1. Digital Electronics
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2. Digital Electronics Number Systems and Logic Electronic Gates
Combinational Logic
Sequential Circuits
ADC – DAC circuits
Memory and Microprocessors
Hardware Description Languages
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3. Weekly Structure Lectures Monday, Tuesday, Wednesday and Thursday
Slides in ppt and pdf format on support website:
(follow link from course website)
Thursday Laboratory 2-5 pm
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4. Course Content
To analyze the circuits that perform computation, you need to know about
Circuit analysis: voltages and currents so you can see how circuits work, resistance and capacitance so you see how long it takes to compute
Electronics: transistors which make complex computation feasible
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5. The lecture today Digital vs Analog data Binary inputs and outputs
Binary, octal, decimal and hexadecimal number systems
Other uses of binary coding.
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6. { = 123 Digital Information
All information inside a computer is represented numerically.
Text: Every letter is represented by a number (ASCII code).
Images: A bitmap image is a table of numbers, with each entry representing the color of a pixel.
{ = 123
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7. Analog/Analogue Systems
V(t) can have any value between its minimum and maximum value
V(t)
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8. Digital Systems
Digital Systems
V(t) must take a value selected from a set of values called an alphabet
Binary digital systems form the basis of almost all hardware systems currently
V(t) 1 1 1
For example, Binary Alphabet: 0, 1.
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9. Slide example Consider a child’s slide in a playground:
a set of discrete steps
continuous movement
levels
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10. Relationship between Analogue and Digital systems
5 Volt
0 Volt
0.8
0.4
2.4
2.8
Input
Range
for 1
for 0
Output
Advantages of Digital Systems
Analogue systems: slight error in input yields large error in output
Digital systems more accurate and reliable
Computers use digital circuits internally
Interface circuits (for instance, sensors and actuators) are often analogue
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11. Exercise Explain whether the following are analog or digital:
A photograph or painting
A scanned image
Sound from a computer’s loud speaker
Sound file stored on disc
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12. Number Representation
There are many ways to represent numbers.
Decimal (base 10, the usual way)
Hexadecimal (base 16, often used in the study of computers)
Binary (base 2)
The ability to represent any number in binary, using 0’s and 1’s, makes computation as we know it possible.
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13. Binary Inputs and Outputs
Coding:
A single binary input can only have two values: True or False (Yes or No) (1 or 0)
A volume control on a stereo requires more than two positions (off and full volume)
2 inputs can represent 4 values 3 inputs can represent 8 values 4 inputs can represent 16 values 5 inputs can represent 32 values ...
Example 1 There would be 210 = 1024 = 1K combinations
Example 2 Need to represent 10 digits Using three bits only allows us to represent 8 and 4 bits allows us to represent 16 - have to use 4 inputs.
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14. Binary More bits = more combinations
Each additional input doubles the number of combinations we can represent i.e. with n inputs it is possible to represent 2n combinations
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15. High and Low Logic Levels
The numbers 0 and 1 are represented by physical quantities:
The number 0 (called logic 0) is represented with a voltage near 0 V.
The number 1 (called logic 1) is represented with a voltage between 2 and 5 V, depending on the technology.
Circuits perform computation by taking voltage inputs and allowing current to flow, creating voltage outputs.
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16. A Logic Gate
The mathematical operation known as “AND” is performed by this circuit: A B S
VDD D
A B
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17. Combinations Example 1: Example 2:
How many combinations are possible with 10 binary inputs?
Example 2:
What is the minimum number of bits needed to represent the digits ‘0’ to ‘9’ as a binary code?”
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18. Decimal systems Number Representation
Difficult to represent Decimal numbers directly in a digital system
Easier to convert them to binary
There is a weighting system: eg
403 = 4 x x x 1
or in, powers of 10:
40310= 4x x x100 =
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19. Binary Inputs and Outputs
Both Decimal and Binary numbers use a positional weighting system, eg: = 1x23+0x22+1x21+0x20 = 1x8 + 0x4 + 1x2 + 0x1 = 1010
decimal
100 (102)
10 (101)
1 (100) 4 3
binary
8 (23)
4 (22)
2 (21)
1 (20) 1
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20. Binary to decimal
Multiply each 1 bit by the appropriate power of 2 and add them together. ?
128 64 32 16 8 4 2 1
= ……………….10 ?
= ……………………10 ?
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21. Binary Inputs and Outputs
Number Representation - Binary to decimal
A decimal number can be converted to binary by repeated division by 2
number /2
remainder
155 77 1
Least Significant Bit 38 19 9 4 2
Most Significant bit
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15510 =

22. Decimal to Binary An alternative way is to use the “placement” method
128 goes into 155 once leaving 27 to be placed
So 64 and 32 are too big (make them zero)
16 goes in once leaving 11
and so on…
128 64 32 16 8 4 2 1 1 1
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23. Representations
There are different ways of representing decimal numbers in a binary coding
BCD or Binary Coded Decimal is one example.
Each decimal digit is replaced by 4 binary digits
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24. Binary Inputs and Outputs
6 of the possible 16 values unused
example = BCD
Note that BCD code is longer than a direct representation in natural binary code:
453 =
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25. Binary Inputs and Outputs
Hexadecimal and Octal
Writing binary numbers as strings of 1s and 0s can be very tedious
Octal (base 8) and Hexadecimal (base 16) notations can be used to reduce a long string of binary digits.
octal
512 (83)
64 (82)
8 (81)
1 (80) 1 2 7
hexadecimal
256 (162)
16 (161)
1 (160) A F
Octal group into lots of 3 binary digits starting from the LSB
Hex group into lots of 4 binary digits starting from the LSB
Notice that hexadecimal requires 15 symbols (each number system needs 0 – base-1 symbols) and therefore A – F are used after 9.
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26. Octal as shorthand for Binary
Each octal digit corresponds to 3 binary bits
binary
octal
000
001 1
010 2
011 3
100 4
101 5
110 6
111 7
To convert a binary string:
Split into groups of 3:
Thus =
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27. Similarly with Hexadecimal
Each hex digit corresponds to 4 binary bits
To convert a binary string:
Split into groups of 4:
Thus = ……………16 ?
binary
hex
0000
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
binary
hex
1000 8
1001 9
1010 A
1011 B
1100 C
1101 D
1110 E
1111 F
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28. Binary inputs and outputs
Color codes
You often see hex used in graphic design programs for the red, blue and green components of a color:
FF0000 represents red, for example.
How many bits are used to represent each color?
How many different colors can be represented?
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29. Binary Inputs and Outputs
Characters
Three main coding schemes used: ASCII (widespread use), EBCDIC (not used often) and UNICODE (new)
ASCII table (in hex) :
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30. Gray Codes Other codes exist for specific purposes
Gray codes provide a sequence where only one bit changes for each increment
Allows increments without ambiguity due to bits changing at different times.
E.g. changing from 3 to 4, normal binary has all three bits changing 011 -> Depending on the order in which the bits change any intermediate value may be created.
Dec
Gray
000 1
001 2
011 3
010 4
110 5
111 6
101 7
100
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31. Summary Support website Analogue and Digital Binary Number Systems
Coding schemes considered were:
Natural Binary
BCD
Octal representation
Hexadecimal representation
ASCII
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32. Exercises
You should practice conversions between binary, octal, decimal and hexadecimal.
You should be able to code decimal to BCD (and BCD to decimal).
You should be able to explain and give examples of digital and analogue data.
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