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SYEN Digital Systems Chapter 1 SYEN 3330 Digital Systems
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Digital System SYEN 3330 DIGITAL SYSTEMS
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Types of Systems SYEN 3330 DIGITAL SYSTEMS
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Digital System Example:
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A Digital Computer Example
Synchronous or Asynchronous? Outputs: CRT, LCD, modem, speakers Inputs: Keyboard, mouse, modem, microphone SYEN 3330 DIGITAL SYSTEMS
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Signals SYEN 3330 DIGITAL SYSTEMS
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Physical Signal Example - Voltage
Threshold Region SYEN 3330 DIGITAL SYSTEMS
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Threshold in the News! Punched = 1 Not punched = 0
What about the rest? SYEN 3330 DIGITAL SYSTEMS
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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
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Signal Examples Over Time
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Number Systems SYEN 3330 DIGITAL SYSTEMS
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Powers of Ten SYEN 3330 DIGITAL SYSTEMS
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Positive Powers of 2 SYEN 3330 DIGITAL SYSTEMS
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Important Powers of 2 SYEN 3330 DIGITAL SYSTEMS
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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
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Converting Binary to Decimal
To convert to decimal, use decimal arithmetic to sum the weighted powers of two: => 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
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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: – 29 = 625 – 512 = 113 => 9 113 – 26 = 113 – 64 = 49 => 6 49 – 25 = 49 – = 17 => 5 17 – 24 = 17 – = => 4 1 – 20 = 1 – = => 0 SYEN 3330 DIGITAL SYSTEMS
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Commonly Occurring Bases
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Numbers in Different Bases
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General Base Conversion
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Radix 10 Example 2,345 6710 => a3a2a1a0 a-1a-2 = * *100 +4* *(1/10) + 7*(1/100) (Integer part) (Fraction part) Term Actual Values Product a3*r3 = * = a2*r2 = * = a1*r1 = * = a0*r0 = * = a-1*r-1 = * 0 = 6 a-2*r-2 = * 0 = 07 Sum => ,34567 SYEN 3330 DIGITAL SYSTEMS
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Conversion Between Bases
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Conversion Details SYEN 3330 DIGITAL SYSTEMS
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Convert To Base 2 SYEN 3330 DIGITAL SYSTEMS
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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 / 2 = 2 remainder = 1 Step / 2 = 1 remainder = 0 Step / 2 = 0 remainder = 1 Result = 23 => SYEN 3330 DIGITAL SYSTEMS
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Convert Fraction 0.687510 to Base 2
Step * 2 = integer = 1 Step * 2 = integer = 0 Step * 2 = integer = 1 Step * 2 = integer = 1 Step * 2 = integer = 0 Result => SYEN 3330 DIGITAL SYSTEMS
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Join Integer and Fraction
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Checking the Conversion
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Octal to Binary and Back
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Octal to Hexadecimal via Binary
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A Final Conversion Note
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Binary Numbers and Coding
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Enumerating elements SYEN 3330 DIGITAL SYSTEMS
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Example: Radix 2, 3 digits SYEN 3330 DIGITAL SYSTEMS
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Representing Numeric Information
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Binary Coding SYEN 3330 DIGITAL SYSTEMS
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Number of Bits Required
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Min. and Max. Digits Required
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Binary Codes for Decimal Digits
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Binary Coded Decimal (BCD)
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Other Decimal Codes SYEN 3330 DIGITAL SYSTEMS
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Warning: Conversion or Coding?
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Binary Addition SYEN 3330 DIGITAL SYSTEMS
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Binary Addition (Extended)
Extending this to multiple digits: Carries Augend Addend 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
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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
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Binary Subtraction (Extended)
Extending this to multiple digits: Borrows 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. 1 1 1 1 1 SYEN 3330 DIGITAL SYSTEMS
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Binary Multiplication
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BCD Arithmetic SYEN 3330 DIGITAL SYSTEMS
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BCD Addition Example Add 1897BCD to 2905BCD 1 1 1 0001 1000 1001 0111
0100 10010 1010 1100 + 0000 + 0110 + 0110 + 0110 0100 1000 0000 0010 SYEN 3330 DIGITAL SYSTEMS
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Error-Detection Codes
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3-Bit Parity Code Example
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ASCII Character Codes SYEN 3330 DIGITAL SYSTEMS
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ASCII Properties SYEN 3330 DIGITAL SYSTEMS
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Other Character Codes SYEN 3330 DIGITAL SYSTEMS
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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 interested SYEN 3330 DIGITAL SYSTEMS
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