CpE358/CS381. Switching Theory and Logical Design. Summer

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1 Switching Theory and Logical Design -

2 Class Schedule Monday Tuesday Wednesday Thursday Friday May Class 2 - Class Class Class Quiz Commencement 3 June 2 - Class Class 6 4 Memorial Day Project defined 7 - Class Class 8 - Class 9 Quiz Class Class 7 - Class 2 8 Quiz Class Class Class 5 25 Quiz 4 Projects due 28 - Class Class 7 July 7/2 Quiz 5-2

3 Course Introduction Logistics: Instructor: Bruce McNair Office: Burchard 26 Phone: Office hours: Class days ~9:3 - ~ Web site: (course notes, solutions, etc. are here) Homework Must be typed or printed hardcopy, or electronic (i.e., not handwritten). is OK with MS/Office (2 or previous), or program (e.g.,.c,.m,...) attachments. Don t me an executable or a macrovirus. pdf is OK. VERIFY THAT PROGRAMMING SUBMISSIONS INCLUDE ENOUGH ENVIRONMENT TO BE BUILT AND RUN (e.g.,.h files, initialization, etc.) Include the problem statement with solution. Keep a copy of your hardcopy or electronic homework (it may not be returned) If you submit homework as an attachment, make sure your name appears in the file. The file name must include your name (or login), course number, and assignment number, e.g.: bmcnair-cpe358-hw2.doc To ensure proper credit for the homework, indicate the due date on the homework Homework will be due at the second class after it is assigned. (E.g., Class homework is due during Class 3) My goal is to grade it and post the solution within a week. Problem solutions will be posted on my web site I do not penalize late homework, but HOMEWORK WILL NOT BE ACCEPTED AFTER THE SOLUTION IS POSTED Grading All items are INDIVIDUAL effort Homework: 25% Project: 25% Weekly tests: % each (5%) Detailed grades and status will be posted on WebCT -3

4 Course Introduction (continued) Course project requirements will be defined in Class 5 Project will be due in Class 5 Project will be an individual (paper) design of a modestly complex digital system using all of the design and analysis techniques covered in the class Specific examples of design problems will be provided (e.g., traffic light controller, digital clock, safety interlocks on door controllers, etc.) Multiple projects can be completed of varying levels of difficulty -4

5 Course Introduction (continued) Course description: Digital systems; number systems and codes; Boolean algebra; application of Boolean algebra to switching circuits; minimization of Boolean functions using algebraic, Karnaugh map, and tabular methods; programmable logic devices; sequential circuit components; design and analysis of synchronous and asynchronous sequential circuits Textbook M. Morris Mano, Digital Design, Third Edition, Prentice Hall, Engelwood Cliffs, NJ, 22. ISBN My approach: Practical, real-world examples Multiple perspectives on an issue -5

6 Topics Fundamental concepts of digital systems Binary codes and number systems Boolean algebra Simplification of switching equations Digital device characteristics (e.g., TTL, CMOS) and design considerations Combinatoric logical design including LSI implementation Hazards, Races, and time related issues in digital design Flip-flops and state memory elements Sequential logic analysis and design Synchronous vs. asynchronous design Counters, shift register circuits Memory and Programmable logic Minimization of sequential systems Introduction to Finite Automata -6

7 ABET Course Objectives By the end of this course, students should be able to: Understand number systems and codes and their application to digital circuits; understand Boolean algebra and its application to the design and characterization of digital circuits (A) Understand the mathematical characteristics of logic gates (4A) Use truth tables, Boolean algebra, Karnaugh maps, and other methods to obtain design equations Use design equations and procedures to design combinatorial and sequential systems consisting of gates and flip-flops (4C) Combine combinatorial circuits and flip-flops to design combinatorial and sequential system (5B) Consider alternatives to traditional design techniques to simplify the design process to yield innovative designs (5E) -7

8 Today s Material Fundamental concepts of digital systems (Mano Chapter ) Binary codes, number systems, and arithmetic (Ch ) Boolean algebra (Ch 2) Simplification of switching equations (Ch 3) Digital device characteristics (e.g., TTL, CMOS)/design considerations (Ch ) Combinatoric logical design including LSI implementation (Chapter 4) Hazards, Races, and time related issues in digital design (Ch 9) Flip-flops and state memory elements (Ch 5) Sequential logic analysis and design (Ch 5) Synchronous vs. asynchronous design (Ch 9) Counters, shift register circuits (Ch 6) Memory and Programmable logic (Ch 7) Minimization of sequential systems Introduction to Finite Automata -8

9 Why Digital? Analog computers were the standard for simulation in the 94s and 5s: v(t) G dv 2 dv G G dt 2 2 G2 dt G 2 dv dv G2 + G 2 + v dt dt Issues: precision, stability, accuracy, aging, noise, Manufacturing and testing are labor intensive processes -9

10 Why Digital? Digital circuits have become the standard for computing, control, and many other applications D Q D Q D Q Q Q Q Functions can be created with a small set of functional elements Designs are stable and repeatable Costs and size are rapidly dropping while speed and functionality increase -

11 Simplicity of Logic Design Any digital logic design can be created with three fundamental building blocks: AND OR NOT Signals are represented by only two states: ON TRUE OFF FALSE -

12 Basic Number Systems - Decimal Integers: ,678 = Rational numbers: 3.42 = In general: aaaa a a = a + a + a + a + a + a Decimal point -2

13 Basic Number Systems Arbitrary Base Integers: = = Rational numbers: 3.42 = = In general: ( a aa a a ) = a n r + + a r + a r + a r + + a r n m r n m Radix point a {,,, r } i m -3

14 Basic Number Systems Base 2 Integers: = = Rational numbers:. = = In general: ( a aa a a ) n = a a 2 + a 2 + a a 2 n m 2 n m binary point a i {,} m -4

15 Numbers to Remember 2 2 = 4 2 = 8 2 = 6 2 = 32 2 = 64 2 = 28 2 = = 52 2 = 24 2 = = = = = = =.25.5 =.25.5 = = =

16 Basic Arithmetic Addition: + + * Multiplication: x x * carry Subtraction: - -6

17 Binary to Decimal The hard way: An easier way: 2 Number Base Conversion = = = = = ((() ) ) = = -7

18 Decimal to Binary: Number Base Conversion = 2i ( 87mod 2) = i = 2i ( 43mod 2) = i = 2i ( 2mod 2) = i + = 2i ( mod 2) = i = 2i ( 5mod2) = i = 2i + ( 2mod2) = 2+ 2 i = 2i + ( mod2) = = -8

19 Number Base Conversion Fractional Numbers Decimal to Binary:.765 =. Switching Theory and Logical Design ( ).765 =.765 i i + i 2 i = i = i2 i i =.53 + i ( ) ( ) ( ) ( ) ( ) ( ) =.53 i i + i 2 i = i = i2 i.3 + i =.6 + i =.6 i i i i =.6 + i = i2 i.6 + i =.2 + i ( ) ( ) ( ) i ( ) i ( ) i ( ) i ( ) i ( ) i = = = = = =

20 Binary Equivalents of Decimal Numbers

21 Other Popular Bases Octal a i {,, 2,3, 4,5,6,7} Binary-Octal mapping: =

22 Other Popular Bases Hexadecimal ai {,,2,3,4,5,6,7,8,9, A, B, C, D, E, F} Binary-Hexadecimal Mapping 7EF9.2C = A B C D E F -22

23 Complements Subtracting by adding: A B = A+ ( B) How can we create -B? -23

24 Complements Subtracting by adding: A B = A+ ( B) How can we create -B? If B is an integer and B < r N Define: x = r x so B = b r + b r + + b r + br + b r Notice N N 2 2 N N 2 2 B+ B+ = r so N B = r + B+ B+ N -24

25 Decimal: 9 s and s Complement 9 s Complement B B = = B + B = s Complement B B = = B + B = Subtraction With s Complement

26 Binary: s and 2 s Complement s Complement b {,} B = B = 2 s Complement 2 2 B = 2 B = + =

27 Working With Negative Binary Numbers Represent +3 in binary as a 6 bit number: How can you represent -3? Signed Magnitude Add use the MSB to represent +/- Useful for multiplication, but not addition/subtraction Signed s Complement Complement the bits Not particularly useful for arithmetic Signed 2 s Complement Complement the bits and add Most widely used means of dealing with signed arithmetic -27

28 Arithmetic With Signed Numbers Add the following numbers (all base ) in binary using 6-bit 2 s complement representation: = Carry out of 6-bit range occurs

29 An Interesting Side Effect of 2 s Complement Add the following numbers (all base ) in binary using 6-bit 2 s complement representation: = +28 (intermediate results cannot be represented in 6-bit 2 s complement! e.g., 7+23=4 > 32) (+4 is out of range) (+32 is out of range) Intermediate overflows can be tolerated, as long as the final result is within the range that can be represented -29

30 Range of Values That Can Be Represented in N-bit 2 s Complement Value bit Representation Value bit Representation

31 Binary Codes Hexadecimal or binary numbers are not easily translated into humanunderstandable forms, e.g.: How old is a person born in ( ) 2? Is it any easier to understand as (7E3) 6? -3

32 Binary Codes Hexadecimal or binary numbers are not easily translated into humanunderstandable forms, e.g.: How old is a person born in ( ) 2? Is it any easier to understand as (7E3) 6? How about: ( ) BCD = () 2 x 3 + () () 2 + () 2-32

33 Binary Codes Typical binary counting order: 2 Transitions in multiple bits 3 may create systems issues: 4 Extra bit errors on communications links 5 Noise pulses in digital 6 systems 7-33

34 Binary Codes Typical binary counting order: 2 Transitions in multiple bits 3 may create systems issues: 4 Extra bit errors on communications links 5 Noise pulses in digital 6 systems 7 Gray code order: 2 3 Adjacent code words 4 differ in only one bit 5 position

35 Binary Codes Errors sometimes occur as data is being stored or transmitted. How can we design a system that is capable of responding to this possibility? Consider: Correct value: Result: -35

36 Binary Codes Errors sometimes occur as data is being stored or transmitted. How can we design a system that is capable of responding to this possibility? Consider: Correct value: Result: Add redundancy bits which convey no information, but protect other bits Correct value: Even Parity - even number of s sent Result: Parity Error odd number of s received Parity can be even or odd. Parity detect a single error in a protected block -36

37 Binary Storage and Registers register Sequential logic Memory Flip-flop Combinatorial logic NOT AND OR -37

38 Binary Storage and Registers Sequential systems require that binary information be stored Storage is in multiples of bit Assume that a storage element holds a value ( or ) until it is changed by a strobe signal input -bit storage write output -38

39 Binary Storage and Registers Sequential systems require that binary information be stored Storage is in multiples of bit Assume that a storage element holds a value ( or ) until it is changed by a strobe signal Multiple storage elements can be used in unison to form a register to store associated binary information, e.g., an N-bit number input -bit storage -bit storage -bit storage -bit storage -bit storage -bit storage write output -39

40 Binary Logic All possible combinatorial logic systems can be implemented with three functions: A B A B A B A AND B Binary Variables Logical Operations C D C D C D C OR D + Truth Tables E E E NOT(E) -4

41 Logic Levels Undefined input levels output V out Impossible output levels V out input V in V in -4

42 Summary Fundamental concepts of digital systems Binary codes, number systems, and arithmetic Boolean algebra Simplification of switching equations Digital device characteristics (e.g., TTL, CMOS)/design considerations Combinatoric logical design including LSI implementation Hazards, Races, and time related issues in digital design Flip-flops and state memory elements Sequential logic analysis and design Synchronous vs. asynchronous design Counters, shift register circuits Memory and Programmable logic Minimization of sequential systems Introduction to Finite Automata -42

43 Homework due in Class 3 Problems -7, -, -3, -2. Show all work -43

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