Chapter 1
Introduction to Digital Systems Design
ECEn 224
Digital vs. Analog
• Analog systems represent information
using physical quantities
– Voltage on a wire, magnetic field strength
• Digital systems represent information
using binary digits, or bits
– 1 or 0, high or low, on or off
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Positional Number Systems
• Two discrete values are insufficient for
most applications
• We combine bits to represent more values
• We use a positional number system for
binary just like we do in decimal
3
Positional Number Systems
• Decimal, base 10, means we have 10 digits (0-9)
• Decimal example:
103210 = 1x103 + 0x102 + 3x101 + 2x100
• Hexadecimal, base 16, means we have 16 digits (0-9, A-F)
• Hexadecimal example:
2A516 = 2x162 + 10x161 + 5x160 = 512 + 160 + 5 = 677
• Binary, base 2, follows the same pattern
• Binary example:
10112 = 1x23 + 0x22 + 1x21 + 1x20 = 8 + 0 + 2 + 1 = 11
• Counting in any base is analogous to counting in decimal
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Digital vs. Analog
• Analog thermometer
– 0V to 10V, could be used to represent 0° to
100° F
– Each 1/10th volt represents 1 degree
• Digital thermometer
– 7-bit binary number could be used to
represent 0° to 127° F
– Seven bits can be used to represent the
numbers 0 to 127
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Digital Precision
•
•
•
How would you represent 10.5° F?
Analog example: 1.05V
Digital example: ????
– 00010102 = 1010
– 00010112 = 1110
– We must either add bits or decrease the
range
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Digital Precision
• 9-bit thermometer, 0° to 127.75° F
– Each discrete number increase represents 0.25° F
– 10.5° F Æ 10.5/0.25 = 42 = 1010102
• 7-bit thermometer, 0° to 12.7° F
– Each discrete number represents 12.7°/127 = 0.1° F
– 10.5° F Æ 10.5/0.1 = 105 = 11010012
• It is not possible to represent all values exactly
using digital representation
– Example: 1/3 can’t be represented in binary, just like
it can’t be represented in decimal
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Example: Analog Photography
• An analog camera uses a chemical
reaction in the film when exposed to light
• The amount of exposure is directly related
to the amount of light that hits the film
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Example: Digital Photography
• A digital camera uses an array of lightsensitive receptors that measure the light
as a binary number
• Image quality is determined mostly by two
factors:
– The number of bits per pixel
– The number of pixels per image
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Example: Digital Photography
1284x897 pixels, 24-bit color
100x70 pixels, 24-bit color
1284x897 pixels, 6-bit color
100x70 pixels, 6-bit color
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Analog vs. Digital Storage
• Analog storage mediums fade over time due to
gradual physical degradation
– Photos turn yellow with time
– Cassette audio tapes lose their clarity
• Digital storage mediums don’t “fade” like analog
– If a 0 or 1 fades it will still be a 0 or 1
– A .jpg image taken 10 years ago is exactly the
same today
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Analog vs. Digital Storage
• Making an analog copy implies measuring the
storage medium
– Always introduces some errors
– Copies of copies are even worse
• Making digital copies implies distinguishing 0’s
from 1’s so copies are exact
– Copies can be made without any error
– Copies of copies are identical
12
Analog vs. Digital Processing
• Modern computers and digital circuits
make it easy to do extremely complex
processing
• Digital processing allows precision and
error to be exactly predicted
13
Combinational vs. Sequential Circuits
• Digital circuits consist of binary inputs and
outputs
A
B
C
D
X
Y
• In combinational circuits, the output is a direct
function of its inputs
• In sequential circuits the output depends on the
current input and previous inputs
– Sequential circuits contain memory that tracks state
– A clock is used to signal when to change states
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Combinational vs. Sequential
• Combinational example: Clock chime
–
–
–
Inputs: seconds and minutes
Output: chime
Behavior: Output chime is 1 if and only if seconds is
0 and minutes is 0, 15, 30, or 45.
• Sequential example: Counter
–
–
–
Inputs: inc and clk
Output: count
Behavior: Increment count on clk edge if and only if
inc is 1
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