CENG 241 Digital Design 1 Lecture 1 Amirali Baniasadi

2 CENG 241: Digital Design 1 Instructor: Amirali Baniasadi (Amir) Office hours: EOW 441, Only by appt. Office Tel: Web Page for this class will be at Text: Digital Design Fifth edition, by Morris Mano, Prentice Hall Publishers

3 Course Structure zLectures: Mostly follow textbook. zReading assignments posted on the web for each week. zHomework: Some from the book some will be posted on the web site. zQuizzes: 3 in class exams. Dates will be announced in advance. zNote that the above is approximate.

4 Course Problems zLate homework 10% penalty per day up to maximum of 5 days (after that Homework will not be accepted) zGuide to completing assignments yStudying together in groups is encouraged yDiscussion (only) yWork submitted must be your own

5 Course Philosophy zBook 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) zRegular Homework (10%) zLab (30%)- Attend ELW A359. zThree Quizzes (30%)- Dates will be announced in advance. zFinal Exam(30%) zTo pass the course you should also pass the lab and the final exam.

6 What are my expectations? zStay Positive and Enjoy. zCommitment: Regular study and homework submission

7 This Lecture zDigital Design? zBinary Systems

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

9 Binary storage & registers

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

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

12 Electrical signals Two values: 0 or 1

13 Symbols for digital logic circuits

14 Input-Output signals for gates

15 Gates with multiple inputs

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

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

18 Boolean Algebra Functions z examples: z F1=x+y’.z zF2=x’.y’.z+x’.y.z+x.y’ z =x’.z(y’+y)+x.y’ zF2=x’.z+x.y’ A Boolean Function can be represented in many algebraic forms We look for the most simple form

19 Boolean Function: Example zTruth table zx y z F1 F2 z z z z z z z z A Boolean Function can be represented in only one truth table forms

20 Boolean Function Implementation y’ Y’.z

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

22 Complement of a function zDeMorgan’s theorem: z(x+y)’=x’.y’ (x.y)’=x’+y’ zWhat about three variables? z(x+y+z)’=? zLet A=x+y (A+z)’=A’.z’=(x+y)’.z’=x’.y’.z’ z (x.y.z)’=x’+y’+z’

23 Canonical & Standard Forms zConsider two binary variables x, y and the AND operation zfour combinations are possible: x.y, x’.y, x.y’, x’.y’ zeach AND term is called a minterm or standard products zfor n variables we have 2 n minterms zConsider two binary variables x, y and the OR operation zfour combinations are possible: x+y, x’+y, x+y’, x’+y’ zeach OR term is called a maxterm or standard sums zfor n variables we have 2 n maxterms zCanonical Forms: zBoolean functions expressed as a sum of minterms or product of maxterms.

24 Minterms z x y z Terms Designation z x’.y’.z’ m0 z x’.y’.z m1 z x’.y.z’ m2 z x’.y.z m3 z x.y’.z’ m4 z x.y’.z m5 z x.y.z’ m6 z x.y.z m7

25 Maxterms z x y z Designation Terms z M0 x+y+z z M1 x+y+z’ z M2 x+y’+z z M3 x+y’+z’ z M4 x’+y+z z M5 x’+y+z’ z M6 x’+y’+z z M7 x’+y’+z’

26 Boolean Function: Exampl How to express algebraically zQuestion: How do we find the function using the truth table? zTruth table example: z x y z F1 F2 z z z z z z z z

27 Boolean Function: Exampl How to express algebraically z1.Form a minterm for each combination forming a 1 z2.OR all of those terms zTruth table example: z x y z F1 minterm z z x’.y’.z m1 z z z x.y’.z’ m4 z z z x.y.z m7 zF1=m1+m4+m7=x’.y’.z+x.y’.z’+x.y.z=Σ(1,4,7)

28 Boolean Function: Exampl How to express algebraically zTruth table example: zx y z F2 minterm z m0 z m1 z m2 z m3 z m4 z m5 z m6 z m7 zF2=m3+m5+m6+m7=x’.y.z+x.y’.z+x.y.z’+x.y.z=Σ(3,5,6,7)

29 Boolean Function: Exampl How to express algebraically z1.Form a maxterm for each combination forming a 0 z2.AND all of those terms zTruth table example: zx y z F1 maxterm z x+y+z M0 z z x+y’+z M2 z x+y’+z’ M3 z z x’+y+z’ M5 z x’+y’+z M6 z zF1=M0.M2.M3.M5.M6 = л(0,2,3,5,6)

30 Boolean Function: Exampl How to express algebraically zTruth table example: z x y z F2 maxterm z x+y+z M0 z x+y+z’ M1 z x+y’+z M2 z z x’+y+z M4 z z z zF=M0.M1.M2.M4=л(0,1,2,4)=(x+y+z).(x+y+z’).(x+y’+z).(x’+y+z)

31 Maxterms & Minterms: Intuitions zMinterms: zIf a function is expressed as SUM of PRODUCTS, then if a single product is 1 the function would be 1. zMaxterms: zIf a function is expressed as PRODUCT of SUMS, then if a single product is 0 the function would be 0. zCanonical Forms: zBoolean functions expressed as a sum of minterms or product of maxterms.

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

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

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

35 Digital Logic Gates

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