ar
X
iv
:c
s/0
50
20
36
v1
[
cs
.IT
]
7 F
eb
20
05
SUBMITTED TO THE ISCTA 2005 CONFERENCE ON 4 FEB 2005 1
Improved Iterative Decoding for Perpendicular
Magnetic Recording
E. Papagiannis∗, C. Tjhai∗, M. Ahmed†, M. Ambroze∗, M. Tomlinson∗
∗Fixed and Mobile Communications Research, University of Plymouth, Plymouth, PL4 8AA, United Kingdom
†Centre for Research in Information Storage Technology, University of Plymouth, PL4 8AA, United Kingdom
email: {epapagiannis,ctjhai,mahmed,mambroze,mtomlinson}@plymouth.ac.uk
Abstract— An algorithm of improving the performance of iter-
ative decoding on perpendicular magnetic recording is presented.
This algorithm follows on the authors’ previous works on the
parallel and serial concatenated turbo codes and low-density
parity-check codes. The application of this algorithm with signal-
to-noise ratio mismatch technique shows promising results in
the presence of media noise. We also show that, compare to the
standard iterative decoding algorithm, an improvement of within
one order of magnitude can be achieved.
I. INTRODUCTION
Longitudinal recording has been the standard method in
magnetic recording for decades. Recent research has shown
that this method seems to reach its physical limits in the
near future due to the superparamagnetic effects. On the
other hand, the technique which has been known prior to the
longitudinal recording–perpendicular recording, has recently
been the centre of research attention. Perpendicular magnetic
recording offers promising increased in recording densities, up
to 1 Terabit per square inch seems feasible [1]. As the areal
density is increased, however, the signal processing aspects of
magnetic recording becomes more difficult. Sources of distor-
tion including media noise, electronics and head noise, jitter
noise, inter-track interference, thermal asperity, partial erasure
and dropouts become more apparent and unless appropriate
mitigation techniques are present, signals cannot be retrieved
reliably from the recording media.
Since the discovery of turbo codes, soft-decision iterative
decoding has been shown to be able to provide significant cod-
ing gain over the conventional detection method on magnetic
recording. The utilisation of iterative decoding on the concate-
nation of partial-response (PR) channel and powerful error-
correcting codes such as low-density parity-check (LDPC) and
turbo codes has been proposed in many literatures. Iterative
decoding is a reduced-complexity method to achieve the
optimum solution–the maximum-likelihood solution and as
such, iterative decoding is sub-optimal.
In this paper, we present a method to improve the sub-
optimality of the iterative decoding and demonstrate its appli-
cations to perpendicular magnetic recording in the presence
of media noise. The improved method, which is known as the
Received-Vector-Coordinate-Modification (RVCM) algorithm,
follows on the previous works of the authors [2],[3],[4], [5],[6].
This work was partly funded by the Overseas Research Students award and
the Data Storage Network, UK.
This method is similar to the works of [7] and [8]. This
paper also investigates the use of signal-to-noise ratio (SNR)
mismatch [9] to mitigate the effect of media noise.
The rest of the paper is organised as follows. Section II
describes the perpendicular recording channel used. The de-
scription of the RVCM algorithm is outlined in Section III
and the performance of this algorithm is demonstrated in
Section IV. Section V concludes this paper.
II. PERPENDICULAR RECORDING CHANNEL MODEL
Fig. 1 shows the block diagram of the perpendicular record-
ing system model used in this paper. The user data, denoted
as ak, is a sequence of of input symbols taking values of
{0, 1}. Some error-protection redundancy is added to the
sequence ak by the error-correcting-codes (ECC) encoder
forming codeword sequence ck. To simulate the write current,
the sequence ck is mapped to {−1,+1} according to 2ck − 1
operation. The scaling factor of 0.5 is to ensure the transition
takes values of {−1, 0,+1}.
We assume that the read head produces zero voltage in the
region of magnetic transitions and some voltage in the region
of constant magnetic polarity. We approximate the single-
transition step response, denoted as s(t), using the hyperbolic-
tangent function [10]:
s(t) = A · tanh
(
ln(3)
t
PW50
)
(1)
where A is the saturation level or the amplitude from zero to
peak (normalised to unity) and PW50 is the time taken for s(t)
to go from −A/2 to +A/2. It is assumed that t and PW50 are
normalised to the symbol period, T . Throughout the paper, it
is assumed that PW50 = 1.4. We define the response of two
adjacent transitions (dibit-response) p(t) as:
p(t) = s(t) − s(t− 1) (2)
and the readback signal r(t) is simply the convolution of ck
and p(t) plus some noise:
r(t) =
∑
k
ckp(t− kT ) + n(t) (3)
where n(t) is the overall noise in the recording system which
consists of media, jitter and electronic noise, i.e. n(t) =
nm(t) + nj(t) + ne(t).
The media noise, nm(t) originates from the imperfections
of the media and its effect is significant in the magnetic

SUBMITTED TO THE ISCTA 2005 CONFERENCE ON 4 FEB 2005 2
ECC
Encoder
Dibit Response
Jitter
Estimate
Saturation
Level
PR
Equaliser
MAP
Decoder
ECC
Decoder
ak ck
0.5
xk
ne(t)
r(t)y
nm(t)
BER/FER
Measurement Point
nj(t)
p(t)
Fig. 1. Perpendicular recording system model
transition regions. Typically, media noise is approximately
four times the electronic noise at transition regions. In our
system model, we consider the media noise as Additive-White-
Gaussian-Noise (AWGN) with mean of 0 and variance of
σ2m, which exists in the transition region only. As shown in
Fig. 1, the media saturation noise depends on the saturation
level and it is evaluated as 1 − (xk/A)2. Unlike media noise
which is media dependent, the jitter noise nj(t) is due to
timing imperfection only. To model the sampling jitter noise,
nj(t), the nth order Taylor approximation is used. The jitter
estimation block shown in Fig. 1 is done with 6th order Taylor
series expansion of s(t). The jitter probability density function
is assumed to be uniform, limited by a maximum value. The
electronic noise, ne(t) is AWGN with mean of 0 and variance
of σe. The recording system in Fig. 1 caters for many different
simulation cases with varying degree of electronics, media and
sampling jitter noise. We define the channel SNR as:
SNR = 10 log10
(
1
2 (σ2e + σ2m)
)
(4)
The noisy readback signal is equalised to (4, 6, 4, 2) PR
target which is only optimal for electronic noise at the con-
sidered PW50 [11]. It serves for comparison purposes only.
The Maximum-a-Posteriori (MAP) decoder of the PR channel
exchanges extrinsic with the ECC decoder to deliver solution
which is used for performance evaluation.
III. RECEIVED-VECTOR-COORDINATE-MODIFICATION
(RVCM) ALGORITHM
It has been shown that the RVCM algorithm provides
considerable coding gain for parallel and serial concatenated
turbo codes [3],[4] and LDPC codes [6]. The algorithm can
be applied directly to perpendicular recording and is described
briefly below.
A. Description of the Algorithm
Let y = {y0, y1, . . . , yn} denote an n−tuple vec-
tor at the output of the MAP decoder, that is the a-
posteriori probability (APP) of the MAP decoder. Let l =
{l0, l1, . . . , ln} denote the reliability sequence of y, where
li = log (Pr(yi| + 1)/Pr(yi| − 1)). Assume that imax is an
integer where 1 ≤ imax ≤ n and p ⊆ {0, 1, . . . , n − 1} is
a vector of length imax.
Step 1. Store the vector l, let the integer i be initialised
to 0.
Step 2a. Set l′pi = lpi and lpi = −∞. Restart the iterative
decoder, store the decoded vector (d−pi).
Step 2b. Set lpi = +∞. Restart the iterative decoder,
store the decoded vector (d+pi) and restore lpi ,
i.e. lpi = l′pi .
Step 3. If i < imax then set i = i + 1 and continue
to Step 2. Otherwise, stop the algorithm and
from the list of all decoded vectors d−pi
⋃
d+pi ,
∀i ∈ {0, 1, . . . , imax − 1}, choose a decoded
vector that has the minimum euclidean distance.
It is assumed that the iterative decoder always
outputs a codeword.
From the steps above, it is clear that the complexity of the
algorithm depends on imax.
B. Critical Symbols
One of the major obstacles concerning the RVCM algorithm
is the difficulty in finding the symbol(s) that, if modified,
can converge the iterative decoder to the maximum-likelihood
solution [3],[4]. These symbols are referred as the critical
symbols and their distribution is uniform with no sign of
vulnerable or favourite symbol positions. On the other hand,
due to their uniform distribution, it is likely that we can find
one of the critical symbols if we confine our search to a small
group, i.e. keeping the value of imax low. In this way, we
can reduce the computational complexity for the price of sub-
optimum performance. As we will show later that, the gain
obtained by confining imax to a small value is still significant
compared to the performance of the standard iterative decoder.
There are various methods for selecting the critical symbols,
see [6] for details. In this paper, we restrict the selection to
one method only, that is the reliability of the APP at the output
of the MAP decoder.

SUBMITTED TO THE ISCTA 2005 CONFERENCE ON 4 FEB 2005 3
10-6
10-5
10-4
10-3
10-2
10-1
100
12 13 14 15 16 17
Er
ro
r R
at
e
SNR, dB
BER - Standard iterative decoder
BER - RVCM (imax = n)
FER - Standard iterative decoder
FER - RVCM (imax = n)
Fig. 2. Error performance of RVCM decoder on the turbo code in the
presence of electronic noise only
10-6
10-5
10-4
10-3
10-2
10-1
100
14 15 16 17 18 19
Er
ro
r R
at
e
SNR, dB
BER - Standard iterative decoder
BER - RVCM (imax = n)
FER - Standard iterative decoder
FER - RVCM (imax = n)
Fig. 3. Error performance of RVCM decoder on the turbo code in the
presence of electronic and media noise
IV. RVCM PERFORMANCE
We evaluate the performance of the RVCM algorithm on
some short-block length turbo and LDPC codes. Fig. 2 and 3
show the error rate performance of the turbo code under
the standard iterative and RVCM decoders. The turbo code
considered is the b/b+1 tail-biting turbo code1, where b = 3,
k = 198 and n = 278 with S−interleaver. Significant
improvement is noticed and the increase in performance gets
better as SNR increases. It is worth noting that the results in
Fig. 3 were obtained using SNR mismatch technique. With
this technique, the improvement in performance over standard
iterative decoder is even greater as SNR increases. In the
presence of media noise, SNR mismatch methods do not
provide the same performance as observed with electronics
noise only [9], however better targets for media noise are being
investigated by the authors.
1C. Berrou once referred this code as duo-binary turbo code.
10-5
10-4
10-3
10-2
10-1
100
12 13 14 15 16 17 18
Er
ro
r R
at
e
SNR, dB
BER - Standard iterative decoder
BER - RVCM (imax = 10)
BER - RVCM (imax = n)
FER - Standard iterative decoder
FER - RVCM (imax = 10)
FER - RVCM (imax = n)
Fig. 4. Error performance of RVCM decoder on the (127, 84) cyclic LDPC
code in the presence of electronic and media noise
10-5
10-4
10-3
10-2
10-1
100
13 14 15 16 17 18
Er
ro
r R
at
e
SNR, dB
BER - (255,175) BP
BER - (255,175) RVCM (imax = n)
BER - (1248,864) BP
FER - (255,175) BP
FER - (255,175) RVCM (imax = n)
FER - (1248,864) BP
Fig. 5. Performance of the (255, 175) cyclic and (1248, 864) codes in the
presence of electronic and media noise
Similar performance improvement is observed for the LDPC
codes, see Fig. 4. The (127, 84) cyclic LDPC code, which
has minimum-distance of 9, was constructed using a method
described in [12]. As mentioned earlier, the RVCM algorithm
allows one to trade off the performance against the com-
putational complexity. From Fig. 4, despite the performance
obtained by setting imax = 10 is approximately 0.3dB inferior
to that by setting imax = n, the coding gain from the standard
iterative decoding is significant. For the case of imax = 10, we
select the critical symbols based on the reliability measure
at the output of the MAP decoder. From the vector l, we
construct a vector p of length imax such that lp0 < lp1 <
lp2 < . . . < lpn−1 . In Fig. 5, we compare the performance of
the (255, 175) cyclic code and that of the (1284, 864) quasi-
cyclic code. We can see that the RVCM algorithm provides
significant gain, within one order of magnitude improvement,
over the BP algorithm. At approximately 3 × 10−5 BER, the
performance of the cyclic code with RVCM is within 0.6dB
away from the longer code under BP decoding.
V. CONCLUSIONS
We have shown that the application of the RVCM algorithm
to perpendicular magnetic recording shows promising results.
Simulation results show that improvement of within one order
of magnitude is possible. Short block length offers an attractive
error-correction scheme in which RVCM algorithm can be
fully exploited by setting imax = n. A bank consisting of
2imax parallel RVCM decoders can be built on chips and
the decoding of short-block length data has low latency. The
performance of longer block-length codes, up to a certain
error-rate, can be outperformed by the application of RVCM
algorithm to shorter codes. The exact point, at which the longer
codes start to perform better, depends on the code structure.
We also extended our investigations on using some non
binary cyclic LDPC codes [13] and we observe similar im-
provement as in the binary cases. Further investigations in
identifying the critical symbols will allow the application of
RVCM algorithm to long powerful codes.
ACKNOWLEDGEMENT
The authors would like to thank Prof. Barry K. Middleton
of University of Manchester for the channel noise discussion.

SUBMITTED TO THE ISCTA 2005 CONFERENCE ON 4 FEB 2005 4
REFERENCES
[1] R. Wood, “The feasibility of magnetic recording at 1 Terabit per square
inch,” IEEE Trans. Magn., vol. 36, pp. 36–42, Jan. 2000.
[2] E. Papagiannis, M. A. Ambroze, and M. Tomlinson, “Received Vector
Codeword Modification (RVCM).” UK Patent Application 0320126.6,
Jul. 2003.
[3] E. Papagiannis, M. A. Ambroze, and M. Tomlinson, “Analysis of non
convergence blocks at low and moderate snr in scc turbo schemes,”
SPSC 2003 8th International worksop on Signal Processing for Space
Communications, Catania, Italy, pp. 121–128, Sep. 2003.
[4] E. Papagiannis, M. A. Ambroze, and M. Tomlinson, “Approaching the
ml performance with iterative decoding,” International Zurich Seminar
on Communications, Zurich, Switzerland, pp. 220–223, Feb. 2004.
[5] C. J. Tjhai, E. Papagiannis, M. Tomlinson, M. A. Ambroze, and M. Z.
Ahmed, “Improved iterative decoder for LDPC codes with performance
approximating to a maximum likelihood decoder.” UK Patent Applica-
tion 0409306.8, Apr. 2004.
[6] E. Papagiannis, M. Ambroze, and M. Tomlinson, “Improved Decoding
of Low-Density Parity-Check Codes with Low, Linearly Increased
Added Complexity.” Submitted to ISIT 2005, Jan. 2005.
[7] H. Pishro-Nik and F. Fekri, “Improved Decoding Algorithms for Low-
Density Parity-Check Codes,” Proc. 3rd Intl. Symp. on Turbo Codes,
Brest, France, pp. 117–120, 2003.
[8] N. Varnica and M. Fossorier, “Belief Propagation with Information
Correction : Imporved Near Maximum-Likelihood Decoding of Low-
Density Parity-Check Codes,” Proc. of IEEE Intl. Symp. Inform. Theory
(ISIT), Chichago, USA, p. 343, July 2004.
[9] W. Tan and J. Cruz, “Signal-to-Noise Ratio Mismatch for Low-
Density Parity-Check Coded Magnetic Recording Channels,” IEEE
Trans. Magn., vol. 40, pp. 498–506, Mar. 2004.
[10] H. Sawaguchi, Y. Nishida, H. Takano, and H. Aoi, “Performance
Analysis of Modified PRML Channels for Perpendicular Recording
Systems,” J. Magn. Magn. Mater., vol. 235, pp. 265–272, 2001.
[11] M. Madden, M. ¨Oberg, Z. Wu, and R. He, “Read Channel for Perpen-
dicular Magnetic Recording,” IEEE Trans. Magn., vol. 40, pp. 241–246,
Jan. 2004.
[12] R. Horan, C. Tjhai, M. Tomlinson, M. Ambroze, and M. Ahmed, “A
Finite-Field Transform Domain Construction of Binary Low-Density
Parity-Check Codes.” Submitted to ITW 2005, Jan. 2005.
[13] C. Tjhai, M. Tomlinson, R. Horan, M. Ambroze, and M. Ahmed,
“GF(2m) Low-Density Parity-Check Codes Derived from Cyclotomic
Cosets.” Submitted to ISIT 2005, Jan. 2005.