ECE/CS 352 Digital System Fundamentals© T. Kaminski & C. Kime 1 ECE/CS 352 Digital Systems Fundamentals Spring 2001 Chapter 1 Tom Kaminski & Charles R. Kime

ECE/CS 352 Digital System Fundamentals Chapter 1 2 Digital System

ECE/CS 352 Digital System Fundamentals Chapter 1 3 Types of Systems

ECE/CS 352 Digital System Fundamentals Chapter 1 4 Digital System Example:

ECE/CS 352 Digital System Fundamentals Chapter 1 5 A Digital Computer Example Synchronous or Asynchronous? Inputs: Keyboard, mouse, modem, microphone Outputs: CRT, LCD, modem, speakers

ECE/CS 352 Digital System Fundamentals Chapter 1 6 Signals

ECE/CS 352 Digital System Fundamentals Chapter 1 7 Physical Signal Example - Voltage Threshold Region

ECE/CS 352 Digital System Fundamentals Chapter 1 8 Threshold in the News! Punched = 1 Not punched = 0 What about the rest?

ECE/CS 352 Digital System Fundamentals Chapter 1 9 What are other physical signals represented by 1 and 0?  CPU Voltage  Disk  CD  Dynamic RAM Other Physical Signals Magnetic Field Direction Surface Pits/Light Electrical Charge

ECE/CS 352 Digital System Fundamentals Chapter 1 10 Signal Examples Over Time

ECE/CS 352 Digital System Fundamentals Chapter 1 11 Number Systems

ECE/CS 352 Digital System Fundamentals Chapter 1 12 Powers of Ten

ECE/CS 352 Digital System Fundamentals Chapter 1 13 Positive Powers of 2

ECE/CS 352 Digital System Fundamentals Chapter 1 14 Important Powers of 2

ECE/CS 352 Digital System Fundamentals Chapter 1 15 Number Digits Decimal number digits are 0 through 9 Binary number digits are 0 through 1 Base (radix) r number digits are 0 through r - 1

ECE/CS 352 Digital System Fundamentals Chapter 1 16 To convert to decimal, use decimal arithmetic to sum the weighted powers of two: => Converting Binary to Decimal 1 x 2 4 = x 2 3 = x 2 2 = x 2 1 = x 2 0 = 0 = __________

ECE/CS 352 Digital System Fundamentals Chapter 1 17 Method 1 (Method 2, Repeated Division Later)  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: Result from the listed powers: Converting Decimal to Binary – = 625 – 512 = 113 => – = – 64 = 49 => 6 49 – = – 32 = 17 => – = – 16 = 1 => 4 – = 1 – 1 = 0 0

ECE/CS 352 Digital System Fundamentals Chapter 1 18 Commonly Occurring Bases

ECE/CS 352 Digital System Fundamentals Chapter 1 19 Numbers in Different Bases

ECE/CS 352 Digital System Fundamentals Chapter 1 20 General Base Conversion

ECE/CS 352 Digital System Fundamentals Chapter 1 21 Radix 10 Example a 3 *r 3 = 2 * 1000 = 2000 a -2 *r -2 = 7 * 0  01 = 0  07 a 2 *r 2 = 3 * 100 = 300 a 0 *r 0 = 5 * 1 = 5 a -1 *r -1 = 6 * 0  1 = 0  6 a 1 *r 1 = 4 * 10 = 40 Sum => 2,345  67 2,345  => a 3 a 2 a 1 a 0  a - 1 a - 2 = 2* *100 +4* *(1/10) + 7*(1/100) (Integer part) + (Fraction part) TermActual Values Product

ECE/CS 352 Digital System Fundamentals Chapter 1 22 Conversion Between Bases

ECE/CS 352 Digital System Fundamentals Chapter 1 23 Conversion Details

ECE/CS 352 Digital System Fundamentals Chapter 1 24 Convert To Base 2

ECE/CS 352 Digital System Fundamentals Chapter 1 25 Convert Integer 46 To Base 2 Step 146 / 2 = 23 remainder = 0 Step 223 / 2 = 11 remainder = 1 Step 311 / 2 = 5 remainder = 1 Step 4 5 / 2 = 2 remainder = 1 Step 5 2 / 2 = 1 remainder = 0 Step 6 1 / 2 = 0 remainder = 1 Result = 23 =>

ECE/CS 352 Digital System Fundamentals Chapter 1 26 Convert Fraction to Base 2 Step * 2 = integer = 1 Step * 2 = integer = 0 Step * 2 = integer = 1 Step * 2 = integer = 1 Step * 2 = integer = 0 Result =>

ECE/CS 352 Digital System Fundamentals Chapter 1 27 Join Integer and Fraction

ECE/CS 352 Digital System Fundamentals Chapter 1 28 Checking the Conversion

ECE/CS 352 Digital System Fundamentals Chapter 1 29 Octal to Binary and Back

ECE/CS 352 Digital System Fundamentals Chapter 1 30 Octal to Hexadecimal via Binary

ECE/CS 352 Digital System Fundamentals Chapter 1 31 A Final Conversion Note

ECE/CS 352 Digital System Fundamentals Chapter 1 32 Binary Numbers and Coding

ECE/CS 352 Digital System Fundamentals Chapter 1 33 Enumerating elements

ECE/CS 352 Digital System Fundamentals Chapter 1 34 Example: Radix 2, 3 digits

ECE/CS 352 Digital System Fundamentals Chapter 1 35 Representing Numeric Information

ECE/CS 352 Digital System Fundamentals Chapter 1 36 Binary Coding

ECE/CS 352 Digital System Fundamentals Chapter 1 37 Number of Bits Required

ECE/CS 352 Digital System Fundamentals Chapter 1 38 Min. and Max. Digits Required

ECE/CS 352 Digital System Fundamentals Chapter 1 39 Binary Codes for Decimal Digits

ECE/CS 352 Digital System Fundamentals Chapter 1 40 Binary Coded Decimal (BCD)

ECE/CS 352 Digital System Fundamentals Chapter 1 41 Other Decimal Codes

ECE/CS 352 Digital System Fundamentals Chapter 1 42 Warning: Conversion or Coding?

ECE/CS 352 Digital System Fundamentals Chapter 1 43 Binary Addition

ECE/CS 352 Digital System Fundamentals Chapter 1 44 Extending this to multiple digits: Carries 0 0 Augend Addend Sum Note: The underlined “0” is a Carry-In to the least digit. Binary Addition (Extended)

ECE/CS 352 Digital System Fundamentals Chapter 1 45 Given two binary digits (X,Y), a borrow in (Z) we get the following difference (S) and borrow (B): Borrow in (Z) of 0: Borrow in (Z) of 1: Binary Subtraction Z X Y BS Z X Y BS

ECE/CS 352 Digital System Fundamentals Chapter 1 46 Extending this to multiple digits: Borrows 0 0 Minuend Subtrahend Difference Note: If the Subtrahend is larger than the Minuend, interchange and append a – to the result. The underlined “0” is a Borrow-In to the least digit. Binary Subtraction (Extended)

ECE/CS 352 Digital System Fundamentals Chapter 1 47 Binary Multiplication

ECE/CS 352 Digital System Fundamentals Chapter 1 48 BCD Arithmetic

ECE/CS 352 Digital System Fundamentals Chapter 1 49 BCD Addition Example Add 1897 BCD to 2905 BCD

ECE/CS 352 Digital System Fundamentals Chapter 1 50 Error-Detection Codes

ECE/CS 352 Digital System Fundamentals Chapter Bit Parity Code Example

ECE/CS 352 Digital System Fundamentals Chapter 1 52 ASCII Character Codes

ECE/CS 352 Digital System Fundamentals Chapter 1 53 ASCII Properties

ECE/CS 352 Digital System Fundamentals Chapter 1 54 Other Character Codes

ECE/CS 352 Digital System Fundamentals Chapter 1 55 Other Character Codes UNICODE extends ASCII to 65,536 universal characters codes  For encoding characters in world languages  Available in many modern applications  2 byte (16-bit) code words  See Supplement in Chapter 1 on Companion Website if you are interestedhttp://