CRC error correction

Started by edim January 6, 2009
Hello,
Can CRC be used for error correction (not just detection)?
In particular, if the CRC-8 generator polynomial is x^8+x^7+x^6+x^4+x^2+1
, can it be used for single error correction ?



edim wrote:

> Hello, > Can CRC be used for error correction (not just detection)?
Yes. CRCs are essentially Hamming or BCH codes, so they can be used for error correction. However by using them for error correction you are compromising the error detection capacity.
> In particular, if the CRC-8 generator polynomial is x^8+x^7+x^6+x^4+x^2+1 > , can it be used for single error correction ?
If this is a prime polynomial, then it can correct a single bit error in a block of up to 255 bits. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
edim <edim0@walla.com> wrote:

> Can CRC be used for error correction (not just detection)? > In particular, if the CRC-8 generator polynomial is x^8+x^7+x^6+x^4+x^2+1 > , can it be used for single error correction ?
It depends on how much data you have. To be able to do single error correction the word size with ECC bits must be less than or equal to 2**N where N is the number of ECC bits. In addition, it is usually desirable to detect (but not correct) double bit errors, which requries one more check bit. (Usually the parity bit for the rest.) For magnetic tapes in the past, one used one parity bit per character, and a CRC (or some other type of check word) at the end of a block. If the CRC failed then the character with the wrong parity bit was likely the one in error. Given that, you might be able to correct the error. -- glen
Thanks,
My message length is 72bits and I add 8bits for CRC using this generator
polynomial.
Is there a way to correct one bit in the received block of 80bits ?
Vladimir,
The polynomial x^8+x^7+x^6+x^4+x^2+1 is not prime because:
x^8+x^7+x^6+x^4+x^2+1 = (x^5+x^4+x^3+x^2+1)(x^2+x+1)(x+1)
edim <edim0@walla.com> wrote:

> My message length is 72bits and I add 8bits for CRC using > this generator polynomial. Is there a way to correct one bit > in the received block of 80bits ?
Yes, absolutely. A plain vanilla single-error-correcting BCH code with 7 check bits can correct a single error for block lengths up to 127 bits. With 8 check bits, properly constructed, you can also detect all 2-error cases. See Sloane et. al, "A Survey of Constructive Coding Theory, and a Table of Binary Codes of Highest Known Rate", in _Discrete Math._, V. 3, pp 265-294, Sept 1972 for the full details for any similar problem, especially if you need to go beyond the BCH limit. Steve
edim <edim0@walla.com> wrote:

>The polynomial x^8+x^7+x^6+x^4+x^2+1 is not prime because: >x^8+x^7+x^6+x^4+x^2+1 = (x^5+x^4+x^3+x^2+1)(x^2+x+1)(x+1)
In this case you should measure the distance of the resulting shortened code; if it is at least three, then you can still do single error correction. This may be most rapidly done by simulation. Steve
On Tue, 06 Jan 2009 10:31:50 -0600, Vladimir Vassilevsky wrote:

> edim wrote: > >> Hello, >> Can CRC be used for error correction (not just detection)? > > Yes. CRCs are essentially Hamming or BCH codes, so they can be used for > error correction. However by using them for error correction you are > compromising the error detection capacity. > >> In particular, if the CRC-8 generator polynomial is >> x^8+x^7+x^6+x^4+x^2+1 , can it be used for single error correction ? > > If this is a prime polynomial, then it can correct a single bit error in > a block of up to 255 bits. > > > Vladimir Vassilevsky > DSP and Mixed Signal Design Consultant http://www.abvolt.com
BUT In the real world errors often come in bursts, so the value of the CRC as an indication that your whole message is probably corrupted gets totally lost. Serious FEC schemes use much higher correction bit/data bit ratios. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
edim <edim0@walla.com> wrote:

> My message length is 72bits and I add 8bits for CRC > using this generator polynomial.
> Is there a way to correct one bit in the received block of 80bits ?
Maybe. Since 2**8 >= 80 it is enough. It would be more usual to use Hamming codes, which directly indicate the bit in error. You need to find out that the CRC values are unique for each possible bit error, including errors in the transmitted CRC. If you find the dependence of each CRC bit on the 72 data bits, you can find out which one has to change based on the received CRC bits. I believe, but haven't verified, that each CRC bit is formed as an exclusive OR of a set of data bits. If you find that set, you can find which bit is in error. (If homework, be sure to reference the newsgroup.) -- glen
Tim Wescott <tim@seemywebsite.com> wrote:
(snip)
 
>>> In particular, if the CRC-8 generator polynomial is >>> x^8+x^7+x^6+x^4+x^2+1 , can it be used for single error correction ?
(snip)
> In the real world errors often come in bursts, so the value of the CRC as > an indication that your whole message is probably corrupted gets totally > lost.
> Serious FEC schemes use much higher correction bit/data bit ratios.
Actually, 8 bits as ECC for 64 bit RAM is fairly common. It is especially convenient as it is the same number of bits needed for byte parity, which may already be available. -- glen
dvsarwate@yahoo.com <dvsarwate@gmail.com> wrote:

>The OP (edim) noted that
>>The polynomial x^8+x^7+x^6+x^4+x^2+1 is not prime because: >>x^8+x^7+x^6+x^4+x^2+1 = (x^5+x^4+x^3+x^2+1)(x^2+x+1)(x+1)
>and Steve Pope said
>> Apparently this polynomial factors, which may mean it looks >> more like a Fire Code, which can correct bursts. &#2013266080;Another
thing
>> to check out.
>Since (x^2+x+1)(x+1) = x^3 + 1 and x^5+x^4+x^3+x^2+1 >is a primitive polynomial of degree 5, the OP's polynomial >x^8+x^7+x^6+x^4+x^2+1 is indeed the generator polynomial >of a Fire code of length (2^5 - 1)x3 = 31x3 = 93. This code >can correct bursts of lengths up to 2, that is, it can correct >single errors and *adjacent* double errors.
I suspected as much. Thank you, Dilip.
>Decoders for Fire codes have simple implementations that >slickly hide the basic idea behind the decoding algorithm. >The implementation works as follows. The received word >is divided by the generator polynomial. Then the syndrome >is massaged to find the burst error and correct it "on the fly" >as the received word is being read out of the decoder.
Right. In the way I have done it, I first compute the syndrome, then shift the syndrome backwards through the FSR until I see the desired burst-error pattern. (In this case, either a single error or a two bit burst.) This is often called error-trapping. Of course with a code this small, one could just use a LUT on the syndrome.
>But, we can *think* of dividing the received word not by the >generator polynomial, but rather by x^3 + 1 and separately >by x^5+x^4+x^3+x^2+1. Now, the 93 received bits can be >viewed as 31 3-bit nybbles. The XOR sum of these nybbles >is, in effect, the result obtained when we divide the received >word by x^3 + 1. The XOR sum of the 31 nybbles can be > >000 (no errors); > >100, 010, 001 (one error); > >110, 011, 101 (double adjacent bit error: note that the last pattern >corresponds to a two-bit >error pattern that crosses a nybble boundary); > >or 111 (undecodable burst error) > >In other words, dividing by x^3 - 1 tells us *what* the burst error >is. >Dividing by x^5+x^4+x^3+x^2+1 tells us *which* nybble the burst >error starts in. Using this, we can correct the burst error. Not >very >efficient for hardware implementation, but might be useful in a >software implementation, especially if the nybble is a 8-bit byte.
I like this. Thanks. Steve
The OP (edim) noted that

>The polynomial x^8+x^7+x^6+x^4+x^2+1 is not prime because: >x^8+x^7+x^6+x^4+x^2+1 = (x^5+x^4+x^3+x^2+1)(x^2+x+1)(x+1)
and Steve Pope said
> Apparently this polynomial factors, which may mean it looks > more like a Fire Code, which can correct bursts. &#2013266080;Another
thing
> to check out.
Since (x^2+x+1)(x+1) = x^3 + 1 and x^5+x^4+x^3+x^2+1 is a primitive polynomial of degree 5, the OP's polynomial x^8+x^7+x^6+x^4+x^2+1 is indeed the generator polynomial of a Fire code of length (2^5 - 1)x3 = 31x3 = 93. This code can correct bursts of lengths up to 2, that is, it can correct single errors and *adjacent* double errors. It *cannot* correct all patterns of two non-adjacent errors in 93 bits. Decoders for Fire codes have simple implementations that slickly hide the basic idea behind the decoding algorithm. The implementation works as follows. The received word is divided by the generator polynomial. Then the syndrome is massaged to find the burst error and correct it "on the fly" as the received word is being read out of the decoder. But, we can *think* of dividing the received word not by the generator polynomial, but rather by x^3 + 1 and separately by x^5+x^4+x^3+x^2+1. Now, the 93 received bits can be viewed as 31 3-bit nybbles. The XOR sum of these nybbles is, in effect, the result obtained when we divide the received word by x^3 + 1. The XOR sum of the 31 nybbles can be 000 (no errors); 100, 010, 001 (one error); 110, 011, 101 (double adjacent bit error: note that the last pattern corresponds to a two-bit error pattern that crosses a nybble boundary); or 111 (undecodable burst error) In other words, dividing by x^3 - 1 tells us *what* the burst error is. Dividing by x^5+x^4+x^3+x^2+1 tells us *which* nybble the burst error starts in. Using this, we can correct the burst error. Not very efficient for hardware implementation, but might be useful in a software implementation, especially if the nybble is a 8-bit byte. Hope this helps --Dilip Sarwate
glen herrmannsfeldt  <gah@ugcs.caltech.edu> wrote:

>Tim Wescott <tim@seemywebsite.com> wrote: >(snip) > >>>> In particular, if the CRC-8 generator polynomial is >>>> x^8+x^7+x^6+x^4+x^2+1 , can it be used for single error correction
?
>(snip)
>> In the real world errors often come in bursts, so the value of the CRC as >> an indication that your whole message is probably corrupted gets totally >> lost.
Apparently this polynomial factors, which may mean it looks more like a Fire Code, which can correct bursts. Another thing to check out.
>> Serious FEC schemes use much higher correction bit/data bit ratios.
>Actually, 8 bits as ECC for 64 bit RAM is fairly common. >It is especially convenient as it is the same number of bits >needed for byte parity, which may already be available.
Yes, "serious" FEC methods come in all manner of block sizes and code rates. Steve
Tim Wescott <tim@seemywebsite.com> wrote:
(snip)
 
>>> In particular, if the CRC-8 generator polynomial is >>> x^8+x^7+x^6+x^4+x^2+1 , can it be used for single error correction ?
(snip)
> In the real world errors often come in bursts, so the value of the CRC as > an indication that your whole message is probably corrupted gets totally > lost.
> Serious FEC schemes use much higher correction bit/data bit ratios.
Actually, 8 bits as ECC for 64 bit RAM is fairly common. It is especially convenient as it is the same number of bits needed for byte parity, which may already be available. -- glen
edim <edim0@walla.com> wrote:

> My message length is 72bits and I add 8bits for CRC > using this generator polynomial.
> Is there a way to correct one bit in the received block of 80bits ?
Maybe. Since 2**8 >= 80 it is enough. It would be more usual to use Hamming codes, which directly indicate the bit in error. You need to find out that the CRC values are unique for each possible bit error, including errors in the transmitted CRC. If you find the dependence of each CRC bit on the 72 data bits, you can find out which one has to change based on the received CRC bits. I believe, but haven't verified, that each CRC bit is formed as an exclusive OR of a set of data bits. If you find that set, you can find which bit is in error. (If homework, be sure to reference the newsgroup.) -- glen
On Tue, 06 Jan 2009 10:31:50 -0600, Vladimir Vassilevsky wrote:

> edim wrote: > >> Hello, >> Can CRC be used for error correction (not just detection)? > > Yes. CRCs are essentially Hamming or BCH codes, so they can be used for > error correction. However by using them for error correction you are > compromising the error detection capacity. > >> In particular, if the CRC-8 generator polynomial is >> x^8+x^7+x^6+x^4+x^2+1 , can it be used for single error correction ? > > If this is a prime polynomial, then it can correct a single bit error in > a block of up to 255 bits. > > > Vladimir Vassilevsky > DSP and Mixed Signal Design Consultant http://www.abvolt.com
BUT In the real world errors often come in bursts, so the value of the CRC as an indication that your whole message is probably corrupted gets totally lost. Serious FEC schemes use much higher correction bit/data bit ratios. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
edim <edim0@walla.com> wrote:

>The polynomial x^8+x^7+x^6+x^4+x^2+1 is not prime because: >x^8+x^7+x^6+x^4+x^2+1 = (x^5+x^4+x^3+x^2+1)(x^2+x+1)(x+1)
In this case you should measure the distance of the resulting shortened code; if it is at least three, then you can still do single error correction. This may be most rapidly done by simulation. Steve
edim <edim0@walla.com> wrote:

> My message length is 72bits and I add 8bits for CRC using > this generator polynomial. Is there a way to correct one bit > in the received block of 80bits ?
Yes, absolutely. A plain vanilla single-error-correcting BCH code with 7 check bits can correct a single error for block lengths up to 127 bits. With 8 check bits, properly constructed, you can also detect all 2-error cases. See Sloane et. al, "A Survey of Constructive Coding Theory, and a Table of Binary Codes of Highest Known Rate", in _Discrete Math._, V. 3, pp 265-294, Sept 1972 for the full details for any similar problem, especially if you need to go beyond the BCH limit. Steve
Vladimir,
The polynomial x^8+x^7+x^6+x^4+x^2+1 is not prime because:
x^8+x^7+x^6+x^4+x^2+1 = (x^5+x^4+x^3+x^2+1)(x^2+x+1)(x+1)
Thanks,
My message length is 72bits and I add 8bits for CRC using this generator
polynomial.
Is there a way to correct one bit in the received block of 80bits ?