Overflow

1
Overflow

• Signed binary is in fixed range
• -2n-1 ? 2n-1
• If the answer for addition/subtraction more than
  the range, it is overflow
• Two situation where overflow can happen
• Positive positive negative (enough n-bit)
• Negative negative positive(more than n-bit)

2
Overflow

• Example Binary number 4-bit (second complement)
• Range -2n-1 ? 2n-1-1
• Range (1000)2s?(0111) 2s
• Range (-8) 10 ? (7) 10
• Two situation where overflow can happen
• Positive positive negative (enough n-bit)
• Negative negative positive(more than n-bit)
• 0101 5 1001 -7
• 0100 4 1010 -6
• ----------- -----------
• 1001 10011

3
Overflow

• Example Binary number 4-bit (second complement)
• Range -2n-1 ? 2n-1-1
• Range (1000)2s?(0111) 2s
• Range (-8) 10 ? (7) 10

(Overflow exist)
(ignore final carry)
(Overflow exist)
4
Fixed Point Number

• Signed number and unsigned number representation
  is given in fixed point number
• Binary point is assumed to have fixed location,
  if it is located at the end of the number
• It can represent integer number between 128 to
  127 (for 8-bit binary complement)

Binary point
5
Fixed Point Number

• Generally, other locations in binary point
  position
• Example If two fraction bit is used, we can
  represent

fraction
Binary point
6
Floating Point Number

• Fixed point number has limited range
• To represent extremely large or extremely small
  number, we use floating point number (like
  scientific number)
• Example
• 0.23X1023(really large number)
• 0.1239X10-10(really small number)

7
Floating Point Number

• Floating point number is divided into three parts
• mantissa, base and exponent
• Base always fixed in number system
• Therefore, only need mantissa and exponent

8
Floating Point Number

• Mantissa always in normalize form
• (base 10) 23X1021 is normalized to 0.23X1023
• (base 10) 0.0017X1021 is normalized to
• -0.17X1019
• (base 10) 0.01101X103 is normalized to
  0.1101X102
• 16-bit floating point number might contain 10-bit
  mantissa and 6-bit exponent
• More exponent, the greater its range
• More mantissa, the greater its persistence

9
Arithmetic with Floating Point Number

• Arithmetic with floating point number is much
  difficult
• MULTIPLICATION
• The steps
• multiply with the mantissa
• Add its exponent
• normalized

10
Arithmetic with Floating Point Number

• Example

(Normalization)
11
Arithmetic with Floating Point Number

• ADDITION
• Steps
• Equalize their exponent
• Add their mantissa
• Normalize them

12
Arithmetic with Floating Point Number

• Example
• (0.12x102)10 (0.0002x104 ) 10
• (0.12x102) 10 (0.02x102) 10
• (0.120.02) 10 x 102
• (0.14x102 ) 10

13
Binary Coded Decimal (BCD)

• Decimal number is normally used by human. Binary
  number is normally used by computer. It is
  expensive to exchange between each other.
• If used only little calculation, we can use
  coding scheme for decimal number.
• One of the scheme is BCD, or also called 8421
  code.
• Which represent every decimal digit with 4-bit
  binary code.

14
Binary Coded Decimal (BCD)

• There are code which is not used, e.g.
  (1010)BCD,(1011)BCD,.,(1111)BCD. This code is
  said to be an error.
• Easy to convert but the arithmetic is hard
• Suitable as interface such as keyboard input and
  digital reading

Decimal Digit
Decimal Digit
15
Binary Coded Decimal (BCD)

• Example
• Notes BCD is not similar to binary
• Example (243)10(11101010)2

Decimal Digit
Decimal Digit
16
Gray Code

• No weight
• Only one bit change from one code number to the
  others
• Suitable for error detection

Decimal Binary Gray Code
Decimal Binary Gray Code
17
Gray Code
18
Gray Code
19
Convert Binary Code to Gray Code

• Fixed MSB
• From left to right, add each coupled binary code
  bit next to each other to get Gray code bit,
  ignore carry
• Example convert binary 10110 to Gray code

20
Convert Gray Code to Binary Code

• Fixed MSB
• From left to right, add each coupled binary code
  executed with Gray code bit at the next position,
  ignore carry
• Example convert Gray 10110 to binary code

21
Other Decimal Code

• Self compliment code excess-3 code, 84-2-1,
  2421
• Error detection code Biquinary code (bitwo,
  quinaryfive)

22
Self Compliment Code

• Example Excess-3, 84-2-1, 2421
• Code represented by coupled compliment-digit
  which compliment each other

23
Alphanumeric Code

• Part of numbers, computer also handle textual
  data
• Set which always used includes
• Letters A,..,Z and a,..,z
• Digits 0,..,9
• Special Characters , ?, !, ,, .,.
• Not Printable SOH, NULL, BELL,.
• Most of the time, it is represented by 7 or 8-bit

24
Alphanumeric Code

• Two standard that are frequently used
• ASCII (American Standard Code for Information
  Interchange)
• EBCDIC (Extended BCD Interchange Code)
• ASCII 7-bit, add with parity bit for error
  detection (odd,even parity)
• EBCDIC 8-bit

25
Alphanumeric Code

• ASCII Table

26
Error Detection Code

• Error can exist in transmission. It must be
  detected so that retransmission can be requested
• With binary number, mostly exist 1-bit error.
  Example 0010 is transmitted incorrectly as 0011,
  or 0000, or 0110, or 1010
• Biquinary using additional 3-bit to detect error.
  For one error detection, only one extra bit is
  needed

27
Error Detection Code

• Parity Bit
• Even parity number of bit 1 is even
• Odd parity number of bit 1 is odd
• Example Odd parity

28
Error Detection Code

• Parity Bit can detect odd error and not even
  error (if odd is set)
• Example For odd parity number
• 10011gt10001 (detected)
• 10011gt10101 (not detected)
• Parity bit can also be used on data block

29
Error Detection Code

• Sometimes, it is not enough to detect code, we
  need to correct it
• Error correction is expensive in practical, we
  only need to use one bit error correction
• Popular technique Hamming Code
• Add k-bit to n-bit number to produce nk bit
• Number the bit 1 on bit nk
• Every parity bit is on the number range

30
Error Detection Code

• E.g For 8-bit number, we need 4 parity bit
• 12 bit number are 0001,0011,,1100. Every 4 bit
  parity is used to detect group of bit. Every
  parity bit is for themselves and has bit 1 on
  certain position bit

31
Error Detection Code

• Therefore
• P1 parity for bit 3,5,7,9,11
• P2 parity for bit 3,6,7,10
• P4 parity for bit 5,6,7,12
• P8 parity for bit 9,10,11,12

32
Error Detection Code

• Given 8-bit number 1100 0100
• Assume even parity is
• P1 parity for bit 3,5,7,9,11 0
• P2 parity for bit 3,6,7,10 0
• P4 parity for bit 5,6,7,12 1
• P8 parity for bit 9,10,11,12 1

33
Error Detection Code

• To check error, execute checking code
• C1 XOR 1,3,5,7,9,11
• C2 XOR 2,3,6,7,10
• C4 XOR 4,5,6,7,12
• C8 XOR 8,9,10,11,12
• If C8 C4 C2 C10000 therefore no error, if
  otherwise C8 C4 C2 C1 show position, there is an
  error for only one bit
• Example