Hamming and Twos Complement

8/14/2019 Hamming and 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 wouldbe without the twos complement

representation.

Examples: 011012=1310 111012=-310


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To Convert apositive number to a negative2s

complement: 001102=610Convert 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 theright), and copy 0s until you get to a 1 (also copy the1): 10

(b) Then flip the rest of the bits:11010


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Why 2s

Complement?

Only one form of 0.

Easyto add:

Just Add!

0100 ( 410)

+1101 (-310)

-----

0001 ( 110)

It works!

Twos Complement Decimal

0111 7

0110 6

0101 5

0100 4

0011 3

0010 2

0001 1

0000 0

1111 -1

1110 -2

1101 -3

1100 -4

1011 -5

1010 -6

1001 -7

1000 -8


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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 thelast two carry bits: iftheyre different, youvegot an overflow condition)


0111 7

0110 6

0101 5

0100 4

0011 3

0010 2

0001 1

0000 0

1111 -1

1110 -2

1101 -3

1100 -4

1011 -5

1010 -6

1001 -7

1000 -8


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Why 2s Complement?

Subtraction is easy, too!

To subtract, simply convert thenumber youre subtractingaway to its 2s complement,and then add:

1011 (-510)

-0010 -( 210)

111

1011 (-510)

+1110 +(-210)

---- -----

1001 (-710)

Ignore the carry out


0111 7

0110 6

0101 5

0100 4

0011 3

0010 2

0001 1

0000 0

1111 -1

1110 -2

1101 -3

1100 -4

1011 -5

1010 -6

1001 -7

1000 -8


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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)

Twos

Complement

Dec.

0111 7

0110 6

0101 5

0100 4

0011 3

0010 2

0001 10000 0

1111 -1

1110 -2

1101 -31100 -4

1011 -5

1010 -6

1001 -7

1000 -8

1111011 (-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 detectan 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 a4-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 aparitybit. This adds an extra bit tothe number, say as the most significant bit. The sendercounts the number of ones in the number, and if it isan odd number, the parity bit becomes a 1 . This iscalled an evenparity bit, because when you look at thewhole string, it is even.

Examples:

1001There is an even number of 1s, so we send 01001.

1110An odd number of 1s, so we send 11110.


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Parity

Even parity bit examples:1001There is an even number of 1s, so we send 01001.

1110An 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 detectedan error.

Notes:

This only detects a single error (e.g., if two ones were bothsent 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 greatanswer)


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Hamming Code

Really, what we want is the ability to detect an

error, and also to correctthe 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 andcorrect

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 45 6 7 89 10


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Hamming CodeNumber: __1 _1 0 1 _0 0

Bit: 1 2 3 45 6 7 89 10

Step 3: Each parity bit calculates the parity for some of thebits in the code word. The position of the parity bitdetermines the sequence of bits that it alternately checksand 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 evenparityaltogether, so we basically count the number of 1s, and if it

is an odd number, the parity bit for that position becomes a1.


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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 45 6 7 89 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 45 6 7 89 10


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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 45 6 7 89 10

There are two ones, so bit 2 becomes a 0 to


Number: 101 _1 0 1 _0 0

Bit: 1 2 3 45 6 7 89 10


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Hamming Code


bits, skip 4 bits, etc. (4,5,6,7,12,13,14,15,...)Number: __1 _1 0 1 _0 0

Bit: 1 2 3 45 6 7 89 10

There are two ones, so bit 4 becomes a 0 to


Number: 101 01 0 1 _0 0

Bit: 1 2 3 45 6 7 89 10


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Hamming Code


bits, skip 8 bits, etc. (8-15,24-31,40-47,...)Number: __1 _1 0 1 _0 0

Bit: 1 2 3 45 6 7 89 10

There are zero ones, so bit 8 becomes a 0 to


Number: 101 01 0 1 00 0

Bit: 1 2 3 45 6 7 89 10

The number we send: 1 1 1 1


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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, 2+4=6, 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 theoriginal number:

1010111000 1010101000 1010101000

The original number sent was 110100

What if only one bit check was incorrect? Then the checkdigit was incorrect.

Ch k 1011101000


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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 weknow 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

0


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110 Step 1: Write number with space for the parity bits:

Number: _ _1 _ 1 0

Bits: 1 23 45 6

Step 2: Perform parity checks:Bit 1: 0_ 1 _ 1 0

Bit 2: 0 11 _ 1 0

Bit 4: 0 1 1 11 0

So, the number to

send is1111


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M H i C d P ti


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More Hamming Code Practice:

Produce a Hamming Code for: 010111What was the message sent if the following Hamming

encoded message was received? 011010010101What is the overhead (i.e., number of extra bits) neededto send a 32-bit binary number using Hamming Code?

Fi l W d


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Final Words:Ive put these notes on Toolkit (under ResourcesClass Notes)

Ive also written and uploaded (to Toolkit) a Python program to

encode and decode a bitstream.

References:

Your Textbook, p.20-21http://users.cs.fiu.edu/~downeyt/cop3402/hamming.html

http://en.wikipedia.org/wiki/Hamming_code

http://www.computing.dcu.ie/~humphrys/Notes/Networks/da

ta.hamming.html

Me: Chris Gregg, [email protected]
http://users.cs.fiu.edu/~downeyt/cop3402/hamming.htmlhttp://en.wikipedia.org/wiki/Hamming_codehttp://www.computing.dcu.ie/~humphrys/Notes/Networks/data.hamming.htmlhttp://www.computing.dcu.ie/~humphrys/Notes/Networks/data.hamming.htmlhttp://www.computing.dcu.ie/~humphrys/Notes/Networks/data.hamming.htmlhttp://www.computing.dcu.ie/~humphrys/Notes/Networks/data.hamming.htmlhttp://en.wikipedia.org/wiki/Hamming_codehttp://users.cs.fiu.edu/~downeyt/cop3402/hamming.html

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Hamming and Twos Complement