Final Oral Defense 040109

April 2009

1

Channel Matched Iterative Decoding for

Magnetic Recording Systems

Final Oral Examination

Hakim Alhussien, PhD Candidate

Adviser: Jae Moon

Communications and Data Storage (CDS) Laboratory

Department of Electrical and Computer Engineering

University of Minnesota

April 06, 2009

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2

Outline

� Perpendicular magnetic recording channel.

• ECC for recording channels.

• Error Pattern Correction Coding (EPCC).

� EPCC enhanced TE (TE-EPCC).

• Error rate analysis of TE-EPCC.

• TE-EPCC and TP-EPCC for PMRC.

� Tensor product parity codes (TPPC).

• Linear-time Encoding of tensor product codes.

• Hard decoding of EPC-RS tensor product codes.

• Error rate analysis of EPC-RS tensor product codes.

� EPC-LDPC tensor product codes.

• Soft-syndrome decoding of EPC-LDPC tensor product code.

• Simulation study of EPC-LDPC.

� Thesis contributions.

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3

Perpendicular Magnetic Recording (PMR) Channel

� Recording channel is “transition-response fixed”

• To achieve the same normalized user density at a lower coding rate, the SNR is degrader by � use high rate codes.

� Saturated-level recording (binary-constrained input)

• Optimal-precoding or SNR water-filling not possible.

� Channel impaired by long error bursts.

• Due to ISI, disk defects, and thermal asperities.

• Symbol-correcting codes effective in burst correction,

such as RS, LDPC over GF(q).

� Data reread is expensive in terms of latency

• Standard frame error rate is very low

� Fixed ISI channel with dominant odd and even error events

• Utilize ECC targeting dominant errors events after ML detection.

� DC full PRML target

• A DC wandering compensation loop is required.

� Transition-dependent medium noise due to zigzag domain boundaries

• Channel detector trellis incorporates PDNP.

( )210110 log

R×∼

13 1410 10− −∼

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4

ECC for PMR Read Channel

� Reed Solomon (RS)

• Minimum distance of RS > LDPC for same block length and rate.

• ML decoding of RS outperforms ML decoding LDPC.

• Iterative Belief propagation decoding approaches ML performance.

• RS parity check matrix very dense – large number of 4-cycles.

• Iterative decoding of LDPC significantly outperforms RS iterative decoding.

� RS with inner LDPC or turbo codes

• Error behavior of LDPC is catastrophic for strong codes.

• Requires high-rate low-column weight LDPC – weak family of codes.

• Convolutional based Turbo: long tail in symbol-error distribution.

� Stand-alone LDPC

• Extensive research on lowering the SER error floor.

• LDPC with sector-wide codeword has low minimum distance.

• Sparse LDPC: improved iterative decoding – larger girth.

• Dense or large-block length LDPC: better Hamming weight spectrum

• Consider: Sparse non-binary LDPC of sector-length codeword!

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The error-pattern correcting code (EPCC)

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6

The Channel Matched ECC Paradigm

High density

perpendicular

recording channel

Write head/medium/read headWrite head/medium/read headStrong general

ECC encoder

Strong general

ECC encoder

Equalizer/DetectorEqualizer/Detector

focuses on correcting a few dominant error patterns

correctsremaining errors

� Premise: for a given ISI channel all dominant error patterns are known a

priori.

• Hyperbolic tangent transition response at a channel density of 1.4,

• 10% AWGN and 90% jitter noise.,

• Target response: 1+0.9D,

• Bit error rates: 2.3276×10-3 (1 PDNP) ,

• Captured # of error patterns: 223,676,

• Edt/N90 = 13.5 dB

Channel-matched

EPC encoder

Channel-matched

EPC encoder

Channel-matched

EPC encoder

Channel-matched

EPC encoderStrong general

ECC decoder

Strong general

ECC decoder

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7

EPCC Design: Target List =5 most dominant errors

� Target the 5 most dominant errors,

which account for 92.04% of possible errors.

� Syndrome set produced by g(x) = 1 + x +x3 + x5 + x6

• Order of g(x)=12.

• Total number of distinct syndrome sets: 5.

• 5 distinct, non-overlapping syndrome sets are utilized to distinguish 5 target error.

� Cyclic generator polynomial used to design a cyclic

(12,6) code of rate=0.5, and code cord length 12.

� Single occurrences of error types {1,2,4,5} decoded

without ambiguity.

� Via channel reliability information and the polarity of

data support, error type 3 can be decoded reliably.

� Unique syndrome-error mapping via channel side

information.

Target Error Polynomial Syndrome

Period

12

12

6

12

12

1 x+

1

2 31 x x x+ + +

21 x x+ +

2 3 41 x x x x+ + + +

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8

EPCC Design: Target List =10 most dominant errors

� Target the 10 most dominant errors,

account for 99.67% of possible errors.

� g1(x) = 1 + x2 +x3 + x5 + x6 +x8

• Order of g1(x)=18.

• 10 distinct syndrome sets.

• Cyclic generator polynomial used to design a cyclic (18,10) code of rate=0.56, and codeword length 18.

� g2(x) = 1 +x3 + x5 + x8

• Order of g2(x)=30.

• 10 distinct syndrome sets.

• Cyclic generator polynomial used to design a cyclic (30,22) code of rate=0.73, and codeword length 30.

� Unique syndrome-error mapping via

channel side information.

Target Error Polynomial Syndrome

Period g1(x)

Syndrome

Period g2(x)

18 30

9 15

18 10

9 15

18 6

9 5

18 30

9 15

2 10

9 32 3 4 5 6 7 8 91 x x x x x x x x x+ + + ++ + ++ +

2 3 4 5 6 7 81 x x x x xx x x+ + + + + + + +

2 3 4 5 6 71 x x x x x xx+ + + + ++ +

2 3 4 5 61 x x x xx x+ + + + ++

2 3 4 51 x x x x x+ + + + +

2 3 41 x x x x+ + + +

1 x+

21 x x+ +

2 31 x x x+ + +

1

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� Syndrome sets produced by g(x) = 1 + x3 + x5 + x8

• Order of g(x): 30 → (30, 22) base cyclic code

• 10+3 extra distinct, non-overlapping syndrome sets are utilized to distinguish 13 target error patterns.

� Multiply g(x) by a degree 6 primitive polynomial which is not a factor of

any target error polynomials :

• The extended code is a (630,616) code of rate 0.98.

Extended periods of syndrome

sets produced by ( )g x′

Approaches to Increase Code Rate of EPCC

� Tensor product coding paradigm.

• Short codeword length (outer ECC symbol length), very high total code rate.

Hakim, April 2009

Error Rate Analysis of TE-EPCC

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1

1

2

Mm

W

m m

m

m

P QM σ′= ≠

′− ≤

∑ ∑

x x� �

,

1 1

1

2

( )2

E

E

E m in

ME

W m d

m d

EE

d d

dP T Q

M

dT d Q

σ

σ

∞

= =

∞

=

≤

=

∑ ∑

∑1x

2x

3x

4x

2 2

E mind d=

WER ML Bound

min1,dT

Decrease number of codewords at

Euclidean minimum distance

(Turbo codes)

Increase Euclidean minimum

distance

(Trellis coded modulation)

Word Error Probability:

11

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1

( , ) ( )Pr( | , )N

E E

d

T d d d d=

= ∑ A� �C C

1

1( , ) ( ) ( )Pr( | , )

( , )

N

E E

dE

w d d d d dT d =

= ∑ A A� ��

C CC

1

( ) ( )Pr( | , )

2E min

NE E

b

d d d

d d d d dP Q

K σ

∞

= =

≤

∑ ∑

A A �C

BER ML Bound

( ) ( )

2E min

E E Eb

d d

T d w d dP Q

K σ

∞

=

≤

∑• Bit Error Probability:

• Average number of codeword sequences of channel noiseless outputs

separated by dE:

• Average Hamming distance between information words that generate codewords

of channel noiseless outputs separated by dE2:

# of codeword sequences of weight dAverage input Hamming weight

of codewords of weight d

12

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13

0

1

0/0

0/ 0

1/1 1/ 1−

Turbo Equalization and Enhancements

Convolutional Encoder (RSCC) ∏

Dicode Channel

(1-D)

TE-EPCC

EPCC

Convolutional Encoder(RSCC)

∏Dicode Channel

(1-D)

TE0

1

0/0

1/0

1/1 0/ 1−


Dicode Channel

(1-D)

Precoded TE

1 1 D⊕

Hakim, April 2009

Partial Response Class-1 (PR1) Channel (1+D)

0

1

0 / 0

1/ 2

1/1 0 /1

21Ed =

2 1Ed =

21Ed =

21Ed =

2 4Ed =

2 0Ed =

…

…

Trellis of Dicode channel

non-dominant error pattern

dominant error pattern

22Ed =

22 4E crd b= + ×

14

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Dicode Channel (1-D)

0

1

0 / 0

1/ 0

1/1 0 / 1−

21Ed =

2 1Ed =

21Ed =

21Ed =

2 0Ed =

2 4Ed =

…

…

Trellis of Dicode channel

dominant error pattern

non-dominant error pattern

22Ed =

22 4E crd b= + ×

15

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…4

11 1

……

1 1

…

1 1

m: # of error patterns in EPCC sub-code

∑ 2

Ed

• Merging branches correspond to

zero error Hamming weight

A Dicode multiple error occurrence

16

Hakim, April 2009

Distribution of dE given d and m

2

2

2

21, 0 integer,2

2 44

1Pr( | , ) , 2 ,

2

0 , otherwsie

dom

d m

EdomE

d m

E E dom dom

d md m

m md m

d d m d m m m

−

−

− − > <−

= = =

# of ways we can have

crossing branches# of crossing branches

d: Hamming weight of multiple error,

m: # of error patterns, mdom: # of dominant error patterns

17

Hakim, April 2009

Π

( )H

d e d=

( )H

d e i=

( )H i

d e d=

A( , )d i

1

N

d

cL N×

K

N

1

1

i i

i i

c

i

N d d

m m

N

d

− −

−

c cN P+

1

1 1

i i

i i

c

i

N d d

m m

N

d

− −

− −

c

i

N

d

Information sequence

RSCC codeword

Interleaved RSCC codeword

EPCC codeword

closed

error patterns

closed

+

open

error patterns

N( )H

d e d=

1

( )L

H i

i

d e d=

=∑

Enumerators for error Hamming weights

18

Hakim, April 2009

1 1

1 1

1 1 1

Pr( | ) Pr( | , ,..., ) Pr( ,..., | )

Pr( | , ,..., , , ,..., )

Pr( , ,..., | ,..., , ) Pr( ,..., | )

E E L L

E L L

L L L

d d d d d d d d d

d d d d m m m

m m m d d d d d d

= ×

=

× ×

( )

( )

1

1

1

1

1

1

0 0

1 0 1

... Pr

... Pr

Pr( | ) ,..., |

| Pr ), ( |

LL

ii

L

LL

ii

d d

E L

d d

d d

d d Ld

E i i

m m m i

m m

d d d d d

d md m d

=

=

= =

=

= = =

=

∑

×

=

∑

∑ ∑

∑ ∑ ∑ ∏

Distribution of Euclidean distance given

Hamming weight of sub-codes

Distribution of sub-code Hamming weights

given Hamming weight of outer code

Distribution of sub-code multiple error patterns

given Hamming weight of sub-codes


19

Hakim, April 2009

1 2

1

...

Pr( ,..., | )

c c c

L

L

N N N

d d dd d d

N

d

× ×

=

1

1Pr( | )

c i i

i i

i i

c

i

N d d

m mm d

N

d

− − ×

− =

Joint distribution of the sub-code Hamming weights:

Distribution of the number of error patterns per sub-code:

# of interleaved RSCC words

of Hamming weight d

# of sub-code words

of Hamming weight dL

# of sub-code words

of Hamming weight di

# of ways di is decomposed

into mi error patterns.

# of ways mi error patterns are

arranged in sub-code i.


20

Hakim, April 2009

1

1 1

21

1

0 0 1 0 0

: 2 0 mod 4,

2

1

1Pr( | ) ...

112

124

...L

L LL L

ii E i

i

d dd d d

E

d d m m m

d dm d m m m

d m Lc j j

Ej jj

d dN

d

d mN d d

d mm m

=

=

= = = = =

= − = =

−

=

= ∑ ∑

− − − × − −

∑ ∑ ∑ ∑ ∑

∏

1

1 1

21

1

min( , ) min( , ) min( ) min( )

0 0 1 0 0

: 2,

1

, ,1

Pr( | , ) ...

11

..

2

.

1

c c c L c

L LL L

ii E i

i

d d d d d dd

E

d d m m m

d dm m d m m

d m Lc

m

j

j

m

j

jj

d dN

d

N d d

m m

=

=

= = = = =

= = =

−

=

= ∑ ∑

− − × −

∑ ∑ ∑ ∑ ∑

∏

C

Pr( | , ) Pr( | ) Pr( | , )E E E

d d d d d d= −�C C

Euclidean distance Enumerator of TE-EPCC, when EPCC is tuned off:

Euclidean distance Enumerator of all correctable TE-EPCC codewords:

Euclidean distance Enumerator of non-correctable TE-EPCC codewords:

21

Hakim, April 2009

Interleaver Gain Exponent of TE

( )1

! !

ddN N d N

d d d

− + >

�

11

11

N d N dm

m mN d

µ

µ µ

− − + − +=

− − +− +

( )1 11 1

1 ( 1)! ( 1)!

mmN d N d N

m m m

µ µ

µ µ µ

− + − +− + − + <

− + − + − + �

2

241

.2 2

Ed

Ed

Q e σ

σ

− ≤

2

2

2

1

4, , ,

1 2 0 1: 2 0mod 4

1

2

ET

E

E

E

dd dm d

b d d m

d d mm d m

P N eK

µ σµ

µµ

∞ −− −

= = = =− + =

< ∑∑∑ ∑ B

2, , ,

1! 1( ) ( ) 2

1( )! 24

E

d m

d d m E

d mdd

d d d mmm

µ µµ

− − − = − + −−

B A A

22

Approximations:

Modified TE bound:

Hakim, April 2009

Interleaver Gain Exponent of

TE-EPCC(dc = 10, mc = 3, L = 1)2

2

2 2

1

4

1 0

min( ) min( , )

, , , , , ,

2 1 2 1: 2 0mod 4 : 2

,

1

2

E

E

T c cT

E E

E E

d

b

d

d d md dm d m d

d d m d d m

d m d mm d m m d

d

m

P eK

N N

σ

µ

µ µµ µ

µ µ

∞ −

= =

− − − −

= = = =− + −==

<

−

∑ ∑

∑ ∑ ∑ ∑B B

23

Hakim, April 2009


TE-EPCC(dc = 10, mc = 3, L = 1)

24

Hakim, April 2009


TE-EPCC(dc = 10, mc = 3, L = 1)

25

Hakim, April 2009


TE-EPCC(dc = 10, mc = 3, L = 1)

26

Hakim, April 2009


TE-EPCC(dc = 10, mc = 3, L = 1)

27

Hakim, April 2009

2 2 2

2 2 2

2

1 1 3

4 2 4

1 5 3

4 2

7

2

4

(2) (2) (2) (2) (2) (2)

(2) (2) 3 (3) (3) (3) (3)

2 (4)

2 2

2 2 2

(4)

bP e e eKN KN KN

e e eK KN K

eKN

O

σ σ σ

σ σ σ

σ

− − −

− − −

−

+ +

+ +

+ +

<

+

A A A A A A

A A A A A A

A A

2 2 2

2

2

2 2

1 1 3

4 2 411 10 10

1 5 3

49

7

4

22

155925 (10) (10) 155925 (10) (10) 779625 (10) (10)

8 8 2

779625 (10) ( (2) (2) (210) ) (2)

2 2

) )

4

2 (4 (4

b

e

e eKN

e e eKN KN K

OKN

N

N

P

Ke

NK

σ σ σ

σ

σ

σ σ

−

−

− −

−

−

−

+ +

+ +

<

+ +

+

A A A A

A A

A A

A

A A

A

AA

Asymptotic BER bound of conventional TE:

Asymptotic BER bound of TE-EPCC(dc = 10, mc = 3, L = 1):


TE-EPCC(dc = 10, mc = 3, L = 1)

28

Hakim, April 2009

29

• TE: K = 4096, punctured R=8/9, (31, 33) RSCC.

• TE-EPCC: (L = 7)EPCC, mc = 3, dc = 10.

• EPCC sub-code: (630, 616), R = 0.98.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25-50

-40

-30

-20

-10

0

10

20

30

d2

E

log

T(d

E)

precoded Dicode, TE

unprecoded Dicode, TE

unprecoded Dicode, TE-EPCC

“Spectral Thinning” of TE-EPCC

Hakim, April 2009

30

0

1

0/0

0/ 0

1/1 1/ 1−

Precoded TE


Dicode Channel

(1-D)

Precoded TE

1 1 D⊕

� Unprecoded Dicode: trellis paths corresponding to different code bits

are at 0 Euclidean distance → long error events have a high

probability of generating low Euclidean distance errors.

� Precoded Dicode: trellis paths corresponding to different code bits

accumulate Euclidean distance → ONLY low Hamming weight

errors generate low Euclidean distance errors.

� The average number of Hamming weight 2 errors that generate dE2 =2

is more for precoded compared to unprecoded Dicode.

• Unprecoded TE achieves a lower error floor compared to precoded TE.

Hakim, April 2009

4 5 6 7 8 9 1010

-8

10-6

10-4

10-2

100

SNR (dB)

BE

R

unprecoded TE, 8/9, sim.

unprecoded TE, 8/9, Bound

precoded TE, 8/9, sim.

precoded TE, 8/9, Bound

616/630 EPCC-TE, sim.

616/630 EPCC-TE, Bound

31

• TE: K = 4096, punctured R=8/9, (31, 33) RSCC.


• EPCC sub-code: (630, 616), R = 0.98.

Hakim, April 2009

4 5 6 7 8 9 10 11 12 13 1410

-10

10-8

10-6

10-4

10-2

100

SNR (dB)

Bo

un

d o

n B

ER

precoded Dicode, TE



4 5 6 7 8 9 10 11 12 13 14 1510

-10

10-8

10-6

10-4

10-2

100

SNR (dB)

Bo

un

d o

n B

ER

precoded Dicode, TE

unprecoded Dicode,TE-EPCC


32

TE

-EP

CC

, R

SC

C (

7,5

)

TE

-EP

CC

, R

SC

C (

5,7

)

interleaver size N = 600,

punctured rate R = 8/9,

and (L = 1)EPCC with

mc = 3 and dc = 10.

N=

10

0, …

, 2

00

0

N=

10

0, …

, 2

00

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25-30

-20

-10

0

10

20

30

d2

E

log

T(d

E)

unprec. Dicode, TE, (7,5) RSCC

unprec. Dicode, TE-EPCC, (7,5) RSCC

prec. Dicode, TE, (7,5) RSCC

unprec. Dicode, TE, (5,7) RSCC

unprec. Dicode, TE-EPCC, (5,7) RSCC

prec. Dicode, TE, (5,7) RSCC

Hakim, April 2009

4 5 6 7 8 9 10 11 12 1310

-10

10-8

10-6

10-4

10-2

100

SNR (dB)

Bo

un

d o

n B

ER

precoded Dicode, TE



33

• TE: N = 1200, punctured R=8/9, (7, 5) RSCC.

• TE-EPCC: (L = 1)EPCC, mc = 1:10, dc = 10.

• EPCC sub-code: (630, 616), R = 0.98.

Hakim, April 2009

34

7 7.5 8 8.5 9 9.510

-10

10-9

10-8

10-7

10-6

10-5

SNR (dB)

BE

R

L=2, m=1

L=3, m=1

L=4, m=1

L=6, m=1

L=2, m=2

L=3, m=2

L=4, m=2

L=6, m=2

L=2, m=3

L=3, m=3

L=4, m=3

L=6, m=3

L=2, m=4

L=3, m=4

L=4, m=4

L=6, m=4

• TE: N = 1200, punctured R=8/9, (7, 5) RSCC.

• TE-EPCC: (L = 1)EPCC, dc = 10.

• EPCC sub-code: (630, 616), R = 0.98.

Hakim, April 2009

5 5.5 6 6.5 7 7.5 8

10-6

10-5

10-4

10-3

10-2

SNR dB

BE

R

TE 1/2

TE 2/3

TE 3/4

TE 5/6

TE 6/7

TE 10/11

TE-EPCC 616/630,5/6 RSCC

TE-EPCC 199/210,5/6 RSCC

35

• TE: N = 4312, (7, 5) RSCC.


• EPCC sub-code: (630, 616) & (210, 199).

Hakim, April 2009

36

2/3 3/4 4/5 5/6 6/7 7/8 8/9 9/106

7

8

9

10

11

12

Punctured Rate

min

imu

m S

NR

(1

×1

0-7 B

ER

)TE, no precoding

TE, precoded

TE-EPCC

• TE: N = 1200, (7, 5) RSCC.


Hakim, April 2009

EPCC Based Turbo Code Performance in PMR

Hakim, April 2009

38

Perpendicular Magnetic Recording (PMR) channel

� Hyperbolic tangent transition response for perpendicular recording

� Channel density Ds ≡ pw50 / T

• pw50 : −50% to 50% width of the transition response

• T : symbol period

50

2( ) tanh

0.5795

th t

pwπ

=

⋅ ⋅

(H. Sawaguchi et al., “Performance analysis of modified PRML channels for perpendicular recording systems,” J. Magn. Magn. Mater., 2001.)

Hakim, April 2009

39

� Continuous-time channel model

• h(t) : Hyperbolic tangent transition response, i.e.,

• s(t) : dibit response, i.e.,

• h'(t) : First-order time derivative of h(t), i.e.,

• p(t) : Front-end band-limiting filter (7th-order butterworth filter)

• n(t) : Additive white Gaussian noise

• jk : Random transition position jitter

• Definition of energy Edt: [ ]2

( )dtE h t dt∞−∞∫ ′=

2 2( ) sech ( )h t tλ λ′ =

( ) tanh( )h t tλ=

PMR Continuous-time Channel Model

[ ]1

( ) ( ) ( )2

s t h t h t T= − −

Hakim, April 2009

40

� Discrete-time channel model

• , ,

• Variance of the additive white Gaussian noise (AWGN) sequence nk :

• Variance of the jitter noise jk :

• Spectral height for the mixed noise: Nα = No + Mo

− Nα signifies α % jitter noise, i.e.,

• SNR can be defined as

[ ]( ) ( )s

k t kTs s t p t

=≡ ∗ [ ]( ) ( )j

k t kTh h t p t

=′≡ ∗ [ ]( ) ( )n n

k k t kTh h p t p t− =

∗ = ∗ −

PMR Discrete-time Channel Model

2

2

on

Nσ =

2

2

oj

Mσ =

100o

o o

M

N Mα = ×

+

SNR dtE

Nα

≡

Hakim, April 2009

41

Partial Response Maximum Likelihood System

0 2 4 6 80

0.1

0.2

0.3

0.4

k

Discrete Time Dibit Response at Ds=1.1

0 5 10 15

-1

0

1

2

3

k

15-tap RLS Equlizer

0 0.5 1-40

-30

-20

-10

0

10

fT

dB

Dibit Vs Taregt (Frequency)

0 0.5 1-40

-30

-20

-10

0

10

fT

dB

Target Vs Dibit*Equalizer (Frequency)

Dibit

Target Equalized Dibit

Target

� Channel density: 1.1

• Mixed noise: 10% AWGN and 90% jitter noise, DC full dibit response.

• Target response: 1+0.85D , optimized to whiten noise for the all-transition input.

Hakim, April 2009

Write head/medium/read head

1+0.9D PR,

90% media noise + 10% electronic noise

(11,10)Convolutional

Encoder (RSCC)

(11,10) RSCCSISO decoder4 state BCJR

RS Encoder

t = 20

RS Decoder

t = 20

SISO Equalizer

4 state BCJR,

1 PDNP tap

∏

∏

∏-1

EPCC enhanced Turbo Equalizer

(630,616)

EPCC

encoder

EPCC SISO

List decoder

rate ≈ 1

Encoder

Decoder

kx

ˆkx

e

kλe

kλ�

EPCC-TE

42

Hakim, April 2009

9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 1510

-7

10-6

10-5

10-4

10-3

10-2

Edt

/ N90

dB

BE

R

Cu.i.d.

EPCC-TE,

[0, 1, 5, 7, & 14 iters]

conv TE, [0, 1, & 5 iters]

QC-LDPC,

[0, 5, & 9 iters]

× 15 LDPC iters

uncoded

BCJR/PDNP

TE-EPCC Performance

43

Hakim, April 2009

Shortened-(330,316) EPCC,

13 codewordsData block

Parity on

parity

Write head/medium/read head

1+0.9D PR

90% media noise + 10% electronic noise

Column

SPC decoder

Column by

column

(14,13) SPC

RS Encoder

t = 20

RS Decoder

t = 20 Channel detector

4 state BCJR

1 PDNP tap

Row by row

EPCC encoder

(330,316)

EPCC list

SISO decoder

TPC decoder

Shortened-EPCC (330,316),

14 codewords

(14,13) SPC,

330 codewords

Columns to

Rows

Rows to

columns

Encoder

Decoder

EPCC based TP

44

Hakim, April 2009

9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 1510

-8

10-7

10-6

10-5

10-4

10-3

10-2

Edt

/ N90

BE

R

TP-EPCC

[0, 2, 4, 6,

9, & 14 iters]

uncoded

BCJR/PDNP

conv TE,

[0, 1, & 5 iters]

Cu.i.d.

TP-EPCC Performance

45

Hakim, April 2009

Tensor Product Parity Codes

EPC-RS

Hakim, April 2009

47

An EPC- Tensor Product Code

� Chaichanavong and Siegel (2006) proposed a tensor product code based on a

single parity code + BCH as an inner code for outer RS ECC.

• Suitable for low density longitudinal recording channels were dominant errors have odd weight of the form , .

• Code combined with MTR for perpendicular recording channels.

• Tensor product code has much higher rate than a short parity code.

• Parity code on the symbol-level – less multiple error occurrences.

� To achieve performance gains with respect to QLDPC we will investigate a

tensor product code based on a short inner multiparty code (EPCC) and

outer QLDPC ECC.

• The EPC multiparty code corrects any single occurrence of a dominant targeted error in a tensor symbol.

• An EPCC sequence of syndromes forms a codeword for QLDPC.

• EPCC is decoded jointly with the channel using post processing techniques that generate a soft “syndrome-codeword” to be decoded by the QLDPC non-binary message-passing decoder.

• Via channel side information, EPCC has a unique syndrome per dominant error single occurrence. A list decoding scheme increases the decoding sphere radius of EPCC to target multiple error occurrences.

[ 2]+ [ 2, 2, 2]+ − +

Hakim, April 2009

48

Introduction to Tensor Product CodesJack K. Wolf, “On Codes Derivable form the Tensor Product of check Matrices,” IT 1965.

� Constituent Codes:

• Binary (3,1) single error correcting code,

• Doubly-extended t=1 (5,3) RS on GF(22),

� The tensor product code parity check matrix in GF(22) is

21 0 1

10 1 1

H α α

= =

2

2

1 0 1

0 1 1H

α α

α α

=

2

2 2 2 2

2 2 2 2(2 )

1 0 0 0 1 1 1

0 0 0 1 1 1 1GFH

α α α α α α α α

α α α α α α α α

=

(2)

101 000 101 011 110

011 000 011 110 101

000 101 101 110 011

000 011 011 101 110

GFH

=

Tensor Symbol

1. This binary (15,11) tensor product code

corrects any single tensor symbol error

provided it contains a single bit error.

2. Binary constituent code rate is 0.34 and

codeword length is 3 bits.

3. Tensor product code rate is 0.74

and codeword length is 15 bits.

Hakim, April 2009

49

Encoding of Tensor Product Codes

� Encoding of a tensor product code of binary code C1: (n1,k1), and non-binary

code C2: (n2,k2)

• Divide n1k2 information bits into k2 columns.

• Encode each column using C1 .

• Convert to .

• Encode intermediate non-binary syndromes using C2 .

• Convert back to .

• Use remaining p2k1 information bits

and the calculated syndromes

bits to calculate p1p2 parity bits using

back substitution and systematic H1.

• Result : If C1 and C2 are linear time

encodable, then

is linear time encodable! 2 2

1 1 0

1 0 1 1 0

0

1 0 0

1

1 1

1 1

0 1

1 1 1

0 1 1

1 0α α α

( )12pGF

( )2GF

1 2C C⊗

transm

itte

d c

odew

ord

Inte

rmed

iate

synd

rom

es

Hakim, April 2009

50

An EPC-RS Tensor Product code

� EPC-RS constituent codes

• (18,10) EPCC over GF(2), Rate=0.556, 8 parity bits.

• (255,195) RS over GF(28), Rate=0.765, t=30, 60 parity symbols.

� EPC-RS tensor product code is a binary (4590,4110) code, Rate=0.895, 480

parity bits.

• Codeword length = 18×255 bits, parity = 8×60 bits.

� Tensor code can correct any combination of 30 or less tensor symbol errors,

given that each 18-bit tensor symbol has a single occurrence of a dominant

error that is correctable by EPCC.

2 3 4 5 6 7 133 134 96 90 82 236 234 217 92 931H α α α α α α α α α α α α α α α α α =

Tensor symbol (1) Tensor symbol (2) Tensor symbol (3) … Tensor symbol (255)

18 bits

Hakim, April 2009

51

Hard Decoding of RS-EPC tensor product code

Tensor symbol (1) Tensor symbol (2) Tensor symbol (3) … Tensor symbol (255)

EPCC Syndrome (1) EPCC Syndrome (2) EPCC Syndrome (3) … EPCC Syndrome (255)

18 bits

Compute EPCC binary

syndromes and convert to GF(28)

RS hard decoding in GF(28)

(or any list soft decoding algorithm ).

EPCC Error Synd (1) EPCC Error Synd (2) EPCC Error Synd (3) … EPCC Error Synd (255)

8 bits

Likely dominant

Error (1)

Likely dominant

Error (2)

Likely dominant

Error (3)…

Likely dominant Error

(255)

18 bits

Convert back to corrected binary

EPCC syndromes.

Find most likely single and

double errors.

Add to ML word

RS symbol (1) RS symbol (2) RS symbol (3) … RS symbol (255)

8 bits or GF(28) symbol

Hakim, April 2009

52

RS-EPC TPPC Residual Errors

� Non-targeted single error occurrences.

� More than double multiple error occurrences.

� Double error occurrences that have a zero EPCC syndrome, since RS generates

syndromes of errors as input to EPCC.

� Residual errors can be corrected by an outer RS code of small correction power,

since the number of residual tensor symbols in error is small.

� EPCC can work as an error locating code: Erasure decoding of outer RS.

…

18 bits

13( )e x

13 bits

…

18 bits3 bits1 bit2 bits

Hakim, April 2009

53

EPC-RS Hard Decoder

BinaryViterbi

−

+

ˆk

c

hk

RS Decoder t=27 GF(28)

kq

kr

EPCC Syndrome Generator

EPCC list decoder

25 test words

Tensor Product Hard Decoder

Modulo 2

RS Decoder t=3 GF(210)

ˆk

b

Hakim, April 2009

Semi-Analytic & Fully-Analytic Multinomial SER estimations

� Step 1: Estimate P1, …, Pm

• Simulation:

1. slide a window of size m symbols over the channel detector’s simulated hard output and count occurrences of 1 to m consecutive symbol errors.

2. divide the m cumulative sums by the number of simulated symbols.

• Analytic:

1. P1=∑ (probability of 1 dominant error-pattern that spans 1 symbol).

2. P2=∑ (probability of 1 dominant error-pattern that spans 2 symbols)+ ∑(probability of 2 dominant error-patterns encapsulated in two separate consecutive symbols).

3. P3=∑ (probability of 1 dominant error-pattern that spans 3 symbols)+ ∑ (probability of 1 dominant error-pattern that spans 2 consecutive symbols)

×(probability of 1 dominant error-pattern that spans a 3rd succeeding symbol )

+ ∑ (probability of 3 dominant error-patterns encapsulated in 3 separate consecutive symbols).

� Step 2:0 1

0 1

0 1

0 1

0

0 0 1

!1

! ! !

: , ; 1 .

m

m

s ss

W m

s s s m

m m m

i i i i

i i i

nP P P P

s s s

s is t s n P P= = =

≥ − … ……

∀ ≤ = = −

∑∑ ∑

∑ ∑ ∑54

Hakim, April 2009

6.5 7 7.5 8 8.5 9 9.510

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

SNR (dB)

Sy

mb

ol

Err

or

Ev

ent

Pro

bab

ilit

y

P1, 10 bit sybmol

P2, 10 bit sybmol

P3, 10 bit sybmol

P1, 18 bit sybmol

P2, 18 bit sybmol

P3, 18 bit sybmol

Symbol Error Event ProbabilitiesSingle-level RS Vs EPC-RS

• ISI channel 5+6D-D3, AWGN.

• Shortened (450, 450-2T) RS

over GF(210).

• (18, 10) EPCC + shortened

(250,250-2Ttp) RS over GF(28).

55

Hakim, April 2009

10 20 30 40 50 60 70 80 90 1007

7.5

8

8.5

9

9.5

10

10.5

11

11.5

12

Correction power, t

SN

R (

dB

)

RS, ~1/R2 penalty

RS, ~1/R penalty

TP-RS, ~1/R2 penalty

TP-RS, ~1/R penalty

Minimum SNR Required for SFR=10-13

Single-level RS Vs EPC-RS



over GF(210).


(250,250-2Ttp) RS over GF(28).

56

Hakim, April 2009

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Rate, R

min

SN

R(R

S)

- m

inS

NR

(TP

RS

)• ISI channel 5+6D-D3, AWGN.


over GF(210).


(250,250-2Ttp) RS over GF(28).

Difference of Minimum SNR Required for SFR=10-13


57

Hakim, April 2009

1/2 K 1 K 3/2 K 2 K 5/2 K 3 K 7/2 K 4 K7.2

7.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9

Sector size

SN

R (

dB

)

Minimum SNR Required for SFR=10-13



• Shortened RS, GF(212), R=0.89.

• (24, 14) EPCC + shortened RS

over GF(210), total R=0.89.

58

Hakim, April 2009

Tensor Product Parity CodesEPC-QLDPC

Hakim, April 2009

60

Non-binary LDPC: Complexity and Performance

� Davey and MacKay (1998) have shown that the near Shannon limit

performance of binary LDPC codes in AWGN can be significantly enhanced

by a move to fields of higher order.

� For monotonic improvement in performance with field order the parity

check matrix for short blocks has to be very sparse

• Column weight 3 codes over GF(q) exhibit worse BER as q increases.

• Column weight 2 codes over GF(q) exhibit monotonically lower BER as q

increases.

• Results confirmed by Hu, Eleftheriou, and Arnold (2005): optimum degree

sequence favors a regular graph of degree-2 in all symbol nodes.

� Chang and Cruz (2008) studied the decoding time complexity of non-binary

LDPC for PR channels

• Moving from binary to non-binary LDPC results in a gain of around 1 dB.

• Size of the Galois field does not affect the decoding complexity.

• The decoding complexity ratios of non-binary to binary LDPC-coded system can be as high as 7.42 (in the number of FLP ops).

• Time complexity ratios are always smaller than the ratios of FLP ops.

Hakim, April 2009

61

Soft Decoding of EPC-LDPC tensor product code

BinaryViterbi

−

+

ˆkc

hk Correlator(e1)

Correlator(e2)

Correlator(elmax

)

kq

kr

Tensor symbol i

::

1 390i≤ ≤

Syndrome j

630 j α≤ ≤

::

List of likely errors and reliabilities

6

( )

(2 )

ch

iSyn j

j GF

γ =

∈

1 390i≤ ≤

LDPC FFT-basedSPA

over GF(26)

Map tobit-level

a priori info

Global iteration

Convolve

EPCC listdecoder

RS decodert=6

ˆk

b

630 j α≤ ≤

6

( )

(2 )

e

iSyn j

j GF

γ =

∈6(2 )GF

kλ kλ

LDPC iteration

Hakim, April 2009

Ho

w t

o g

ener

ate

Sy

nd

rom

e p

.m.f

. fo

r

each

Ten

sor

Sy

mb

ol?

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

j

Pr[

Syndro

me(

i)]=

j

p.m.f. of Tensor Symbol i

j=127

j=66

j=233

62

Hakim, April 2009

3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.810

-5

10-4

10-3

10-2

10-1

100

Eb / N

o dB

Secto

r E

rro

r R

ate

CU.I.D

GF(256) LDPC, Col wt 2.




63

SFR Comparison of Single-level LDPC Systems

• (4550, 4095) GF(2)-LDPC,

col. wt.= 5, cycle size= 91, and

binary BCJR. 10×50 TE.

• (570, 510) GF(28)-LDPC, 4560

bits, col. wt.= 2, cycle size= 15

symbols, and GF(28)-BCJR.

0×50 TE.

•(760, 684) GF(26)-LDPC, 4560



0×50 TE.

• (775, 700) GF(26)-LDPC, 4650



0×50 TE.

Hakim, April 2009

4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.710

-5

10-4

10-3

10-2

10-1

100

Eb / N

o dB

Sect

or

Err

or

Rat

e



Binary LDPC, Col wt 5.

1/2 K T-EPCC-GF(64)LDPC.

1 K T-EPCC-GF(64)LDPC.

64

SFR of Ideal T-EPCC-QLDPC

• (4550, 4095) GF(2)-LDPC,

col. wt.= 5, cycle size= 91, and

binary BCJR. 10×50 TE.

_____________________________

• (570, 510) GF(28)-LDPC, 4560



0×50 TE.

_____________________________

• (775, 700) GF(26)-LDPC, 4650



0×50 TE.

_____________________________

EPCC (12, 6), R=1/2:

• 1/2KB T-EPCC-QLDPC:

(4680, 4212) TPPC, R=0.9,

(390, 312) PEG-QC GF(26) LDPC,

R=0.8, col. wt.= 3, cycle size 26.

• 1KB T-EPCC-QLDPC:

(936, 8424) TPPC, R=0.9,

(780, 624) PEG-QC GF(26) LDPC,


Hakim, April 2009

65

SFR of Practical T-EPCC-QLDPC in 1/R penalty

4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.710

-5

10-4

10-3

10-2

10-1

100

Eb / N

o dB

Sec

tor

Err

or

Rat

e




Ideal, 1/2K T-EPCC-GF(64)LDPC.

Real, 1/2K T-EPCC-GF(64)LDPC.

Ideal, 1 K T-EPCC-GF(64)LDPC.

Real, 1 K T-EPCC-GF(64)LDPC.


(4680, 4212) TPPC, R=0.9.

(390, 312) PEG-QC GF(26) LDPC,


EPCC (12, 6), R=1/2.

3×50 TE.

Outer RS, t=6.


(936, 8424) TPPC, R=0.9.

(780, 624) PEG-QC GF(26) LDPC,


EPCC (12, 6), R=1/2.

3×50 TE.

Outer RS, t=12.

Hakim, April 2009

66

SFR of Practical T-EPCC-QLDPC in 1/R2 penalty

4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.710

-5

10-4

10-3

10-2

10-1

100

Eb / N

o dB

Secto

r E

rror

Rate




Ideal, 1/2K T-EPCC-GF(64)LDPC.

Real, 1/2K T-EPCC-GF(64)LDPC.

Ideal, 1 K T-EPCC-GF(64)LDPC.

Real, 1 K T-EPCC-GF(64)LDPC.


(4680, 4212) TPPC, R=0.9.

(390, 312) PEG-QC GF(26) LDPC,


EPCC (12, 6), R=1/2.

3×50 TE.

Outer RS, t=6.


(936, 8424) TPPC, R=0.9.

(780, 624) PEG-QC GF(26) LDPC,


EPCC (12, 6), R=1/2.

3×50 TE.

Outer RS, t=12.

Hakim, April 2009

Thesis Contributions

67

� Proposed a channel matched turbo equalization scheme based on the SISO list

decoder of EPCC, termed TE-EPCC.

� Demonstrated the “Spectral Thinning” effect achieved by incorporating EPCC in

TE of the Dicode channel.

� Derived an upper bound on the ML BER of TE-EPCC.

� Proposed a turbo-product code based on EPCC.

� Proposed an error-pattern correcting tensor product code that is linear time

encodable.

� Derived a fully analytic multinomial method to estimate the SER of RS in ISI.

� Designed a two-level coding scheme based on the tensor product of EPCC and

QLDPC that achieves a better complexity-performance trade-off compared to

single-level QLDPC.

� Designed a soft iterative decoder of T-EPCC-QLDPC.

Thank you !

Hakim, April 2009

69

6.5 7 7.5 8 8.5 9 9.510

-7

10-6

10-5

10-4

10-3

10-2

10-1

SNR (dB)S

ym

bo

l E

rro

r E

ven

t P

rob

abil

ity

P1, simulation

P2, simulation

P3, simulation

P1, analytic

P2, analytic

P3, analytic

6.5 7 7.5 8 8.5 9 9.510

-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

SNR (dB)

Sy

mb

ol

Err

or

Ev

ent

Pro

bab

ilit

y

P1, simulation

P2, simulation

P3, simulation

P1, analytic

P2, analytic

P3, analytic

Hakim, April 2009

6.5 7 7.5 8 8.5 9 9.5 1010

-7

10-6

10-5

10-4

10-3

10-2

10-1

SNR (dB)

SE

R

RS, t=5, analytic

RS, t=5, count

TP-RS, t=6, analytic

TP-RS, t=6, count

RS, t=10, analytic

RS, t=10, count

TP-RS, t=12, analytic

TP-RS, t=12, count

70

Hakim, April 2009

7 7.5 8 8.5 9 9.5 1010

-20

10-15

10-10

10-5

SNR (dB)

SE

R

Analytic t=5:5:40.

Semi-Analytict = 5:5:40.

71

Hakim, April 2009

Non-binary LDPC: Hamming Weight Spectrum Hu(2004)

72

Final Oral Defense 040109

Documents

Transcript of Final Oral Defense 040109