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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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
Hakim, April 2009
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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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.
� 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]+ − +
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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.
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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
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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
RS symbol (1) RS symbol (2) RS symbol (3) … RS symbol (255)
8 bits or GF(28) symbol
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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
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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
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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
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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
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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
• 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).
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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.
• Shortened (450, 450-2T) RS
over GF(210).
• (18, 10) EPCC + shortened
(250,250-2Ttp) RS over GF(28).
Difference of Minimum SNR Required for SFR=10-13
Single-level RS Vs EPC-RS
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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
Single-level RS Vs EPC-RS
• ISI channel 5+6D-D3, AWGN.
• Shortened RS, GF(212), R=0.89.
• (24, 14) EPCC + shortened RS
over GF(210), total R=0.89.
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Tensor Product Parity CodesEPC-QLDPC
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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.