ELE 336 Digital Systems Laboratory

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ELE 336Digital Systems Laboratory

• 1st Meeting
• Noon Today!
• Anderson 231

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Digital vs. Analog

• An analog system has a continuous range of values
• A mercury thermometer
• Vinyl records
• AM radio
• The set of real numbers
• A digital system has a set of discrete values
• CDs
• DSS TV
• The set of integers

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Benefits of Digital

• Cheap electronic circuits
• Easier to calibrate and adjust
• Resistance to noise

An analog signal
The digital equivalent
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Resistance to Noise

An analog signal
The digital equivalent
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The Binary Number System

• Numbers have positional importance
• 349.2510

2 x 10-1 2/10
9 x 100 9 x 1 9
5 x 10-2 5/100
4 x 101 4 x 10 40
3 x 102 3 x 100 300
In the binary system, positional importance
follows powers of 2
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Why Binary?

• All binary digits (bits) are 0 or 1
• Any device with two states can be used to
  represent binary digits

Open
Early computers used switches
Closed
Modern computers use transistors which can
be Saturated On Cutoff Off
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Binary Numbers

• 1 0 1 1. 0 1 2
  

1 x 2-2 1/4
0 x 2-1 0
1 x 20 1 x 1 1
1 x 21 1 x 2 2
0 x 22 0 x 4 0
1 x 23 1 x 8 8
1 0 1 1. 0 1 2 11-1/410 11.2510
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Counting in Base 2

• Base 10 Base 2
• 0 0
• 1 1
• 2 10
• 3 11
• 4 100
• 5 101
• 6 110
• 7 111

Base 10 Base 2 8 1000 9 1001 10 1010 11
1011 12 1100 13 1101 14 1110 15 1111

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Powers of 2

• 20 1
• 21 2 2-1 1/2
• 22 4 2-2 1/4
• 23 8 2-3 1/8
• 24 16 2-4 1/16
• 25 32 2-5 1/32
• 26 64 2-6 1/64
• 27 128 2-7 1/128
• 28 256 2-8 1/256
• 29 512 2-9 1/512
• 210 1024 2-10 1/1024

210 1024 1K
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A Heuristic Way to Convert Decimal to Binary

• Consider that you have the powers of 2 (like pool
  balls). Put a 1 in the right position to
  include them and put a 0 in the right position
  to omit them
• 25 24 23 22 21 20 . 2-1 2-2
• 32, 16, 8, 4, 2, 1 . 1/2, 1/4
• What is the binary representation of 40.7510?

40 32 8 0.75 1/2 1/4 40.7510 101000.112
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A Structured Way to ConvertDecimal to Binary

• X10 abcdef2
• X10 a25 b24 c23 d22 e21 f20
• Divide by two

X/2 a24 b23 c22 d21 e20
f
Quotient (call it Y)
Remainder
The first remainder after dividing by 2 is the
least significant bit of the binary representation
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Continued division

• X/2 a24 b23 c22 d21 e20
  f

Quotient (call it Y)
Remainder

• Y/2 a23 b22 c21 d20 e

The second remainder after dividing by 2 is the
coefficient of 21 in the binary representation
? Continue dividing and saving remainders
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An Example

• What is the binary representation of 6110?
• 61/2 30 with a remainder of 1
• 30/2 15 with a remainder of 0
• 15/2 7 with a remainder of 1
• 7/2 3 with a remainder of 1
• 3/2 1 with a remainder of 1
• 1/2 0 with a remainder of 1

Read remainders from last to first 6110 1111012
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Check your work

• 6110 1111012 Is this the right answer?
• 1111012
• 1 4 8 16 32 61

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Homework for Monday, Jan. 10

• 1) (10 points)
• What decimal number is represented by the
  following binary numbers?A) 1001101.0012
• B) 11100. 1002
• 2) (10 points)
• What is the binary representation of the
  following decimal numbers?
• A) 91.510
• B) 65.2510

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For Monday, Jan. 11

• Review Homework Problems
• Addition, Subtraction
• 2s Complement
• 16 / 8 / 2hexadecimal, octal, binary