More About Twos Complement

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More About Twos Complement

• Twos Complement is used to represent both
  positive and negative numbers, but the positive
  numbers are the same as they would be without the
  twos complement representation.
• Examples 0110121310
• 111012-310

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To Convert a positive number to a negative 2s
complement

• 001102610? Convert to -6 in 2s complement
• There are two methods
• Method 1
• (a) Flip all the bits
• 11001 (this is the 1s complement)
• (b) Add 1
• 11010
• Method 2 (shortcut!)
• (a) Start at least significant bit (the farthest
  to the right), and copy 0s until you get to a 1
  (also copy the 1) 10
• (b) Then flip the rest of the bits11010

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Why 2s Complement?

• Only one form of 0.
• Easy to add
• Just Add!
• 0100 ( 410)
• 1101 (-310)
• -----
• 0001 ( 110)
• It works!

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Why 2s Complement?

• Just Add!
• 1011 (-510)
• 0010 ( 210)
• -----
• 1101 (-310)
• It works!
• Wow.
• (by the way, you detect overflow by looking at
  the last two carry bits if theyre different,
  youve got an overflow condition)

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Why 2s Complement?

• Subtraction is easy, too!
• To subtract, simply convert the number youre
  subtracting away to its 2s complement, and then
  add
• 1011 (-510)
• -0010 -( 210)
• 111
• 1011 (-510)
• 1110 (-210)
• ---- -----
• 1001 (-710)
• Ignore the carry out

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Why 2s Complement?

• Subtraction is easy, too!
• To subtract, simply convert the number youre
  subtracting away to its 2s complement, and then
  add
• 1011 (-510)
• -0010 (-210)

111 1011 (-510) 0010 (210) ---- -----
1101 (-310) Ignore the carry out
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Hamming Code

• Lets say you want to send a series of binary
  digits from one location to another, via a wire.
• Question Is the message guaranteed to get from
  sender to receiver perfectly?
• Answer No. In the physical world, errors
  sometimes occur in transmission. Generally, the
  faster the transmission speed, the greater the
  number of errors that will occur.

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Hamming Code

• Obviously, we want to be able to detect an error
  if there is one.
• If we detect an error, what can we do about it?
• Lets say we send a 4-bit number, 1010, and it
  arrives as 1011. Can we detect if it is wrong?
• Answer In this case, not really. All we know is
  that were expecting a 4-bit number, and we got a
  4-bit number. But we have no idea if it is
  correct or not.

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Parity

• Well, then, what can we do?
• One idea is to add a parity bit. This adds an
  extra bit to the number, say as the most
  significant bit. The sender counts the number of
  ones in the number, and if it is an odd number,
  the parity bit becomes a 1 . This is called an
  even parity bit, because when you look at the
  whole string, it is even.
• Examples
• 1001 ? There is an even number of 1s, so we send
  01001.
• 1110 ? An odd number of 1s, so we send 11110.

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Parity

• Even parity bit examples
• 1001 ? There is an even number of 1s, so we send
  01001.
• 1110 ? An odd number of 1s, so we send 11110.
• Now, if we receive a number that has the wrong
  parity (e.g., we count the 1s, and there isnt an
  even number of them), then we have detected an
  error.
• Notes
• This only detects a single error (e.g., if two
  ones were both sent as zeros, there would still
  be even parity, and we wouldnt know there was an
  error)
• What do we do once we detect the error? Throw
  away the number? Ask the sender to resend it?
  (neither is a great answer)

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Hamming Code

• Really, what we want is the ability to detect an
  error, and also to correct the error. How can we
  do this?

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Hamming Code

• In 1950, Richard Hamming came up with a clever
  method to both detect and correct single errors
  in a series of bits.
• This is how the code works
• Step one. The sender decides on how many message
  bits to use. Well use a 6-bit number
• 110100

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Hamming Code

• Our number 110100
• Step 2 Write the number down, but leave the
  powers of two bits empty, starting with the
  highest order bit
• Number _ _ 1 _ 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10

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Hamming Code

• Number _ _ 1 _ 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10
• Step 3 Each parity bit calculates the parity for
  some of the bits in the code word. The position
  of the parity bit determines the sequence of bits
  that it alternately checks and skips.
• Position 1 check 1 bit, skip 1 bit, check 1 bit,
  skip 1 bit, etc. (1,3,5,7,9,11,13,15,...)
• Position 2 check 2 bits, skip 2 bits, check 2
  bits, skip 2 bits, etc. (2,3,6,7,10,11,14,15,...)
• Position 4 check 4 bits, skip 4 bits, check 4
  bits, skip 4 bits, etc. (4,5,6,7,12,13,14,15,20,21
  ,22,23,...)
• Position 8 check 8 bits, skip 8 bits, check 8
  bits, skip 8 bits, etc. (8-15,24-31,40-47,...)
• We want the bits that weve checked to be even
  parity altogether, so we basically count the
  number of 1s, and if it is an odd number, the
  parity bit for that position becomes a 1.

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Hamming Code

• Position 1 check 1 bit, skip 1 bit, check 1 bit,
  skip 1 bit, etc. (1,3,5,7,9)
• Number _ _ 1 _ 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10
• There are three ones, so bit 1 becomes a 1 to
  make the parity even
• Number 1 _ 1 _ 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10

16
Hamming Code

• Position 2 check 2 bits, skip 2 bits, check 2
  bits, skip 2 bits, etc. (2,3,6,7,10)
• Number _ _ 1 _ 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10
• There are two ones, so bit 2 becomes a 0 to make
  the parity even
• Number 1 0 1 _ 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10

17
Hamming Code

• Position 4 check 4 bits, skip 4 bits, check 4
  bits, skip 4 bits, etc. (4,5,6,7,12,13,14,15,...)
• Number _ _ 1 _ 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10
• There are two ones, so bit 4 becomes a 0 to make
  the parity even
• Number 1 0 1 0 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10

18
Hamming Code

• Position 8 check 8 bits, skip 8 bits, check 8
  bits, skip 8 bits, etc. (8-15,24-31,40-47,...)
• Number _ _ 1 _ 1 0 1 _ 0 0
• Bit 1 2 3 4 5 6 7 8 9 10
• There are zero ones, so bit 8 becomes a 0 to make
  the parity even
• Number 1 0 1 0 1 0 1 0 0 0
• Bit 1 2 3 4 5 6 7 8 9 10
• The number we send 1010101000

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Hamming Code

• The number we send 1010101000
• The receiver then receives the bits and checks
  the number by verifying each check bit. Lets
  say the following number was received
• 1010111000
• Check
• Bit 1 1 0 1 0 1 1 1 0 0 0 Even parity ?
• Bit 2 1 0 1 0 1 1 1 0 0 0 Odd parity ?
• Bit 4 1 0 1 0 1 1 1 0 0 0 Odd parity ?
• Bit 8 1 0 1 0 1 1 1 0 0 0 Even parity ?

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Hamming Code

• 1010111000
• Check
• Bit 1 1 0 1 0 1 1 1 0 0 0 Even parity ?
• Bit 2 1 0 1 0 1 1 1 0 0 0 Odd parity ?
• Bit 4 1 0 1 0 1 1 1 0 0 0 Odd parity ?
• Bit 8 1 0 1 0 1 1 1 0 0 0 Even parity ?

• The best part about this code is the next check
• Add the check bit numbers that were incorrect,
  and you get the incorrect bit!
• In this case, 246, so bit six is incorrect, it
  should be a 0.

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Hamming Code

• The last step is to fix the incorrect bit, and
  then extract the original number
• 1010111000 ? 1010101000 ? 1010101000
• The original number sent was 110100
• What if only one bit check was incorrect? Then
  the check digit was incorrect.

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• Check 1011101000
• Bit 1 1 0 1 1 1 0 1 0 0 0 Even parity ?
• Bit 2 1 0 1 1 1 0 1 0 0 0 Even parity ?
• Bit 4 1 0 1 1 1 0 1 0 0 0 Odd parity ?
• Bit 8 1 0 1 1 1 0 1 0 0 0 Even parity ?

• In this case, only bit 4 had a bad parity check,
  so we know that it was the digit that was
  incorrect (of course 4 0 4, so we can also
  use our add the incorrect parity bits solution,
  too)

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Hamming Code

• Practice Apply a Hamming Code to the following
  3-digit binary number
• 110

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• 110 Step 1 Write number with space for the
  parity bits
• Number _ _ 1 _ 1 0
• Bits 1 2 3 4 5 6

Step 2 Perform parity checks Bit 1 0 _ 1 _ 1
0 Bit 2 0 1 1 _ 1 0 Bit 4 0 1 1 1 1 0
So, the number to send is 011110
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The number sent was 011110 Lets say the number
received was 011111 Check the parity bits Bit 1
0 1 1 1 1 1 Even parity ? Bit 2 0 1 1 1 1 1
Odd parity ? Bit 4 0 1 1 1 1 1 Odd parity
? Which bit is incorrect? 2 4 bit 6 is
incorrect.
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• More Hamming Code Practice
• Produce a Hamming Code for 010111
• What was the message sent if the following
  Hamming encoded message was received?
  011010010101
• What is the overhead (i.e., number of extra bits)
  needed to send a 32-bit binary number using
  Hamming Code?

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• Final Words
• Ive put these notes on Toolkit (under
  Resources?Class Notes)
• Ive also written and uploaded (to Toolkit) a
  Python program to encode and decode a bitstream.
• References
• Your Textbook, p.20-21
• http//users.cs.fiu.edu/downeyt/cop3402/hamming.h
  tml
• http//en.wikipedia.org/wiki/Hamming_code
• http//www.computing.dcu.ie/humphrys/Notes/Networ
  ks/data.hamming.html
• Me Chris Gregg, chg5w_at_virginia.edu