1. Number Systems and Codes
Chapter 1
Number Systems and Codes
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2. Digital Versus Analog Digital Analog See Figure 1-1 ON and OFF 0 and 1
continuously varying
temperature, pressure, velocity
See Figure 1-1
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3. Figure 1-1
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4. Digital Representations of Analog Quantities
Audio Recording
CD or DAT
Conversions
digital-to-analog
analog voltage to 8-bit Digital equivalent
See Figure 1-2 and 1-3
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5. Figure 1-2
Figure 1-3
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6. Figure 1–4 Adding unwanted electrostatic noise to: (a) an analog waveform; (b) a digital waveform.
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7. Decimal Numbering System (Base 10)
10 different possible digits
Least significant position
rightmost
Most significant digit
leftmost
Weighting factor of 10
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9. Binary Numbering System (Base 2)
Only 0 and 1
Weighting factor of 2
Conversion techniques
digit times weighting factor
successive division
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10. Table 1–1 Powers-of-2 Binary Weighting Factors
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12. Figure 1– Successive division by 2 to develop fractional binary weighting factors and show that 20 is equal to 1.
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14. Decimal-to-Binary Conversion
Subtracting weighting factors
Successive division
Least Significant Bit (LSB)
Most Significant Bit (MSB)
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18. Octal Numbering System (Base 8)
Allowable digits
0,1,2,3,4,5,6,7
Number after 7 is 10
Weighting factor of 8
Used by computers that utilize 3-bit codes to indicate instructions or operations
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19. Table 1–2 Octal Numbering System
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20. Octal Conversions Binary to Octal Octal to Binary Octal to Decimal
group binary positions in groups of three
write the octal equivalent
Octal to Binary
reverse the process
Octal to Decimal
multiply by weighting factors
Decimal to Octal
successive division
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24. Hexadecimal Numbering System (Base 16)
4-bit groupings
Allowable digits
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Number after F is 10
Two hex digits are used to represent 8 bits
a byte
4 bits are a nibble
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25. William Kleitz Digital Electronics with VHDL, Quartus® II Version
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26. Hexadecimal Conversions
Binary to Hexadecimal
group the binary in groups of four
write the equivalent hex digit
Hexadecimal to Binary
reverse the process
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28. Hexadecimal Conversions
Hexadecimal to Decimal
multiply by weighting factors
Decimal to Hexadecimal
successive division
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32. Binary-Coded-Decimal System BCD
Each of the 10 decimal digits has a 4-bit binary code
Conversion
convert each decimal digit to its 4-bit binary code
BCD to Decimal - reverse the process
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34. William Kleitz Digital Electronics with VHDL, Quartus® II Version
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35. The ASCII Code American Standard Code for Information Interchange
Represents alphanumeric data
Uses 7 bits
128 different code combinations
See Table 1-5
3-bit group is most significant
4-bit group is least significant
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36. William Kleitz Digital Electronics with VHDL, Quartus® II Version
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37. Summary
Numerical quantities occur in analog form but must be converted to digital form to be used by computers or digital circuitry.
The binary numbering system is used in digital systems because the 1’s and 0’s are easily represented by ON or OFF transistors, which output 0V for 0 and 5V for 1.
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38. Summary
Any number system can be converted to decimal by multiplying each digit by its weighting factor.
The weighting factor for the least significant digit in any number system is always 1.
Binary numbers can be converted to octal by forming groups of 3 bits and to hexadecimal by forming groups of 4 bits.
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39. Summary
The successive division procedure can be used to convert from decimal to binary, octal or hexadecimal
The binary-coded-decimal system uses groups of 4 bits to drive decimal displays such as those in a calculator.
ASCII is used by computers to represent all letters, numbers and symbols in digital form.
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