CSCE 211: Digital Logic Design

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CSCE 211Digital Logic Design

• Chin-Tser Huang
• huangct_at_cse.sc.edu
• University of South Carolina

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Chapter 1 Introduction
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Digital Systems

• They are everywhere!
• They are usually binary operating on two-valued
  signals
• Take an arbitrary number of inputs and produce an
  arbitrary number of outputs
• Some systems require a timing signal called clock

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Examples

• A system with three inputs, A, B, and C, and one
  output Z, such that Z 1 if and only if two of
  the inputs are 1
• A system with eight inputs, representing two
  4-bit binary numbers, and one 5-bit output,
  representing the sum

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Examples

• A system with one input, A, plus a clock, and one
  output, Z, which is 1 iff the input was one at
  the last three consecutive clock times
• A traffic controller on two streets the light is
  green on each street for a fixed period of time,
  then goes to yellow for another fixed period and
  finally to red. The only input to this system is
  the clock

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Truth Table

• Describe the behavior of a digital system in
  tabular form

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A Brief Review of Number Systems

• Integers are usually written using a positional
  number system
• N an-1rn-1 an-2rn-2 a2r2 a1r a0
• where 0 ? ai lt r

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Conversion between Number Systems

• How to convert from binary to decimal? Evaluate
  the power series
• Example 1010112 ?

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Conversion between Number Systems

• How to convert from decimal to binary? Two
  algorithms
• Repeatedly subtract from the number the largest
  power of 2 less than that number and put a 1 in
  corresponding position
• Repeatedly divide the number by 2 and put the
  remainder from right to left

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Hexadecimal

• Radix r 16
• Why use hexadecimal? Shorthand notation for
  binary
• Grouping 4 bits in binary to get 1 digit in
  hexadecimal

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Binary Addition
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One-bit Adder
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4-bit Adder
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Signed Numbers

• Signed-magnitude human friendly, but causes
  complexity of arithmetic
• Twos complement store a as the binary
  equivalent of 2n a in an n-bit system

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Twos Complement

• An easier way to get twos complement
  representation for a negative number
• Find binary equivalent of the magnitude
• Complement each bit (change 0s to 1s, and
  change 1s to 0s)
• Add 1

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Addition using Twos Complement

• Its simple!
• Do the binary addition as usual
• If there is a carry out of the most significant
  bit, just ignore it
• But watch out for overflow!
• Overflow occurs when the sum is out of range,
  which indicates an error

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Binary Subtraction

• Subtraction can be accomplished by
• Taking the twos complement of second operand
• Adding the first operand and the twos complement
  of second operand

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Overflow in Binary Subtraction

• For unsigned numbers, overflow occurs when the
  second number is larger than the first number,
  and is indicated by a carry out of 0
• For signed numbers, overflow may occur if we
  subtract a negative number from a positive one or
  subtract a positive number from negative one

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Binary Coded Decimal (BCD)

• Most computers operate on binary numbers
• However, for computers to interface with humans,
  the mode of communication is generally decimal
• Convert from decimal to binary on input
• Convert from binary to decimal on output
• But the decimal output still needs to be coded
  into binary, digit by digit

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Binary-Coded Decimal Codes
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Other Codes

• ASCII used to transmit alphanumeric information
• Gray code consecutive numbers differ in only one
  bit
• Particularly useful in coding the position of a
  continuous device and error detection

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