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 SchemeQuizzes/ 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

Quizzes

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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 ComputersDIGITAL : 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: ENIACElectrical Numerical Integrator And

Computer (ENIAC): developed by Professor John Mauchly and his

graduate student John Presper Eckert For army’s 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 ComputerMemory

Central processing unit

I/O ProcessorInputOutput Devices

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• Structure and behavior of the computer as seen by

the user. It includes:1. Instruction set.2. Information formats.3. Techniques for addressing memory.

The course is an introduction to the three aspects.

Computer Architecture:

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 Boole’s variables "the Universe" and "Nothing" “True” and “False” 1 and 0Boole thought: Boole’s Algebra: representedmathematical systemization of human thought. Not true. But thinking about machines, which thinklike a human being powerful machines.

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Hollerith IBM 1881:Mechanical 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: Hollerith’s Tabulating Machines

Company International Business Machines

Logic gates:

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Buffer gate

Inverter or NOT gateLogic Symbol Truth Table Algebraic Expression

A Out01

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A, A’–A

Out

NOT gate

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Logic Symbol Truth Table Algebraic Expression

Gates (continued): AND and OR GATES

A B Out0 00 11 01 1

0001

A B Out0 00 11 01 1

0111

A.B, AB, A*B, A^B

A+B, AvB

AB

Out

AND

AB

Out

OR

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Gates (continued): XOR and NAND gates

Logic symbol Truth Table Algebraic ExpressionA B Out

0 00 11 01 1

0110

A B Out0 00 11 01 1

1110

A B

A.B, (AB)’

AB

Out

XOR

OutAB

NAND

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Gates (continued): NOR and XNOR gates

Logical Symbol Truth Table Algebraic ExpressionA B Out

0 00 11 01 1

1000

A B Out0 00 11 01 1

1001

A + B

A B

AB

Out

NOR

AB

Out

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.

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.

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/Identity_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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1 x + 0 = x x . 0 = 02 x + 1 = 1 x . 1 = x3 x + x = x

(Idempotency) x . x = x (Idempotency)

4 x + x’ = 1 x . x’ = 05 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)

Boolean identities:

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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 Morgan’s Theorem)

(x.y)’ = x’ + y’(De Morgan’s Theorem)

9 (x’)’ = x10

x + xy = x x (x + y) = x

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x + x’y = x + y x (x’ + y) = xy

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xy + xy’ = x (x + y).(x + y’) = x

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xy + x’z + yz = xy + x’z (Consensus Theorem)

(x+y)(x’+z)(y+z)=(x+y)(x’+z) (Consensus Theorem)

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A few comments:7(b) x + yz = (x + y)(x + Z) RHS = x + xy + xz + yz = x(1 + y + z) + yz = x + yz = LHS

Example 7(b):

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De Morgan’s Theorems8(a) (x + y)’ = x’.y’

De Morgan’s Theorem:

Xy

F

NOR

XY F

NOR

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x y x + y

(x+y)’

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

Proof:

Continuation of the proof: De’ Morgan’s theorem

Another Method for proving De Morgan’s 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 Morgan’s theorem …2Proof: (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) = 0Hence (x + y)’ = x’.y’.

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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

Examples 8(b) and 10 (b):

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Examples 11(a):11 (a) LHS = x + x’yUse 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

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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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 + x’z = RHS13 (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’ + x’y + yz + xz = x’.(x + y) + z.(x + y) = (x + y).(x’ + z) = RHS

Example 13:

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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

Boolean cont…

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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