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Microprocessors & Interfacing
XX F241
Number systemsKCS Murt i
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Structure
Number systems
Decimal
Binary
Hexa Decimal
Operations
BCD Codes
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XX F241 MicroprocessorsKCS Murti
Number Systems
Decimal (0,1,2,3,4,5,6,7,8,9) Octal (0,1,2,3,4,5,6,7)
Hexadecimal (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)
Binary (0,1)
A number anan-1a2a1a0a-1a-m expressed in base-rsystem, has coefficients multiplied by powers of r
an.rn+an-1.r
n-1+.+a2.r2+a1.r
1+a0.r0+a-1.r
-1+a-
2.r-2+a-m.r-
m
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XX F241 MicroprocessorsKCS Murti
Convert to decimal
(B65F)16= 11X 163 + 6X 162 +5X 161 +15= (46687)10
(4021.2)5=4X 53+0 X 52+ 2X 51 +1 X 50 +2 X 51 = (511.4)10
(1010.011)2= 23 + 21+ 2-2+ 2-3= (10.375)10
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Binary Octal and Hexadecimal
(10 1100 0110 1011 . 1111 0010 )2= (2C6B.F2)16
(10 110 001 101 011 . 111 100 000 110)2= (26153.7460)8
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XX F241 MicroprocessorsKCS Murti
Repeated division steps: Divide the decimal number by 2
Write the remainder after eachdivision until a quotient of zero is
obtained. The first remainder is the LSB
and the last is the MSB
Decimal to Binary
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XX F241 MicroprocessorsKCS Murti
Decimal to binary..
10
Convert (153)10to Octal
0
5 0
41
20 1
1 0
2 1
Ans=101001
Integer ReminderConvert (41)10to binary
2 3
0 2
153
19 1
Ans=(231)8
0 1
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XX F241 MicroprocessorsKCS Murti
Decimal to hexadecimal
0 E
227
14 3
Ans=0xE3H
Integer ReminderConvert (227)10toHexadecimal
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XX F241 MicroprocessorsKCS Murti
Decimal fractions to octal
0.513 X 8= 4.104
0.104 X 8= 0.832
0.832 X 8= 6.656
0.656 X 8= 5.2480.248 X 8= 1.984
0.984 X 8= 7.872
(0.513)10= (0.406517..)8
Convert (0.513)10to octal
4 0 6 5 1 7
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XX F241 MicroprocessorsKCS Murti
Two types namely radix and diminished radix complement Diminished Radix or (r-1)s complement : For a number N
with n digits,
it is defined as (rn-1)-N
Radix or rscomplement : For a number N with n digits, it
is defined as
(rn-1)-N+1
Complements
9s complement of 546700 is (999999-546700)=453299
9s complement of 012398 is (999999-012398)=987601
10s complement of 012398 is 987602
10s complement of 246700 is 753300
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XX F241 MicroprocessorsKCS Murti
1s and 2s complement
1s complement of 1011000 is
2s complement of 1101011 is
0100111
10100101s complement of 0101101 is
0010101
2s complement of 0110111 is 1001001
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XX F241 MicroprocessorsKCS Murti
Signed Binary Numbers
Signed magnitude of +7 00000111
Signed magnitude of -7 10000111
Represent +7 in 2s complement form
Place 0 in sign bit 0
Place magnitude in remaining 7 bits 00000111
Represent -7 in 2s complement form
Start with 8 bit code for +7 00000111
Invert each bit including the MSB 11111000
Add 1 11111001
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XX F241 MicroprocessorsKCS Murti
Signed binary numbers
Get the magnitude of -7
Its 2s complement is 11111001
MSB is 1. So magnitude is in 2s
complement.Invert all bits including sign. 00000110
Add 1 00000111
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XX F241 MicroprocessorsKCS Murti
Range of numbers in 2s complement
Signed binary Decimal
01111111 +127
------
00000001 +100000000 Zero
11111111 -1
-------
10000001 -127
10000000 -128
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XX F241 MicroprocessorsKCS Murti
Addition
+13 00001101
+9 00001001
+22 00100110
+13 00001101
-9 11110111
+4 000001001
Ignore carry
+9 00001001
-13 11110011
-4 11111100
-9 11110111
-13 11110011
-22 11101010
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XX F241 MicroprocessorsKCS Murti
Hexa decimal addition
7A 0111
3F
1010
0011 1111
B9 1011 1001
7 A
3 F
1110 2510
B16 916
1 Carry
B16 916
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XX F241 MicroprocessorsKCS Murti
Overflow
01110011
11010001
01000100
115 115
209 -47
68 68
Interpreted as
unsigned binary
(Incorrect)
Interpreted as
2s complement
(correct)
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XX F241 MicroprocessorsKCS Murti
Each decimal digit is represented using 4 bits. Ex: convert 87410to BCD:
8 7 4
0100 0111 0100 = 010001110100BCD
Reverse the process to convert BCD to decimal
BCD code
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Advantages
1. Ease of conversion
2. Easy to design the logic circuit.
3. Only the 4 bit groups for the decimal digits 0 to 9 need to
be remembered.
Disadvantages:
1. BCD requires more bits
BCD code..
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Other binary codes
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Binary Gray Code000 000001 001010 011011 010100 110101 111110 101111 100
Grey code
Only one bit changes between successive values
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XX F241 MicroprocessorsKCS Murti
Gn = BnGn-1= Bn Bn-1
Gn-2 = Bn-1 Bn-2
..
..
G1 = B2 B1
Binary to Grey code conversion
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Binary 1 0 0 1
Gray 1 1 0 1
Gray to Binary
Gray 1 1 0 1
Binary 1 0 0 1
Example
Binary to Gray
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KCS Murti
Grey codes
Angular position Measurement where eachsegment is assigned a binary number
Drive
Load
Shaft
encoder
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KCS Murti
Shaft encoder
Shaft Encoder
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Bytes : A byte is a string of eight bits.
Nibble: A nibble is a string of four bits.
Word : The word size can be defined as the
number of bits in the binary word that a
digital system operates on.
i.e., it depends on the data pathway ofthe system.
The byte, Nibble and Word
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KCS Murti
ASCII American Standard Code forInformation Interchange
A binary code for letters, numerals, specialcharacters and control characters.
Seven bit code: 27= 128 possible codegroups
94 Graphic Characters and 34 non printing
control characters
Alphanumeric Code
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Alphanumeric Code
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Parity:
An extra bit (parity bit) is added to each word
being transmitted.
Even parity: P = 0 or 1 at Tx, such that no . of 1s
in the code including parity bit is an
evennumber.
Odd Parity: P= 0 or 1 , such that no. 1s in thecode includingparity bit is an odd
number.
Error detection codes
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Let the code group is 1000001 (ASCII code forA),if evenparity is used, P=0.
so, the new code 10000010
ifodd parityis used, P=1
so, the new code 10000011
Where the last bit in the new code is parity bit (P)
Example
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1. Position of error cannot be detected.2. Parity method would not work if two bits
were in error.
Limitations:
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