DIGITAL DESIGN

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DIGITAL DESIGN

• ICS 30/CS 30

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DIGITAL DESIGN

• logic design
• switching circuits
• digital logic
• digital systems

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DIGITAL DESIGN

• digital computers
• electronic calculators
• digital control devices
• digital communication equipment
• many other applications that require electronic
  digital hardware

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DIGITAL DESIGN

• SSI circuits contain several gates or flip-flops
  in a single package
• MSI devices provide specific digital functions
• LSI devices provide complete computer modules

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DIGITAL SYSTEMS

• general-purpose digital computer
• Best-known example of a digital system
• telephone switching exchanges
• digital voltmeters
• frequency counters
• calculating machines
• teletype machines

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DIGITAL SYSTEMS

• Its manipulation of discrete elements of info.
• electric impulses
• decimal digits
• letters of an alphabet
• arithmetic operations
• punctuation marks
• any other set of meaningful symbols

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DIGITAL SYSTEMS

• A sequence of discrete elements forms a language,
  i.e. a discipline that conveys info.
• discrete information processing system
• Discrete elements of info. are represented in a
  digital system by physical quantities called
  signals.
• Electrical signals such as voltages and currents
  are the most common.

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DIGITAL SYSTEMS

• Digital systems that are constrained to take
  discrete values are further constrained to take
  binary values.
• An analog computer performs a direct simulation
  of a physical system
• The term analog signal is sometimes substituted
  for continuous signal because analog computer
  has come to mean a computer that manipulates
  continuous variables.

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DIGITAL SYSTEMS
Processor or Arithmetic Unit
Control Unit
Block Diagram of a Digital Computer
Storage or Memory Unit
Input Devices and Control
Output Devices and Control
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DIGITAL SYSTEMS

• An electronic calculator is a digital system
  similar to a digital computer, why?
• A digital computer, however, is a more powerful
  device than a calculator, how?
• A digital computer is an interconnection of
  digital modules.

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DIGITAL SYSTEMS

• A processor when combined with the control unit,
  forms a component referred to as a central
  processing unit or CPU
• A CPU enclosed in a small IC package is called a
  microprocessor
• A CPU combined with memory and interface control
  to form a small-size computer is called a
  microcomputer

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BINARY SYSTEM

• Operands used for calculations may be expressed
  in the binary number system.
• Other discrete elements, including the decimal
  digits, are represented in binary codes
• Data processing is carried out by means of binary
  logic elements using binary signals
• Quantities are stored in binary storage elements

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NUMBER BASE CONVERSIONS

• Decimal to Binary and Vice versa
• Hexadecimal System
• Provides another convenient and simple method for
  expressing values represented by binary numerals.
• It uses a base, or radix, of 16 and the place
  values are the powers of 16.

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NUMBER BASE CONVERSIONS
DECIMAL AND HEXADECIMAL NUMBERS
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BINARY SYSTEM

• Binary Subtraction by Complementing
• Complement arithmetic has particular appeal
  in the case of the binary system since to obtain
  the inverse of binary numbers no subtraction is
  required, only bit inversion.

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BINARY SYSTEM

• Subtraction is accomplished by adding the
  complement of the number to be subtracted.
• The equivalent of the 9s complement is the 1s
  complement.
• The 1s complement of a binary number is simply
  the difference between 1 and each of the digits
  of the number.

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BINARY SYSTEM

• Complementing in the binary system simply
  consists of putting down a 1 for a 0 and a 0 for
  a 1.
• Thus the complement of 11010 is 00101, and the
  complement of 11110 is 000001.
• To subtract by complementing in the binary
  system, you add the 1s complement plus 1, and
  ignore the initial 1.

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BINARY SYSTEM

• Binary Multiplication
• The rule for decimal multiplication also holds
  for binary multiplication.
• In fact, binary multiplication is simpler since
  multiplying a number by the bit 0 or 1 yields
  respectively 0 or the number.
• In any number system, multiplication consists
  of adding a number to itself as many times as is
  specified by the multiplier.

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BINARY SYSTEM

• Binary Division
• As in the decimal system, division is the
  inverse of multiplication and division by zero is
  similarly meaningless.
• Division is defined as the process of
  determining how many times one number, the
  divisor, can be subtracted from another number,
  the dividend, while still leaving a positive
  remainder.
• The number of times this can be done is the
  result, or quotient.

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COMPLEMENTS

• Complements are used in digital computers for
  simplifying the subtraction operation and for
  logical manipulations
• 2 types for each base-r system
• rs complement
• (r-1)s complement

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COMPLEMENTS

• The rs Complement
• Given a positive number N in base r with an
  integer part of n digits, the rs complement of N
  is defined as rn N for N ? 0 and 0 for N0.

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COMPLEMENTS

• The (r-1)s Complement
• Given a positive number N in base r with an
  integer part of n digits and a fraction part of m
  digits, the (r-1)s complement of N is defined as
  rn r -m N.

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COMPLEMENTS

• The rs complement of N is rn N and the
  complement of (rn N) is rn (rn N) N
• Subtraction w/ rs complements
• The subtraction of two positive numbers ( M -
  N), both of base r, may be done as follows
• 1. add the minuend M to the rs complement of
  the subtrahend N.

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COMPLEMENTS

• 2. Inspect the result obtained in step 1
  for an end carry
• (a) If an end carry occurs discard it.
• (b) If an end carry does not occur, take
  the rs complement of the number
  obtained in step 1 and place a negative
  sign in front.

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COMPLEMENTS

• Subtraction w/ (r-1)s complements
• The subtraction of ( M - N), both positive
  numbers in base r, may be done as follows
• 1. add the minuend M to the (r-1)s complement
  of the subtrahend N.

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COMPLEMENTS

• 2. Inspect the result obtained in step 1 for an
  end carry
• (a) If an end carry occurs, add 1 to the
  least significant digit (end-around carry).
• (b) If an end carry does not occur, take
  the (r-1)s complement of the number
  obtained in step 1 and place a negative
  sign in front.

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COMPLEMENTS

• Comparison between 1s and 2s Complements
• 1s complement
• easier to implement by digital components since
  the only thing that must be done is to change 0s
  to 1s and 1s into 0s.

• 2s complement
• may be obtained in two ways
• (1) By adding 1 to the least significant digit of
  the 1s complement

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COMPLEMENTS

• Comparison between 1s and 2s Complements
• 2s complement
• (2) by leaving all leading 0s in the least
  significant positions and the first 1 unchanged,
  and only then changing all 1s into 0s and all
  0s into 1s.

• During subtraction of two numbers by
  complements, the 2s complement is advantageous
• only one arithmetic addition operation is
  required

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COMPLEMENTS

• Comparison between 1s and 2s Complements
• 1s complement
• - requires two arithmetic additions when an
  end-around carry occurs.

• 1s complement has the additional disadvantage
  of possessing two arithmetic zeros one w/ all
  0s and one w/ all 1s.

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COMPLEMENTS

• Comparison between 1s and 2s Complements
• Consider the subtraction of two equal binary
  numbers 1100-1100 0
• Using 1s complement
• 1100
• 0011
• 1111

• complement again to obtain 0000
• Using 2s complement
• 1100
• 0100
• 0000

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COMPLEMENTS

• Comparison between 1s and 2s Complements
• While the 2s complement has only one arithmetic
  zero, the 1s complement zero can be positive or
  negative, which may complicate matters.

• 1s complement is also useful in logical
  manipulations, since the change of 1s to 0s and
  vice versa is equivalent to a logical inversion
  operation.

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COMPLEMENTS

• Comparison between 1s and 2s Complements
• The 2s complement is used only in conjunction
  with arithmetic applications.
• Consequently, it is convenient to adopt the ff.
  convention
• When the word complement, w/o mention of the
  type, is used in conjunction with a nonarithmetic
  application, the type is assumed to be the 1s
  complement.