Chapter 4: The Building Blocks: Binary Numbers, Boolean Logic, and Gates

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Chapter 4 The Building Blocks Binary Numbers,
Boolean Logic, and Gates

• Invitation to Computer Science,
• C Version, Third Edition

2
Objectives

• In this chapter, you will learn about
• Boolean logic and gates
• The binary numbering system
• Building computer circuits
• Control circuits

3
Introduction

• Chapter 4 focuses on hardware design (also called
  logic design)
• How to represent and store information inside a
  computer
• How to use the principles of symbolic logic to
  design gates
• How to use gates to construct circuits that
  perform operations such as adding and comparing
  numbers, and fetching instructions

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The Reliability of Binary Representation
Sec 4.2.3

• Electronic devices are most reliable in a
  bistable environment
• Bistable environment
• Distinguishes only two electronic states
• Current flowing or not
• Direction of flow
• Easiest way to make computers reliable was to use
  a two-state design. Hence they are designed to
  store and process binary information.

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Binary Storage Devices
Sec 4.2.4

• Magnetic core
• Historic device for computer memory
• Tiny magnetized rings flow of current sets the
  direction of magnetic field
• Binary values 0 and 1 are represented using the
  direction of the magnetic field

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Sec 4.2.4

• Figure 4.9
• Using Magnetic Cores to Represent Binary Values

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Binary Storage Devices (continued)
Sec 4.2.4

• Transistors
• Solid-state switches either permits or blocks
  current flow
• A control input causes state change
• Constructed from semiconductors

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Sec 4.2.4

• Figure 4.11
• Simplified Model of a Transistor

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Sec 4.2.4
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Sec 4.2.4
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Boolean Logic and Gates Boolean Logic
Sec 4.3.1

• Boolean logic is a branch of mathematics which
  describes operations on objects with two possible
  values, True or False
• True/False values map easily onto a two-state
  electronic environment.
• Boolean logic operations may be performed on
  electronic signals using devices built out of
  transistors and other electronic devices.

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Boolean Logic (continued)
Sec 4.3.1

• Let a and b represent objects with two possible
  values, true and false
• Boolean logic operations
• a AND b
• True only when a is true and b is true
• a OR b
• True when either a is true or b is true, or both
  are true
• NOT a
• True when a is false, and vice versa

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Boolean Logic (continued)
Sec 4.3.1

• Boolean expressions
• Constructed by combining together Boolean
  operations
• Example (a AND b) OR ((NOT b) AND (NOT a))
• Truth tables capture the output/value of a
  Boolean expression
• A column for each input plus the output
• A row for each combination of input values

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Boolean Logic (continued)
Sec 4.3.1

• Example (a AND b) OR ((NOT b) and (NOT a))
  _ _alternate
  expression (a b) ( b a )

a b Value
0 0 1
0 1 0
1 0 0
1 1 1
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Gates
Sec 4.3.2

• Gates
• Hardware devices built from transistors to mimic
  Boolean logical operations on digital electrical
  signals
• AND gate
• Two input lines, one output line
• Outputs a 1 when both inputs are 1

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Gates (continued)
Sec 4.3.2

• OR gate
• Two input lines, one output line
• Outputs a 1 when either input is 1
• NOT gate
• One input line, one output line
• Outputs a 1 when input is 0 and vice versa

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Sec 4.3.1

• Figure 4.15
• The Three Basic Gates and Their Symbols

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Sec 4.3.2
NAND gate
AND gate
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Sec 4.3.2
NOR gate
OR gate
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Gates (continued)
Sec 4.3.2

• Abstraction in hardware design
• Transform hardware devices to Boolean logical
  descriptions
• Design more complex devices in terms of logic,
  not electronics
• Conversion from logic to hardware design may be
  automated

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Sec 4.4.1

• Figure 4.19
• Diagram of a Typical Computer Circuit

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Building Computer Circuits Introduction
Sec 4.4.1

• A circuit is a collection of logic gates
• Transforms a set of binary inputs into a set of
  binary outputs
• Values of the outputs depend only on the current
  values of the inputs
• Combinational circuits have no cycles in them (no
  outputs feed back into their own inputs)

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Building Computer Circuits Intro Contd.
Sec 4.4.1
inputs inputs output
a b c
0 0 1
0 1 0
1 0 0
1 1 0
We want a 1 only when both inputs are 0. Is there
a gate which will do this? An AND gate gives a
1 only when both inputs are 1. An OR gate gives
a 1 whenever at least one input is 1 Could we
use an AND gate or an OR gate somehow?
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A Circuit Construction Algorithm
Sec 4.4.2

• Sum-of-products algorithm is one way to design
  circuits
• Truth table to Boolean expression to gate layout

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Sec 4.4.2

• Figure 4.21
• The Sum-of-Products Circuit Construction Algorithm

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A Circuit Construction Algorithm (continued)
Sec 4.4.2

• Sum-of-products algorithm
• Truth table captures every input/output possible
  for circuit
• Repeat process for each output line
• Build a Boolean expression using AND and NOT for
  each 1 of the output line
• Combine together all the expressions with ORs
• Build circuit from whole Boolean expression

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The Binary Numbering System

• A computers internal storage techniques are
  different from the way people represent
  information in daily lives
• Information inside a digital computer is stored
  as a collection of binary data

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Binary Representation of Numeric and Textual
Information

• Binary numbering system
• Base-2
• Built from ones and zeros
• Each position is a power of 2
• 1101 1 x 23 1 x 22 0 x 21 1 x 20
• Decimal numbering system
• Base-10
• Each position is a power of 10
• 3052 3 x 103 0 x 102 5 x 101 2 x 100

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• Figure 4.2
• Binary-to-Decimal
• Conversion Table

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Taxonomy of Integers
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Binary Representation of Numeric and Textual
Information (continued)

• Representing integers
• Decimal integers are converted to binary integers
• Given k bits, the largest unsigned integer is 2k
  - 1
• Given 4 bits, the largest is 24-1 15
• Signed integers must also represent the sign
  (positive or negative)

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Binary to Decimal Conversion
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Decimal to Binary Conversion
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Twos Complement Representation of Signed Integers

• Used by almost all computers today
• All places hold the value they would in binary
  except for the leftmost place which is negative
• 8 bit integer -128 64 32 16 8 4 2 1
• Range -128,127
• If last bit (leftmost)
• is 0, then positive ? looks just as before
• is 1, then negative ?add values for first 7
  digits and then subtract 128
• 1000 1101 148-128 -115

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Twos Complement Representation of Signed Integers

• Converting from decimal to 2s complement
• For positive numbers find the binary
  representation
• For negative numbers
• Find the binary representation for its positive
  equivalent
• Flip all bits
• Add a 1
• 43
• 43 0010 1011
• -43
• 43 0010 1011 ? 1101 0100 ? 1101 0101
• 1101 0101 -128641641 -43
• We can use the usual laws of binary addition

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Twos Complement Representation

• 125 - 19 106
• 0111 1101 1110 1101 ------------- 0110 1010
  106 !
• What happened to the one that we carried at the
  end?
• Got lost but still we got the right answer!
• We carried into and carried out of the leftmost
  column

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Binary Representation of Numeric and Textual
Information (continued)

• Representing real numbers
• Real numbers may be put into binary scientific
  notation a x 2b
• Example 101.11 x 20
• Number then normalized so that first significant
  digit is immediately to the right of the binary
  point
• Example .10111 x 23
• Mantissa and exponent then stored

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Binary Representation of Numeric and Textual
Information (continued)
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Converting Decimal Fractions To Binary
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Converting Decimal Fractions To Binary

• Note that the binary representation of 0.410 is a
  repeating base two fraction, .01102
• This just says that 0.4 cannot be exactly
  represented as a sum of inverse powers of two.
• .875
• .400

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Binary Representation of Textual Information

• Many transformations exist and all are arbitrary
• Two popular
• EBCDIC (Extended Binary Coded Decimal Interchange
  Code) by IBM
• ASCII (American Standard Code for Information
  Interchange) by American National Standards
  Institute (ANSI)
• Most widely used
• Every letter/symbol is represented by 7 bits
• How many letters/symbols do we have in total?
• A-Z (26) , a-z (26) , 0-9 (10), symbols (33),
  control characters (33)
• If using 1 byte/character ? we have one extra bit
• Extended ASCII-8 (more mathematical and graphical
  symbols or phonetic marks) --- 256

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Binary Representation of Numeric and Textual
Information (continued)

• Characters are mapped onto binary numbers
• ASCII code set
• 8 bits per character 256 character codes
• UNICODE code set
• 16 bits per character 65,536 character codes
• Text strings are sequences of characters in some
  encoding

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Binary Representation of Images

• Representing image data
• Images are sampled by reading color and intensity
  values at even intervals across the image
• Each sampled point is a pixel
• Image quality depends on number of bits at each
  pixel

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Binary Representation of Images
Divide the screen into a grid of cells each
referred to as a pixel 512256 image ? grid has
512 columns and 256 rows Pixel values and sizes
depend on the type of the image For black
white images, we can use 1 bit for every pixel
such that 1 ? black and 0 ? white For grayscale
images, we use 1 byte where 255 ? black and 0 ?
white and anything in between is gray
(higher/lower values are closer to
black/white) For color images, we need three
values per pixel (depends on the color scheme
used) Red/Green/Blue We need 1 byte per value
? 3 bytes per pixel For an 512x256 image
512x256x3 384K bytes
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Binary Representation of Movies

• Image movies are built from a number of images
  (or frames) that are displayed in a certain
  sequence at high speeds
• 30 frames per second
• 2-hr movie needs (assume previous image used)
• 384K 30 60 120 83 GB (billion) bytes!
• People use compression to reduce large movie
  sizes
• Usually the change between two consecutive images
  is small ? store only difference between frames
  (Temporal compression)
• Large areas with the same color can be stored by
  saving the boundary pixels only (everything
  within the boundaries has the same value)
  (Spatial compression)

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Binary Representation of Sound

• Sound/Audio Data
• An object produces sound when it vibrates in
  matter (e.g. air)
• A vibrating object sends a wave of pressure
  fluctuations through the atmosphere
• We hear different sounds from different vibrating
  objects because of variations in the sound wave
  frequency
• Higher wave frequency means air pressure
  fluctuation switches back and forth more quickly
  during a period of time
• We hear this as a higher pitch (tone)
• When there are fewer fluctuations in a period of
  time, the pitch is lower
• The level of air pressure in each fluctuation,
  the wave's amplitude or height, determines how
  loud the sound is

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Binary Representation of Sound

• Numbers used to represent the amplitude of sound
  wave
• Analog is continuous and we need digital
• Digitize the sound signal
• Measure the voltage of the signal at certain
  intervals (40,000 per sec)
• Reconstruct wave
• Compression can also be used for audio files
• MP3 reduces size to 1/10th
• ? faster transfer over the Internet

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• Figure 4.5
• Digitization of an Analog Signal
• (a) Sampling the Original
• Signal
• (b) Recreating the
• Signal from the Sampled
• Values

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Binary Representation of Sound and Images

• Storage required
• Text 300 page novel
• 100,000 words x 5 char/word x 8 bits/char 4
  million bits
• Music WAV
• 44,100 samp/sec x 16bits/samp x 60 sec/min 42
  Mbits / min
• For stereo there are left and right channels gt
  84 Mbits / min
• Digital Photo 3 Megapixel camera, 24 bit color
• 3,000,000 pixels/photo x 24 bits/pixel 72
  Mbits 7 Mbytes

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Binary Representation of Sound and Images

• Data Compression
• Lossless
• Run-length Encoding (RLE)http//en.wikipedia.org/
  wiki/Run-length_encoding
• Variable length code sets (Huffman
  code)http//en.wikipedia.org/wiki/Huffman_coding
  
• Lossy
• JPEG pictureshttp//en.wikipedia.org/wiki/JPEG
• MP3, ATRAC audiohttp//en.wikipedia.org/wiki/MP
  3

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Binary Representation of Sound and Images

• Digital image and audio have a lot of advantages
  over non-digital ones
• Can easily be modified by changing the bit
  pattern
• Image enhancement, noise/distortion removal, etc
  
• Superimpose one sound on another or image on
  another results in newer ones
• http//en.wikipedia.org/wiki/Digital_audio_editor
• http//en.wikipedia.org/wiki/Video_editing_softwar
  e

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Back To Circuit Design And Construction

• Compare-for-equality circuit
• Addition circuit
• Both circuits can be built using the
  sum-of-products algorithm

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A Compare-for-equality Circuit

• Compare-for-equality circuit
• CE compares two unsigned binary integers for
  equality
• Built by combining together 1-bit comparison
  circuits (1-CE)
• Integers are equal if corresponding bits are
  equal (AND together 1-CE circuits for each pair
  of bits)

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A Compare-for-equality Circuit (continued)

• 1-CE circuit truth table

a b Output
0 0 1
0 1 0
1 0 0
1 1 1
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A Compare-for-equality Circuit (continued)

• 1-CE Boolean expression
• First case (NOT a) AND (NOT b)
• Second case a AND b
• Combined
• ((NOT a) AND (NOT b)) OR (a AND b)

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• Figure 4.22
• One-Bit Compare for Equality Circuit

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N-bit Compare for Equality Circuit
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Binary Addition
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An Addition Circuit

• Addition circuit
• Adds two unsigned binary integers, setting output
  bits and an overflow
• Built from 1-bit adders (1-ADD)
• Starting with rightmost bits, each pair produces
• A value for that order
• A carry bit for next place to the left

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An Addition Circuit (contd)
Full 4-bit Adder Circuit
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An Addition Circuit (continued)

• 1-ADD truth table
• Input
• One bit from each input integer
• One carry bit (always zero for rightmost bit)
• Output
• One bit for output place value
• One carry bit

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An Addition Circuit (continued)
ai bi ci Ci1 Sumi
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1
Truth Table
Full 1-bit Adder
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• Figure 4.24
• The 1-ADD Circuit and Truth Table

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An Addition Circuit (continued)

• Building the full adder
• Put rightmost bits into 1-ADD, with zero for the
  input carry
• Send 1-ADDs output value to output, and put its
  carry value as input to 1-ADD for next bits to
  left
• Repeat process for all bits

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Circuit Analysis Introduction

• To analyze what a circuit does there are three
  approaches.
• Convert the circuit into its corresponding truth
  table by hand
• Simulate the circuit and generate its truth table
  from observations
• Build the circuit and study its behavior fro
  various inputs

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Circuit Analysis Intro. Contd.
Generate truth table by hand ? Remove
Internal Lines
inputs inputs internal lines internal lines internal lines Output
a b c d e z
0 0 0 1 0 0
0 1 0 1 1 1
1 0 0 1 1 1
1 1 1 0 1 0
inputs inputs output
a b z
0 0 0
0 1 1
1 0 1
1 1 0

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Control Circuits

• Do not perform computations
• Choose order of operations or select among data
  values
• Major types of controls circuits
• Multiplexors
• Select one of inputs to send to output
• Decoders
• Sends a 1 on one output line, based on what input
  line indicates

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Control Circuits Contd.
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Control Circuits (continued)

• Decoder
• Form
• N input lines
• 2N output lines
• N input lines indicate a binary number, which is
  used to select one of the output lines
• Selected output sends a 1, all others send 0

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Control Circuits Decoders
A 1-to-2 Decoder
Input Out 0 Out 1
0 1 0
1 0 1
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Control Circuits Decoders
Input Lines Input Lines OutputLines OutputLines OutputLines OutputLines
in1 in2 out0 out1 out2 out3
0 0 1 0 0 0
0 1 0 1 0 0
1 0 0 0 1 0
1 1 0 0 0 1
A 2-to-4 Decoder
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• Figure 4.29
• A 2-to-4 Decoder Circuit

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Control Circuits (continued)

• Decoder purpose
• Given a number code for some operation, trigger
  just that operation to take place
• Numbers might be codes for arithmetic add,
  subtract, etc.
• Decoder signals which operation takes place next

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Control Circuits (continued)

• Multiplexor form
• 2N regular input lines
• N selector input lines
• 1 output line
• Multiplexor purpose
• Given a code number for some input, selects that
  input to pass along to its output
• Used to choose the right input value to send to a
  computational circuit

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Control Circuits (continued)
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• Figure 4.28
• A Two-Input Multiplexor Circuit

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A Two-Input Multiplexor Circuit Using A 1-to-2
Decoder
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A 4-Input Multiplexor Circuit Using A 2-to-4
Decoder
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Decoder Example 1
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Multiplexor Example 1
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Summary

• Digital computers use binary representations of
  data numbers, text, multimedia
• Binary values create a bistable environment,
  making computers reliable
• Boolean logic maps easily onto electronic
  hardware
• Circuits are constructed using Boolean
  expressions as an abstraction
• Computational and control circuits may be built
  from Boolean gates