 8 9 1001
9 A 1010 10 B 1011 11 C 1100
12 D 1101 13 E 1110 14 F 1111 15

• Binary-to-Hex Conversion
• 1 1 1 0 1 0 0 1 1 02 0 0 1 1 1 0 1 0
  0 1 1 0
• 3
  A 6
• 3A616

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• Binary-Coded-Decimal Code
• Each digit of a decimal number is represented by
  its binary equivalent
• 8 7 4
  (Decimal)
• 1000 0111 0100
  (BCD)
• Only the four bit binary numbers from 0000
  through 1001 are used
• Comparison of BCD and Binary
• 13710 100010012
  (Binary) - require only 8 bits
• 13710 0001 0011
  0111BCD (BCD) - require 12 bits
• Alphanumeric Representation
• Alphanumeric character set(Tab. 3-4)
• 10 decimal digits, 26 letters, special
  character(, , ,.)
• A complete list of ASCII p. 384, Tab. 11-1
• ASCII(American Standard Code for Information
  Interchange)
• Standard alphanumeric binary code uses seven bits
  to code 128 characters

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• 3-2 Complements
• Complements are used in digital computers for
  simplifying the subtraction operation and for
  logical manipulation
• There are two types of complements for base r
  system
• 1) rs complement 2) (r-1)s
  complement
• Binary number 2s or 1s complement
• Decimal number 10s or 9s complement
• (r-1)s Complement
• (r-1)s Complement of N (rn-1)-N
• 9s complement of N546700
• (106-1)-546700 (1000000-1)-546700
  999999-546700
• 453299
• 1s complement of N101101
• (26-1)-101101 (1000000-1)-101101 111111-101101
• 010010
• rs Complement
• rs Complement of N rn-N
• 10s complement of 2389 76101 7611
• 2s complement of 1101100 00100111 0010100

N given number r base n digit number
546700(N) 453299(9s com) 999999
101101(N) 010010(1s com) 111111
rs Complement (r-1)s Complement 1
(rn-1)-N1 rn-N
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• Subtraction of Unsigned Numbers
• 1) M (rn-N)
• 2) M ? N Discard end carry, Result M-N
• 3) M ? N No end carry, Result - rs
  complement of (N-M)
• Decimal Example)
• 72532(M) - 13250(N) 59282
• 72532
• 86750 (10s complement of 13250)
• 1 59282
• Result 59282
• Binary Example)
• 1010100(X) - 1000011(Y) 0010001
• 1010100
• 0111101 (2s complement of 1000011)
• 1 0010001
• Result 0010001

(M-N), N?0

• 13250(M) - 72532(N) -59282
• 13250
• 27468 (10s complement of 72532)
• 0 40718
• Result -(10s complement of 40718)
• -(592811) -59282

M ? N
M ? N
Discard End Cary
No End Carry

• 1000011(X) - 1010100(Y) -0010001
• 1000011
• 0101100 (2s complement of 1010100)
• 0 1101111
• Result -(2s complement of 1101111)
• -(00100001) -0010001

X ? Y
X ? Y
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Numeric Data 1) Fixed Point 2) Floating Point
32.25 1) 0.25, 2) 32.0, 3) 32.25

• 3-3 Fixed-Point Representation
• Computers must represent everything with 1s and
  0s, including the sign of a number and
  fixed/floating point number
• Binary/Decimal Point
• The position of the binary/decimal point is
  needed to represent fractions, integers, or mixed
  integer-fraction number
• Two ways of specifying the position of the binary
  point in a register
• 1) Fixed Point the binary point is always fixed
  in one position
• A binary point in the extreme left of the
  register(Fraction 0.xxxxx)
• A binary point in the extreme right of the
  register(Integer xxxxx.0)
• The binary point is not actually present, but the
  number stored in the register is treated as a
  fraction or as an integer
• 2) Floating Point the second register is used
  to designate the position of the
• binary point in
  the first register(refer to 3-4)
• Integer Representation
• Signed-magnitude representation
• Signed-1s complement representation
• Signed-2s complement representation

MSB for Sign 0 is plus 1 is minus
-
14 -14 0 0001110
1 0001110 0 0001110 1 1110001
0 0001110 1 1110010
Most Common
61
(-12) (-13) -25 (12) (13) 25
(25) (-37) 37 - 25 -12

• Arithmetic Addition
• Addition Rules of Ordinary Arithmetic
• The signs are same sign common sign, result
  add
• The signs are different sign larger sign,
  result larger-smaller
• Addition Rules of the signed 2s complement
• Add the two numbers including their sign bits
• Discard any carry out of the sign bit position
• Arithmetic Subtraction
• Subtraction is changed to an Addition
• ( A) - ( B) ( A) (- B)
• ( A) - ( - B) ( A) ( B)
• Overflow
• Two numbers of n digits each are added and the
  sum occupies n1 digits
• n 1 bit cannot be accommodated in a register
  with a standard length of n bits(many computer
  detect the occurrence of an overflow, and a
  corresponding F/F is set)

Addition Exam) 6 00000110 - 6
11111010 13 00001101 13 00001101
19 00010011 7 00000111 6
00000110 - 6 11111010 - 13
11110011 - 13 11110011 - 7
11111001 - 19 11101101
Subtraction Exam) (- 6) - ( - 13)
7 11111010 - 11110011 11111010 2s comp of
11110011
11111010 00001101
1 00000111 7
Discard End Carry
62

• Overflow
• An overflow may occur if the two numbers added
  are both positive or both negative
• When two unsigned numbers are added
• an overflow is detected from the end carry out of
  the MSB position
• When two signed numbers are added
• the MSB always represents the sign
• - the sign bit is treated as part of the
  number
• - the end carry does not indicate an
  overflow

Overflow Exam) out in
out in carries 0 1
carries 1 0 70 0 1000110 - 70
1 0111010 80 0 1010000 - 80
1 0110000 150 1 0010110 - 150 0
1101010

• Overflow Detection
• Detected by observing the carry into the sign bit
  position and the carry out of the sign bit
  position
• If these two carries are not equal, an overflow
• condition is produced(Exclusive-OR gate 1)
• Decimal Fixed-Point Representation
• A 4 bit decimal code requires four F/Fs
• for each decimal digit
• The representation of 4385 in BCD requires 16
  F/Fs (0100 0011 1000 0101)
• The representation in decimal is wasting a
  considerable amount of storage space and the
  circuits required to perform decimal arithmetic
  are more complex

Decimal Exam) (375) (-240) 375 (10s comp
of 240) 375 760 0 375 (0000 0011 0111
0101) 9 760 (1001 0111 0110 0000) 0 135
(0000 0001 0011 0101)
Advantage Computer I/O data are generated by
people who use the decimal system
63
Decimal 6132.789 Fraction
Exponent 0.6132789 4

• 3-4 Floating-Point Representation
• The floating-point representation of a number has
  two parts
• 1) Mantissa signed, fixed-point number
• 2) Exponent position of binary(decimal) point
• Scientific notation m x re (0.6132789 x 104)
• m mantissa, r radix, e exponent
• Example m x 2e (.1001110)2 x 24
• Normalization
• Most significant digit of mantissa is nonzero
• 3-5 Other Binary Codes
• Gray Code
• Gray code changes by only one bit (Tab. 3-5
  4-bit Gray Code )
• ??
• The data must be converted into digital form
  before they can be used by a digital
  computer(Analog to Digital Converter)
• The analog data are represented by the continuous
  change of a shaft position(Rotary Encoder of
  Motor)

Fraction Exponent 01001110
000100
64

• Other Decimal Codes
• Binary codes for decimal digits require four
  bits. A few possibilities are shown in Tab. 3-6
• Excess-3 Gray Code
• Gray code? BCD ?? ?, 9 ?? 0 ?? ??? 1101?? 0000??
  ?? 3 ??? ??? ???? Tab. 3-5 ?? 3 ?? 12?? ???? 0010
  ?? 1010?? 1??? ??.
• Self-Complementing excess-3 code
• 9s complement of a decimal number is easily
  obtained by 1s complement(changing 1s to 0s
  and 0s to 1s)
• Weighted Code 2421 code
• The bits are multiplied by the weights, and the
  sum
• of the weighted bits gives the decimal digit
• Other Alphanumeric Codes
• ASCII Code?? Tab. 3-4 ?? p. 384, Tab. 11-1
• Format effector Functional characters for
  controlling the layout of printing or display
  devices(carriage return-CR, line feed-LF,
  horizontal tab-HT,)
• Data communication flow control(acknowledge-ACK,
  escape-ESC, synchronous-SYN,)
• EBCDIC(Extended BCD Interchange Code)
• Used in IBM equipment(?? ??? ?? ??)

Self-Complement Exam) 410 0111 (3-excess)
1000 ( 1s comp) 510 (3-excess
in Tab. 3-6) 510( 9s comp of 4)
65

• 3-6 Error Detection Codes
• Binary information transmitted through some form
  of communication medium is subject to external
  noise
• Parity Bit
• An extra bit included with a binary message to
  make the total number of 1s either odd or
  even(Tab. 3-7)
• Even-parity method
• The value of the parity bit is chosen so that the
  total number of 1s (including the parity bit) is
  an even number
• 1 1 0 0 0 0 1 1
  
• Odd-parity method
• Exactly the same way except that the total number
  of 1s is an odd number
• 1 1 0 0 0 0 0 1

Added parity bit
Added parity bit
66
1 1 0 0 0 0 1 1 C 1
1 0 0 0 0 1 0 B (Even-parity Generator)
(Even-parity Checker)

• Parity Generator/Checker
• At the sending end, the message is applied to a
  parity generator
• At the receiving end, all the incoming bits are
  applied to a parity checker
• Can not tell which bit in error
• Can detect only single bit error(odd number of
  errors)
• 3 bit data line example Fig. 3-3
• 4 bit data line example

67

• Odd Parity Generator/Checker
• Truth Table

68
Chapter 2 Data Representation
CS140 Computer Organization
These slides are derived from those of Null
Lobur the work of others.
69
Chapter 2 Objectives

• Understand the fundamentals of numerical data
  representation and manipulation in digital
  computers both integer and floating point.
• Master the skill of converting between various
  radix systems.
• Understand how errors can occur in computations
  because of overflow and truncation.
• Gain familiarity with the most popular character
  codes.
• Understand the concepts of error detecting and
  correcting codes.

70
2.1 Introduction

• REVIEW
• A bit is the most basic unit of information in a
  computer.
• It is a state of on or off in a digital
  circuit.
• Sometimes these states are high or low
  voltage instead of on or off..
• A byte is a group of eight bits.
• A byte is the smallest possible addressable (can
  be found via its location) unit of computer
  storage.
• A word is a contiguous group of bytes.
• Words can be any number of bits (16, 32, 64 bits
  are common).
• A group of four bits is called a nibble (or
  nybble).
• Bytes, therefore, consist of two nibbles a
  high-order nibble, and a low-order nibble.

71
2.2 Positional Numbering Systems

• Bytes store numbers using the position of each
  bit to represent a power of 2.
• The binary system is also called the base-2
  system.
• Our decimal system is the base-10 system. It
  uses powers of 10 for each position in a number.
• Any integer quantity can be represented exactly
  using any base (or radix).
• The decimal number 947 in powers of 10 is
• The decimal number 5836.47 in powers of 10 is

9 ? 10 2 4 ? 10 1 7 ? 10 0
5 ? 10 3 8 ? 10 2 3 ? 10 1 6 ? 10 0
4 ? 10 -1 7 ? 10 -2
72
2.2 Positional Numbering Systems

• Binary works the same as decimal
• The binary number 11001 in powers of 2 is
• When the radix of a number is something other
  than 10, the base is denoted by a subscript.
• Sometimes, the subscript 10 is added for
  emphasis
• 110012 2510

1 ? 2 4 1 ? 2 3 0 ? 2 2 0 ? 2 1 1 ?
2 0 16 8 0
0 1 25
73
2.3 Decimal to Binary Conversions
Subtraction Method

• Suppose we want to convert the decimal number 190
  to base 3.
• We know that 3 5 243 so our result will be
  less than six digits wide. The largest power of
  3 that we need is therefore 3 4 81, and
  81 ? 2 162.
• Write down the 2 and subtract 162 from 190,
  giving 28.

Note The book also talks about the Division
Method which you may prefer.
74
2.3 Decimal to Binary Conversions
Subtraction Method

• Converting 190 to base 3...
• The next power of 3 is 3 3 27.
  Well need one of these, so we subtract 27 and
  write down the numeral 1 in our result.
• The next power of 3, 3 2 9, is too large, but
  we have to assign a placeholder of zero and carry
  down the 1.

75
2.3 Decimal to Binary Conversions
Subtraction Method

• Converting 190 to base 3...
• 3 1 3 is again too large, so we assign a zero
  placeholder.
• The last power of 3, 3 0 1, is our last
  choice, and it gives us a difference of zero.
• Our result, reading from top to bottom is
• 19010 210013

76
2.3 Decimal to Binary Conversions

• Fractional values can be approximated in all base
  systems.
• Fractions do not necessarily have exact
  representations under all radices.
• The quantity ½ is exactly representable in
  binary and decimal, but not in the ternary (base
  3) numbering system.

• Fractional decimal values have nonzero digits to
  the right of the decimal point.
• Fractional values of other radix systems have
  nonzero digits to the right of the radix point.

0.4710 4 ? 10 -1 7 ? 10 -2 0.112 1 ? 2
-1 1 ? 2 -2 ½ ¼
0.5 0.25 0.75
77
2.3 Decimal to Binary Conversions
Subtraction Method

• This shows the subtraction method to convert the
  decimal 0.8125 to binary.
• Our result, reading from top to bottom is
• 0.812510 0.11012
• This method works with any base, not just binary.

Note The book also talks about the
Multiplication Method which you may prefer.
78
2.3 Decimal to Binary Conversions
HEX

• The binary numbering system is the most important
  radix system for digital computers.
• But, its difficult to read long strings of
  binary numbers-- and even a modest decimal number
  becomes a very long binary number.
• For example 110101000110112 1359510
• For compactness and ease of reading, binary
  values are usually expressed using the
  hexadecimal, or base-16, numbering system.

• The hexadecimal numbering system uses the
  numerals 0 through 9 and the letters A through F.
• The decimal number 12 is C16.
• The decimal number 26 is 1A16.
• It is easy to convert between base 16 and base 2,
  because 16 24.
• Thus, to convert from binary to hexadecimal, all
  we need to do is group the binary digits into
  groups of four.

A group of four binary digits is called a hextet
79
2.3 Decimal to Binary Conversions
HEX

• Using groups of hextets, the binary number
  110101000110112 ( 1359510) in hexadecimal is
• Practice Convert 100010 to Hex

In hexadecimal, we cant write numbers 10-16 as
single digits, so we use the letters A F. The
number FA316 is F (15) in the 162 column, A (10)
in the 161 column and 3 in the 160 column
80
2.4 Signed Integer Representation

• To represent negative values, computer systems
  allocate the high-order bit to indicate the sign
  of a value.
• The high-order bit is the leftmost bit in a byte.
  Its also called the most significant bit.
• The remaining bits contain the value of the
  number.
• There are three ways in which signed binary
  numbers may be expressed
• Signed magnitude,
• Ones complement ? we wont do much with this
  here
• Twos complement.

81
2.4 Signed Integer Representation

• In an 8-bit word, signed magnitude representation
  places the absolute value of the number in the 7
  bits to the right of the sign bit.

• In 8-bit signed magnitude,
• Positive 3 is 00000011
• Negative 3 is 10000011
• Computers perform arithmetic operations on signed
  magnitude numbers the same way as humans do
  pencil and paper arithmetic.
• Humans ignore the signs of the operands while
  doing a calculation, applying the appropriate
  sign at the end.

82
2.4 Signed Integer Representation

• Binary addition is as easy as it gets. You need
  to know only four rules
• 0 0 0 0 1 1
• 1 0 1 1 1 10
• The simplicity of this system makes it possible
  for digital circuits to carry out arithmetic
  operations.
• We will describe these circuits in Chapter 3.

Lets see how the addition rules work with signed
magnitude numbers . . .
83
2.4 Signed Integer Representation

• Example
• Using signed magnitude binary arithmetic, find
  the sum of 75 and 46.
• Convert 75 and 46 to binary, and arrange as a
  sum. Separate the (positive) sign bits from the
  magnitude bits.

• As in decimal arithmetic, find the sum starting
  with the rightmost bit and work left.

• In the second bit, we have a carry, so we note it
  above the third bit.

84
2.4 Signed Integer Representation

• Example
• Using signed magnitude binary arithmetic, find
  the sum of 75 and 46.
• The third and fourth bits also give us carries.
• Once we have worked our way through all eight
  bits, we are done.

In this example, we were careful to pick two
values whose sum would fit into seven bits. If
thats not the case, we have a problem.

• Example
• Using signed magnitude binary arithmetic, find
  the sum of 107 and 46.
• The carry from the seventh bit overflows and is
  discarded, giving us the erroneous result 107
  46 25.

85
2.4 Signed Integer Representation

• The signs in signed magnitude representation work
  just like the signs in pencil and paper
  arithmetic.
• Example Using signed magnitude binary
  arithmetic, find the sum of -46 and -25.

Because the signs are the same, all we do is add
the numbers and supply the negative sign when we
are done.

• Mixed sign addition (or subtraction) is done the
  same way.
• Example Using signed magnitude binary
  arithmetic, find the sum of 46 and -25.

The sign of the result gets the sign of the
number that is larger. Note the borrows from
the second and sixth bits.
86
2.4 Signed Integer Representation

• Signed magnitude representation is easy for
  people to understand, but needs complicated
  computer hardware.
• Another disadvantage it allows two different
  representations for zero positive zero and
  negative zero.
• So computers systems employ complement systems
  for numeric value representation.

4-bit Binary Decimal 1000
-8 1001 -7 1010 -6
1011 -5 1100 -4 1101
-3 1110 -2 1111
-1 0000 0 0001 1 0010
2 0011 3 0100
4 0101 5 0110 6 0111
7
87
2.4 Signed Integer Representation

• To express a value in twos complement
• If the number is positive, just convert it to
  binary and youre done.
• If the number is negative, find the ones
  complement (change each of the bits) of the
  number and then add 1.
• Example
• In 8-bit ones complement, positive 3 is
• 00000011
• Negative 3 in ones complement is
• 11111100
• Adding 1 gives us -3 in twos complement form
• 11111101.

88
2.4 Signed Integer Representation

• With twos complement arithmetic, all we do is
  add our two binary numbers. Just discard any
  carries emitting from the high order bit.

Example Using twos complement binary
arithmetic, find the sum of 48 and - 19.
We note that 19 in ones complement is
00010011, so -19 in ones complement is
11101100, and -19 in twos complement is
11101101.
89
2.4 Signed Integer Representation

• When we use any finite number of bits to
  represent a number, we run the risk of the result
  of calculations being too large to be stored in
  the computer.
• While we cant always prevent overflow, we can
  always detect overflow.

Example Using twos complement binary
arithmetic, find the sum of 107 and 46 (we did
this before in slide 17) We see that the nonzero
carry from the seventh bit overflows into the
sign bit, giving us the erroneous result 107
46 -103.
Rule for detecting signed twos complement
overflow When the carry in and the carry
out of the sign bit differ, overflow has
occurred.
90
2.4 Signed Integer Representation

• Signed and unsigned numbers are both useful.
• For example, memory addresses are always
  unsigned.
• Using the same number of bits, unsigned integers
  can express twice as many values as signed
  numbers.
• Trouble arises if an unsigned value wraps
  around.
• In four bits 1111 1 0000.
• Good programmers stay alert for this kind of
  problem.

91
2.4 Signed Integer Representation

• There are a number of algorithms that improve the
  performance of computer arithmetic.
• Booths algorithm as discussed in the book is one
  of these enhancements.
• Many of these methods are based on the fact that
  multiplication is HARD for a computer, but
  addition and shifting is SIMPLE. So techniques
  that optimize the simple operations make
  hardware, firmware, software go faster.

92
2.5 Floating-Point Representation

• Integer formats are not useful in scientific or
  business applications that deal with real number
  values.
• Floating-point representation solves this
  problem.
• Most of todays computers are equipped with
  specialized hardware that performs floating-point
  arithmetic with no special programming required.

93
2.5 Floating-Point Representation

• Detour on biased notation.

Excess-8 Notation 0000 -8 0001 -7 0010 -6
0011 -5 0100 -4 0101 -3 0110 -2 0111 -1
1000 0 1001 1 1010 2 1011 3 1100 4 1101
5 1110 6 1111 7

• Use unsigned magnitude to represent both positive
  and negative numbers by using a bias, or excess,
  representation
• The entire numbering system is shifted up some
  positive amount.
• To convert a number from excess-8 to decimal.
• For a number in excess-8, we subtract 8 from the
  number to get the real value (1001 in excess-8 is
  1001 1000 00012 1)
• To convert a number from decimal to excess-8.
• 6 01102. Add on the bias like this 0110
  1000 1110 ? the is the excess-8 number.
• To use the representation
• numbers have to be shifted, then stored, and then
  shifted back when retrieved
• Unnecessarily complicated for regular use.
• Useful to represent exponents in our floating
  point representation (shown next)

94
2.5 Floating-Point Representation

• Floating-point numbers allow an arbitrary