improved hamming code for error detection

8/14/2019 Improved Hamming Code for Error Detection

1/20

1

Improved Hamming Code forError Detection and Correction

U. K. Kumar, Member, IEEE, and B. S.

Umashankar2007 2nd International Symposium on

Wireless Pervasive Computing


2/20

2

Outline

Introduction

Proposed Improved Hamming Code

Conclusion

Improved Hamming Code for Error Detection and Correction


3/20

3

well known for its single-bit error detection &correction capability.

based on the of adding r redundancy bits to n databits such that 2r n + r + 1.

2r -1 bitsHamming Code r redundancy bits (2r -1) -rdata bits redundancy bitsbe interspersed at bit positions

2n (n = 0, 1, 2, 3) with the original data bits. Data bitsAll other bit positions are for the data to be

encoded. (positions 3, 5, 6, 7, 9..etc.)

Hamming code

Introduction


4/20

4

Rule & How to Remember

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. (8,9,10,11,12,13,14,15,24,25,26,27,...)

total number of ones in the positions is odd -> set check

bit 1. total number of ones in the positions is even -> set

check bit 0.

Hamming code

Introduction


5/20

5

An example for 7-bit Hamming code is given

below( for comparison with the proposedimprovement.)

the 7-bit data to be transmitted is 0110010.

the 11-bit data actually transmitted is011P001P0PP

the Ps refer to the Hamming bits that are to be

calculated and interspersed at bit positions 1,2, 4, & 8.

Hamming code

Introduction


6/20

6

An example for 7-bit Hamming code 0110010.

the 11-bit data actually transmitted is

011P001P0PP bit position 1is even parity at bit positions 1, 3, 5, 7, 9,

& 11. Thus the Hamming bit at position 1 will be 0.

bit position 2

is even parity at bit positions 2, 3, 6, 7,

10, & 11. Thus the Hamming bit at position 2 will be 1.

bit position 4is even parity at bit positions 4, 5, 6, & 7.Thus the Hamming bit at position 4 will be 1.

bit position 8is even parity at bit positions 8, 9, 10, &11. Thus the Hamming bit at position 8 will be 0.

Hamming code

Introduction


7/20

7

Thus the 11-bit data actually transmitted is01100011010.

For the calculation of Hamming bits atpositions 1, 2, 4, & 8, even-parity checkswere performed on 6, 6, 4, & 4 bitsrespectively.

Thus a total of 20 bits are involved in theprocess of Hamming bits calculations.

Hamming code

Introduction


8/20

8

The error-detection and correction process in Hamming code isas illustrated in Table I.

11-bit data actually transmitted is 01100011010

S0-bit position 1 : 1357911S1-bit Position 2 : 23671011S2-bit position 4 : 4567S3-bit position 8 : 891011

TABLE I

error-detection and correction using hamming code

Hamming code

Introduction


9/20

9

After error detection & correction, if any, the data bitshave to be reassembled by removing the

redundancy bits. Hamming code is normally used for transmission of 7-bit

data item.

Scaling it for larger data lengths results in a lot of

overhead due to interspersing the redundancy bitsand their removal later.

Hamming code

Introduction


10/20

10

IMPROVED HAMMING CODE

The number of redundancy bits, r to be appended to n-bit data is obtained such that the relation (2r-11) n issatisfied.

The number of redundancy bits in this method is sameas that for Hamming code for some values of n.

But in some cases, it will be just one more redundancybit than needed in the Hamming code.

PROP OSED IMPROVED HAMMING CODE


11/20

11


consider the same example. The 7-bit data

to be transmitted is 0110010. For 7-bit data, the number of redundancy

bits required will be 4.

the 11-bit data actually transmitted isPPPP0110010.

the Ps refer to the redundancy bits that are

to be calculated and appended at bitpositions 8, 9, 10, & 11.



12/20

12

the 11-bit data actually transmitted is PPPP0110010.

bit position 8is even parity at bit positions 1, 3, 5,7 and bit position 8. Thus the bit at position 8 willbe 1.

bit position 9is even parity at bit positions 2, 3, 6,

7 and bit position 9. Thus the bit at position 9 willbe 0.

bit position 10is even parity at bit positions 4, 5, 6,7 and bit position 10. Thus the bit at position 10will be 0.

bit position 11 is even parity considering only theredundancy bits8,9,10,11. Thus the bit atposition 11 will be 1.




13/20

13

Thus the 11-bit data actually transmitted is10010110010.

For the calculation of bits at positions 8, 9,10, & 11, even parity checks were performed

on 5, 5, 5, & 4 bits respectively. Thus a total of 19 bits are involved in the

process of calculation of redundancy bits.




14/20

14

The error-detection and correction process in the improvedmethod is as illustrated in Table II.

the 11-bit data actually transmitted is 10010110010.

S0-bit position 8 : 13578S1-bit Position 9 : 23679S2-bit position 10: 456710S3-bit position 11: 891011


TABLE IIerror-detection and correction using improved hamming code



15/20

15

Suppose the received information is 10010100010, with anerror at bit position 5.

the status of parity check S0 is shown as Feven parityerror at bit positions 1, 3, 5, 7 & 8. the status of parity check S1 is shown as T no even

parity error at bit positions 2, 3, 6, 7 & 9.

the status of parity check S2 is shown as F even parityerror at bit positions 4, 5, 6, 7& 10

the status of parity check S3 is shown as T evenparity occurs at bit positions 8, 9, 10 & 11.

Interpreting F as 1 and T as 0 in the status of parity check,we find that the error is at bit position 0101 i.e. 5. Bit5,which was received as 0, is corrected as 1.




16/20

16

Suppose the received information is 10000110010, withan error at redundancy bit position 8.

Then the status of parity check will turn out to beFTTF.

If status of parity check is FXXX, it implies thatthere is an error only in a redundancy bit.

The actual redundancy bit in error can easily beidentified by the position of the of F. However, this isnot a data bit there is no need for any correction.

Suppose the received information is 10010110010, withno error. Then the status of parity check will turn outto be TTTT, implying error at bit position 0000 i.e. 0.This is treated as no error.




17/20

17

After error detection & correction, if any, the redundancy

bits can be removed easily and there is no need to

reassemble the data bits.

This improved method can be scaled easily for larger

data lengths. The redundancy bits just get appended to

the data.




18/20

18

As an example, let us consider a 56-bit data stream, which

can contain eight 7-bit ASCII characters or seven bytes.

This needs 7 redundancy bits, which are selected as in theprevious example.

In spite of an extra redundancy bit when compared to

the Hamming code, the number of bits involved in the

process of calculation of redundancy bits is much less

than in the Hamming code.

It can be shown that it involves a total of 172 bits in the

process of calculation of these 7 redundancy bits. In comparison, in the Hamming code it involves a total

of 182 bits in the process of calculation of 6 redundancy

bits.




19/20

19

In the proposed improvementtheredundancy bits are appended at the end of

data bits.

This eliminates the overhead of interspersing

the redundancy bits at the sender end and their

removal at the receiver end after checking for

single-bit error and consequent correction, if any.

Further the effort needed in identifying the values

of the redundancy bits is lower in the proposed

novel method.

Conclusion

Proposed Improvement vs. Hamming Code


20/20

20

Hamming code is normally used for transmission of 7-bitdata item.

Scaling it for larger data lengths results in a lot ofoverhead due to interspersing the redundancy bits andtheir removal later.

In contrast, the proposed method is highly scalable

without such overhead.

there is only 7 bit overhead for a 56-bit data stream, which ismuch less compared to 4 bit overhead for a 7-bit data.

Because of this feature this new method is suitable fortransmission of large size data bit-streams as long asthere is likelihood of at the most single-bit error duringtransmission.

Proposed Improvement vs. Hamming Code

Conclusion

improved hamming code for error detection

Documents