Computer Engineering

1
Computer Engineering

• By
• Dr. M. Moustafa Hassan
• Elec. Power Engineering
• Cairo University, Giza, Egypt

2
Computer Engineering

• Topics
• Binary Representation
• Reliability thereof
• Binary Storage Devices
• Learning Objectives
• Understand why binary is used, to translate
  between decimal and binary and to do simple
  binary calculations
• Know the conditions of a bistable environment and
  the way light switches, magnetic cores and
  transistors meet them.

3
Data Representation

• Notational conventions for representing
  information
• Decimal representation for numerical values
• 0,1,2,3,4,5,6,7,8,9
• Signed notation ,- (unsigned means positive
  only)
• Decimal Notation for real numbers
• Decimal point separates whole number part from
  fractional part
• The 26 letters A,B,C,X,Y,Z for text
• Plus lowercase and punctuation
• Digitizing Sound and Images

4
Data Representation
5
Decimal Representation

• In decimal, the following symbols are used to
  represent different values 0, 1, 2, 3, 4, 5, 6,
  7, 8, 9.
• Heres four thousand, seven hundred and
  sixty-three in decimal.
• 1000s  100s   10s    1s 103    102   
  101    100   4     7     6      3
• or, it can be written this way
• (4 x 1000) (7 x 100) (6 x 10) (3 x 1)
  4763

6
Binary Representation

• In the binary system, just two symbols are used
  0 and 1.
• Therefore any number must be represented by 0s
  and 1s only
• Lets examine a 6 digit (bit) binary number
• 1 1 1 0 0 1
• Position value 25 24 23 22 21 20
• 32 16 8 0 0 1
• 57
• This kind of a table makes conversions easy

7

• Binary-to-Decimal
• Conversion Table

8
Converting Decimal Numbers to Binary

• 53 32 16 4 1
• 25 24 22 20
• 125 124 023 122 021 120
• 110101 in binary
• 00110101 as a full byte in binary
• 211 128 64 16 2 1
• 27 26 24 21 20
• 127 126 025 124 023 022
• 121 120
• 11010011 in binary

9
Converting Decimal Numbers to Binary

• What is 10011010 in decimal?
• 10011010 127 026 025 124 123
  
• 022 121 020
• 27 24 23 21
• 128 16 8 2
• 154
• What is 00101001 in decimal?
• 00101001 027 026 125 024 123
• 022 021 120
• 25 23 20
• 32 8 1
• 41

10
Max. Integer

• Every computer has a maximum number of bits used
  to store an integer
• Given k bits, the largest unsigned integer is 2k
  - 1
• 16 bit 216 1 65,535
• 32 bit 232 1 4,294,967,296
• Adding just one more (than max) causes an
  arithmetic overflow error.

11
Neg. Integer Sign/Magnitude

• We desire the availability of negative integers
• Let the most significant bit will represent
  negation negative if on, positive if off
• Our previous 6 digit binary now needs an extra
  bit to make it negative
• Here it is in full 1 byte representation
• Can you see a problem with this scheme?

1 0 1 1 1 0 0 1 Position value neg. 26 25 24
23 22 21 20 - 0 32 16 8 0 0
1 - 57
12
Neg. Integer Twos Complement

• The most common way of storing integers
• The most significant bit will represent negative
  128 (-128)
• Heres 57 and -57 in full 1 byte representation
• Whats the range of a 1 byte integer ?

1 1 0 0 0 1 1 1 Position value -27 26 25 24 2
3 22 21 20 -128 64 0 0 0
4 2 1 -
57 0 0 1 1 1 0 0 1 Position
value -27 26 25 24 23 22 21 20 -0 0
32 16 8 0 0 1
57
13
Representing Real Numbers

• Real numbers may be put into binary scientific
  notation for storage a x 2b
• First, convert from decimal to binary
• Example
• 5.75 is 101.11 in binary
• 5 is 101 and
• .75 is .11
• in decimal there are 10ths, 100ths, 1000ths, etc
  while in binary
• there are halves, quarters, eighths, etc
• So .75 is made up of a half and a qtr written .11

14
Real Numbers
Step 3) Normalize 5.75 101.11 x 20 5.75
10.111 x 21 5.75 1.0111 x 22 5.75 .10111 x
23 Which is (1/2 1/8 1/16 1/32) x 8 5.75

• Second, use scientific binary notation
• 101.11 is 101.11 x 20
• Third, normalize so that first significant digit
  is immediately to the right of the binary point
  (see next for further explanation)
• .10111 x 23
• Fourth, store mantissa and exponent

0 1 0 1 1 1 0 0 0 0 0 0 0 0 1 1
Mantissa
Exponent
Sign of Mantissa
Sign of Exponent
Binary Point
15
Negative Real Numbers

• First, convert to binary
• -5/16 is -(1/4 1/16) is -0.0101
• Second, use scientific binary notation
• -0.0101 x 20
• Third, normalize
• -.101 x 2-1
• Fourth, stored mantissa and exponent

1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1
Mantissa
Exponent
Sign of Mantissa
Sign of Exponent
Binary Point
16
Lets Practice

• Convert the following to binary using the
  standard notation
• 78, 255
• Convert the following to binary using twos
  complement notation
• -78, -105, 118
• Convert several bit patterns to decimal
• HINT use windows calc to check your answers
• Convert -3.0625 to binary

17
Binary Representation of Characters

• Each character is represented by a unique
  (binary) number
• The ASCII table is the most common scheme
• Uses 8 bits 28 or 256 different characters
• First 7 bits only (actually 8-bit ASCII is a
  misnomer)
• Unicode
• Uses 16 bits 216 or 65,536 different characters
• Go to http//www.cs.tut.fi/jkorpela/chars.html
  for an excellent tutorial on character code
  issues
• IMPORTANT NOTE the numeric value of number
  characters is NOT the same as the usual
  arithmetic value.
• ie The character 7 has the numeric value 55

18
Examples of ASCII Codes
19
(No Transcript)
20
IMPORTANT NOTE

• The computer does not know what a specific bit
  pattern in memory means
• The meaning is imposed only as a result of how
  some process has been designed to interpret the
  pattern
• Strong Typing In writing programs we declare
  how we will use locations in memory (java c)
• boolean b means 1 bit that can be true or false
• byte number means a signed 8 bit integer
• int number means a signed 32 bit integer
• char ch means a 16 bit uni-code character
  (note we only discuss the 7 bit portion used in
  the lower ASCII char set)
• Loose Typing Some programming (scripting)
  languages are able to do this automatically (php,
  perl)
• Which is better? it depends! Read More !!.

21
Binary Representation of Sound

• Multimedia data is sampled to store in digital
  form, with or without detectable differences
• Representing sound data
• Sound data must be digitized for storage in a
  computer
• Digitizing means periodic sampling of amplitude
  values

22
Sound (continued)

• From samples, original sound may be approximated
• To improve the approximation
• Sample more frequently
• Use more bits for each sample value

23

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

24
Representing Image Data

• Images are sampled by reading color and intensity
  values at even intervals (a grid) across the image

25
Representing Image Data

• The size of the grid is called resolution
• 1024 x 768 or 1600 x 1200
• Each sampled point is a pixel
• Image quality depends on number of bits at each
  pixel
• True-Color (24bit) uses 1 byte for each of red,
  green blue to represent over 16 million colours
  at each pixel
• What is the storage requirement (size) of a
  True-Color image with a resolution of 1600 x 1200?

26
Putting it into Perspective

• Encoding the text (no images) like a text book
• 100,000 words x 5 char/word x 8 bits/char
  4,000,000 bits
• 1 minute of sound using MP3 (44,100 samples/sec
  using a 16 bit depth
• 44,100 samples/sec x 16 bits/sample x 60
  sec/minute 42,000,000 bits
• 1 photo using a 3 mega pixel camera
• 3,000,000 pix/photo x 24 bits/pixel 72,000,000
  bits

27
Data Compression

• Compression Ratio
• Ratio Size of Uncompressed / Size of Compressed
• Lossless
• No data is lost during compression
• Used when compressing text
• 1.5 to 1.7 compression ratios
• Lossy
• The higher the compression the higher the loss
• Used for sound and images
• 10, 20 higher compression ratios

28
Reliability

• Q Why binary ?
• A Electrical systems work best in a bistable
  environment.
• That is, an electrical component can remain
  accurate as it ages (and drifts).
• If it only needs to deal with voltage or no
  voltage circuit on/circuit off
• The text points out that nearly a 50 drift in
  voltage is still tolerable when interpreting a
  stored value.

29
Binary Storage Devices

• Binary Hardware Criteria
• 2 stable energy states
• A large energy barrier between them
• Ability to read the state without destroying it
• Ability to change the state as needed
• Two approaches
• Magnetic Cores historic
• Transistors current

30
Magnetic Cores

• An Iron oxide-coated donut can be magnetized by
  running current through a wire strung through its
  whole
• The direction of the resulting magnetic field is
  dependant upon the direction of the current
• right-hand rule

Even though these mag cores where small (1/50
dia) it would take a cube 230 per side to
construct a modern 512 mb memory !!!
31
Magnetic Cores to Represent Binary Values
32
Transistor

• Just a switch its turned on/off electrically
• Current technology (at text book printing)
• Switch state 1 billionth of a sec
• Density 30 to 50 million per cm2
• Made from semiconductors (silicon or gallium
  arsenide)
• Transistors are printed photographically onto a
  wafer of silicon using a mask.
• This is used to produce many copies of the
  circuit (chip).

33
Relationship of Transistors, Chips
34
Transistor Model
35
Interesting Links

• Binary Numbers
• http//www.math.grin.edu/rebelsky/Courses/152/97F
  /Readings/student-binary.html
• Character Issues
• http//www.cs.tut.fi/jkorpela/chars.html
• Core Memory
• http//www.psych.usyd.edu.au/pdp-11/core.html
• Transistor
• http//www.pbs.org/transistor/
• Semiconductor
• http//www.answers.com/topic/semiconductor

36
George Boole (1815 1864)

• Created a new form of logic containing the values
  true and false and the operators AND OR NOT
• Developed a set of rules to interpret and
  manipulate expressions that contain these values
• 100 years later his theories became the framework
  for designing computer systems

37
Boolean Logic

• Boolean Logic deals with manipulating the values
  true and false
• Anything stored in a sequence of binary digits
  can also be viewed as a sequence of logical
  values, true and false
• The operators AND, OR, NOT map a set of True
  False values into a single (true/false) result

38
Boolean Logic

• Any expression can be either true or false
• IE x y 25
• A boolean expression combines arithmetic
  expressions using the boolean operators
• AND (x y 25 AND y lt 100)
• OR (x y OR x lt 10)
• NOT (NOT (x y))

39
AND rule

• a AND b, also written as a?b, is true only if the
  value of a and the value of b is true
• Otherwise the expression a?b has the value of
  false
• To check a test score S in the range 90 to 100
  inclusive
• Sgt90 AND Slt100

40
Truth Table for AND Operation
Input Input Output
a b a AND b
 
FALSE FALSE FALSE
FALSE TRUE FALSE
TRUE FALSE FALSE
TRUE TRUE TRUE
41
OR rule

• a OR b, also written as ab, is true if the value
  of a is true, if value of b is true, or both are
  true
• Otherwise the expression ab has the value of
  false
• To check a students major
• (major math) OR (major comp science)

42
Truth Table for OR Operation
Input Input Output
a b a OR b
 
FALSE FALSE FALSE
FALSE TRUE TRUE
TRUE FALSE TRUE
TRUE TRUE TRUE
43
NOT rule

• NOT reverses or complements the value of the
  Boolean expression
• Not a, also written as a, is true if a has the
  value of false and false if a has the value of
  true
• To compare a students gpa
• (gpa gt 3,5) is true if your gpa is greater than
  3.5
• NOT (gpa gt 3.5) is true only if your gpa 3.5

44
Truth Table for NOT Operation
Input Output
a NOT a
 
FALSE TRUE
TRUE FALSE


45
Precedence

• The order of precedence rule for boolean
  operations is NOT first, AND second and OR third.
• So
• x AND NOT y OR z
• Means
• (x AND (NOT y)) OR z
• Parentheses can be used to change the order and
  are recommended where needed to enhance
  readability

46
Other Boolean Logic Symbols
English Math Java
TRUE 1 true
FALSE 0 false
NOT ? !
(Negation NOT false is true and NOT
true is false)
AND ?
(Conjunction P AND Q is true only when both P
Q are true)
OR ?
(Disjunction P OR Q is true when P is true,
or Q is true, or both)
47
Truth Table 1

• Shows the truth or falsity of a logical
  expression for every possible logical value
  assignment.

P (Q R)
0 0 0 1 0 0 0 1
0 0 0 1 1 1 1 1
48
Truth Table 2
P !(Q R)
0 0 0 0 1 0 0 0
0 1 1 1 0 1 1 1
1 0 0 0 1 0 0 0
49
Review of Boolean Algebra

• Just like Boolean logic
• Variables can only be 1 or 0
• Instead of true / false

50
Review of Boolean Algebra

• Not is a horizontal bar above the number
• 0 1
• 1 0
• Or is a plus
• 00 0
• 01 1
• 10 1
• 11 1
• And is multiplication
• 00 0
• 01 0
• 10 0
• 11 1

_
_
51
Review of Boolean Algebra
_ _ _

• Example translate (xyz)(xyz) to a Boolean
  logic expression
• (x?y?z)?(?x??y??z)
• We can define a Boolean function
• F(x,y) (x?y)?(?x??y)
• And then write a truth table for it

x y F(x,y)
1 1 0
1 0 0
0 1 0
0 0 0
52
Basic Logic Gates

• Not
• And
• Or
• Nand
• Nor
• Xor

53
Question 1

• Find the output of the following circuit
• Answer (xy)y Or (x?y)??y

xy
__
54
Question 2

• Find the output of the following circuit
• Answer xy Or ?(?x??y) x?y

55
Question 3

• Draw the circuits for the following Boolean
  algebraic expressions
• xy

__
56
Question 4

• Draw the circuits for the following Boolean
  algebraic expressions
• (xy)x

_______
xy
57
Writing XOR Using AND/OR/NOT

• p ? q ? (p ? q) ? (p ? q)
• x ? y ? (x y)(xy)

x y x?y
1 1 0
1 0 1
0 1 1
0 0 0
____
xy
(xy)(xy)
xy
xy
58
Gates AND, OR, NOT
A gate operates on binary inputs to produce
binary outputs
59
The NOT Gate
0
1
1
0
60
NAND and AND Gates
61
NOR and OR Gates
62
Important Note

• Gates are another abstraction in Computer Science
• we dont have to deal with transistors,
  resistors and capacitors or voltages, current and
  resistance the building block of circuit
  construction (for us) will be gates and the rules
  of Boolean logic!

63
Building Computer Circuits

• A combinational circuit (just circuit to us) is
  made up of logic gates
• That transform a set of binary inputs into a set
  of binary outputs,
• Where the values of the outputs depend only on
  the values of the inputs (no feedback loops)
• Sequential circuits (containing feedback loops)
  feed the output from a gate back as input to an
  earlier gate not discussed here

64
Diagram of a Typical Computer Circuit
65
Our First Circuit

• c (a OR b)
• d NOT ( (a OR b) AND (NOT b ) )

66
Circuit Construction Algorithm

• Construct the truth table
• Every output of 1 denotes a sub-expression
• Create sub-expressions using AND NOT
• each sub-expression becomes a sub-circuit of the
  completed circuit
• Combine sub-expressions using OR
• sub-circuits are joined with OR gates
• Construct the circuit
• Repeat 2 through 4 for each output column

67
The Sum-of-Products Circuit Construction Algorithm
68
Truth Table
input input output output
a b o1 o2
0 0 1 0
0 1 0 1
1 0 0 1
1 1 1 0

• What general concept do you think that each
  output represents?
• Create the sub-expressions
• Combine them
• Construct the circuit

69
Circuit Construction

• IMPORTANT The Sum of Products Algorithm
  doesnt always produce an optimal circuit
• The circuit on the right has the same output as
  the one on the left.

70
Addition Circuit

• Binary addition
• 11 (the carry bit)
• 001101 (bin value 13)
• 001110 (bin value 14)
• 011011 (bin value 27)
• Relate this to the truth table required for the
  circuit (next slide)

71
How to Add Binary Numbers

• Consider adding two 1-bit binary numbers x and y
• 00 0
• 01 1
• 10 1
• 11 10
• Carry is x AND y
• Sum is x XOR y
• The circuit to compute this is called a half-adder

x y Carry Sum
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0
72
The Half-Adder

• Sum x XOR y
• Carry x AND y

73
Using Half Adders

• We can then use a half-adder to compute the sum
  of two Boolean numbers

0
0
1
1 1 0 0 1 1 1 0
0
1
0
?
74
How to Fix This

• We need to create an adder that can take a carry
  bit as an additional input
• Inputs x, y, carry in
• Outputs sum, carry out
• This is called a full adder
• Will add x and y with a half-adder
• Will add the sum of that to the carry in
• What about the carry out?
• Its 1 if either (or both)
• xy 10
• xy 01 and carry in 1

x y c carry sum
1 1 1 1 1
1 1 0 1 0
1 0 1 1 0
1 0 0 0 1
0 1 1 1 0
0 1 0 0 1
0 0 1 0 1
0 0 0 0 0
75
The Full Adder

• The HA boxes are Half-Adders

76
1-Add Circuit and Truth Table

• Note how the truth table supports binary addition
  on the previous slide
• Create sub-expressions for Si
• Write the expression for Si

77
Sum Output for 1-Add
78
1-Add
79
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

80
The Full Adder Add Circuit
81
The Process of Abstraction

• Full Adder as a circuit
• Full Adder using 1-Add
• 1-Add using gates
• Gates using transistors

82
The Full Adder

• The full circuitry of the full adder

83
Adding Bigger Binary Numbers

• Just chain of full adders together

84
Adding Bigger Binary Numbers

• A half adder has 4 logic gates
• A full adder has two half adders plus a OR gate
• Total of 9 logic gates
• To add n bit binary numbers, you need 1 HA and
  n-1 FAs
• To add 32 bit binary numbers, you need 1 HA and
  31 FAs
• Total of 4931 283 logic gates
• To add 64 bit binary numbers, you need 1 HA and
  63 FAs
• Total of 4963 571 logic gates

85
More About Logic Gates

• To implement a logic gate in hardware, you use a
  transistor.
• Transistors are all enclosed in an Integrated
  Circuit IC.
• The current Intel Pentium IV processors have 55
  million transistors!

86
Pentium Math Error 1

• Intels Pentiums (60Mhz 100 Mhz) had a
  floating point error
• Graph of z y/x
• Intel reluctantlyagreed to replace them in 1994
• Graph from http//kuhttp.cc.ukans.edu/cwis/units/I
  PPBR/pentium_fdiv/pentgrph.html

87
Pentium Math Error 2

• Top 10 reasons to buy a Pentium
• 10 Your old PC is too accurate
• 8.9999163362 Provides a good alibi when the IRS
  calls
• 7.9999414610 Attracted by Intel's new "You don't
  need to know what's inside" campaign
• 6.9999831538 It redefines computing--and
  mathematics!
• 5.9999835137 You've always wondered what it
  would be like to be a plaintiff
• 4.9999999021 Current paperweight not big enough
• 3.9998245917 Takes concept of "floating point"
  to a new level
• 2.9991523619 You always round off to the nearest
  hundred anyway
• 1.9999103517 Got a great deal from the Jet
  Propulsion Laboratory
• 0.9999999998 It'll probably work!!

88
Flip-Flops

• Consider the following circuit
• What does it do?

89
Latches And Flip Flops
RS Flip Flop

• What are Flip Flops?
• Flip Flops are digital circuits that have
  feedback connections.
• Thus, can be used as memory devices, oscillators,
  etc.

An AND-OR gate used as a ones catching'' latch
and its timing diagram
90
In vivo digital circuits
91
Memory

• A flip-flop holds a single bit of memory
• The bit flip-flops between the two NAND gates
• In reality, flip-flops are a bit more complicated
• Have 5 (or so) logic gates (transistors) per
  flip-flop
• Consider a 1 Gb memory chip
• 1 Gb 8,589,934,592 bits of memory
• Thats about 43 million transistors!
• In reality, those transistors are split into 9
  ICs of about 5 million transistors each

92
The Digital Advantage!

• In a nut shell Digital signals are more reliable
  compared with analog signals!
• Positive Negative logic
• We will study - logic and use it in the
  design of digital circuits.
• Combinational vs. Sequential circuits.
• We will study these two types of circuits and use
  them to design complex systems.
• Asynchronous vs. Synchronous.
• Sequential circuits can be further classified as
  synchronous or asynchronous.
• We will study both types.

93
Transistors and Physical Size

• How many transistors are needed in a 32-Bit
  Adder?
• NOT 32 x 3 96 NOT gates _at_ 1 tran/gate
  96
• AND 32 x 16 512 AND gates _at_ 3
  tran/gate 1536
• OR 32 x 6 192 OR gates _at_ 3
  tran/gate 576
• Total 2208
• How big is this addition circuit?
• Built with vacuum tubes
• Built with magnetic cores
• Built with transistors
• NOTE optimized 32-bit addition circuits can be
  constructed with FAR fewer transistors (500 600)

94
Control Circuits

• Control circuits used to
• determine the order in which operations are
  performed
• select the correct data values to process
• Definitions (no other meaning matters to us)
• Multiplexor
• From 2N input, select 1 output, using N selector
  lines
• Decoder
• N input lines determine which ONE of 2N output
  lines

95
Multiplexor

• N selector lines can select one output from 2N
  input lines
• Useful to select data

96
Multiplexor Example
97
Decoder

• N input lines can determine which one (and ONLY
  one) of 2N output lines will be turned on
• Useful to select the correct instruction

98
Decoder Example
99
Truth Tables, Expressions Circuits

• You need to be able to create two of these given
  the third. Thats any two!
• Always use the sum of products algorithm
• Make sure you practice the process enough that
  you can FULLY answer an exam question.
• What do I mean by fully?
• You may also have to combine circuits to create
  more complex circuits.

100
Interesting Links

• Binary Numbers
• http//www.math.grin.edu/rebelsky/Courses/152/97F
  /Readings/student-binary.html
• Character Issues
• http//www.cs.tut.fi/jkorpela/chars.html
• Core Memory
• http//www.psych.usyd.edu.au/pdp-11/core.html
• Transistor
• http//www.pbs.org/transistor/
• Semiconductor
• http//www.answers.com/topic/semiconductor

101
Misconceptions And Difficulties

• The distinction between the ASCII representation
  of a number as a character and the binary
  representation of that number as an integer or
  float.
• Confusion with construction of gates is common
• Translating the goal of binary addition into the
  steps used in creating the adder circuit.
• Review the binary addition algorithm
• Function and design of both multiplexors and
  decoders is difficult. (roles/quantity of input,
  output and selector lines)

102
Analysis and Design Tools

• De Morgans Laws
• The duality principle
• Karnaugh maps provide an alternative technique
  for representing Boolean functions.
• Equivalent circuits (AND-OR and NAND-NAND
  circuits)

4-signal Karnaugh map
Karnaugh map representation of a 2-input AND gate
103
Karnaugh maps software
http//karnaugh.shuriksoft.com/
Interactive Karnaugh maps http//maui.theoinf.tu-
ilmenau.de/sane/projekte/karnaugh/embed_karnaugh.
html