Joint Detection and Decoding

Schemes for Two Dimensional

Data Storage Systems

Joseph A. O’Sullivan

Naveen Singla

Joint work with Y. Wu and R. S. Indeck

Electronics Systems and Signals Research Laboratory

Magnetic Information Science Center

Washington University

Enabling Technology: Disk Drives

Magnetic disk storage areal density

vs. year of IBM product introduction

(From D. A. Thompson)

~10,000,000x increase in 45 years!

Problem Description

• Reliable data retrieval from channels having

2-D ISI

Advanced storage media: Patterned media

As bit aspect ratio reduces inter-track

interference becomes significant

Optical memories

Science enables:

6 million-million

bits/square inch!

Conventional PRML System

Receive

Filter

(low pass)

Adaptive

Filter

Viterbi

Detector

Magnetic

Recording

Channel

Timing &

Tap Updating

Wide-Band

Noise

nT+

e

ny nxnxny

,....3,2,1,)1)(1()( nDDDP n

No generalization to 2D

Joint Equalization and Decoding

Schemes

• General 2D ISI

Using 2D MMSE equalization and decoding

Using novel message-passing algorithms that

take advantage of the 2D dependence

• Separable 2D ISI

Using turbo equalization

Singla et al., “Iterative decoding and equalization for 2-D recording channels,” IEEE Trans.

Magn., Sept. 2002.

Channel Model

• x(i,j) {+1,-1}

• Channel ISI is 2D and linear

• Noise assumed to be AWGN

LDPC

EncoderEqualizer

LDPC

Decoder

Channel

ISI

Channel

),( jia ),( jix ),( jir ),( jix ),( jia

),( jiw

2D Intersymbol Interference

1111

11

11

11

1111

21

21

11211

kkkk

k

xxx

x

xxx

1111110

110

12

1111110

100100

kkkkkk

kkkkkk

k

kk

k

rrrr

rrrr

r

rrrr

rrr

GUARD BAND

),(),(),(),(2

0,

jiwnmhnjmixjirnm

),( jiw

cbc

bab

cbc

h

MMSE Equalization and

Decoding

• Equalizer designed assuming inputs to be

Gaussian

LDPC

EncoderEqualizer

LDPC

Decoder

Channel

ISI

),( jia ),( jix ),( jir ),( jix ),( jia

),( jiw

Performance

MMSE Equalization and Decoding

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7

SNR (dB)

Bit

Err

or

Ra

te,

Lo

g B

as

e 1

0

BIAWGN Capacity

LDPC No ISI

Wiener

220.0366.0220.0

366.0819.0366.0

220.0366.0220.0

h

Iterative MMSE Equalization and

Decoding

• Soft information, estimated mean of the codeword,

passed from LDPC decoder to equalizer

]][***[*][

)1Pr()1Pr(][

xEhrWxEx

xxxE

Extrinsic Information

LDPC

EncoderEqualizer

LDPC

Decoder

Channel

ISI

),( jia ),( jix ),( jir ),( jix ),( jia

),( jiw

Performance

Iterative MMSE Equalization and Decoding

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7

SNR (dB)

Bit

Err

or

Ra

te,

Lo

g B

as

e 1

0

BIAWGN Capacity

LDPC No ISI

Itrw iener_10

Wiener

220.0366.0220.0

366.0819.0366.0

220.0366.0220.0

h

Full Graph Message-Passing

Check Nodes

Codeword Bit Nodes

Data Nodes

jijijijijiji wxxxxr ,1,11,,1,, 25.05.05.025.05.0

5.01h

Full Graph

x(i+2,j)

x(i+1,j)

x(i,j)

x(i+2,j+1)

x(i+1,j+1)

x(i,j+1)

x(i+2,j+2)

x(i+1,j+2)

x(i,j+2)

r(i+1,j+1)

r(i,j+1)r(i,j)

r(i+1,j)

From

Check

Nodes

Performance

Full Graph Message-Passing

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7

SNR (dB)

Bit

Err

or

Ra

te,

Lo

g B

as

e 1

0

BIAWGN Capacity

LDPC No ISI

Full graph_50

Itrw iener_10

Wiener

220.0366.0220.0

366.0819.0366.0

220.0366.0220.0

h

Full Graph Analysis

• Length 4 cycles present which degrade

performance of message-passing algorithm

x(i+2,j) x(i+2,j+1)

x(i+1,j) x(i+1,j+1)

x(i,j) x(i,j+1)

x(i+2,j+2)

x(i+1,j+2)

x(i,j+2)

r(i+1,j+1)

r(i,j+1)r(i,j)

r(i+1,j)

From

Check

Nodes

Kschischang et al., “Factor graphs and the sum-product algorithm,” IEEE Trans. Inform.

Theory, Feb. 2001.

Ordered Subsets Message-Passing

• From Imaging – Data set is grouped into

subsets to increase rate of convergence

• For Decoding – Observed data is grouped

into subsets to eliminate short length cycles

in the Channel ISI graph

H. M. Hudson, and R. S. Larkin, “Accelerated image reconstruction using ordered subsets

of projection data,” IEEE Trans. Medical Imaging, Dec. 1994

Grouped ISI Graph

• Labeling of data nodes into 4 subsets

• For each iteration use data nodes of one label only

J. A. O’Sullivan, and N. Singla, “Ordered subsets message-passing,” Submitted to Int’l

Symp. Inform. Theory 2003.

Performance

Ordered Subsets Message-Passing

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7

SNR (dB)

Bit

Err

or

Ra

te,

Lo

g B

as

e 1

0

BIAWGN Capacity

LDPC No ISI

Ord Subsets_100

Full graph_50

Itrw iener_10

Wiener

220.0366.0220.0

366.0819.0366.0

220.0366.0220.0

h

Performance

Joint Equalization and Decoding Schemes

-7

-6

-5

-4

-3

-2

-1

0

0 0.5 1 1.5 2 2.5

SNR (dB)

Bit

Err

or

Ra

te,

Lo

g B

as

e 1

0

BIAWGN Capacity

LDPC No ISI

Ord Subsets_100

Full graph_50

Itrw iener_10

Wiener

019.0098.0019.0

098.0980.0098.0

019.0098.0019.0

h

Performance

Joint Equalization and Decoding Schemes

-7

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5 6 7

SNR (dB)

Bit

Err

or

Ra

te,

Lo

g B

as

e 1

0

BIAWGN Capacity

LDPC No ISI

Ord Subsets_100

Full graph_50

Itrw iener_10

Wiener

0353.00

353.0707.0353.0

0353.00

h

Performance

Joint Equalization and Decoding Schemes

-7

-6

-5

-4

-3

-2

-1

0

0 0.5 1 1.5 2 2.5

SNR (dB)

Bit

Err

or

Ra

te,

Lo

g B

as

e 1

0

BIAWGN Capacity

LDPC No ISI

Ord Subsets_100

Full graph_50

Itrw iener_10

Wiener

0098.00

098.0981.0098.0

0098.00

h

Consider Separable 2D ISI

Wu et al., “Iterative detection and decoding for separable two-dimensional intersymbol

interference” Submitted to IEEE Trans. Magn., June. 2002.

A Separable 2D ISI

• Advantages of Separable 2D ISIApply existing one-dimensional equalization methodsReduced Detector Complexity

5.015.0

1

25.05.0

5.01h

x(i, j)Row ISI Column ISI

Channel

Detector

r(i, j)

w(i,j)

y(i, j)

Separable Channel Response

Tüchler et al., “Turbo equalization: principles and new results,” IEEE Trans. On Comm.,

May 2002.

Row-by-Row Decoder

• MMSE and Zero-forcing criteria used for Equalization

Column

Equalization

MAP

Detector

LDPC

Decoder

r(i, j)

Conventional One-Dimensional Iterative Decoder

Extrinsic Information

Hard Decision

Performance

Row-by-Row Decoder

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5

SNR (dB)

Bit-E

rro

r-R

ate

In

lo

g1

0

LDPC No ISI

Row -by-Row : MMSE_iter10

Row -by-Row : ZF_iter10

25.05.0

5.01h

Row-and-Column Decoder

• Inputs to column detector are not binary

Column MAP

Detector

Row MAP

Detector

LDPC

Decoder

r(i, j)

Conventional One-Dimensional Iterative Detector

Extrinsic Information

Hard Decision

Performance

Row-and-Column Decoder

-6

-5

-4

-3

-2

-1

0

1 2 3 4 5

SNR (dB)

Bit

Err

or

Rate

, Log B

ase 1

0

LDPC No ISI

Row -and-Column_10

Ordered Subsets_100

Full Graph_50Itrw iener_10

Wiener

Row -by-Row : ZF_iter10

25.05.0

5.01h

Conclusions

• MMSE equalization and decoding

Good Performance with Iterative Equalization

• Message-passing algorithms

Full graph algorithm performance deteriorated due to short cycles

Ordered subsets message-passing gives best performance for general 2D ISI

• Separable ISI decoding

Best performance for separable 2D ISI

Low complexity

Approximate channel response by separable response

Performance

Ignoring ISI

-6

-5

-4

-3

-2

-1

0

0 2 4 6 8 10 12 14

SNR [dB]

Bit E

rror R

ate

in log10

ISI-free

ISI ignored

Block length 10000 regular (3,6) LDPC code

MMSE Equalization and Decoding

-6

-5

-4

-3

-2

-1

0

0 2 4 6 8 10 12 14

SNR [dB]

Bit E

rro

r R

ate

in

lo

g1

0

ISI-free

Wiener

ISI ignored

Performance

Iterative MMSE and Decoding

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5

SNR [dB]

Bit E

rror

Rate

in log10

ISI-free

Itr Wiener_10

Wiener

Performance

Full Graph Message-Passing

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5

SNR [dB]

Bit E

rro

r R

ate

in

lo

g1

0

ISI-free

Full Graph_50

Itr Wiener_10

Wiener

Performance

Ordered Subsets Message-Passing

-6

-5

-4

-3

-2

-1

0

0 1 2 3 4 5

SNR [dB]

Bit E

rro

r R

ate

in

lo

g1

0

ISI-free

Ordered Subsets_100

Full Graph_50

Itr Wiener_10

Wiener

Performance

Advanced Media