SYEN 3330 Digital Systems Chapter 1 SYEN 3330 Digital Systems

Digital System SYEN 3330 DIGITAL SYSTEMS

Types of Systems SYEN 3330 DIGITAL SYSTEMS

Digital System Example: SYEN 3330 DIGITAL SYSTEMS

A Digital Computer Example Synchronous or Asynchronous? Outputs: CRT, LCD, modem, speakers Inputs: Keyboard, mouse, modem, microphone SYEN 3330 DIGITAL SYSTEMS

Signals SYEN 3330 DIGITAL SYSTEMS

Physical Signal Example - Voltage Threshold Region SYEN 3330 DIGITAL SYSTEMS

Threshold in the News! Punched = 1 Not punched = 0 What about the rest? SYEN 3330 DIGITAL SYSTEMS

Other Physical Signals What are other physical signals represented by 1 and 0? CPU Voltage Disk CD Dynamic RAM Magnetic Field Direction Surface Pits/Light Electrical Charge SYEN 3330 DIGITAL SYSTEMS

Signal Examples Over Time SYEN 3330 DIGITAL SYSTEMS

Number Systems SYEN 3330 DIGITAL SYSTEMS

Powers of Ten SYEN 3330 DIGITAL SYSTEMS

Positive Powers of 2 SYEN 3330 DIGITAL SYSTEMS

Important Powers of 2 SYEN 3330 DIGITAL SYSTEMS

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 SYEN 3330 DIGITAL SYSTEMS

Converting Binary to Decimal To convert to decimal, use decimal arithmetic to sum the weighted powers of two:  110102 => 1 x 24 = 16 + 1 x 23 = 8 + 0 x 22 = 0 + 1 x 21 = 2 + 0 x 20 = 0 __________ = 2610 SYEN 3330 DIGITAL SYSTEMS

Converting Decimal to Binary 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: 62510 Result from the listed powers: 10011100012 – 29 = 625 – 512 = 113 => 9 113 – 26 = 113 – 64 = 49 => 6 49 – 25 = 49 – 32 = 17 => 5 17 – 24 = 17 – 16 = 1 => 4 1 – 20 = 1 – 1 = 0 => 0 SYEN 3330 DIGITAL SYSTEMS

Commonly Occurring Bases SYEN 3330 DIGITAL SYSTEMS

Numbers in Different Bases SYEN 3330 DIGITAL SYSTEMS

General Base Conversion SYEN 3330 DIGITAL SYSTEMS

Radix 10 Example 2,345  6710 => a3a2a1a0  a-1a-2 = 2*1000 + 3*100 +4*10 + 5 + 6*(1/10) + 7*(1/100) (Integer part) + (Fraction part) Term Actual Values   Product a3*r3 = 2 * 1000 = 2000 a2*r2 = 3 * 100 = 300 a1*r1 = 4 * 10 = 40 a0*r0 = 5 * 1 = 5 a-1*r-1 = 6 * 01 = 06 a-2*r-2 = 7 * 001 = 007 Sum => 2,34567 SYEN 3330 DIGITAL SYSTEMS

Conversion Between Bases SYEN 3330 DIGITAL SYSTEMS

Conversion Details SYEN 3330 DIGITAL SYSTEMS

Convert 46.687510 To Base 2 SYEN 3330 DIGITAL SYSTEMS

Convert Integer 46 To Base 2 Step 1 46 / 2 = 23 remainder = 0 Step 2 23 / 2 = 11 remainder = 1 Step 3 11 / 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 4610 = 23 => 1011102 SYEN 3330 DIGITAL SYSTEMS

Convert Fraction 0.687510 to Base 2 Step 1 0.6875 * 2 = 1.3750 integer = 1 Step 2 0.3750 * 2 = 0.7500 integer = 0 Step 3 0.7500 * 2 = 1.5000 integer = 1 Step 4 0.5000 * 2 = 1.0000 integer = 1 Step 5 0.0000 * 2 = 0.0000 integer = 0 Result 0.687510 => 0.101102 SYEN 3330 DIGITAL SYSTEMS

Join Integer and Fraction SYEN 3330 DIGITAL SYSTEMS

Checking the Conversion SYEN 3330 DIGITAL SYSTEMS

Octal to Binary and Back SYEN 3330 DIGITAL SYSTEMS

Octal to Hexadecimal via Binary SYEN 3330 DIGITAL SYSTEMS

A Final Conversion Note SYEN 3330 DIGITAL SYSTEMS

Binary Numbers and Coding SYEN 3330 DIGITAL SYSTEMS

Enumerating elements SYEN 3330 DIGITAL SYSTEMS

Example: Radix 2, 3 digits SYEN 3330 DIGITAL SYSTEMS

Representing Numeric Information SYEN 3330 DIGITAL SYSTEMS

Binary Coding SYEN 3330 DIGITAL SYSTEMS

Number of Bits Required SYEN 3330 DIGITAL SYSTEMS

Min. and Max. Digits Required SYEN 3330 DIGITAL SYSTEMS

Binary Codes for Decimal Digits SYEN 3330 DIGITAL SYSTEMS

Binary Coded Decimal (BCD) SYEN 3330 DIGITAL SYSTEMS

Other Decimal Codes SYEN 3330 DIGITAL SYSTEMS

Warning: Conversion or Coding? SYEN 3330 DIGITAL SYSTEMS

Binary Addition SYEN 3330 DIGITAL SYSTEMS

Binary Addition (Extended) Extending this to multiple digits: Carries 0 0 Augend 01100 10110 Addend +10001 +10111 Sum Note: The underlined “0” is a Carry-In to the least digit. 1 1 1 1 1 1 1 1 1 1 1 SYEN 3330 DIGITAL SYSTEMS

Binary Subtraction Z X 1 - Y -0 -1 BS 0 0 1 1 0 1 Z 1 X - Y -0 -1 BS 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: Z   X 1 - Y -0 -1 BS 0 0 1 1 0 1 Z   1 X - Y -0 -1 BS 11 1 0 0 0 1 1 SYEN 3330 DIGITAL SYSTEMS

Binary Subtraction (Extended) Extending this to multiple digits: Borrows 0 0 Minuend 10110 10110 Subtrahend - 10010 - 10011 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. 1 1 1 1 1 SYEN 3330 DIGITAL SYSTEMS

Binary Multiplication SYEN 3330 DIGITAL SYSTEMS

BCD Arithmetic SYEN 3330 DIGITAL SYSTEMS

BCD Addition Example Add 1897BCD to 2905BCD 1 1 1 0001 1000 1001 0111 0001 1000 1001 0111 + 0010 1001 0000 0101 0100 10010 1010 1100 + 0000 + 0110 + 0110 + 0110 0100 1000 0000 0010 SYEN 3330 DIGITAL SYSTEMS

Error-Detection Codes SYEN 3330 DIGITAL SYSTEMS

3-Bit Parity Code Example SYEN 3330 DIGITAL SYSTEMS

ASCII Character Codes SYEN 3330 DIGITAL SYSTEMS

ASCII Properties SYEN 3330 DIGITAL SYSTEMS

Other Character Codes SYEN 3330 DIGITAL SYSTEMS

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 http://www.prenhall.com/mano if you are interested SYEN 3330 DIGITAL SYSTEMS