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CENG 241 Digital Design 1 Lecture 1 Amirali Baniasadi

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1 CENG 241 Digital Design 1 Lecture 1 Amirali Baniasadi

2 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 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 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 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 6 What are my expectations? zStay Positive and Enjoy. zCommitment: Regular study and homework submission

7 7 This Lecture zDigital Design? zBinary Systems

8 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 9 Binary storage & registers

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

11 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 12 Electrical signals Two values: 0 or 1

13 13 Symbols for digital logic circuits

14 14 Input-Output signals for gates

15 15 Gates with multiple inputs

16 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 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 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 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 20 Boolean Function Implementation y’ Y’.z

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

22 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 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 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 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 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 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 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 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 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 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 32 Standard Forms Standard From: Sum of Product or Product of Sum

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

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

35 35 Digital Logic Gates

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


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