60 -265 COMPUTER ARCHITECTURE I Digital Design Akshai

60 -265 COMPUTER ARCHITECTURE I: Digital Design Akshai Aggarwal 1

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 2

Grading Scheme Quizzes/ Individual Assignments Quizzes: to be conducted at the beginning of some of 10% 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 1 STJune 3

Quizzes Day and Date Time of Quiz for Section 51 Time of Quiz for Section 52 Wednesday, 13 th May 4 PM 5. 30 PM Wednesday, 20 th May 4 PM 5. 30 PM Wednesday, 27 th May 4 PM 5. 30 PM Wednesday, 3 rd June 4 PM 5. 30 PM Wednesday, 10 th June 4 PM 5. 30 PM 4

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. 5

ARRAYS OF BITS: 1001011 may represent • 75 in decimal, or • K in ASCII code or • some control code. 6

History: ENIAC Electrical Numerical Integrator And Computer (ENIAC): n n n 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 k. W power, 30 ton, 1500 sq ft of floor space Decimal machine; programming manually by setting switches and plugging and unplugging cables 7

1945: A New Project: Electronic Discrete Variable Computer (EDVAC) n n n 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 n Arithmetic Logic Unit and Control Unit CPU § Registers in CPU: Accumulator, Address Register, Program Counter, Data Register and other Registers n Input/Output n n Used binary numbers 8

FUNCTIONAL PARTS: - hardware: CPU, memory, I/O units -software • System software • Application program Logical view vs. Physical components 9

Architecture of a computer: n those properties, which directly affect the logical working of a program; n the attributes, which are apparent to a programmer Examples: instruction set, number of bits used to represent data 10

Von Neumann Architecture of a digital computer: Von Neumann Architecture of a Digital Computer Memory Central processing unit Input I/O Processor Output Devices 11

Computer Architecture: • 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. 12

Organization n 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 13

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. 14

Objectives of Boole n n n 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. 15

The Values of Boole’s variables n n n "the Universe" and "Nothing" “True” and “False” 1 and 0 Boole thought: Boole’s Algebra: represented mathematical systemization of human thought. Not true. But thinking about machines, which think like a human being powerful machines. 16

Hollerith IBM n 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 n 1882 -83: Instructor at MIT n 1896: Hollerith’s Tabulating Machines Company International Business Machines 17

Logic gates: Buffer gate Inverter or NOT gate Logic Symbol Truth Table A A Out NOT gate Out 0 1 1 0 Algebraic Expression – A’ A, 18

Gates (continued): AND and OR GATES Logic Symbol Truth Table A B A B Out AND Out OR Out 0 0 1 1 0 1 A 0 0 1 1 B 0 1 0 0 0 1 Out 0 1 1 1 Algebraic Expression A. B, A*B, A^B A+B, Av. B 19

Gates (continued): XOR and NAND gates Logic symbol Truth Table Algebraic Expression A B Out XOR Out NAND 0 0 1 1 0 1 0 1 1 0 A 0 0 1 1 B 0 1 Out 1 1 1 0 A B A. B, (A B)’ 20

Gates (continued): NOR and XNOR gates Logical Symbol Truth Table Algebraic Expression A B Out NOR A B Out XNOR 0 0 1 1 0 0 0 A B Out 0 0 1 0 1 0 0 1 1 1 A+B A B 21

Boolean Gates: AND, OR, NOT n XOR, XNOR n NAND, NOR n 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 n n n Fan-in. 22

Boolean Algebra n n 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 n 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) n OR operation: 0 is the Identity Element for OR: 1 + 0 = 1; 0 + 0 = 0 n AND operation: 1 is the Identity Element for AND: 1. 1 = 1; 0. 1 = 0 24

Boolean identities: 1 2 3 x+0=x x+1=1 x+x=x (Idempotency) 4 5 x + x’ = 1 x+y=y+x (Commutativity) 6 x + (y + z) = (x + y) + z (Associativity) x. 0=0 x. 1=x x. x= x (Idempotency) x. x’ = 0 x. y=y. x (Commutativity) x. (y. z) = (x. y). z (Associativity) 25

Continuation of Boolean Identities: 7 8 x. (y + z) = x. y + x. z (Distributive) (x + y)’ = x’. y’ (De Morgan’s Theorem) 9 10 11 12 x + (y. z)=(x + y). (x + z) (Distributive) (x. y)’ = x’ + y’ (De Morgan’s Theorem) (x’)’ = x x + xy = x x + x’y = x + y xy + xy’ = x 13 xy + x’z + yz = xy + x’z (Consensus Theorem) x (x + y) = x x (x’ + y) = xy (x + y). (x + y’) = x (x+y)(x’+z)(y+z)=(x+y)(x’+z) (Consensus Theorem) 26

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 27

De Morgan’s Theorem: De Morgan’s Theorems 8(a) (x + y)’ = x’. y’ F X y NOR X Y F NOR 28

Continuation of the proof: De’ Morgan’s theorem Proof: x y x + y (x+y)’ x’ y’ x’. y’ 0 0 0 1 1 0 1 0 1 1 1 0 0 29

Another Method for proving De Morgan’s theorem … 1 Identity 4(a): A + A’ = 1 n Identity 4(b): A. A’ = 0 n 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. n 30

Another Method for proving De Morgan’s 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 = 0. y’ + x’. 0 by using Th. 4(b) =0 Hence (x + y)’ = x’. y’. 31

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 32

Examples 11(a): 11 (a) LHS = x + x’y Use x = x. (1 + y) by using 1 + y = 1 and x. 1 = x + x. y = x. x + x. y by using 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 33

Examples 11(b) and 12: 11(b) 12 (b) x. (x’ + y) = x. x’ + x. y = x. y by using x. x’ = 0 and (x + y). (x + y’) = x + x. y’ + x. y + y. y’ = x. (1 + y’ + x) 0+A=A by using x. x’ = 0 and 0 + A =x by using 1 + y’ + x = 1 and x. 1 = x 34

Example 13: 13 (a) 13 (b) 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 = RHS LHS = (x + y). (x’ + z). (y + z) = (x. y + x. z + 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 35

Boolean Algebra n n n 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 36

Boolean cont… Decimal A B C D 0 0 0 1 0 2 0 1 0 0 3 0 1 1 0 4 1 0 0 0 5 1 0 6 1 1 0 0 7 1 1 37