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SYEN 3330 Digital Systems. Chapter 1. Digital System. Types of Systems. Digital System Example:. A Digital Computer Example. Synchronous or Asynchronous?. Outputs: CRT, LCD, modem, speakers. Inputs: Keyboard, mouse, modem, microphone. Signals. Physical Signal Example - Voltage.
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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 – 24 = 17 – 16 = 1 => 4 17 – 20 = 1 – 1 = 0 => 0 1 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
Term Actual Values Product 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) 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 * 01 = 06 a-2*r-2 = 7 * 001 = 007 Sum => 2,34567 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 00 Augend 01100 10110 Addend +10001+10111 Sum Note: The underlined “0” is a Carry-In to the least digit. 1 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 1 1 1 SYEN 3330 DIGITAL SYSTEMS
Z Z 0 1 0 1 1 0 1 0 X X 0 0 0 0 1 1 1 1 BS BS 11 0 0 1 1 1 0 0 1 0 0 0 0 1 1 - Y - Y -0 -0 -1 -1 -0 -0 -1 -1 Binary Subtraction • 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: SYEN 3330 DIGITAL SYSTEMS
Binary Subtraction (Extended) • Extending this to multiple digits: Borrows00 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. 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 0 0 SYEN 3330 DIGITAL SYSTEMS
Binary Multiplication SYEN 3330 DIGITAL SYSTEMS
BCD Arithmetic SYEN 3330 DIGITAL SYSTEMS
BCD Addition Example • Add 1897BCD to 2905BCD 1 0 1 1 0001 1000 1001 0111 + 0010100100000101 1100 0100 10010 1010 + 0110 + 0110 + 0000 + 0110 0100 0010 1000 0000 SYEN 3330 DIGITAL SYSTEMS
Error-Detection Codes SYEN 3330 DIGITAL SYSTEMS