1

PERFORMANCE ESTIMATION OF CRC WITH LDPC CODES

Presenters: Rich Prodan and BZ Shen

2

PACKET ERROR DETECTION WITH CYCLIC REDUNDANCY CHECK (CRC)

Add n-k bits of extra data (the CRC field) to an k-bit message to provide error detection function (i.e. an (n,k) binary cyclic code)

For efficiency, n-k << n

e.g., n-k = 32 for Ethernet and k = 12,000 (1500 bytes)

Header Body

8 16 32 8

CRC Beginning sequence

Ending sequence

3

A PICTORIAL VIEW OF CRC CODING

2n vectors 2k code words

Replace k information bits by a unique n bit code word

4

ERROR DETECTION

2n vectors transmitted

received

Received vector not equal to one of the 2k code words

5

ERROR DETECTION CAPABILITY

code words differ in at least dmin positions

2k vectors

up to dmin –1

errors are detectable

0xx1x0x0

4 differences

1xx0x1x1

≤ 3 errors can be detected

6

MISDETECTION

2n vectors 2k code words

transmitted

received

For a random vector:

the probability of false

acceptance is (2k-1)/2n

≈ 2-(n-k) for 2k >> 1

(e.g. 2-32 = 2.3e-10)

= probability of hitting the wrong code word (misdetection)

7

PROPER CODES

Linear block codes do not necessarily obey the 2-(n-k) bound

A code is proper if PU is monotonically increasing in p for 0 ≤ p ≤ 0.5

Hamming codes and primitive double error-correcting BCH codes are proper

2-16 = 1.5e-5

8

PROBABILITY OF UNDETECTABLE ERROR

C: binary [n, k] linear code

n: the codeword length

k: the number of information bits

p : bit error probability on every received bit

Pue(C, p): probability of undetectable error

Among 2n binary random strings of size n (i.e. p=1/2) there are 2k strings are codewords of C → the fraction of such strings that are codewords of C is 2k/2n. Thus

)(2)2

1,( kn

ue CP

In general, it is not necessary true that 2

1for )

2

1,(),( pCPpCP ueue

9

EXISTENCE UPPER BOUND

Levenshtein bound (1977) For a given n and k, if 0≤p≤1/2 and

R=k/n≤1-H2(p), then there is a binary [n, k] linear codes C such that

RRH

pppppH

pppCP RnnRnue

1))((such that

1/2(R)0 and )1(log)1()(log)( where

)1(),(

2

222

)()(

The bound gives the best a CRC code can achieve. However, not all CRC codes have such upper bound.

10

GOOD CRC CODES

Philip Koopman, “32-Bit Cyclic Redundancy Codes for Internet Applications,”

The International Conference on Dependable Systems and Networks (DSN) 2002

11

ADDING CRC BITS TO LDPC INFORMATION BITS

On long size downstream/upstream (16200,14400) LDPC code

After adding 32 bit CRC, the actual number of information bits is 14368

Need a (14400,14368) CRC code

On medium size upstream (5940, 5040) LDPC code

After adding 32 bit CRC, the actual number of information bits is 5008

Need a (5040,5008) CRC code

On short size upstream (1120, 840) LDPC code

After adding 32 bits CRC, the actual number of information bits is 808

Need a (840,808) CRC code

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ON DOWNSTREAM LDPC CODE (4096 QAM)

0001579.0 when )(19050.00015792)(0.0222221

0.99778 :rate CRC)14368,14400(),( :CRC bits 32

14368:ninformatio code LDPC

2

ppHRRR

R kn

bits

BER (bit error rate)

(after max. 30 iterations LDPC decoding)

Best Pue

34.7 34.75 34.8 34.85 34.9 34.95 35 35.05 35.110

-20

10-18

10-16

10-14

10-12

10-10

10-8

SNR(dB)

Pue

34.7 34.75 34.8 34.85 34.9 34.95 35 35.05 35.110

-9

10-8

10-7

10-6

10-5

10-4

10-3

SNR (dB)

BE

R

DS LDPC(16200,14400)

4096QAM, 30it

4096QAM

13

ON LONG SIZE UPSTREAM LDPC CODE (4096 QAM)

0001579.0 when )(19050.00015792)(0.0222221

0.99778 :rate CRC)14368,14400(),( :CRC bits 32

14400:ninformatio code LDPC

2

ppHRRR

R kn

bits

BER (bit error rate)

(after max. 30 iterations LDPC decoding)

Best Pue

34.8 34.85 34.9 34.95 35 35.05 35.1 35.15 35.2

10-20

10-18

10-16

10-14

10-12

10-10

SNR(dB)

Pue

34.8 34.85 34.9 34.95 35 35.05 35.1 35.15 35.2

10-8

10-7

10-6

10-5

10-4

SNR (dB)

BE

R

US LDPC(16200,14400)

4096QAM, 30it

4096QAM

14

29.2 29.25 29.3 29.35 29.4 29.45 29.5 29.55 29.6 29.65

10-8

10-7

10-6

10-5

10-4

10-3

SNR (dB)

BE

R

US LDPC(16200,14400)

41024 QAM, 30it

ON LONG SIZE UPSTREAM LDPC CODE (1024 QAM)

0001579.0 when )(19050.00015792)(0.0222221

0.99778 :rate CRC)14368,14400(),( :CRC bits 32

14400:ninformatio code LDPC

2

ppHRRR

R kn

bits

BER (bit error rate)

(after max. 30 iterations LDPC decoding)

Best Pue

1024QAM

29.3 29.35 29.4 29.45 29.5 29.55 29.6

10-18

10-16

10-14

10-12

10-10

SNR(dB)

Pue

15

ON MEDIAN SIZE UPSTREAM LDPC CODE (1024 QAM)

0005133.0 when )(10180.00051326)(63492060.006349201

36507940.99365079 :rate CRC)5008,5040(),( :CRC bits 32

5040:ninformatio LDPC

2

ppHRRR

R kn

bits

BER (bit error rate)

(after max. 30 iterations LDPC decoding)

Best Pue

28 28.2 28.4 28.6 28.8 29

10-8

10-7

10-6

10-5

10-4

10-3

10-2

SNR (dB)

BE

R

US LDPC(5940,5040)

1024 QAM, 30it

28.3 28.4 28.5 28.6 28.7 28.8 28.9 29

10-20

10-18

10-16

10-14

10-12

SNR(dB)

Pue

16

ON SHORT SIZE UPSTREAM LDPC CODE (1024 QAM)

004059.0 when )(168390.00405948)(80952380.038095231

19047620.96190476 :rate CRC)808,840(),( :CRC bits 32

840 :coden informatio LDPC

2

ppHRRR

R kn

bits

BER (bit error rate)

(after max. 30 iterations LDPC decoding)

Best Pue

26 26.5 27 27.5 28 28.5

10-8

10-7

10-6

10-5

10-4

10-3

10-2

SNR (dB)

BE

R

US LDPC(1120,840)

1024 QAM, 30it

26.4 26.6 26.8 27 27.2 27.4 27.6 27.8 28 28.2 28.4

10-25

10-20

10-15

10-10

SNR(dB)

Pue

17

CONCLUSION

A bound for probability of an undetected error using CRC has been presented

The bound is calculated for the proposed long, medium, and short codes for active plant for adding a 32 bit CRC to each codeword

The overhead in adding a 32 bit CRC is only 0.2%, 0.6%, and 3.8% for the long, medium, and short size codes respectively

The probability of an undetected error is less than 1e-18 above the threshold probability of error for these codes