ECE 20B, Winter Purpose of Course. Introduction to Electrical Engineering, II. Administration

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1 ECE 20B, Winter 2003 Introduction to Electrical Engineering, II Instructor: Andrew B Kahng (lecture) abk@eceucsdedu Telephone: office, cell Office: 3802 AP&M Lecture: TuThu 3:30pm 4:50pm, HSS, Room 2250 Discussion: Wed 6:00pm-6:50pm, Peterson Hall, 108 Class Website: Login: ece20b Password: b02ece Purpose of Course Introduction to design of digital systems and computer hardware Basic to CS, EE, CE Major topics Information representation and manipulation Logic elements and Boolean algebra Combinational Logic Arithmetic Logic Sequential Logic Registers, Counters, Memories Control 1 2 Administration Lab instructor: Prof Mohan Trivedi Textbook: Mano and Kime, 2 nd edition (updated) Goal: cover MK Chapters 1-5, (6), 8 Rizzoni (Sections 121-2) used only for op amps Labs This week: (1) show up and verify partner (2) if you need a partner, talk to Prof Trivedi If you need to switch lab sections, go to undergrad office Adding ECE 20B Must get stamp from undergrad office in EBU I Prerequisites rigidly enforced Course Structure Homework: assigned but not collected All problems and solutions posted on web Exams are based on homework problems Do problems before looking at solutions! Discussion: Wed 6:00-6:50pm, Peterson 108 Will typically go over the previous lectures and problems Grading 40%: 2 in-class midterms (Jan 30, Feb 25), 40%: 1 final 20%: un-announced in-class quizzes Exams cover both lecture (~3/4) and lab (~1/4) For each lab, a set of prelab questions will be assigned These must be included in your lab notebook 3 4 Course Conduct Resources Discussion session (TA) Lab sessions (readers) Questions about course see me Broader consultation see academic advisor Academic misconduct: do not let this happen 5 Introduction Assigned reading: Chapters 1, 2 of MK (see website for specific sections) Homework: Check website for problems/solutions Today Concept of digital Number systems Next lecture Binary logic Boolean algebra We will spend ~3 weeks going through the first 3 chapters of MK 6

2 Digital System Takes a set of discrete information inputs and discrete internal information (system state) and generates a set of discrete information outputs Discrete Inputs Discrete Information Processing System Discrete Outputs Types of Systems With no state present Combinational logic system Output = Function (Input) With state present State updated at discrete times (eg, once per clock tick) Synchronous sequential system State updated at any time Asynchronous sequential system System State 7 8 Example: Digital Counter (eg, Odometer) Example: Digital Computer UP RESET Inputs: Count Up, Reset Outputs: Visual Display State: Value of stored digits Is this system synchronous or asynchronous? Inputs: keyboard, mouse, modem, microphone Outputs: CRT, LCD, modem, speakers Is this system synchronous or asynchronous? 9 10 Signals Information variables mapped to physical quantities In digital systems, the quantities take on discrete values Two-level, or binary, values are the most prevalent values in digital systems Binary values are represented abstractly by digits 0 and 1 Signal examples over time: Analog Physical Signal Example - Voltage Threshold Region Asynchronous (Time) Synchronous 11 Other physical signals representing 1 and 0 CPU Voltage Disk Magnetic field direction CD Surface pits / light Dynamic RAM Charge 12

3 Threshold in the News Punched = 1 Not punched = 0 Number Systems Decimal Numbers What does 5,634 represent? Expanding 5,634: 5 x 10 3 = 5, x 10 2 = x 10 1 = x 10 0 = 4 5,634 What is 10 called in the above expansion? The radix What is this type of number system called? Decimal What are the digits for decimal numbers? 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 What are the digits for radix-r numbers? 0, 1,, r-1 What about the rest? Powers of 2 Noteworthy powers of 2: 2 10 = kilo- = K 2 20 = mega- = M 2 30 = giga- = G 2 40 = tera- = T 2 50 = peta- = P General Base Conversion Number Representation Given a number of radix r of n integer digits a n-1,,a 0 and m fractional digits a -1,,a -m written as : 15 a n-1 a n-2 a n-3 a 2 a 1 a 0 a -1 a -2 a -m has value: i = n-1 j = -1 i j (Number) = r ( Σ a ) + i r ( aj r ) Σ i = 0 j = -m (Integer Portion) + (Fraction Portion) 16 Commonly Occurring Bases Name Radix Digits Binary 2 0,1 Octal 8 0,1,2,3,4,5,6,7 Decimal 10 0,1,2,3,4,5,6,7,8,9 Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Decimal (Base 10) Binary (Base 2) Octal (Base 8) Hexadecimal (Base 16) D E F Converting Binary to Decimal To convert to decimal, use decimal arithmetic to sum the weighted powers of two Converting to N 10 N 10 = = 26 18

4 Converting Decimal to Binary Method 1 (Method 2 repeated division next slide) Subtract the largest power of 2 that gives a positive result and record the power Repeat, subtracting from the prior result, until the remainder is zero Place 1 s in the positions in the binary result corresponding to the powers recorded; in all other positions place 0 s Example: = = = = = 0 1 Place 1 s in the the positions recorded and 0 s elsewhere Converting binary to decimal: sum weighted powers of 2 using decimal arithmetic, eg, = 625 Conversion Between Bases Convert the Integral Part Repeatedly divide the number by the radix you want to convert to and save the remainders The new radix digits are the remainders in reverse order of computation Why does this work? This works because, the remainder left in the division is always the coefficient of the radix s exponent If the new radix is > 10, then convert all remainders > 10 to digits A, B, Convert the Fractional Part Repeatedly multiply the fraction by the radix and save the integer digits that result The new radix fraction digits are the integer numbers in computed order Why does this work? To convert fractional part, it should be divided by reciprocal of radix, which is same as multiplying with radix If the new radix is > 10, then convert all integer numbers > 10 to digits A, B, Join together with the radix point Example: Convert To Base 2 Convert 46 to Base 2 46/2 = 23 remainder = 0 23/2 = 11 remainder = 1 11/2 = 5 remainder = 1 5/2 = 2 remainder = 1 2/2 = 1 remainder = 0 1/2 = 0 remainder = 1 Read off in reverse order: Convert to Base 2: * 2 = int = * 2 = int = * 2 = int = * 2 = int = Read off in forward order: Converting Among Octal, Hexadecimal, Binary Octal (Hexadecimal) to Binary: Restate the octal (hexadecimal) as three (four) binary digits, starting at radix point and going both ways Binary to Octal (Hexadecimal): Group the binary digits into three (four) bit groups starting at the radix point and going both ways, padding with zeros as needed in the fractional part Convert each group of three (four) bits to an octal (hexadecimal) digit Example: Octal to Binary to Hexadecimal = = (000) 2 (regrouping) = 1 9 D 3 F 8 16 (converting) Join together with the radix point: Non-numeric Binary Codes Given n binary digits (called bits), a binary code is a mapping from a subset of the 2 n binary numbers to some set of represented elements Binary Number Color Example: A 000 Red binary code 001 Orange for the seven 010 Yellow colors of the 011 Green rainbow 100 (Not mapped) 101 Blue 110 Indigo 111 Violet Flexibility of representation: can assign binary code word to any numerical or non- numerical data as long as data uniquely encoded 23 Number of Bits Required Given M elements to be represented by a binary code, the minimum number of bits, n, needed satisfies the following relationships: 2 n M > 2 n 1 n = ceil(log 2 M) where ceil(x) is the smallest integer greater than or equal to x Example: How many bits are required to represent decimal digits with a binary code? M = 10 n = 4 24

5 Number of Elements Represented Given n digits in radix r, there are r n distinct elements that can be represented But, can represent m elements, m < r n Examples: Can represent 4 elements in radix r = 2 with n = 2 digits: (00, 01, 10, 11) Can represent 4 elements in radix r = 2 with n = 4 digits: (0001, 0010, 0100, 1000) This code is called a "one hot" code Binary Codes for Decimal Digits There are over 8,000 ways that you can chose 10 elements from the 16 binary numbers of 4 bits A few are useful: Decimal 8,4,2,1 Excess3 8,4,-2,-1 Gray Binary Coded Decimal (BCD) The BCD code is the 8,4,2,1 code This code is the simplest, most intuitive binary code for decimal digits and uses the same weights as a binary number, but only encodes the first ten values from 0 to 9 Example: 1001 (9) = 1000 (8) (1) How many invalid code words are there? What are the invalid code words? 27 Excess-3 Code and 8, 4, 2, 1 Code Decimal Excess , 4, 2, What property is common to these codes? These are reflected codes; complementing is performed simply by replacing 0 s by 1 s and vice-versa 28 Gray Code Gray Code: Optical Shaft Encoder Decimal 8,4,2,1 Gray What property does this Gray code have? Counting up or down changes only one bit at a time (including counting between 9 and 0) B 2 B 1 B (a) Binary Code for Positions 0 through G 0 G 1 G (b) Gray Code for Positions 0 through 7 Shaft encoder: Capture angular position (eg, compass) For binary code, what values can be read if the shaft position is at boundary of 3 and 4 (011 and 100)? For Gray code, what values can be read? 30

6 Warning: Conversion or Coding? Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE = (This is conversion) (This is coding) Binary Arithmetic Single Bit Addition with Carry Multiple Bit Addition Single Bit Subtraction with Borrow Multiple Bit Subtraction Multiplication BCD Addition Single Bit Binary Addition with Carry Given two binary digits (X,Y), a carry in (Z) we get the following sum (S) and carry (C): Carry in (Z) of 0: Z Y Carry in (Z) of 1: C S Z Y C S Multiple Binary Addition Extending this to two multiple bit examples: Carries Augend Addend Sum Note: The 0 is the default Carry-In to the least significant bit Single Bit Binary Subtraction with Borrow Given two binary digits (X,Y), a borrow in (Z) we get the following difference (S) and borrow (B): Borrow in (Z) of 0: Z Borrow in (Z) of 1: -Y BS Z Y BS Multiple Bit Binary Subtraction Extending this to two multiple bit examples: Borrows Minuend Subtrahend Difference Notes: The 0 is a Borrow-In to the least significant bit If the Subtrahend > the Minuend, interchange and append a to the result 36

7 Binary Multiplication The binary multiplication table is simple: 0 0 = = = = 1 Extending multiplication to multiple digits: Multiplicand 1011 Multiplier x 101 Partial Products Product Error-Detection Codes Redundancy (eg extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1 s odd or even Parity can detect all single-bit errors and some multiple-bit errors A code word has even parity if the number of 1 s in the code word is even A code word has odd parity if the number of 1 s in the code word is odd Bit Parity Code Example Fill in the even and odd parity bits: Even Parity Odd Parity Message - Parity Message - Parity The binary codeword "1111" has even parity and the binary code "1110" has odd parity Both could be used to represent data 39 ASCII Character Codes American Standard Code for Information Interchange This code is the most popular code used to represent information sent as character-based data It uses 7- bits to represent: 94 Graphic printing characters 34 Non-printing characters Some non-printing characters are used for text format (eg BS = Backspace, CR = carriage return) Other non-printing characters are used for record marking and flow control (eg STX and ETX start and end text areas) ASCII is a 7-bit code, but most computers manipulate 8-bit quantity called byte To detect errors, the 1 st bit is used as a parity bit Eg, ASCII A = (7 bits) ASCII A with parity bit = (8 bits) (Note: even parity is used) 40