60-265 COMPUTER ARCHITECTURE I: Digital Design

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60-265 COMPUTER ARCHITECTURE I
Digital Design

• Akshai Aggarwal

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• Course Outline
• Binary, Octal and Hexadecimal number system
• Digital logic and Boolean Algebra
• Combinational and sequential circuit design
• Digital components decoders, multiplexers,
  registers,
• counters, memory etc. Digital Integrated
  Circuits
• Register transfer and micro-operations
• Basic computer organization and design
• CPU structure, control unit design, interrupt
  handling
• ? HOW DOES A COMPUTER WORK?
• Elementary assembly language instruction set and
  the execution process

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Grading Scheme
Quizzes/ Individual Assignments 10 Quizzes to be conducted at the beginning of some of the labs For dates, please see the web-site.
Midterm 1 15 Saturday May 30, 12.30 PM, Venue Erie 1118
Midterm 2 15 Saturday June 13, 12.30 PM, Venue Erie 1118
Final 40 As scheduled by the university
Labs 10 Attendance is mandatory
Group Assignment 10 Submission of Assignment in the Lab on Monday 1STJune
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Quizzes
Day and Date Time of Quiz for Section 51 Time of Quiz for Section 52
Wednesday, 13th May 4 PM 5.30 PM
Wednesday, 20th May 4 PM 5.30 PM
Wednesday, 27th May 4 PM 5.30 PM
Wednesday, 3rd June 4 PM 5.30 PM
Wednesday,10th June 4 PM 5.30 PM
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• Digital Computers
• DIGITAL information represented by variables
  that take a limited number of values
• BINARY - reliability of hardware
• binary nature of human logic
• DIGITAL COMPUTER A discrete information
  processing system
• A SYSTEM An organized collection of components,
  that interact through communication links among
  themselves and with their environment to provide
  a predefined functionality.

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• ARRAYS OF BITS
• 1001011 may represent
• 75 in decimal, or
• K in ASCII code or
• some control code.

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History ENIAC

• Electrical Numerical Integrator And Computer
  (ENIAC)
• developed by Professor John Mauchly and his
  graduate student John Presper Eckert
• For armys Ballistic Research Lab for calculating
  range and trajectory tables for new weapons
• Start 1943 development completed in 1946 used
  for Nuclear bomb computations
• 18000 vacuum tubes, 140 kW power, 30 ton, 1500 sq
  ft of floor space
• Decimal machine programming manually by setting
  switches and plugging and unplugging cables

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1945 A New Project Electronic Discrete
Variable Computer (EDVAC)

• EDVAC planned by John Von Neumann as a Stored
  Program machine
• Developed at Princeton Institute for Advanced
  Studies as the IAS computer from 1946 -1952
• consisted of
• main memory
• Arithmetic Logic Unit and Control Unit ? CPU
• Registers in CPU Accumulator, Address Register,
  Program Counter, Data Register and other
  Registers
• Input/Output
• Used binary numbers

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• FUNCTIONAL PARTS
• - hardware CPU, memory, I/O units
• -software
• System software
• Application program
• Logical view vs. Physical components

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Architecture

• Architecture of a computer
• those properties, which directly affect the
  logical working of a program
• the attributes, which are apparent to a
  programmer
• Examples instruction set, number of bits used
  to represent data

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Von Neumann Architecture of a digital computer
Von Neumann Architecture of a Digital Computer
Memory
Central processing unit
Input
Output Devices
I/O Processor
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Computer Architecture

• Structure and behavior of the computer as seen by
  
• the user.
• It includes
• Instruction set.
• Information formats.
• Techniques for addressing memory.
• The course is an introduction to the three
  aspects.

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Organization

• Organization operational units and their
  interconnection for realizing the architectural
  specifications
• Determination of which hardware should be used
  and
• how the parts should be connected together

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Logic Gates and Boolean Algebra

• 1832 17 years old Boole inspired to put logical
  statements in a mathematical form
• 1854 The Laws of thought - George Boole.
• An investigation of the laws of thought, on
  which are founded the Mathematical Theories of
  Logic and Probabilities
• Binary Variables
• Logic Operations Truth Table, logic diagram,
• Boolean Expressions.

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Objectives of Boole

• to investigate the fundamental laws of those
  operations of the mind by which reasoning is
  performed
• to give expression to them in the symbolic
  language of a Calculus
• to deduce, from these studies, some probable
  imitations concerning the nature and constitution
  of the human mind.

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The Values of Booles variables

• "the Universe" and "Nothing"
• True and False
• 1 and 0
• Boole thought Booles Algebra represented
• mathematical systemization of human thought.
• ? Not true.
• But thinking about machines, which think
• like a human being ? powerful machines.

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Hollerith ? IBM

• 1881Mechanical Engineer at American Census
  Bureau, Washington devised a punched card system
  for processing the census data of 1890 the idea
  from his boss, John Shaw Billings
• 1882-83 Instructor at MIT
• 1896 Holleriths Tabulating Machines Company ?
  International Business Machines

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Logic gates
Buffer gate
Inverter or NOT gate Logic Symbol Truth Table
Algebraic Expression
A
Out

A, A
Out
0 1
1 0
A
NOT gate
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Gates (continued) AND and OR GATES
Logic Symbol Truth Table Algebraic
Expression
A B
Out
A B

• 0 0
• 0 1
• 0
• 1 1

0 0 0 1
A.B, AB, AB, AB
Out
AND
Out
A B

• 0 0
• 0 1
• 0
• 1 1

0 1 1 1
AB, AvB
A B
Out
OR
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Gates (continued) XOR and NAND gates
Logic symbol Truth Table Algebraic
Expression
A B
Out

• 0 0
• 0 1
• 0
• 1 1

0 1 1 0
A B
A ? B
Out
XOR
A B
Out
A.B, (A?B)

• 0 0
• 0 1
• 0
• 1 1

1 1 1 0
A B
Out
NAND
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Gates (continued) NOR and XNOR gates
Logical Symbol Truth Table Algebraic
Expression
A B
Out

• 0 0
• 0 1
• 0
• 1 1

1 0 0 0
A B
Out
A B
NOR
Out
A B

• 0 0
• 0 1
• 0
• 1 1

1 0 0 1
A B
Out
A ? B
XNOR
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Boolean Gates

• Gates
• AND, OR, NOT
• XOR, XNOR
• NAND, NOR
• Buffer
• NOT and Buffer are single-input gates.
• All the others are multi-input gates.
• The number of inputs to a gate is known as its
  Fan-in.

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Boolean Algebra

• Boolean Algebra uses Boolean variables.
• A Boolean variable can have only two possible
  values 0 or 1.
• Boolean operations any of the operations defined
  by logic gates
• Property of CLOSURE If A and B are Boolean
  variables, A B and A.B are also Boolean
  variables.

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Identity Element

• identity element (or neutral element) is a
  special type of element of a set with respect to
  a binary operation on that set. It leaves other
  elements unchanged when combined with them. This
  is used for groups and related concepts.
• (Ref http//en.wikipedia.org/wiki/Id
  entity_element)
• OR operation 0 is the Identity Element for OR
• 1 0 1 0 0 0
• AND operation 1 is the Identity Element for AND
• 1.1 1 0.1 0

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Boolean identities
1 x 0 x x . 0 0
2 x 1 1 x . 1 x
3 x x x (Idempotency) x . x x (Idempotency)
4 x x 1 x . x 0
5 x y y x (Commutativity) x . y y . x (Commutativity)
6 x (y z) (x y) z (Associativity) x . (y . z) (x . y) . z (Associativity)
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Continuation of Boolean Identities
7 x.(y z) x.y x.z (Distributive) x (y.z)(x y).(x z) (Distributive)
8 (x y) x.y (De Morgans Theorem) (x.y) x y (De Morgans Theorem)
9 (x) x
10 x xy x x (x y) x
11 x xy x y x (x y) xy
12 xy xy x (x y).(x y) x
13 xy xz yz xy xz (Consensus Theorem) (xy)(xz)(yz)(xy)(xz) (Consensus Theorem)
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Example 7(b)
A few comments 7(b)
x yz (x y)(x Z) RHS
x xy xz yz
x(1 y z) yz x
yz LHS
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De Morgans Theorem
De
Morgans Theorems 8(a) (x y) x.y
F
X Y
X y
F
NOR
NOR
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Continuation of the proof
De Morgans theorem
Proof
x y x y (xy) x y x . y
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0
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Another Method for proving
De Morgans theorem 1

• Identity 4(a) A A 1
• Identity 4(b) A.A 0
• To prove A is the complement of B ?
• prove that (i) A B 1 and (ii) A.B 0
• To prove (x y) x.y,
• consider A x.y and B x y
• Prove that (i) x.y (x y) 1 and
• (ii) x.y.(x y) 0 ?
• A is the
  complement of B.

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Another Method for proving
De Morgans theorem 2

• Proof (i) LHS x.y (x y)
• (x.y x) y
• x y y by using
  Th. 11(a)
• x 1 by
  using Th. 4(a)
• 1 by
  using Th. 2(a)
• (ii) LHS x.y.(x y) x.x.y x.y.y
• 0.y x.0 by using Th.
  4(b)
• 0
• Hence (x y) x.y.

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Examples 8(b) and 10 (b)
8(b) (x y) x y The proof is similar to
that for 8(a).


10(b) LHS x (x y)
x xy
x (1 y)
x RHS
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Examples 11(a)
11 (a) LHS x xy Use x x.(1 y)
by using 1 y 1 and x.1 x x
x.y x.x x.y by using x x.x
x.x x.y x.x
by using x.x 0 and A 0 A
LHS (x.x x.y x.x) x.y
x.(x y) x.(x y) (x
x).(x y) x y by using x
x 1 and 1.B B
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Examples 11(b) and 12

• 11(b) x.(x y)
• x.x x.y
• x.y by using x.x 0 and 0
  A A
• 12 (b) (x y).(x y)
• x x.y x.y y.y
• x.(1 y x)
• by using x.x 0
  and 0 A A
• x by using 1 y x 1 and
  x.1 x

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Example 13
13 (a) LHS x.y x.z y.z
x.y x.z (x x).y.z
x.y.(1 z)
x.z.(1 y) xy
xz RHS 13 (b) LHS (x y).(x
z).(y z) (x.y
x.z y y.z).(x z)
(x.z y.(1 x z)). (x z)
(y x.z).(x z)
x.y y.z x.x.z x.z
xx xy yz xz
x.(x y) z.(x y)
(x y).(x z) RHS
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Boolean Algebra

• Closure X Y, X.Y
• Identity Element 0, 1
• Commutation X Y Y X, X.Y
  Y.X
• Distribution X.(Y Z) X.Y X.Z,
• X (Y.Z) (X Y).(X
  Z)
• Complements X X 1,
  X.X 0
• Idempotency
• X X X, X.X X

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Boolean cont
Decimal A B C D
0 0 0 0 0
1 0 0 1 0
2 0 1 0 0
3 0 1 1 0
4 1 0 0 0
5 1 0 1 0
6 1 1 0 0
7 1 1 1 1