Media Defect Recovery Using Full-Response Reequalization in Magnetic Recording Channels

Abstract

Conventional method for media defect recovery in a low-density parity-check (LDPC) coded magnetic recording channel is erasure decoding. In erasure decoding, read back signal in media defect region is erased and not used in channel detection or iterative decoding. In this paper, a new method based on full-response reequalization, in which media defect corrupted signal is first reequalized to full response then partially used in iterative decoding, is proposed. Simulation results show significant performance improvement over conventional erasure decoding in both non-precoded and preceded channels.

Full text

Page 1

Media Defect Recovery Using Full-Response
Reequalization in Magnetic Recording Channels
Weijun Tan1, Shaohua Yang2, Kelly Fitzpatrick3, Hao Zhong2, Li Du1, Yuanxing Lee2
1 LSI Corporation, Longmont, CO 80503
2 LSI Corporation, Milpitas, CA 95035
3 LSI Corporation, Andover, MA, 01810


Abstract – Conventional method for media defect recovery in a
low-density parity-check (LDPC) coded magnetic recording
channel is erasure decoding. In erasure decoding, read back
signal in media defect region is erased and not used in channel
detection or iterative decoding. In this paper, a new method
based on full-response reequalization, in which media defect
corrupted signal is first reequalized to full response then partially
used in iterative decoding, is proposed. Simulation results show
significant performance improvement over conventional erasure
decoding in both non-precoded and precoded channels.
Keywords –Erasure decoding, iterative decoding, full-response,
low-density parity-check (LDPC) codes; magnetic recording
channels, media defect, partial-response
I. INTRODUCTION
INCE the rediscovery in 1990's, low-density parity-check
(LDPC) codes have been intensively studied due to their
outstanding performance. In recent years, a lot of efforts have
been put in how to implement LDPC codes in the magnetic
recording read channel. Toward this goal, two major problems
to overcome are the performance and complexity. In terms of
performance, LDPC codes must be guaranteed to be superior
to the traditionally used Reed Solomon (RS) codes. In terms
of complexity, the prohibitive high complexity in both
encoding and decoding of the LDPC codes must be reduced.
This paper focus on the performance of LDPC codes in
magnetic recording channels. This performance includes both
random error performance and burst error performance.
While it is well known that LDPC codes have superb random
error performance, the burst error performance is still a
concern. And this is why RS codes are still widely used in
magnetic recording read channels, because they are optimum
for burst error correction.
Two major physical distortions that cause burst errors in
magnetic recording are thermal asperities (TAs) and media
defects (MDs). In the viewpoint of error correction, TAs are
easy to handle, because they are easy to detect and mapped out
in flaw scan and they are usually short. On the other hand,
MDs are more challenging, because their signatures are more
various therefore more difficult to detect, and they can be very
long. Of the different types of MDs, the most common one is
drop-out, in which the signal amplitude is attenuated by some
extent in the MD region. Although the detection of MDs is
also a challenging problem, this paper focus on how to
recovery MD, i.e., to retrieve the information bits buried under
the MD.
The rest of paper is organized as following. Chapter II
formulates the problem. Chapter III described full-response
re-equalization techniques. Chapter IV presents the media
defect recovery techniques and simulation results in magnetic
recording channels. Finally, Chapter V concludes the paper.
0100200 300 400500600 7008009001000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
bit index
media defect profile g
rect
exp+rand
exp

Fig.1. Profiles of MD drop-out ratio g.
II. PROBLEM FORMULATION
The system model under investigation is very simple and
conventional. On the write side, user data are encoded by a
LDPC code. On the read side, the read back signal is first
equalized to a pre-set partial response (PR) target. Then a soft-
input soft-output (SISO) data-dependent noise-predictive
maximum-likelihood (NPML) detector and a LDPC decoder
work in an iterative and turbo equalization manner. No run-
length limited (RLL) or other modulation codes are considered
in this work.
A. Media Defect Model
MDs in this paper refer to the sudden complete or partial
loss of signal during read back, which we also call drop-outs.
Due to the nature of the MDs in magnetic recording, the bits in
MD are assumed contiguous. We further assume a rectangular
S
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window (other windows will be considered later) for the MDs,
and use two parameters to model them: g is used to represent
the drop-out ratio, ranging from zero (all drop-out) to one (no
drop-out), and the length L is the number of bits in MD out of
all N bits in the codeword. Let the data bit recorded on the
channel be
the read back signal,

⎧
++
=
∑
=
0
k

where
in Fig. 1 are three MD profiles with g ≈ 0.25used in this paper.
These three profiles, the rectangular one, the exponential one,
and the exponential plus random variable one will be used in
numerical simulations. In the system performance evaluation,
we always assume the presence of such an MD with a random
starting location in every codeword.
ix , and the PR target be
],...,,[
110
−
=
M
ffff
, then

⎪
⎩
⎪
⎨
++
∑
=
k
g
−
−
−
−
1
bits MDfor ,)(
bits MD-nonfor ,
1
0
ii
M
k
k
ii
ii
M
k
k
i
i
ejfx
ejfx
y, (1)
ij is the jitter noise,
ie is the electronic noise. Shown
B. Erasure Decoding for Media Defect Recovery
Conventional technique for MD recovery is erasure
decoding [1-2]. In LDPC coded channels, this simply means
to zero out the soft input to the LDPC decoder in all global
iterations. In terms of the branch metric of the SISO NPML
detector (e.g.,[1]),

⎧
Λ
),(
,ch
iai
xx
(2)

where
)(
,chi
Λ
is the a priori log-likelihood ratio (LLR) for
y
,
LDPC coded magnetic recording channel where global
iteration is available,
)(
,ch
i
a
x
the global iteration number. Please note that for simplicity the
NP filters on
i
y
,
An enhancement to (2) is to use partial intrinsic LLR from
previous global iteration as extrinsic LLR for MD bits,


⎧
Λ⋅+
)1 (
ch
i
x
α
.
(3)

This way, the extrinsic LLR
Λ
the BER of the hard decisions for MD bits in this NPML
detector is not 0.5. This is sometimes very useful if the hard
⎪⎩
⎪⎨
−−Λ
=
−
bits MDfor
bits MD-nonfor ,2/)()(
),(
22
,ideal, ch
1
iiiai
kki
yyxx
SS
σ
γ
ax
ix , and
iideal is the ideal channel output for that branch. In
)(
extldpc,
1
i
jj
x
Λ=Λ+
, where j is the
iy and
ideal are not included in (2).
⎪⎩
⎪⎨
−−Λ
=
−
bits MDfor ),(
bits MD-nonfor ,2/)()(
),(
,
22
,ideal
x
,ch
1
ia
iiiai
kki
yyxx
SS
σ
γ
),()(
ch, ext,ch
iaii
xxx
Λ⋅=
α
and
decisions of the NPML detector are used for some other
purposes. Numerical experiments (results not shown in this
paper) show that this enhancement improves the performance
of erasure decoding, almost in all cases. So the enhanced
erasure decoding is the conventional erasure decoding in this
paper.
C. Problem Formulation
In Equations (2)-(3), we see that the read back signals
are not used at all in the MD region. This is apparently not
optimum, because the signals are just dropped-out, but not
completely wiped out, for η >0. If it was a memoryless
AWGN channel, it is not hard to see that in the MD region, the
LLRs based on the attenuated signals are still useful in LDPC
decoding. This is because, most likely, the drop-out only
reduces the amplitude of the LLRs, but does not change the
sign, particularly for high SNRs, when the electronic noise is
very small.
In magnetic recording channels, however, due to the PR
nature and strong ISI, we cannot use directly the LLRs
calculated by the SISO NPML detector from the distorted
signals. This fact stimulated us to consider full-response
equalization techniques to convert the PR channel to a
memoryless channel, such that the LLRs can be used in a way
similar to that in a memoryless AWGN channel.
iy
III. FULL-RESPONSE EQUALIZATION
A. ZFE Equalization
Typical equalization technique to remove ISI is zero-
forcing equalization (ZFE). Let the ZFE filter be
,,,,[w
11KKK
wwww
−+−−
=
?
filter is

∑∑
−=−=
kKk

where
minimized by forcing the equalizer response to be

⎛
≠
, 0j

the so-called ZFE criterion. One condition to meet this
criterion is to impose

{
)(
−
ii
xzxE

The ZFE can be implemented as follows,


)(
−Δ+=
zxww
iikk
]
K
, the output of the ZFE

∑
−=
k
−
−+
≠
−
−
++==
K
K
kik
KM
kK
kik
0
i
K
k
k
ii
nwxqxqwyz
1
,
0
, (4)
in is the overall noise. The ZFE equalizer output is

⎜⎜
⎝
=
=
0
0, 1j
qk
, (5)
} 0
=
−ki
. (6)
?
, 2 , 1 , 0,
1
=
−
+
ix
ki
ii
(7)
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In decision driven mode,
ix is unknown and can be replaced
by decision
ix ˆ .
B. MMSE Equalization
Another equalization technique to remove ISI is minimum
mean-square error (MMSE) equalizer. MMSE equalizer can
be use to equalize the channel to any channel response. In this
work, a special channel response is described in (5). However,
the MMSE criterion to minimize

{
⎪⎩

is used instead. Please note that the desired output being
indicates this is a full-response equalizer. Accordingly, the
adaptation is different from (7), written as follows,


(
−Δ+=
zxww
iikk

MMSE equalization is more commonly used than ZFE
equalization. Firstly, ZFE equalization typically has severe
noise enhancement. Secondly, the convergence condition for
ZFE is tighter than for MMSE equalization. Generally
speaking, if the eye diagram of input to the ZFE equalizer is
not open, the ZFE equalizer usually has convergence problem.



}
⎪⎭
⎪⎬
⎫
⎪⎨
⎧
−=−=
∑
−=
k
−
22
||||
K
K
kkiiii
wyxEzxEJ (8)
ix
?
, 2 , 1 , 0,)
1
=
−
+
iy
ki
ii
(9)


Fig.2. Full-response reequalization in magnetic recording
channel.

C. Full-Response Reequalization in PR Channel
In magnetic recording channel, the outputs of the analog-
to-digital converter (ADC) are first equalized to a pre-set PR
target. This adaptive PR equalizer is usually driven by the first
NPML (or simplified NPML) detector hard decision
full-response reequalizer is added upon the PR equalizer, as
shown in Fig. 2. This equalizer can also be driven by the first
NPML detector hard decision
training mode is also available in Fig.2, which uses know data
ix and
i
y
,
A different option is to use this extra equalizer on the ADC
output. But there is no benefit for doing so and is not
ix ˆ . The
ix ˆ . In addition, a known data
ideal to adapt the equalizer in (7) or (9).
considered in this paper. That is why in the title of this paper,
we use “full-response reequalization”, because we reequalize
already PR-equalized signal
Applying the full-response equalization techniques to the
problem of MD recovery, we expect that the ISI is removed to
some extent such that the gain of having correct LLR signs in
the MD bits exceeds the negative effect of noise enhancement
and misequalization. We tested both ZFE and MMSE
equalizations in this work and observed that the MMSE
equalizer worked better than the ZFE equalizer.
iy to full-response.
IV. MEDIA DEFECT RECOVERY
A. Non-Precoded Channel
We first consider a non-precoded channel, as shown in Fig.
3. The encoding side is not shown. On the decoding side,
along with the normal SISO NPML detector, there is a parallel
data path using the full-response reequalization. After the ISI
is removed, an LLR estimator is used to convert the equalized
sample
i
Λ . This converter is actually just a scalar
β , scaling down
Λ . After the NPML detector
and the LLR-converter, a multiplexer is used to select which
input, either from the NPML detector or from the LLR-
converter, is used in the LDPC decoder. Similarly to the
enhancement in erasure decoding, an enhancement of using
LDPC extrinsic LLRs can be used as well. So overall, the soft
input to the LDPC decoder is as follows,

⎧
Λ
=Λ+
)(
ldpc,
i
z
αβ

Other than this new soft input, the LDPC decoder works the
same as in a conventional LDPC coded channel. The defect
detector provides a flag indicating where a bit is in MD region.
In this work an ideal detector is assumed. For a actual
detector, please refer to [3].


iz to LLR
iz to obtain
i
⎪⎩
⎪⎨
Λ⋅+⋅
bits MD for ),(
bits MD- nonfor
x
),(
ext
extch,1
aldpc,
i
j
i
j
i
j
x
x
. (10)


Fig.3. Media defect recovery using the full-response
reequalizer in an LDPC coded channel.

In the first test we did we used a rate R=8/9 LDPC code for
a rectangular MD drop-out window of length L=300 and 350.
ADC
Full-response
equalizer
PR
equalizer
yi
Detector
hard
decision
xi
PR
target
yi
xi
ix ˆ
Known
data
yideal,i
PR
equalize
SISO
NPML
detector
LDPC
decoder
Defect
detector
Soft LLR
estimator
extrinsic-LLR
intrinsic-LLR
Full
response
equalizer
α
••
• •
mux
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The full-response reequalizer is a 6-tap FIR filter. The user
information density IBD=2.0, and jitter percentage is 90%.
Shown in Fig. 4 are the simulation results. From the plot, we
see that with erasure decoding, L=300 is correctable, but
L=350 is out of this code’s erasure correction capability.
However, this capability is significantly improved by the full-
response reequalization. While the SNR gain for L=300 is only
0.6 dB, the improvement for L=350 is big. Although at high
SNR, the curves becomes flat like an error floor, it is fine
considering the chance of MD happening in real magnetic
recording system.

17181920 21 222324
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR
SFR
L=300,erasure decoding
L=350,erasure decoding
L=300,full-resonse
L=350,full-resonse

Fig. 4. Performance of full-response reequalization for a
rectangular drop-out window of length L=300 and 350 bits.

1718 19202122 2324
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR
SFR
rect,erasure decoding
rect
exp
exp+rand

Fig.5. Performance of full-response reequalization for different
drop-out windows of length L=350.


In the second test we used the same settings as in the first
test, but for L=350 only. However, we used several different
MD drop-out profiles to demonstrate the robustness of the full-
response reequalization technique. These profiles are the ones
shown in Fig. 1. The results in Fig. 5 show that the
performance is even better for the other two profiles. This is
mainly due to the better effects on the boundaries of the
window. In the rectangular window, the full-response
reequalized samples around the two boundaries of the window
are very unreliable. However, in the other two windows, since
the samples are changing slowly, the full-response reequalized
samples are much more reliable now.
B. Precoded Channel
Now we consider a precoded channel. One may think it is
a simple extension from a non-precoded channel to a precoded
channel. It turns out not. The main reason is in a precoded
channel, the SISO NPML detector only provides LLRs for the
data before the precoder. However, the full-response equalizer
is on the data after the precoder. Therefore, the soft LLR
estimator only provides LLRs for the MD bits after the
precoder. However, the LDPC decoder takes LLRs for the
data before the precoder as input So we need a LLR converter
to convert the LLRs from after the precoder to before the
precoder. As shown in Fig. 5, let the data before and after the
precoder be
soft LLR estimator, we need a converter to convert
)(i
Λ
.


ui
iu and
ix . After the full-response equalizer and
)(iax
Λ
to
au


Fig.6. Media defect recovery using the full-response
reequalizer in a LDPC precoded channel.


A simple method is to use the min-sum approximation
assuming
1
are independent. With this assumption,

sign())(sign()(
Λ⋅Λ=Λ
aiaia
xu
(11)

It is not hard to imagine that this approximation, although very
simple, does not have good performance. Firstly, the
independence assumption is not true. Secondly, error
propagation in (11) is severe. Every one bit error in
two bit errors in
ix and
− ix
)}( ),(min{))(
11
−−
ΛΛ
iaiai
xxx.
ix causes
iu . No need to mention that in the full-
PR
equalize
SISO
NPML
detector
LDPC
decoder
Defect
detector
Soft LLR
estimator
intrinsic-LLR
Full
response
equalizer
α
••
••
mux
LDPC
Encoder
1/(1+D)
xi
2-state map
deprecoder
Λa(x)
• •
Λa(u)
Λext(u)
Λext(u)
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response reequalization, the misequalization is large and bit
error rate is high.
A better method is to use a 2-state MAP deprecoder, as
already shown in Fig. 6. This 2-state MAP deprecoder is
nothing else but a conventional SISO convolutional-code
decoder, however for the precoder in this work. This
deprecoder takes soft inputs
generates soft outputs (
ext
Λ
needed). This deprecoder has very low complexity, compared
with the SISO NPML detector. It only has 2-state for the
precoder 1/1+D. It does not have any data dependency
therefore does not have noise prediction. It does not take
channel input in its branch metric. In other words, the branch
metric only handle soft inputs
Λ
details, the readers are referred to [4] and [5].
In addition to the 2-state MAP deprecoder, the
enhancement of using LDPC extrinsic LLRs is used in the
same way as in Fig. 3. The total soft input to the LDPC
decoder is,

⎧
Λ
=Λ+
,2(map
extldpc,
i
z
β
. (12)

where map2( ) implements the function of the 2-state MAP
deprecoder. If the min-sum approximation is used, the first
term in the second equation of (12) is replaced by (11).
In this test, we used a lower-rate R=7/8 LDPC code. This
code has much better erasure correction capability. We tested
this code for a rectangular MD drop-out window of length
L=500 bits. The results are shown in Fig. 7. Compare with
the curve for L=350 in Fig. 4, the slope of this curve is still
sharp. More importantly, the performance improvement of the
full-response reequalization is promising. Even with the min-
sum approximation in (11), the performance is a lot better than
the erasure decoding. The 2-state MAP deprecoder in (12)
gives more improvement. If we want to quote SNR gain at
SFR=1e-3, the gain from erasure decoding to full-response
reequalization with 2-state MAP deprecoder is around 1.3 dB.
It is worth pointing out that this improvement is better for
larger drop-out ratio g. While the erasure decoding is not
sensitive to g, meaning when g is under a certain value, e.g.,
0.5, the erasure decoding performance is almost the same, the
full-response reequalization is sensitive to g. The larger g, the
better the performance.

)(i
(and
ax
Λ
)
and
Λ
)(i
)
au
ix
Λ
(
, and
but not
iu
ext
)(iax and )(iau
Λ
. For more
⎪⎩
⎪⎨
Λ⋅+Λ⋅
bits MDfor ),( ))(
bits MD-non
j
for
α
),(
)(
extldpc,
extch,
1
aldpc,
ii
j
i
j
i
j
uu
u
x
18.618.81919.219.419.6
SNR
19.82020.220.420.6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SFR
erasure decoding
full-response, min-sum
full-response,2-state map

Fig.7. Performance of full-response reequalization for a
rectangular drop-out window of length L=500 bits in a
precoded channel.

VI. CONCLUSION
We proposed a new technique to improve the correction of
MD in magnetic recording channels. In this technique, MD
dropped-out samples are reequalized to full response. By
doing so, we can partially use the residue useful signals in
these samples to correct the MD bits. A couple of equalization
techniques, and LLR estimating techniques were tested.
Simulation results indicate this is a very promising method and
can be used in real magnetic recording systems to correct MD.
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[3] W. Tan and J. R. Cruz, “Detection of Media Defect in Perpendicular
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2958, Oct. 2005.
[4] W. Ryan, A Turbo Code
www.ece.arizona.edu/~ryan/publications/turbo2c.pdf.
[5] T. Souvignier, A. Friedmann, M. Oberg, P. H. Siegel, R. E. Swanson,
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Tutorial, [online] available at
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.
978-1-4244-2324-8/08/$25.00 © 2008 IEEE.