CENG 241 Digital Design 1 Lecture 1 Amirali

CENG 241 Digital Design 1 Lecture 1 Amirali Baniasadi amirali@ece. uvic. ca

CENG 241: Digital Design 1 Instructor: Amirali Baniasadi (Amir) Office hours: EOW 441, Only by appt. Email: amirali@ece. uvic. ca Office Tel: 721 -8613 Web Page for this class will be at http: //www. ece. uvic. ca/~amirali/courses/CENG 241/ceng 241. html Text: Digital Design Fourth edition, by Morris Mano, Prentice Hall Publishers 2

Course Structure z Lectures: Mostly follow textbook. z Reading assignments posted on the web for each week. z Homework: Some from the book some will be posted on the web site. z Quizzes: 3 in class exams. Dates will be announced in advance. z Note that the above is approximate. 3

Course Problems z Late homework 10% penalty per day up to maximum of 5 days (after that Homework will not be accepted) z Guide to completing assignments y Studying together in groups is encouraged y Discussion (only) y Work submitted must be your own 4

Course Philosophy z Book to be used as supplement for lectures (If a topic is not covered in the class, or a detail not presented in the class, that means I expect you to read on your own to learn those details) z Regular Homework (10%) z Lab (30%)- Attend orientation @ ELW A 359. z Two Midterms (30%)- Dates will be announced in advance. z Final Exam(30%) z To pass the course you should also pass the lab and the final exam. 5

What are my expectations? z Stay Positive and Enjoy. z Commitment: Regular study and homework submission 6

This Lecture z Digital Design? z Binary Systems 7

Binary storage & registers z How do we store binary information? z Binary cell : place to store one bit of information. 0 or 1. z Register: a group of binary cells. z Register transfer: An operation in a digital system 8

Binary storage & registers 9

Binary information processing Example: Add two 10 -bit binary numbers 10

Binary logic z Binary logic deals with variables that take on two discrete values and operations that assume logical meaning. z Logic gates: electronic circuits that operate on one or more input signals to produce an output signal. z Example x y x AND y 0 0 1 11

Electrical signals Two values: 0 or 1 12

Symbols for digital logic circuits 13

Input-Output signals for gates 14

Gates with multiple inputs 15

Boolean Algebra z Basic definitions: z z z x+0=0+x=x x. 1=1. x=x x. (y+z)=(x. y)+(x. z) x+(y. z)=(x+y). (x+z) x+x’=1 x. x’=0 16

Boolean Algebra Theorems z z z x+x=x x+1=1 x. 0=0 x+x. y=x x. (x+y)=x 17

Boolean Algebra Functions z examples: z F 1=x+y’. z z F 2=x’. y’. z+x’. y. z+x. y’ z =x’. z(y’+y)+x. y’ z F 2=x’. z+x. y’ A Boolean Function can be represented in many algebraic forms We look for the most simple form 18

Boolean Function: Example z Truth table z z z z z x 0 0 1 1 y 0 0 1 1 z 0 1 0 1 F 1 0 0 1 1 F 2 0 1 1 1 0 0 A Boolean Function can be represented in only one truth table forms 19

Boolean Function Implementation y’ Y’. z 20

Boolean Function Implementation X’. y’. z X’. y. z X. y’ X’. z 21

Complement of a function z De. Morgan’s theorem: z (x+y)’=x’. y’ (x. y)’=x’+y’ z What about three variables? z (x+y+z)’=? z Let A=x+y (A+z)’=A’. z’=(x+y)’. z’=x’. y’. z’ z (x. y. z)’=x’+y’+z’ 22

Canonical & Standard Forms z Consider two binary variables x, y and the AND operation z four combinations are possible: x. y, x’. y, x. y’, x’. y’ z each AND term is called a minterm or standard products z for n variables we have 2 n minterms z Consider two binary variables x, y and the OR operation z four combinations are possible: x+y, x’+y, x+y’, x’+y’ z each OR term is called a maxterm or standard sums z for n variables we have 2 n maxterms z Canonical Forms: z Boolean functions expressed as a sum of minterms or product of maxterms. 23

Minterms z z z z z x 0 0 1 1 y 0 0 1 1 z 0 1 0 1 Terms x’. y’. z’ x’. y’. z x’. y. z’ x’. y. z x. y’. z’ x. y’. z x. y. z’ x. y. z Designation m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 24

Maxterms z z z z z x 0 0 1 1 y 0 0 1 1 z 0 1 0 1 Designation M 0 M 1 M 2 M 3 M 4 M 5 M 6 M 7 Terms x+y+z’ x+y’+z’ x’+y+z’ x’+y’+z’ 25

How to express algebraically z Question: How do we find the function using the truth table? z z z z z Truth table example: x y z 0 0 0 1 1 1 0 0 1 1 1 F 1 0 0 1 1 F 2 0 1 1 1 0 0 26

How to express algebraically z 1. Form a minterm for each combination forming a 1 z 2. OR all of those terms z z z z z Truth table example: x y z 0 0 0 1 1 1 0 0 1 1 1 F 1 0 0 1 minterm x’. y’. z m 1 x. y’. z’ m 4 x. y. z m 7 z F 1=m 1+m 4+m 7=x’. y’. z+x. y’. z’+x. y. z=Σ(1, 4, 7) 27

How to express algebraically z z z z z Truth table example: x y z 0 0 0 1 1 1 0 0 1 1 1 F 2 0 0 0 1 1 1 minterm m 0 m 1 m 2 m 3 m 4 m 5 m 6 m 7 z F 2=m 3+m 5+m 6+m 7=x’. y. z+x. y’. z+x. y. z’+x. y. z=Σ(3, 5, 6, 7) 28

How to express algebraically z 1. Form a maxterm for each combination forming a 0 z 2. AND all of those terms z z z z z Truth table example: x y z 0 0 0 1 1 1 0 0 1 1 1 F 1 0 0 1 maxterm x+y+z M 0 x+y’+z’ M 2 M 3 x’+y+z’ x’+y’+z M 5 M 6 z F 1=M 0. M 2. M 3. M 5. M 6 = л(0, 2, 3, 5, 6) 29

How to express algebraically z z z z z Truth table example: x y z 0 0 0 1 1 1 0 0 1 1 1 F 2 0 0 0 1 1 1 maxterm x+y+z M 0 x+y+z’ M 1 x+y’+z M 2 x’+y+z M 4 z F=M 0. M 1. M 2. M 4=л(0, 1, 2, 4)=(x+y+z). (x+y+z’). (x+y’+z). (x’+y+z) 30

Maxterms & Minterms: Intuitions z Minterms: z If a function is expressed as SUM of PRODUCTS, then if a single product is 1 the function would be 1. z Maxterms: z If a function is expressed as PRODUCT of SUMS, then if a single product is 0 the function would be 0. z Canonical Forms: z Boolean functions expressed as a sum of minterms or product of maxterms. 31

Standard Forms Standard From: Sum of Product or Product of Sum 32

Nonstandard Forms Nonstandard From: Neither a Sum of Product nor Product of Sum 33

Implementations Three-level implementation vs. two-level implementation Two-level implementation normally preferred due to delay importance. 34

Digital Logic Gates 35

Summary? • • Read textbook & readings Be up-to-date Solve exercises Come back with your input & questions for discussion • Binary systems, Binary logic. 36