Prepared by, Mr. K. P. Jagtap
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Prepared by, Mr. K. P. Jagtap
Principal of Digital Techniques
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WEL-COME TO DIGITAL WORLD
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ADVANTAGES OF DIGITAL SYSTEM:
1.Easy to Design.
2. It can Store Information
3. Higher Accuracy And Precision
4. Highly Reliable
5. Less Noisy.
6. Longer Life
7. Small Size.
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NUMBER SYSTEM : Group of Individual
(different) symbols used for any quantitative
representation
Number of SYMBOLS used in NUMBER
SYSTEM is called as Base or Radix.
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System Base Symbols
Binary 2 0, 1
Decimal 10 0, 1, … 9
Octal 8 0, 1, … 7
Hexa-Decimal
16 0, 1, … 9,
A, B, … F
1. Number Systems
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Decimal Binary Octal
Hexa-decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7
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Decimal Binary Octal
Hexa-decimal
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F
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Decimal Binary Octal
Hexa-decimal
16 10000 20 10
17 10001 21 11
18 10010 22 12
19 10011 23 13
20 10100 24 14
21 10101 25 15
22 10110 26 16
23 10111 27 17 Etc.
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EACH DIGIT IN BINARY IS CALLED AS “BIT”
FOUR BIT = 1 NIBBLE
EIGHT BIT = 1 BYTE
SIXTEEN BIT = 1 WORD
THIRTY TWO BIT = DOUBLE WORD
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The possibilities:
Hexadecimal
Decimal Octal
Binary
pp. 40-46
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2510 = 110012 = 318 = 1916
Base
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Hexadecimal
Decimal Octal
Binary
Next slide…
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12510 => 5 x 100 = 52 x 101 = 201 x 102 = 100
125
Base
Weight
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Hexadecimal
Decimal Octal
Binary
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Technique Multiply each bit by 2n, where n is the
“weight” of the bit The weight is the position of the bit,
starting from 0 on the right Add the results
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1010112 => 1 x 20 = 11 x 21 = 20 x 22 = 01 x 23 = 80 x 24 = 01 x 25 = 32
4310
Bit “0”
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Hexadecimal
Decimal Octal
Binary
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Technique Multiply each bit by 8n, where n is the
“weight” of the bit The weight is the position of the bit,
starting from 0 on the right Add the results
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7248 => 4 x 80 = 42 x 81 = 167 x 82 = 448
46810
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Hexadecimal
Decimal Octal
Binary
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Technique Multiply each bit by 16n, where n is the
“weight” of the bit The weight is the position of the bit,
starting from 0 on the right Add the results
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ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560
274810
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Hexadecimal
Decimal Octal
Binary
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Technique Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-
significant bit) Second remainder is bit 1 Etc.
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12510 = ?22 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1
12510 = 11111012
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Hexadecimal
Decimal Octal
Binary
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Technique Convert each octal digit to a 3-bit
equivalent binary representation
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7058 = ?2
7 0 5
111 000 101
7058 = 1110001012
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Hexadecimal
Decimal Octal
Binary
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Technique Convert each hexadecimal digit to a 4-bit
equivalent binary representation
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10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112
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Hexadecimal
Decimal Octal
Binary
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Technique Divide by 8 Keep track of the remainder
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123410 = ?8
8 1234 154 28 19 28 2 38 0 2
123410 = 23228
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Hexadecimal
Decimal Octal
Binary
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Technique Divide by 16 Keep track of the remainder
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123410 = ?16
123410 = 4D216
16 1234 77 216 4 13 = D16 0 4
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Hexadecimal
Decimal Octal
Binary
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Technique Group bits in threes, starting on right Convert to octal digits
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10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278
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Hexadecimal
Decimal Octal
Binary
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Technique Group bits in fours, starting on right Convert to hexadecimal digits
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10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16
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Hexadecimal
Decimal Octal
Binary
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Technique Use binary as an intermediary
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10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16
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Hexadecimal
Decimal Octal
Binary
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Technique Use binary as an intermediary
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1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148
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Don’t use a calculator!
Decimal Binary Octal
Hexa-decimal
33
1110101
703
1AF
Skip answer Answer
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Decimal Binary Octal
Hexa-decimal
33 100001 41 21
117 1110101 165 75
451 111000011 703 1C3
431 110101111 657 1AF
Answer
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Binary to decimal
pp. 46-50
10.1011 => 1 x 2-4 = 0.06251 x 2-3 = 0.1250 x 2-2 = 0.01 x 2-1 = 0.50 x 20 = 0.01 x 21 = 2.0 2.6875
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Decimal to binary
p. 50
3.14579
.14579x 20.29158x 20.58316x 21.16632x 20.33264x 20.66528x 21.33056
etc.11.001001...
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Don’t use a calculator!
Decimal Binary Octal
Hexa-decimal
29.8
101.1101
3.07
C.82
Skip answer Answer
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Decimal Binary Octal
Hexa-decimal
29.8 11101.110011… 35.63… 1D.CC…
5.8125 101.1101 5.64 5.D
3.109375 11.000111 3.07 3.1C
12.5078125 1100.10000010 14.404 C.82
Answer
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GENERAL
KNOWLEDGE
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Power Preface Symbol
10-12 pico p
10-9 nano n
10-6 micro
10-3 milli m
103 kilo k
106 mega M
109 giga G
1012 tera T
Value
.000000000001
.000000001
.000001
.001
1000
1000000
1000000000
1000000000000
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Base 2Power Preface Symbol
210 kilo k
220 mega M
230 Giga G
Value
1024
1048576
1073741824
• What is the value of “k”, “M”, and “G”?• In computing, particularly w.r.t. memory, the base-2 interpretation generally applies
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/ 230 =
In the lab…1. Double click on My Computer2. Right click on C:3. Click on Properties
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Binary Arithmetic
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Two 1-bit values
pp. 36-38
A B A + B
0 0 0
0 1 1
1 0 1
1 1 10“two”
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6620/04/23
0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 (carry 1)
1+1+1 = 1 (carry 1)
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Two n-bit values Add individual bits Propagate carries
10101 21+ 11001 + 25 101110 46
11
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Binary, two 1-bit values
A B A B
0 0 0
0 1 0
1 0 0
1 1 1
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Binary, two n-bit values As with decimal values E.g.,
1110 x 1011 1110 1110* 0000** 1110***10011010
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Code is the symbolic representation of discrete information, which can be represented in the form of numbers and letters.
Codes are classified in five groups: Weighted binary codes: These codes follow the positional
weighting principle in which the position of the numbers represent the weight. The different types of weighted codes are
Codes
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71
Binary Coded Decimal (BCD) is the combination of four digits, 8421, which represent the binary numbers.
Example: BCD equivalent of 1 is 0001 BCD equivalent of 2 is 0010 BCD equivalent of 5 is 0101 BCD equivalent of 9 is 1001
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Binary Coded Decimal
0 : 00001 : 00012 : 00103 : 00114 : 01005 : 01016 : 01107 : 01118 : 10009 : 1001
e.g. 1) 37=
0011 0111
2) 125
0001 0010 0101
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9’s complement It can be obtained by subtracting a number from 9
e.g. 9’s complement of 5 is = 9 - 5= 4 9’s complement of 0101 is 1001 - 0101 = 0100
10’s complement It can be obtained by adding 1 to 9’s complement 10’s complement of 0101 is 1001 – 0101+1 = 0100+1=0101
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Excess-3 CodesThese codes are obtained by adding binary three into its original binary number
Binary
0000
0001
0010
0011
0100
0101
0110
0111
Excess-3
0+ 0011
1+0011
10+0011
11+0011
100+0011
101+0011
110+0011
111+0011
Decimal
0
1
2
3
4
5
6
7
Excess-3
0011
0100
0101
0110
0111
1000
1001
1010
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