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Detection and Decoding for Magnetic Storage Systems
Item Type text; Electronic Dissertation
Authors Radhakrishnan, Rathnakumar
Publisher The University of Arizona.
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Download date 09/06/2018 01:00:12
Link to Item http://hdl.handle.net/10150/194396
DETECTION AND DECODING FOR MAGNETIC STORAGE SYSTEMS
by
Rathnakumar Radhakrishnan
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2 0 0 9
2
THE UNIVERSITY OF ARIZONAGRADUATE COLLEGE
As members of the Dissertation Committee, we verify that we have read the dis-sertation prepared by Rathnakumar Radhakrishnan entitled Detection andDecoding for Magnetic Storage Systems and recommend that it be acceptedas fulfilling the dissertation requirement for the Degree of Doctor of Philosophy
Date: May 26, 2009
Bane Vasic, Ph.D.Dissertation Director
Date: May 26, 2009
William E. Ryan, Ph.D.
Date: May 26, 2009
Michael W. Marcellin, Ph.D.
Final approval and acceptance of this dissertation is contingent upon the candi-date’s submission of the final copies of the dissertation to the Graduate College. Ihereby certify that I have read this dissertation prepared under my direction andrecommend that it be accepted as fulfilling the dissertation requirement.
Date: May 26, 2009
Dissertation Director: Bane Vasic, Ph.D.
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirementsfor an advanced degree at The University of Arizona and is deposited in the Uni-versity Library to be made available to borrowers under rules of the library.
Brief quotations from this dissertation are allowable without special per-mission, provided that accurate acknowledgment of source is made. Requests forpermission for extended quotation from or reproduction of this manuscript in wholeor in part may be granted by the head of the major department or the Dean of theGraduate College when in his or her judgment the proposed use of the material isin the interests of scholarship. In all other instances, however, permission must beobtained from the author.
SIGNED:Rathnakumar Radhakrishnan
4
ACKNOWLEDGEMENTS
Almost all doctoral students experience ups and downs in their research career. Itwas no different for me. It has truly been a roller coaster ride for me, but I amfinally grateful that the ride is over! I strongly believe that most people withoutsufficient support from their family and friends would find it extremely difficult tocome out of their trough. Fortunately, I had no dearth of support from my nearand dear ones. I cannot be more thankful to my dear parents for, well, everything.Their complete unconditional love has been a constant pillar in my life. My dearbig-sister Sudha and big-brother Suresh are a big part of my success. They havesupported me throughout and have shielded me from all responsibilities so far!
I would like to express my sincerest gratitude to my advisor Dr. Bane Vasic withoutwhom, none of this would have been possible. His constant support, help and advicehas been an essential ingredient to my success. It is amazing how he makes himselfalways (and I mean always) available for his students. It has been nothing short ofcomplete pleasure and honor to have worked with him. I’ll always be grateful tohim for all he has done for me.
Lab-mates are a big part of any PhD students career. It has been a fruitful expe-rience to have worked with Sundar, Ananth, Shiva, Shashi and Dzung. I’ll alwaysremember the numerous discussions we have had and the numerous pointless tripswe have made to the student union! Many colleagues over the years have helpedme in my research and I am thankful to them and for the opportunity to work withthem. Specifically, I would like to acknowledge Dr. Erden, Dr. Venkataramani andDr. Kuznetsov from Seagate for many stimulating discussions we have had. Thefinancial support from Seagate has been the backbone of my research career and Iam very grateful to them.
I would also like to thank my committee members Dr. Marcellin and Dr. Ryan forbeing on my dissertation committee. Finally, the importance of having good friendscannot be measured. My wonderful friends have been and will be a big part of mylife and they have contributed to my success more than they realize. I’ll alwayscherish the enjoyable experiences I have shared with them while I was at UofA.
Above all, I thank Him.
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To my beloved parents, sister and brother.
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TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . 14
CHAPTER 2 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . 21
2.1 Principle of Magnetic Recording . . . . . . . . . . . . . . . . . . . . 21
2.2 Magnetic Recording as a Communication System . . . . . . . . . . 24
CHAPTER 3 DETECTOR WITH AUXILIARY LDPC PARITYCHECKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Sub-Optimality of Turbo-Equalization . . . . . . . . . . . . . . . . 33
3.2 Optimal Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Trellis with Parity Checks . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 Performance of Detector . . . . . . . . . . . . . . . . . . . . 43
3.4.2 Performance of Turbo-Equalizer . . . . . . . . . . . . . . . . 48
3.5 Disjoint Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.6 Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6.1 Minimal Span Matrices . . . . . . . . . . . . . . . . . . . . . 56
3.6.2 Graphical Representation . . . . . . . . . . . . . . . . . . . 59
3.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
CHAPTER 4 JOINT MESSAGE-PASSING SYMBOL DECODER 66
4.1 Message-Passing Detection Algorithm . . . . . . . . . . . . . . . . . 68
4.1.1 Channel Graph . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.2 Message-Passing Symbol Detector . . . . . . . . . . . . . . . 72
4.2 Joint Message-Passing Symbol-Decoding Algorithm . . . . . . . . . 75
4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
CHAPTER 5 TERNARY MAGNETIC RECORDING SYSTEM 88
5.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2 Optimal Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Equalization and Detection . . . . . . . . . . . . . . . . . . . . . . 95
5.3.1 Equivalent Discrete Time System . . . . . . . . . . . . . . . 95
TABLE OF CONTENTS — Continued
7
5.3.2 MIMO Equalizer . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.3 GPR Targets . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4 Application to High-Radial Density Systems . . . . . . . . . . . . . 101
5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
CHAPTER 6 HEAT-ASSISTED MAGNETIC RECORDING SYS-TEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1 Principle of Heat-Assisted Magnetic Recording . . . . . . . . . . . . 111
6.2 Thermal Williams-Comstock Model . . . . . . . . . . . . . . . . . . 113
6.2.1 Longitudinal HAMR . . . . . . . . . . . . . . . . . . . . . . 117
6.2.2 Perpendicular HAMR . . . . . . . . . . . . . . . . . . . . . . 118
6.3 Microtrack Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Unique Characteristics of HAMR . . . . . . . . . . . . . . . . . . . 123
6.4.1 Effects of Laser-Spot Position . . . . . . . . . . . . . . . . . 126
6.4.2 Non-Linear Transition Shift (NLTS) . . . . . . . . . . . . . . 135
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
CHAPTER 7 TWO-DIMENSIONAL MAGNETIC RECORDINGSYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.1.1 Voronoi-grain Model . . . . . . . . . . . . . . . . . . . . . . 152
7.1.2 Random-grain Model . . . . . . . . . . . . . . . . . . . . . . 154
7.2 TDMR as BEEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3 Capacity of TMDR . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
CHAPTER 8 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . 163
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
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LIST OF FIGURES
2.1 Illustration of write and read process in a magnetic recording drive . . 22
2.2 Grains of the medium . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Magnetic transition in a medium . . . . . . . . . . . . . . . . . . . . . 23
2.4 Magnetic recording communication system . . . . . . . . . . . . . . . . 25
3.1 Block diagram of a partial response system. . . . . . . . . . . . . . . . 32
3.2 Tree representing parity check constraints of a LDPC code. . . . . . . . 35
3.3 State diagram of a PR4 channel. . . . . . . . . . . . . . . . . . . . . . 39
3.4 State diagram of a single parity check code. . . . . . . . . . . . . . . . 40
3.5 Composite state diagram of a single parity check code and the PR4 channel. 41
3.6 Trellis of a: (a) PR4 channel; and, (b) trellis of a single parity check code. 42
3.7 Trellis of the single parity check coded PR4 channel . . . . . . . . . . . 43
3.8 BER detector performance with auxiliary LDPC checks . . . . . . . . . 47
3.9 Performance of PG(273,191) LDPC code at the output of the decoderfor the EEPR4 channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.10 Histogram of error weights in incorrectly detected codewords at the out-put of the detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.11 Performance of (273,191) random LDPC code at the output of the de-coder for the EEPR4 channel . . . . . . . . . . . . . . . . . . . . . . . 52
3.12 Performance of (495,433) random LDPC code, rate 0.87, with 4 randomparity checks in trellis for PR1 channel . . . . . . . . . . . . . . . . . . 54
3.13 Graph representing the MSPM . . . . . . . . . . . . . . . . . . . . . . 61
3.14 Two sets of disjoint checks of the PG(273,191) LDPC code. . . . . . . 63
3.15 Performance of the PG(273,191) LDPC code over the PR1 channel . . 64
4.1 Block diagram of a PR system. Decoder is either turbo-equalizer (upperbranch) or joint decoder (lower branch) and noise is modeled as additivewhite Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Generalized factor graph representation of a PR channel trellis. . . . . 71
4.3 A graph that represents constraints imposed by the channel on the noise-less channel output sequences. . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 A graph that represents constraints imposed by the channel and theparity checks of the LDPC code on the noiseless channel output sequences 76
4.5 Trellis diagram of a single parity check code of length 3 . . . . . . . . . 78
4.6 Expanded code trellis of Fig. 3.6(b) . . . . . . . . . . . . . . . . . . . . 80
LIST OF FIGURES — Continued
9
4.7 Bit error rate comparison of (1908,212) random LDPC code on a PR4channel when decoded using turbo-equalizer and the joint decoder. . . 82
4.8 Performance of column-permuted (1908,212) random LDPC code thatdoes not have any four or six-cycles in the combined graph (excludingcycles in the bipartite code graph). . . . . . . . . . . . . . . . . . . . . 83
4.9 Performance of (273,82) random LDPC code on a EPR4 channel whendecoded using turbo-equalizer and joint decoder. . . . . . . . . . . . . . 85
4.10 Performance of column-permuted (273,82) random LDPC code that doesnot have any four or six-cycles in the combined graph (excluding cyclesin the bipartite code graph). . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1 Ternary magnetic recording channel . . . . . . . . . . . . . . . . . . . . 92
5.2 Optimal receiver for the ternary magnetic recording channel. . . . . . . 93
5.3 Discrete time equivalent system of the ternary magnetic recording system. 96
5.4 MIMO equalizer: a set of four FIR filters that equalizes all four channels. 97
5.5 GPR calculation for ternary magnetic recording system with MIMOequalizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.6 PRML binary magnetic recording system with ITI. . . . . . . . . . . . 102
5.7 BER comparison between a binary system with ITI (λ = 0.25) andternary system with no ITI . . . . . . . . . . . . . . . . . . . . . . . . 107
5.8 BER comparison between a binary system with ITI (λ = 0.2) andternary system with no ITI . . . . . . . . . . . . . . . . . . . . . . . . 108
6.1 Example of an MH loop, illustrating its dependence on temperature . . 112
6.2 MH loop with effective field . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 Longitudinal head and field . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 Perpendicular head and field: actual head configuration and equivalentset-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5 Microtrack modeling of HAMR channel . . . . . . . . . . . . . . . . . . 122
6.6 Magnetization of the medium after a transition is made . . . . . . . . . 125
6.7 Transition center curvature across the track for various deep gap fields 126
6.8 Readback signal width at various peak temperatures for different gap fields128
6.9 PW50 at various laser spot positions in Longitudinal HAMR . . . . . . 129
6.10 Coercivity for various laser spot positions . . . . . . . . . . . . . . . . . 130
6.11 Transition centers across the track at various laser spot positions in thedown-track direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.12 Transition parameter across the track at various laser spot positions inthe down-track direction . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.13 Transition centers across the track at various laser spot positions in theup-track direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
LIST OF FIGURES — Continued
10
6.14 Transition parameters across the track at various laser spot positions inthe up-track direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.15 Transition centres at various position of laser to the left of gap centre . 135
6.16 Transition parameter at various position of laser to the left of gap centre 136
6.17 Effect of demagnetizing field of past transitions . . . . . . . . . . . . . 137
6.18 NLTS variation across track at various normalized densities for HAMR 140
6.19 NLTS variation across track at ND 3.0 for conventional longitudinalrecording system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.20 Average NLTS at various bit lengths . . . . . . . . . . . . . . . . . . . 142
6.21 PW50 variation due to NLTS at various bit lengths . . . . . . . . . . . 143
6.22 NLTS variation for a series of transitions at ND 3 . . . . . . . . . . . . 144
6.23 Depiction of transition curvatures of 3 consecutive transitions before andafter NLTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.24 NLTS variation for a series of transitions at ND 4 . . . . . . . . . . . . 146
6.25 Average NLTS variation in a series of transitions at various normalizeddensities for HAMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.26 Average NLTS variation in a series of transition at various normalizeddensities for conventional systems . . . . . . . . . . . . . . . . . . . . . 148
7.1 Two-dimensional magnetic recording (TDMR) system model. . . . . . 151
7.2 Voronoi-grain model: (a) ideal medium and (b) an instance of a non-idealmedium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.3 Random-grain model: (a) input bits, and (b) the resultant magnetizedmedium after writing the input bits onto the medium (with write-errors). 156
7.4 Pdf of effective magnetization determined by simulations using theVoronoi-grain model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.5 Pdf of effective magnetization determined by simulations using theRandom-grain model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.6 Error and erasure rates for various thresholds and variances of the grainarea (Voronoi-grain model) . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.7 Capacity for BEEC corresponding to the error and erasure rates shownin Fig. 7.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.8 Error and erasure rates for various thresholds (random-grain model) . . 161
7.9 Capacity for BEEC corresponding to the error and erasure rates shownin Fig. 7.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
11
LIST OF TABLES
3.1 Error event distribution at 12 dB for PG(273,191) . . . . . . . . . . . . 46
5.1 Binary-to-Ternary Mapping Code . . . . . . . . . . . . . . . . . . . . . 106
6.1 System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
12
ABSTRACT
The hard-disk storage industry is at a critical time as the current technologies are
incapable of achieving densities beyond 500 Gb/in2, which will be reached in a few
years. Many radically new storage architectures have been proposed, which along
with advanced signal processing algorithms are expected to achieve much higher
densities. In this dissertation, various signal processing algorithms are developed to
improve the performance of current and next-generation magnetic storage systems.
Low-density parity-check (LDPC) error correction codes are known to pro-
vide excellent performance in magnetic storage systems and are likely to replace or
supplement currently used algebraic codes. Two methods are described to improve
their performance in such systems. In the first method, the detector is modified to
incorporate auxiliary LDPC parity checks. Using graph theoretical algorithms, a
method to incorporate maximum number of such checks for a given complexity is
provided. In the second method, a joint detection and decoding algorithm is de-
veloped that, unlike all other schemes, operates on the non-binary channel output
symbols rather than input bits. Though sub-optimal, it is shown to provide the
13
best known decoding performance for channels with memory more than 1, which
are practically the most important.
This dissertation also proposes a ternary magnetic recording system from
a signal processing perspective. The advantage of this novel scheme is that it is
capable of making magnetic transitions with two different but predetermined gradi-
ents. By developing optimal signal processing components like receivers, equalizers
and detectors for this channel, the equivalence of this system to a two-track/two-
head system is determined and its performance is analyzed. Consequently, it is
shown that it is preferable to store information using this system, than to store
using a binary system with inter-track interference. Finally, this dissertation pro-
vides a number of insights into the unique characteristics of heat-assisted magnetic
recording (HAMR) and two-dimensional magnetic recording (TDMR) channels. For
HAMR channels, the effects of laser spot on transition characteristics and non-linear
transition shift are investigated. For TDMR channels, a suitable channel model is
developed to investigate the two-dimensional nature of the noise.
14
CHAPTER 1
INTRODUCTION
Magnetic recording has been the most widely used method to store information
for several decades. It is the principle behind hard-disk drives that are synonymous
with computer storage and other consumer electronic devices requiring large storage
capacities. The ever-increasing demand from such devices for increased storage
density has so far been met by the storage industry. Over the last decade, there
has been a phenomenal increase in maximum achievable storage density every year.
The current state-of-the-art hard disk drives achieve a density of over 500 Gb/in2.
A hard-disk drive consists of a disk made of magnetic material and a record-
ing/writing head. The disk is divided into many equally spaced tracks where the
data is stored as a binary sequence utilizing the two directions of magnetization
possible. The tremendous increases in storage capacity of a hard disk drive were
almost always achieved by scaling down the area used to store one bit of informa-
tion, along with the use of sophisticated signal processing algorithms. The bit area
is scaled down by reducing the volume of the magnetic grains of the medium. As a
15
result of this decrease in grain volume, their stability also decreases. The magnetic
industry has reached a critical point where the grain volume can no longer be re-
duced. Consequently, the storage density has hit a road-block and to increase the
density to 1 Tb/in2 and beyond is a significant challenge.
In order to obtain densities beyond 1 Tb/in2, the magnetic recording tech-
nology needs to be fundamentally altered. It is widely acknowledged that along
with new recording technologies, more sophisticated signal processing algorithms
and stronger error-correction codes and decoders would be required. This disser-
tation presents several methods to improve the performance of current and future
magnetic storage systems. Its contribution spans many system components, includ-
ing the channel, detector and decoder.
Low-density parity-check (LDPC) codes, invented by Gallager [1] and redis-
covered by Mackay and Neal [2] have been shown to be capacity achieving on mem-
oryless channels. It has also been shown to achieve excellent error rate performance
over channels with memory, such as magnetic storage [3] and optical communication
channels [4]. They are likely to replace or supplement currently used algebraic codes
in next-generation magnetic storage systems. For reasons of complexity, magnetic
storage channels, which are characterized as inter-symbol interference (ISI) chan-
nels, are equalized to a partial-response (PR) target with relatively small memory
16
compared to the unequalized channel. Any LDPC decoder must cope with the con-
trolled amount of ISI introduced by the PR. For an uncoded system, the channel
input sequence is optimally detected in the presence of ISI by the Viterbi algorithm.
For an LDPC coded system, the optimal maximum a posteriori (MAP) decoding
is impractical, but good error rate performance can be achieved by using the turbo
principle [5]. In this technique, information is iteratively passed back and forth be-
tween a soft-input soft-output (SISO) detector and a SISO LDPC decoder. The ISI
in the channel output is eliminated by a detector using the channel information un-
known to the decoder, and subsequently, the output is decoded by a LDPC decoder
using the code structure information unknown to the detector. Since, Viterbi algo-
rithm produces only hard decisions, algorithms like soft-output Viterbi algorithm
(SOVA) [6] and Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm [7] are used for SISO
detection and the well-known sum-product algorithm (SPA) [8] is used for SISO de-
coding. This iterative algorithm is known as turbo-equalizer. Though sub-optimal,
it is the best and most widely used decoder known today.
The first part of the dissertation focuses on methods to improve the per-
formance of turbo-equalizer. We provide a tutorial on the conventional magnetic
recording system currently used in Chapter 2, which also explains the turbo-
equalizer in more detail. In Chapter 3, we present a simple method to improve
17
the turbo-equalizer performance by pruning the channel trellis on which the detec-
tor like SOVA and BCJR operate. The trellis is pruned by incorporating certain
parity checks of the LDPC code. By simulations, we show that this method provides
significant performance improvement over turbo-equalizer. It provides a practical
way to improve error rate performance without any significant change in the de-
coder architecture. However, due to complexity constraints, the number of parity
checks that can be incorporated is limited. Nevertheless, we determine that certain
parity checks, termed as disjoint checks, can be incorporated into the trellis with-
out an exponential increase in complexity. Further, we present a method to identify
such parity checks and theoretically prove that this method yields the biggest set
of parity checks that satisfy this property [9].
Significant advances have been made in the general understanding and the
improvement of iterative message-passing algorithms. In [10], it is shown that many
such algorithms used in different applications in computer science and engineering
are special cases of the generalized belief-propagation algorithm. It also provides
a general method to improve such algorithms. Unfortunately, this method has not
yet yielded performance that is significantly better than the performance of turbo-
equalizer, which is an iterative belief-propagation algorithm [11]. It is a common
knowledge that a natural way to improve performance of turbo-equalizer is to use
both the channel and code structure information simultaneously to make decisions
18
on the transmitted bits. This is known as joint decoding. Currently, a number of
researchers are focused on this subject, but there has so far not been a significant
headway in this direction as well [12][13][14]. In Chapter 4, we present a joint de-
coder that achieves a significant performance improvement over the turbo-equalizer.
Moreover, we show that this algorithm can be applied over channels with memory
more than 1, which are practically the most important. The fundamental advantage
of this decoder stem from the fact that it attempts to make a decision on the channel
output symbols rather than on the channel binary inputs. In order to design such
a decoder, we combine a graph-based detection algorithm and a modified version
of the LDPC message-passing decoder that provides information on the channel
output symbols. The superior performance obtained by this decoder is shown using
simulations for various LDPC codes and under various channel conditions [15].
The second part of the dissertation focuses on channel modeling, analysis
and system development of next-generation storage systems. As already noted, it
is necessary to developed fundamentally new magnetic recording technologies in or-
der to keep increasing the storage density. In Chapter 5, a novel ternary magnetic
recording architecture is proposed and analyzed from a read-channel perspective.
Specifically, we explore the advantages of a recording channel that is capable of
making magnetic transitions with two different but predetermined gradients. For
19
this channel, we develop optimal signal processing components like receivers, equal-
izers and detectors and show that it is equivalent to the two-track/two-head system.
Consequently, we show by simulations that it is preferable to store information using
the ternary system, than to store using a binary system with inter-track interference
[16].
In the last few years, many novel storage architectures have been pro-
posed that are radically different from the conventional architectures. Three such
important technologies are heat-assisted magnetic recording (HAMR) [17], two-
dimensional magnetic recording (TDMR) [18] and bit-patterned media (BPM)
recording [19]. HAMR relies on sophisticated write-process, BPM relies on sophis-
ticated production of patterned medium and TDMR relies on sophisticated signal
processing and coding algorithms. Significant resources are being devoted to each
of these technologies, as it is not clear yet, which one would turn out to be the
most suitable for achieving ultra-high densities. Each of these technologies have
certain unique advantages and have to overcome significant challenges, before their
potential can be fully realized. In Chapters 6 and 7, we focus on channel modeling
and analysis of HAMR and TDMR. Specifically, in Chapter 6, we introduce HAMR,
explain the techniques used to numerically model the magnetization process in lon-
gitudinal HAMR and extend it to perpendicular HAMR. Utilizing these models, we
investigate the unique transition characteristics that depend, among other factors,
20
on various thermal parameters. In particular, we determine the effects of change in
laser spot position on transition length and center. Also, we determine the unique
effects of the thermal profile on the characteristics of the non-linear transition shift
[20][21][22].
In Chapter 7, we introduce TDMR and present a method to view this sys-
tem as a binary error and erasure channel. Using this method and a simple detec-
tor, we experimentally determine some bounds on capacity for the TDMR channel.
We introduce and analyze two different TDMR read-channel models, Voronoi-grain
model and random-grain model, to determine the effects of write-errors on the ca-
pacity bounds [23][24][25].
21
CHAPTER 2
PRELIMINARIES
2.1 Principle of Magnetic Recording
There are many ways digital data can be stored. Two of the most popular ones
are optical and magnetic storage. Over the last 40 years, magnetic recording, the
technology behind hard-disk drives, have clearly emerged as the primary means
of data storage. Magnetic disk drives are one of the most complicated consumer
electronic devices used today. They are mainly composed of five parts - magnetic
read/write head mechanism, magnetic disks, read-channel electronics and interface.
Binary data can be stored in the magnetic disk, by magnetizing the disk to one of
two possible polarities.
The process of storing and retrieving information using magnetic recording
is illustrated in Fig. 2.1. The write current to the recording head is modulated
according to the binary information to be stored, which induces a series of magnetic
transitions in the medium. During the read process, magnetic transitions in the
22
Medium
Write current
Readback signal
Data 1 1 0 1 0 0
Figure 2.1: Illustration of write and read process in a magnetic recording drive
medium induces a readback signal in the head flying over it. There will be no output
signal if there is no transition under the head. The read-channel electronics attempts
to detect this signal in order to make decisions on the stored binary information.
Magnetic disk consists of tiny irregularly-shaped grains, which are the
smallest magnetizable component of the disk. Fig. 2.2 shows a two-dimensional
depiction of the medium. A collection of such grains are magnetized to one of two
polarities in order to store one bit of information. Fig. 2.3 depicts the storage of
one bit of information. The areal density, defined as the number of bits stored in
a unit area of the medium, is determined by the average size of the grains and the
number of grains used to store one bit of information.
Over the years, there has been a constant increase in demand for smaller
hard-disk drives with higher storage capacity. Through many innovations, it has so
23
Figure 2.2: Grains of the medium
Figure 2.3: Magnetic transition in a medium
far been possible to satisfy this demand. The underlying theme of these innovations
has been to reduce the average grain size of the medium. This coupled with advances
in read-channel electronics has been a major factor in the success of hard-disk drives.
Huge advances in read-channel algorithms and electronics have been possible since
the magnetic disk drives were viewed as a communication system with inter-symbol
interference (ISI).
24
2.2 Magnetic Recording as a Communication System
The medium can be magnetized either in the longitudinal or perpendicular direction
with respect to its surface. They are known as longitudinal and perpendicular mag-
netic recording, respectively. Hard-disks with perpendicular recording technology
has only recently been introduced and has shown to provide better areal densities
than longitudinal recording. The response of the read-head to an isolated transition,
known as transition response, for the longitudinal recoding is given as,
hl(t, w) =V
1 +(
2t/w) , (2.1)
and for the perpendicular recording is given as,
hp(t, w) = V · erf(
2√
ln2
w· t)
(2.2)
where, V is the peak voltage of the response, erf(x) is the error function and t can
be interpreted as either time or distance, since the disk rotates at a constant speed.
The quantity w (also denoted as PW50) is defined as the width at half of the peak
amplitude of hl(t, w) for longitudinal recording and the derivative of hp(t, w) for
perpendicular recording.
Fig. 2.4 shows the various components of a conventional magnetic recording
system. The binary input sequence u = {ui} is encoded using a linear block error
correction encoder. The encoded sequence x = {xi} is stored in the medium. If
25
Encoder
Decoder Detector
u x y
z
x
Turbo-Equalizer
w)h(t, LPFT
Equalizerr
Figure 2.4: Magnetic recording communication system
the input bit pattern represents the magnetization pattern of the medium onto
which it was written, the corresponding noiseless readback signal is given as y(t) =
∑
k(xk − xk−1) · h(t − kT, w), where T is the bit period.
There are many types of noise sources in a practical magnetic disk drive,
which can be classified as linear and non-linear noise. The linear noise is primarily
due to electronics and is modeled as additive white Gaussian noise (AWGN). The
non-linear noise sources are primarily due to the imperfections of the grain sizes
and shapes and are, in general, termed as medium noise. Grain imperfections result
in a small shift of the transitions from their intended location during the process
of writing. They also result in the broadening of the readback signal. These are
known as jitter and pulse broadening, respectively. The received signal that suffers
from such noise sources are commonly modeled as y(t) =∑
k(xk − xk−1) · h(t −
kT +∆tk, w +∆wk)+ z(t), where ∆tk and ∆wk are modeled as truncated Gaussian
26
random variables and z(t) denotes the AWGN.
For the purpose of software development, the channel is simulated in an
oversampled domain and the channel output is subsequently downsampled. The
optimal receiver for this channel can either be implemented using a filter matched
to the channel response, or equivalently using a low-pass filter with a bandwidth of
1/2T .
Apart from the various noise sources mentioned above, magnetic recording
systems suffer from inter-symbol interference (ISI), as the PW50 of the channel
response usually spans more than one bit period. Normalized density, defined as
the ratio of PW50 over T , is a measure of the interference. It is common for current
longitudinal recording systems to operate at normalized densities in the range of 2
to 3. As a result, the output of the channel at any instant is dependent on several
bits in the neighborhood. Since, the length of the ISI determines the complexity of
the optimal detector, it is important to use filtering methods to reduce this length.
This is known as equalization, which is implemented using a finite-impulse response
(FIR) digital filter. The goal of the equalizer is not to remove the ISI completely,
but to limit the span of ISI. It has been shown that this yields a better error rate
performance and is known as a partial response (PR) equalizer and the resultant
channel response is known as the target response. It has been shown that optimal
equalization is possible by jointly optimizing the equalizer filter coefficients and the
27
target response.
In Fig. 2.4, the sampled output of the equalizer is denoted as r = {ri}.
The purpose of the detector is to make a decision on x based on its input r. The
two conventional detectors that are optimal for AWGN are - maximum likelihood
(ML) sequence detector and maximum a posteriori (MAP) symbol detector. The
ML detector estimates,
x = arg max p (r|x) , ∀x (2.3)
whereas, the MAP detector estimates,
xi = arg max p (xi|r) , ∀xi ∈ {0, 1}. (2.4)
The ML and MAP detector minimizes the sequence error rate and bit error rate re-
spectively. They are efficiently implemented using the Viterbi and BCJR algorithm
that operate on the channel trellis. More details on these algorithms is given in [7]
and [26].
The output of the detector is sent to the decoder in order to make a de-
cision on u. The decoding algorithms depends on the error correction code used.
Traditionally, algebraic codes like Reed-Solomon (RS) codes have been used, for
which hard-decision bounded distance decoders are available. Recently, low-density
parity check (LDPC) codes have been shown to be capacity achieving in memoryless
channel. Although, a practical ML or MAP decoder for LDPC codes has not been
28
invented yet, the use of graph-based message-passing (MP) decoder has proved to
yield excellent error rate performance. Application of LDPC codes to channels with
memory (ISI channels), like the magnetic recording channel, has also yielded very
good results. The error rate performance of magnetic recording system is improved
further by using the turbo-principle. The turbo principle was first introduced with
the invention of turbo-codes, where two serially (or parallely) concatenated codes
are decoded iteratively by exchanging information between each other. It has sub-
sequently been applied to many systems. As shown in Fig. 2.4, the detector and
decoder exchange soft extrinsic information on the channel input bits. This im-
proves the error rate performance because, the channel information is not available
to the decoder and the code structure information is not available to the detector.
So far, this is the best decoder that is available today.
The turbo-principle is a powerful technique that can be used to improve
performance of any ISI system. Application of this technique requires soft-input
and soft-output detectors and decoders. Viterbi algorithm is a soft-input and hard-
output detector, but can be modified to provide soft-output information on the chan-
nel bits. This modified algorithm is known as soft-output Viterbi algorithm (SOVA).
Alternatively, BCJR algorithm can also be used as the soft-detector, though it is
more complex than SOVA. The soft MP decoding algorithm for LDPC codes makes
29
it very attractive for use in magnetic recording systems. Although, some soft-
decoders for RS codes have been developed, they are generally very complex and
have not proved to be as effective as the MP decoder for binary LDPC codes.
Even though LDPC codes provide superior performance over RS codes,
they cannot yet replace them. On one hand, the use of LDPC codes is advantageous
because they can be decoded using the turbo-principle, but on the other hand, the
dynamics of the decoder is so complex that a clear understanding on conditions
of decoder failures are not known. In other words, a bounded distance decoder is
not yet available for LDPC codes that can guarantee error correction of a certain
length. It is known that MP decoder does not possess this property and in fact
results often in an error floor at high signal-to-noise ratios (SNRs).
30
CHAPTER 3
DETECTOR WITH AUXILIARY LDPC PARITY CHECKS
LDPC codes are likely to supplement algebraic codes currently used in magnetic
storage systems. They are already in use in wireless communications systems. As
a result, designing decoders that perform better than turbo-equalizer has received
a lot of attention, though with little success. As mentioned in Chapter 1, it is the
best decoder available today for LDPC coded channels with memory. This chapter
presents one method to improve over the performance of the turbo-equalizer.
Generally, any attempt to improve the turbo-equalizer performance involves
making available the code information to the detector and the channel information
to the decoder. A popular approach in achieving this has been to combine detector
and the LDPC decoder in order to jointly decode using both channel and code
information [11], [12], [13]. If the graph on which the joint message-passing decoder
operates is cycle free, then the decoder is optimal. Usually, such algorithms are more
complex than turbo-equalizer. In this chapter, we present a technique to improve
the performance of turbo-equalizer, that will serve as a middle-ground between the
31
turbo-equalizer and the joint decoder in terms of performance and complexity. In
this technique, some LDPC code structure information is made available to the
detector by incorporating some parity checks of the LDPC code in the channel
trellis. These parity checks may not be part of the parity-check matrix of the code.
A well-known method to improve detector performance is to prune the
trellis in order to remove certain unwanted sequences to be wrongly detected. This
technique has been studied in the past and is routinely employed for various pur-
poses. For instance, many storage systems employ some constraint codes in order
to impose certain characteristics on the channel input sequences. These constraint
codes can be decoded optimally by pruning the channel trellis to remove all se-
quences restricted by the constraint code [27]. In some cases, specific codes are
employed for the purpose of increasing the minimum distance of the channel trellis
[28]. Incorporating parity checks of a linear block code used in the system is an
another way to systematically prune the channel trellis. In fact, some simple codes
like single-parity check codes have been used to improve the detector performance
[29][30]. These codes are designed to facilitate a post-processor in identifying and
correcting detected sequences that contain some particular error events. But, since
these codes are employed in addition to an error correction code, the overall code
rate decreases and consequently increases coding loss. Also, such techniques can not
always be used along with turbo-equalizer, since they may hinder the soft-iterative
32
LDPCEncoder
h(D)Detector(BCJR)
MPdecoder
k1u n
1xmn+
1y mn+1r
( )2,0 σN
n1x
Turbo-Equalizer
Figure 3.1: Block diagram of a partial response system.
information exchange between the detector and the error correction decoder.
In this work, instead of designing new parity checks, we simply use some
parity checks of the LDPC error correction code used in the system. The addition
of LDPC checks does not necessarily increase the minimum distance of the trellis,
but may improve the a posteriori probability (APP) estimates of the detector. This
technique does not entail any reduction in the overall code rate. Addition of parity
checks in the trellis reduces the number of valid sequences or paths in the trellis,
thereby improving the performance of the detector. But, the effects on performance
of turbo-equalization itself is not obvious. It is difficult to precisely determine the
change in behavior of turbo-equalizer in the presence of these additional checks
in the detector. Nevertheless, by focusing on how the parity checks in the trellis
affects the sub-optimality of the turbo-equalizer, we provide an intuition on why
its performance may improve. Also, we discuss how it is related to the optimal
decoder.
33
3.1 Sub-Optimality of Turbo-Equalization
We first provide a clear understanding of the underlying cause for the sub-optimality
of the turbo-equalizer. The general magnetic recording system was discussed in
Chapter 2, but we discuss it very briefly again for clarity and completeness. Fig.
3.1 shows the binary magnetic recording system, where the channel response is
represented by a polynomial h(D) or the corresponding coefficient vector h. A
sequence u = {ui} of k binary symbols is encoded by an LDPC encoder in to
a codeword x = {xi} of n binary bits. The codeword is transmitted through the
ideal partial response (PR) channel, implemented by a finite impulse response filter,
whose non-binary output is corrupted by additive white Gaussian noise (AWGN).
The noiseless channel output, y = x ∗ h, is of length n + m where, ∗ represents the
convolution operator and m is the length of the channel memory. The noisy channel
output is given by r = y + n. The noisy channel output sequence is decoded by a
turbo-equalizer comprising of a detector and a message-passing decoder. Though,
optimal detector is efficiently implemented by the BCJR algorithm, soft-output
Viterbi algorithm (SOVA) is preferred for their low complexity.
For the ensuing discussion alone, assume that the detector is implemented
by BCJR and the LDPC code is such that its bipartite graph is a tree. There-
fore, both detector and the decoder are optimal. However, turbo-equalization is
34
still sub-optimal. In this scenario, it is interesting to investigate how the channel
memory affects the optimality of turbo-equalizer. For simplicity, we focus on the
first iteration of turbo-equalization where, the detector calculates P (xi/r) ∀ i =
1, . . . , n assuming independent and identically distributed (i.i.d) channel inputs.
This is used as a priori probabilities by the decoder in its attempt to calculate
P (xi/r, S) ∀ i = 1, . . . , n where, S is the event that all parity checks of the code
are satisfied. In order to analyze the decoding operation further, consider Fig. 3.2,
which shows the tree structure of the code with node xd as its root. During the first
iteration, the node operation at xd estimates P(
xd/r, S(0)d
)
, where S(i)d is the event
that all checks in the ith tier of the tree shown in Fig. 3.2 are satisfied. Assuming
a (j, k)-regular code, we know that [1],
P(
xd = 0/r, S(0)d
)
P(
xd = 1/r, S(0)d
) =1 − Pd
Pd
j∏
i=1
1 +∏k−1
l=1
(
1 − P(1)il
)
1 −∏k−1
l=1
(
1 − P(1)il
) (3.1)
where, Pd is the probability of bit xd conditional on the received output at position
d. Similarly, P(1)il is the probability of lth bit node connected to the ith check in the
1st tier of the tree conditional on the received output at the corresponding position.
These quantities are estimated by the detector and passed on to the decoder. It is
well-known that Eqn. 3.1 is valid only when for each check in the 1st tier, all k nodes
connected to them are independent of each other. In a partial response system with
channel memory length m and i.i.d inputs, the k bits connected to each check in
the 1st tier are independent if at least m bits occur between each other. During the
35
dx
1st tier check nodes
1st tier variable nodes
ith tier check nodes
ith tier variable nodes
Root node
Figure 3.2: Tree representing parity check constraints of a LDPC code.
second iteration, the node operation at xd estimates P(
xd/r, S(0)d , S
(1)d
)
, by using
P(
xil/r, S(1)d
)
instead of Pil in Eqn. 3.1. For this equation to be valid, the k nodes
connected to every check in both the 1st and 2nd tier should have at least m bits
between each other. Moreover, there should be at least m bits occurring between
node xd and all of the bits in the 2nd tier. Similarly, after n iterations of the
decoder, for the independence assumption to be valid, among other requirements,
there should be m bits occurring between node xd and all nodes in the nth tier or
lower. Consequently, the independence assumption is bound to fail, since eventually
nodes xd−m, . . . , xd−1, xd+1, . . . , xd+m will be included. Therefore, the decoder does
not converge to the true APP estimate for node xd. Hence, turbo-equalization
is sub-optimal even when the detector and the decoder are optimal. In the next
section, we will discuss the steps necessary to make turbo-equalizer optimal.
36
3.2 Optimal Decoding
Let us first assume that the goal of turbo-equalizer is to optimally decode only
bit xd. Further, let us also assume that the k bits connected to every check of
the code are independent of each other, i.e. their position in the channel input
sequence are such that they are separated by at least m bits between any two of
them. Now, the APP estimate of node xd is inaccurate only due to the presence
of nodes xd−m, . . . , xd−1, xd+1, . . . , xd+m in the code graph. For optimal decoding of
xd, these nodes should somehow be removed from the code graph for the decoding
operation, but at the same time the information they provide on node xd should
be taken into account in estimating its APP. This can be achieved by removing
all the checks attached to these nodes from the code graph and including them in
the channel trellis of the detector. By including the checks in the channel trellis,
we mean that the trellis is modified such that the sequence corresponding to any
path through the trellis will satisfy these checks. By our assumption, these checks
cannot be connected to xd and therefore node xd still remains as the root of the
tree. The detector operating on the modified trellis estimates the probability of
transmitted bits conditional on the received sequence and also on the event that
the checks included in the trellis are satisfied. When these quantities are used as a
priori probabilities by the decoder, which operates on the modified tree, the APP
37
estimate of xd will be optimal.
If the goal of the turbo-equalizer is to optimally decode all bits of the
codeword, then the above procedure needs to be carried out for all bits. It is
easy to see that this would require all checks of the code to be removed from
the code graph and included in the channel trellis. This leads to the well known
optimal decoder that operates on the trellis that combines both channel and code
constraints. Obviously, this is impractical even for small codes due to the size of
the code trellis.
In this work, we add only a small number of LDPC checks in the channel
trellis. Further, the checks are selected by random combinations of the checks from
the code graph (i.e. parity-check matrix) and no checks are removed from the code
graph. Random combination of checks will generally result in checks with high
weight and since, only a few checks are added to the channel trellis, this will ensure
that they influence the APP estimate for most of the bits in the codeword. These
checks are generally called auxiliary checks since, they are not used by the LDPC
message-passing decoder.
Serial detection algorithms like BCJR and SOVA efficiently operate on the
channel trellis. In this chapter, we first discuss the construction of channel trellis
when LDPC parity checks are incorporated and discuss the associated cost in terms
38
of complexity. Some simulation results are then presented for structured and ran-
dom codes that shows the performance of the detector operating on the modified
trellis and also the performance of the turbo-equalizer.
3.3 Trellis with Parity Checks
All possible states of a magnetic recording channel and all possible transitions be-
tween those states are represented by a state diagram. A state diagram D = (V,E)
is a graph consisting of vertex set V that represents the channel states and edge set
E that represents the state transitions. For a channel with memory m, the current
state of the channel is completely defined by the current input and m previous in-
puts. Since, in magnetic recording only two symbols can be transmitted, only 2m
states are possible, i.e. |V | = 2m. If u, v ∈ V , then an edge e ∈ E connecting u and
v, denoted by e : u → v, represents the channel state transition in consecutive time
instants by either a transmission of 0 or 1. An example of a state diagram is shown
in Fig. 3.3 for the PR4 channel with m = 2. The channel state at time instant
k is determined by the channel input at time k, xk and the channel state at time
k−1, which is represented as (xk−2, xk−1). Therefore, the channel state at time k is
represented as (xk−1xk). Since channel inputs are binary, the graph in Fig. 3.3 has
4 possible channel states with two incoming and two outgoing edges for each state.
The edge labels denote the channel input and the corresponding noiseless channel
39
00
10 01
11
1/1
1/10/-1
0/-1
1/0
0/0
1/0
0/0
Figure 3.3: State diagram of a PR4 channel.
output for the transition.
State diagrams can not only represent states and transitions for a channel
but can also represent states of an error control code. For example, consider the
state diagram for a single parity check code shown in Fig. 3.4. Here, the states
represent the parity of the sequence and the edges represent the change in parity of
the sequence when a 0 or 1 is appended to the sequence. Therefore, if the codeword
length is n then, all sequences of length n that begin in state 0 and end in state 0
constitute all possible codewords for this code. Such a state diagram can be defined
for any code. If the code has Nc independent parity checks, then the state diagram
can be defined by 2Nc states.
We’ll now consider representation of the channel when the input to the
channel are encoded by an LDPC code. The state diagram of such a channel can
40
0 101
10
Figure 3.4: State diagram of a single parity check code.
be defined as the cross-product of the channel state diagram and the code state
diagram. Therefore, if Dc = (Vc, Ec) and Ds = (Vs, Es) denote the state diagram
of channel and LDPC code respectively, then the states of the input constrained
channel are defined as {vivj : vi ∈ Vc, vj ∈ Vs} ∀ i, j. Further, an edge e connects
states (vivj) and (vkvl) if and only if e : vi → vj ∈ Ec and e : vk → vl ∈ Es.
The total number of states in the composite state diagram is |Vc| × |Vs| where, | · |
represents the order of the set operator and each state still has two incoming and
two outgoing edges. As an example, if the PR4 channel is encoded by a single parity
check code, then the composite state diagram is as shown in Fig. 3.5, with 4×2 = 8
states.
Trellis is another way of representing the channel states and the transitions
between them. It incorporates the concept of time in the state diagram. For
example, the trellis for the PR4 channel for three time instants is shown in Fig.
3.6(a) and the trellis for the single parity check is shown in Fig. 3.6(b). Sequences
corresponding to paths from state 0 in the first time instant to state 0 in the third
time instant are all possible codewords. Similar to the construction of the state
diagram, the trellis for the channel whose inputs are encoded by some LDPC code
41
1/1
1/10/-1
0/-1
1/0
0/0
1/0
0/0
110
011
010
001000
101
100
111
1/00/-1
0/-1
1/0
0/0
1/1
1/1
0/0
Figure 3.5: Composite state diagram of a single parity check code and the PR4channel.
can be determined by the cross-product of the channel trellis and the LDPC code
trellis. The cross-product operation is defined in the same way as before. Therefore,
the trellis for the single parity check coded PR4 channel is as shown in Fig. 3.7.
The trellis shows only valid paths of length 3. That is, it shows only edges starting
and ending in one of the 0 parity states.
Note that the number of states in the trellis shown in Fig. 3.7 is increased
by a factor of 2 compared to the uncoded channel trellis since, the code consists of
only one check. If an LDPC code with Nc parity checks is used, then the number
of states of the trellis increases by a factor of 2Nc . As Nc increases, the exponential
increase in the number of states makes it impractical to construct the trellis. As
mentioned in the last chapter, if all Nc checks are incorporated in the trellis, then
the detector operating on it is optimal. But, in practice, only a small number
42
00
10
01
11
00
10
01
11
0
1
01
-10
-1
0
00
10
01
11
00
10
01
11
0
1
01
-10
-1
0
00
10
01
11
00
10
01
11
0
1
01
-10
-1
0
(a)
0
1
0 0
1
0
1
0
11
0
00
111
(b)
Figure 3.6: Trellis of a: (a) PR4 channel; and, (b) trellis of a single parity checkcode.
of checks can be incorporated. In our work, we typically restrict the number of
checks incorporated in the trellis to 4. In the rest of the chapter, we discuss some
simulation performance of the turbo-equalizer when the detector operates over the
modified trellis.
43
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
000
001
010
011
100
101
110
111
Parity bit
Figure 3.7: Trellis for the single parity check coded PR4 channel constructed fromthe cross-product of PR4 channel trellis and the single parity check code trellis.
3.4 Simulation Results
3.4.1 Performance of Detector
Consider the system model for a LDPC coded partial response system shown in Fig.
3.1. We use soft-output Viterbi algorithm (SOVA) instead of BCJR as the detection
algorithm since, it has been shown to be a good trade-off between complexity and
performance [31]. Before determining the performance of the turbo-equalizer, we
44
determine by simulation the performance at the output of the detector, when some
LDPC parity checks are incorporated in the channel trellis.
SOVA operating on the channel trellis estimates APP’s of the input bits.
When some parity checks are included in the trellis, the APP estimates of SOVA
will be closer to the actual value. The inclusion of parity checks will not benefit
all bits of the codeword equally. In general, for a bit’s APP to improve, either
itself or some of its neighboring bits should be involved in one or more of the parity
checks included in the trellis. Usually, checks in the parity check matrix H of LDPC
codes have low-row weight compared to the codeword length and as a result are not
suitable for inclusion in the trellis. Therefore, the checks that are to be included in
the trellis are obtained by randomly combining two or more rows of the H-matrix.
If the column weight of the H-matrix of the code is odd, then adding (modulo 2)
all checks in the H-matrix will result in a single parity check involving all variable
nodes. Such a check has been used [29],[30] to improve performance of the detector,
since it aids in avoiding error events that are of odd length.
The single parity check involving all bit nodes is not particularly interesting
in our case. Our goal is to improve bit APP estimates of the detector and not to
target any specific error event. Absence of any particular error event at the output
of the detector need not necessarily improve the performance of the message-passing
decoder that follows the detector.
45
In order to benefit from the checks included in the trellis, the detector’s
decision delay should be equal to the slength of the codeword. Only when the
entire codeword sequence is available can it be determined whether they satisfy the
parity check or not. This increases the detector memory of the system. But, they
may not affect the overall decision delay of the turbo-equalizer, since the decoder
operates only after the APP estimates of the entire codeword is available at the
detector output.
For the LDPC code (273,191) constructed from projective geometry [32],
four random checks were constructed by randomly adding (modulo 2) checks from
the H matrix. The four checks obtained were each of weight 45. Increasing the
number of checks added in the trellis will obviously improve the performance, but
the number of checks added is limited to 4 as a trade-off between complexity and
performance. For the EEPR4 (1 2 0 − 2 − 1) channel, these four checks were
incorporated in the channel trellis and the error rate performance after the detector
in the first iteration of the turbo-equalizer was determined by simulation and is
shown in Fig. 3.8(a). At a bit error rate of about 10−5, the SNR gain is about 1 dB
compared to the performance of the detector when no parity checks are incorporated
in the trellis. This gain is not achieved by correcting any particular error event.
Table 3.1 shows the contribution of various error events, as a percentage of the total
error events detected, at 12 dB. It can be seen that in both cases, the distribution
46
of error events does not vary significantly, and are dominated by the occurrence of
minimum distance error event (±[+−+]). The same experiment was performed for
a random LDPC code (273,191) [33] by again constructing four random checks in
a similar way as before. The checks were of weight 34, 36, 38 and 40. The bit error
rate at the output of detector in the first iteration of turbo-equalization for EEPR4
channel is shown in Fig. 3.8(b). In this case too, there is a significant gain of about
0.5 dB at a bit error rate of 10−5.
Table 3.1: Error event distribution at 12 dB for PG(273,191)Error Events % contribution
Without checks in trellis With checks in trellis±[+ − +] 87.4 81.8±[+ − + − +] 3 0±[+ − +−] 2.4 4±[+] 1.2 7±[+0 + 0+] 1.2 0.6±[+0+] 0.6 2.9±[− + −00 − +−] 0.6 1.2
From our experiments, it was determined that performance for different sets
of randomly constructed checks were similar. These simulation results suggest that
by incorporating only a small number of checks in the channel trellis, significant
gain can be obtained at the detector output. But, this technique will only be useful
when the detector gain improves performance of the LDPC decoder, which is the
focus of the next section.
47
4 5 6 7 8 9 10 11 12 1310
−5
10−4
10−3
10−2
10−1
100
SNR(dB)
BE
R
PG (273,191) − Without parity checksPG (273,191) − With 4 parity checks
(a)
4 5 6 7 8 9 10 11 12 13 1410
−6
10−5
10−4
10−3
10−2
10−1
100
SNR(dB)
BE
R
Random (273,191) − Without parity checksRandom (273,191) − With 4 parity checks
(b)
Figure 3.8: Bit error rate performance of a (a) PG(273,191) LDPC code at theoutput of the detector; and, (b) a random (273,191) LDPC code at the output ofthe detector. The detector trellis contains four checks randomly chosen from thecode space.
48
3.4.2 Performance of Turbo-Equalizer
When determining the performance of turbo-equalizer, it is important to carefully
choose the number of iterations of the LDPC decoder (local) and the number of
times information is exchanged between the detector and the decoder (global) for
best performance. In general, increasing the number of global iterations will improve
the performance. For the PG(273,191) code, it was determined that the performance
improvement beyond 1 global iteration was insignificant. In this case, increasing the
number of local iterations will improve the performance, but we fix the number of
local iterations to 7. When checks are included in the trellis, the overall complexity
of the turbo-equalizer increases, but the performance is compared for the same
number of local and global iterations. For the EEPR4 channel, the performance
at the output of the LDPC decoder is shown in Fig. 3.9. The figure also includes
detector performance previously shown in Fig. 3.8(a). It can be seen that when
parity checks are not included in the channel trellis, the LDPC decoder suffers from
error floor. But, when they are included, the decoder is able to overcome the error
floor and as a result, at a frame error rate of 10−5, the SNR gain is 0.5 dB.
In the previous section, it was mentioned that the improvement in detector
performance cannot be attributed to avoiding any particular error event. Usually,
dominant error events at the detector output are of small length, therefore the
49
4 5 6 7 8 9 10 11 1210
−6
10−5
10−4
10−3
10−2
10−1
100
SNR(dB)
BE
R/F
ER
Without parity checks in detectorWith parity checks in detector
BER
After SOVA
FER
Figure 3.9: Bit and frame error rate performance of PG(273,191) code at the outputof LDPC decoder for the EEPR4 channel. Number of local iterations is 7 andnumber of global iterations is 1.
probability of being included in many checks among those incorporated in the trellis
are less. But, at the same time the Euclidean distance of such events are small and
therefore, if they are included in many checks, probability of correcting them is
high. Similarly, longer error events are more likely to be included in many checks
and are less likely to be corrected. But, since they usually have large Euclidean
distance, they occur rarely. However, multiple error events of small length are more
likely to occur in a codeword and are also more likely to be affected by the checks
included in the trellis. But, the effect of checks on the multiple error-event cannot
be easily determined. In order to understand this effect better, we study the weight
of errors in incorrectly detected codewords. Fig. 3.10(a) and Fig. 3.10(b) shows the
50
histogram of error weights at the output of SOVA for PG(273,191) code at 9 dB, for
both when checks are and are not included in the trellis respectively. In both cases,
over 300 codewords that were in error after detection were captured by simulation.
The histograms, in general, suggest that the checks incorporated in the trellis may
help in reducing the number of errors in the detected codewords. When the trellis
does not include any checks, the maximum error weight in a detected codeword is
13, whereas when checks are included, the maximum error weight is 9. It also has
significantly lesser number of detected codewords with 4 and 6 errors. Interestingly,
it results in significantly more number of dibit errors.
Reduction in error weight of codewords at the output of the detector will
aid the performance of LDPC decoder in two ways. Firstly, if the number of errors
in the sector received by the decoder is less, there is a better chance of correcting it
and secondly, the number of iterations required to correct a codeword may reduce
if it has lesser number of errors to begin with.
For the random (273,191) code, the performance of turbo-equalizer is shown
in Fig. 3.11. For this code, the number of local iterations is 3 and number of global
iterations is 2, beyond which the performance did not significantly improve. The
figure shows the error rate performance after every iteration. It was determined
in our simulations that the effect of checks in the trellis is insignificant after the
first global iteration. Therefore, in all our simulations, checks are included in the
51
1 2 3 4 5 6 7 8 9 10 11 12 130
50
100
150
200
250
Number of bit errors in a codeword
Num
ber
of ti
mes
occ
urre
d
(a)
1 2 3 4 5 6 7 8 9 10 11 12 130
50
100
150
200
250
Number of bit errors in a codeword
Num
ber
of ti
mes
occ
urre
d
(b)
Figure 3.10: Histogram of error weights in incorrectly detected codewords at theoutput of the detector: (a) When parity checks are not included in the trellis and(b) when parity checks are included in the trellis.
52
4 5 6 7 8 9 10 1110
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
SNR(dB)
BE
R/F
ER
Without parity checks in detectorWith parity checks in detector
FER
BER
Figure 3.11: Bit and frame error rate performance of random (273,191) code at theoutput of LDPC decoder for the EEPR4 channel after every global iteration of theturbo-equalizer. Number of local iterations is 3 and number of global iterations is2.
trellis only during the first time SOVA operates on the channel trellis. The figure
shows that the performance of turbo-equalizer with checks in trellis is better after
the first iteration. However, the performance are similar after the second iteration,
although the slope of the curves suggest that it will have some gain after the second
iteration at SNR’s higher than shown here. Another interesting feature from the
figure is that at some SNR, the performance of the turbo-equalizer with checks in
the trellis after one iteration will be same as the performance of turbo-equalizer
without checks in the trellis after two iterations. This would significantly reduce
the overall delay and complexity of the turbo-equalizer.
53
The phenomenon observed above is more prominent in Fig. 3.12(a) and
Fig. 3.12(b), which shows the bit and frame error rate for random code of length
495 and rate 0.87 for the PR1 channel. As before, four parity checks from the code
space are incorporated in the channel trellis. Number of local iterations is 5 and
the number of global iterations is 7. The figure shows the error rate performance
after every global iteration. Observe that the general characteristics are similar
as before. The gain after the first iteration is about 0.5 dB at a bit error rate
of 2 × 10−6, but decreases for subsequent iterations. Also, note that at 8.5 dB,
one and two global iterations of turbo-equalizer with checks in trellis results in the
same performance as two and four iterations of the turbo-equalizer without checks
in trellis, respectively. Eventually, the simulation suggests that the performance of
both these turbo-equalizers converge to the same error rates, but the checks in the
trellis helps in converging quickly.
From the various results shown in this chapter, it is apparent that adding
parity checks in the channel trellis is beneficial to the decoding process. The im-
provement is limited by the doubling of number of states for every check added in
the trellis. But, by careful selection of checks, it is possible to add more checks in the
trellis without increasing the complexity further. Determining their characteristics
and identifying such checks in the code space is the focus of the next chapter.
In this chapter, we study a technique that allows us to add more checks in
54
2 3 4 5 6 7 8 910
−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
SNR(dB)
BE
R
Without parity checks in detectorWith parity checks in detector
(a)
2 3 4 5 6 7 8 910
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
SNR(dB)
FE
R
Without parity checks in detectorWith parity checks in detector
(b)
Figure 3.12: Performance of random code (495,433), rate 0.87, with 4 randomparity checks in trellis for PR1 channel after every global iteration. Number oflocal iterations is 5 and global iterations is 7: (a) Bit error rate, and (b) Frame orcodeword error rate.
55
trellis with only a small increase in complexity. In this technique, a set of checks
called disjoint checks, when added to the channel trellis would require only one
additional bit for every state to track the parity for all checks in the set. We first
define the property of this check set and discuss an algorithm that identifies such
checks in the code space.
3.5 Disjoint Checks
If xi denote the ith bit node of an LDPC codeword of length n, then two checks,
cm = xi1 + xi2 + . . . + xip , i1 < i2 < . . . < ip
cn = xj1 + xj2 + . . . + xjq, j1 < j2 < . . . < jq,
(3.2)
are said to be disjoint if j1 > ip or i1 > jq. A set of checks is said to be disjoint
if all pairs of checks in the set are disjoint. Incorporating all checks of a disjoint
set in trellis would require the number of states to increase only by a factor of 2.
For instance, if both cm and cn are included in the trellis and if j1 > ip, then the
bit that represents the parity of check cm can be re-used after time instant ip to
represent parity of check cn. Therefore, only one additional bit is required to track
the parity of both checks in the trellis.
Though disjoint checks are useful for the purpose of incorporating in the
trellis, it is usually difficult to find such checks in the code space. A (n, k) code has
2n−k checks and an exhaustive search is improbable. But, we show that they can be
56
efficiently identified in parity check matrices that possess a special property, known
as minimal span. Any parity check matrix can be modified without altering the
code space, to satisfy this property. In the next section, the minimal span property
is defined and the search algorithm to determine disjoint checks is explained.
3.6 Search Algorithm
3.6.1 Minimal Span Matrices
Minimal span generator matrices (MSGM) have been well-studied and are known to
be useful in minimizing the complexity of the corresponding code trellis [34],[35],[36].
Some of those results, particularly those found in [36] are useful in finding disjoint
checks when applied to parity check matrices. We first begin with some definitions
and list certain important results associated with MSGM. Then, we discuss their
applicability to parity check matrices. For the following discussion, notations from
[36] are closely followed.
Let X = [x1, x2, . . . , xn] be a non-zero binary vector of length n. The left
index L(X) of vector X is defined as the smallest index i such that xi 6= 0 and
similarly, the right index R(X) is defined as the largest index i such that xi 6= 0.
The span of a vector X is defined as the set {L(X), L(X) + 1, . . . , R(X)} and the
span length is defined as the number of elements in the span set. Span length of a
57
matrix is defined as the sum of the span length of all rows of the matrix.
A set of binary vectors {X1, X2, . . . , Xk} is said to have the left-right (LR)
property if L(Xi) 6= L(Xj) and R(Xi) 6= R(Xj), whenever i 6= j, and is said to have
the predictable span property if,
span
(
∑
j∈J
Xj
)
=⋃
j∈J
span (Xj) , ∀ J ⊆ {1, 2 . . . , k}. (3.3)
We state two useful results without proof. Please refer to [36] and [35] for
proof and a detailed discussion.
Lemma 3.6.1. A set of non-zero binary vectors {X1, X2, . . . , Xk} has the pre-
dictable span property if and only if it has the LR property.
Theorem 3.6.2. A generator matrix G is an MSGM if and only if it has the LR
property. Any two MSGM’s have the same span set.
Therefore, there can be more than one MSGM for a code, but they all
have the same span length. The property that any generator matrix of a code can
be derived from some linear combination of rows of any other generator matrix,
enables proving the above result. The set of all possible parity check matrices that
can represent a code also possesses this property and therefore the above result can
also be stated as,
Theorem 3.6.3. A parity check matrix H is a minimal-span parity check matrix
58
(MSPM) if and only if it has the LR property. Any two MSPM’s have the same
span set.
We now state and prove the main result of this chapter.
Theorem 3.6.4. A (n, k) binary LDPC code C has disjoint checks among the 2n−k
checks of the code if and only if there are disjoint checks in the MSPM of the code.
Further, if one MSPM of the code has disjoint checks, then all MSPM’s of the code
will have disjoint checks.
Proof. If the MSPM of the code has disjoint checks, then obviously there are disjoint
parity checks in the code space. Now, suppose there are two checks cm and cn that
are disjoint and are not part of the MSPM. These checks can be formed from certain
linear combinations of rows from MSPM,
cm =∑
i∈I
hi , I ⊆ {1, 2, . . . , n − k}
cn =∑
j∈J
hj , J ⊆ {1, 2, . . . , n − k}.(3.4)
Where, hi and hj are the ith and jth row of MSPM H respectively and the sums in
the above equation are modulo 2. From Lemma 3.6.1 and Theorem 3.6.3, we know
that rows of MSPM has the predictable span property. Therefore,
span (cm) =⋃
i∈I
span (hi) ,
span (cn) =⋃
j∈J
span (hj) .
(3.5)
59
Consequently, span(hi) ⊆ span(cm), ∀ i ∈ I and span(hj) ⊆ span(cn), ∀ j ∈ J .
Since, cm and cn are disjoint, hi and hj are disjoint for any i ∈ I and j ∈ J .
Therefore, there exists disjoint checks in the MSPM. Further, since all MSPM’s
have the same span set, either all or none of the MSPM’s have disjoint checks.
3.6.2 Graphical Representation
One of the algorithms that provide an efficient method to convert any parity checks
matrix to an MSPM is described in [35], known simply as the greedy algorithm.
The pseudo-code of the algorithm is as follows:
1. While LR property of the matrix is not satisfied, do the following steps.
2. Choose two rows that does not satisfy LR property, and
3. Replace one of the rows with the sum (modulo 2) of the two rows. Go to step
1.
If the dimensions of the MSPM are small, then disjoint checks in the matrix
can be determined by a simple exhaustive search. But often, codes of practical
interest are much larger and an exhaustive search becomes inefficient. We describe
a simple algorithm that is guaranteed to determine the biggest disjoint set. In this
technique, the MSPM is first represented as a directed graph in which, every check
60
of the MSPM is represented by a node, known as the check node. It also consists of
a source and a destination node. So, if MSPM H is of dimension (n− k)× n, then
the graph consists of n−k +2 nodes. There exists a directed edge from node i to j,
if the corresponding checks in H are disjoint and if j > i. The edge is given a weight
of -1. Further, there exists a directed edge with weight -1 from the source node to
every check node and from every check node to the destination node. Therefore,
any path from the source to the destination node gives a list of checks in the MSPM
that are disjoint. Thus, the longest among those paths gives the disjoint set with
the largest number of checks. The longest path can be determined by applying the
Dijkstra’s algorithm, which actually determines the path with the smallest weight.
But, since edge weights are negative and the graph is acyclic, the minimum weight
path is the longest path in the graph.
As an example, consider the MSPM H shown below for a (15,8) LDPC
code. The corresponding graph for this matrix is as shown in Fig. 3.13. It is easy
to see that the longest path from source to destination is s→1→7→d, indicating
that check 1 and 7 are disjoint and this is the largest disjoint set of the code. In order
to determine more disjoint sets, the Dijkstra’s algorithm can be applied again to
the graph after removing the disjoint checks already identified and their associated
61
s
d
1 2 3 4 5 6 7
Figure 3.13: Graph representing the MSPM H
edges.
H =
0 0 0 0 0 0 0 1 1 0 1 0 0 0 1
0 0 0 0 0 0 1 1 0 1 0 0 0 1 0
0 0 0 0 0 1 1 0 1 0 0 0 1 0 0
0 0 0 0 1 1 0 1 0 0 0 1 0 0 0
0 0 0 1 1 0 1 0 0 0 1 0 0 0 0
0 0 1 1 0 1 0 0 0 1 0 0 0 0 0
1 1 0 1 1 0 0 0 0 0 0 0 0 0 0
3.7 Simulation Results
Existence of disjoint checks is difficult to determine from the code characteristics.
However, they are more probable in high-column weight codes and structured codes.
In general, it has been noticed that disjoint checks do not exist in good random
62
codes.
For the high-column weight PG(273,191) code, MSPM was determined us-
ing the greedy algorithm. By applying the search algorithm described above, two
sets of disjoint checks, one with 9 checks and the other with 8 checks, were de-
termined. Fig. 3.14 shows the two check sets, where the black area indicates the
variable nodes included in the checks. Note the symmetry in the check connections
to the variable nodes. The 17 checks from the two disjoint sets can be included in
the channel trellis with only a 4 times increase in the number of states. In compar-
ison, including 17 randomly chosen checks will require an increase in the number of
states by 217. However, it should be noted that including randomly chosen checks
may be more beneficial than including disjoint checks, since it allows some flexibil-
ity in choosing certain check characteristics, like the weight of checks. The lack of
flexibility in choosing check characteristics when including disjoint checks is a dis-
advantage of this technique. Note that for the (273,191) code, all checks comprising
the two disjoint sets are of low weight. This reduces their influence on the decoding
process. However, the combined benefit of including more number of checks in the
trellis provides a net gain as shown in Fig. 3.15. The figure shows three bit error and
frame error performance curves for the PR1 channel - 1) corresponding to the usual
turbo-equalizer, 2) when four randomly chosen checks are included in the trellis and
3) when two of those checks are replaced by the two disjoint check sets. Similar to
63
Variable nodes
Che
cks
50 100 150 200 250
2
4
6
8
10
12
14
16
Set 2
Set 1
Figure 3.14: Two sets of disjoint checks of the PG(273,191) LDPC code.
the results for the EEPR4 channel shown in the last chapter, parity checks in the
trellis helps the decoder avoid the error floor. Without change in complexity of the
trellis, two of the four random checks is replaced with the two sets of disjoint checks
providing an additional gain of 0.5 dB.
3.8 Summary
A method to improve the performance of turbo-equalizer for LDPC coded partial
response system was discussed. Particularly, the benefits of incorporating 4 or less
parity checks of the LDPC code in the channel trellis was evaluated. Bit error and
frame error rates for various LDPC codes were determined by simulation. It was
64
1 2 3 4 5 6 7 8 9 10 1110
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
SNR(dB)
BE
R/F
ER
Without parity checks in detectorWith random parity checks in detectorWith disjoint parity checks in detector
FER
BER
Figure 3.15: Bit error and frame error performance of the PG(273,191) LDPC codeover the PR1 channel.
observed that this method significantly improved the detector performance and con-
sequently, the performance of the turbo-equalizer, in some cases even lowering error
floors. Although, adding checks to a trellis increases the complexity exponentially,
this scheme is practical since the checks are included in the trellis only during the
first global iteration. Also, this method may require less number of global iterations
of the turbo-equalizer to achieve the same performance, thus reducing the overall
latency. Unlike other methods, this method requires only a simple change in the
existing turbo-equalizer architecture.
The improvement to the performance of the turbo-equalizer is limited by
65
the small number of checks that can be added to the trellis until the complexity
becomes too high. But, in some cases it was shown that it is possible to add more
checks, if they are disjoint. An efficient and simple algorithm has been presented
to find such checks in a code. This algorithm was used to find two sets of disjoint
checks in the PG(273,191) code. Incorporating these checks in the trellis resulted
in an additional gain of up to 0.5 dB compared to the case when all checks included
in the trellis were randomly chosen.
The inherent complexity in the turbo-equalizer operation makes it difficult
to analyze the effects of checks added in the trellis. In other words, it is difficult
to identify checks that may be more beneficial to the turbo-equalizer than other
checks. Future efforts to improve this method should focus on this issue. Some
innovation needs to be incorporated in selecting checks to be included in the trellis.
One possible area of focus could be in the error floor region. If error patterns
responsible for error floor of a code is known, then in that region, the checks included
in the trellis may be selected so as to avoid the occurrence of those particular error
patterns.
66
CHAPTER 4
JOINT MESSAGE-PASSING SYMBOL DECODER
The importance of turbo-equalizer to the current and future storage and wireless
systems cannot be over-emphasized. As mentioned in the last chapter, they are
the best and most widely used decoder today, and inventing methods to improve
their performance is an important problem. In the previous chapter, we presented a
simple but effective method to improve the performance of the turbo-equalizer. We
also made the general observation that the performance of LDPC coded systems
with memory can be improved beyond what is achieved by turbo-equalizer, if both
channel and the code information are simultaneously used to make decisions on the
channel inputs. This is referred to as joint detection and decoding or simply as joint
decoding, and is the focus of this chapter. It has also been the focus of some recent
works in [12], [13], [11] and [14]. A practical and popular approach in this direction
has been to design message-passing (MP) decoding algorithms that operate on a
graph, which represents the constraints imposed by both the channel and the LDPC
code.
67
In turbo-equalizer, the channel constraints are represented by the channel
trellis and the code constraints are represented by the bipartite graph of the code,
known as the Tanner graph. Detectors like SOVA and BCJR operate serially on
the trellis, while the LDPC decoder operates parallely on the Tanner graph. While
aiming to conceive a joint MP decoding algorithm operating on a graph, it is logical
to first consider the problem of representing the channel constraints as a graph and
then to design a parallel MP detection algorithm that operates on this graph. This
problem has been addressed by Kurkoski et. al. in [12], who introduced parallel
bit-based and state-based MP algorithms for channel detection. The bit-based MP
algorithm is useful only in channels with unit memory length, but has been modified
for use in channels with longer memory by Colavolpe et. al. [13]. Even though in
[12], the state-based MP detector was combined with the LDPC MP decoder to
obtain a joint MP decoder, it achieved at best the same performance as the turbo-
equalizer. This is due to the fact that the joint MP decoder is simply a parallel
schedule of the turbo-equalizer.
The challenge in jointly using both the channel and code information arises
from the fact that the channel imposes constraints on the channel output sequences,
whereas the code imposes constraints on the channel input sequences. The idea
motivating our approach is the observation that by imposing constraints on the
channel input sequences, the code also imposes certain constraints on the noiseless
68
channel output sequences. In this chapter, we modify the LDPC MP decoder to
produce information on the noiseless channel output symbols rather than on channel
inputs. This enables us to combine it with a modified version of the state-based MP
detector to design a joint decoder, that significantly surpasses the performance of
the turbo-equalizer. The joint decoder estimates a posteriori probabilities (APPs)
of channel output symbols, from which APPs of channel inputs are derived. Also,
as it will be shown, this algorithm can be used irrespective of the channel memory
length, although as described in Section 4.2, it may perform relatively better for
channels with small memory.
The rest of the chapter is organized as follows. We describe a graphical
model used to represent the channel and describe an optimal MP detection algo-
rithm operating on this graph in Section 4.1. This model is extended to include
code constraints and a joint MP decoding algorithm that operates on this combined
graph is described in Section 4.2. Bit error simulated performance for an LDPC
code is shown in Section 4.3 and finally, the chapter is concluded in Section 4.4.
4.1 Message-Passing Detection Algorithm
Fig. 4.1 shows the system model considered, where the channel response is repre-
sented by a polynomial h(D) or the corresponding coefficient vector h. A sequence
69
LDPCEncoder
h(D)Detector(BCJR)
MPdecoder
Jointdecoder
k1u n
1xmn+
1y mn+1r
z
n1x
n1x
Turbo-Equalizer
Figure 4.1: Block diagram of a PR system. Decoder is either turbo-equalizer (upperbranch) or joint decoder (lower branch) and noise is modeled as additive whiteGaussian.
u of k binary bits is encoded by an LDPC code into a codeword x of n binary
bits. The codeword is transmitted through the PR channel, whose non-binary out-
put is corrupted by additive white Gaussian noise (AWGN). The noiseless channel
output, y = x ∗ h, is of length n + m, where ∗ denotes the convolution operator
and m denotes the channel memory length. The noisy channel output is given by
r = y + z. We first consider an uncoded system, where the optimal detector is the
one that estimates MAP probability of the bits p (xi|r), ∀ i = 0, 1, . . . , k − 1, where
xi is the ith element of the vector x and xi ∈ {0, 1}. These quantities are efficiently
determined by operating the BCJR algorithm on the channel trellis.
4.1.1 Channel Graph
The trellis of a channel represents the constraints imposed on the range of noiseless
channel output sequences. A channel trellis can be given as a factor graph [8] shown
in Fig. 4.2, where q0, q1, . . . , qn+1 represents state (hidden) variables, x0, x1, . . . , xn
70
represents channel inputs and y0, y1, . . . , yn represents noiseless channel outputs.
The factor graph can be divided into n sections, where the ith section denoted by
Ti is defined by all valid triples {qi, yi, qi+1}. Therefore, each section acts as a local
constraint of the channel. Consequently, a sequence of state and channel output
variables {q0, q1, . . . , qn+1, y0, y1, . . . , yn} is valid if and only if it satisfies all local
constraints T0, T1, . . . , Tn.
When a global channel constraint is factored into local channel constraints,
numerous scheduling schemes for implementing the BCJR algorithm are possible.
Typically, one instance of a BCJR algorithm is operated on one section of the trellis
at any time instant and is progressively moved to other sections. This scheduling is
known as fully-serial [8]. On the other extreme, n instances of the BCJR algorithm
can operate on each section of the trellis simultaneously, exchanging information
through the state variables during every iteration. This scheduling is known as
fully-parallel, and is referred to as the parallel state-based MP algorithm in [12].
Naturally, intermediate scheduling schemes are possible. For example, the factor
graph shown in Fig. 4.2 can be divided into p sections (p < n), and during every
iteration, p instances of the BCJR algorithm can operate on each of these sections,
exchanging information with the adjacent ones, but within each section, the BCJR
can operate serially.
In this chapter, we consider the fully-parallel MP scheduling, but modify
71
q0 qi+1
y0 y1 yi
x0 x1 xi
q1 q2 qi
iT
Figure 4.2: Generalized factor graph representation of a PR channel trellis.
the factor graph representing the channel constraints to remove the need for state
variables and correspondingly alter the detection algorithm. This algorithm is es-
sentially the same as the parallel state-based MP algorithm and with some efficient
computation it can be shown that it also has the same complexity. But, we prefer
this version since it allows us to treat both the channel constraint and the LDPC
constraint in the same way. The modified factor graph is now simply represented
as a bipartite graph G shown in Fig. 4.3, where circles correspond to the noiseless
channel symbols yi, and squares correspond to the local channel constraints. These
sets of nodes are referred to as symbol nodes and channel nodes respectively. Every
channel node represents two sections of the trellis. For example, a channel node
si acts as a local constraint and represents all valid 5-tuples {qi, yi, qi+1, yi+1, qi+2}.
Therefore, unlike the factor graph in Fig. 4.3, every section of the trellis is repre-
sented by two channel nodes in this graph. As will be described in the next section,
information pertaining to state transition probabilities is exchanged between sym-
bol nodes through the channel nodes. Like the factor graph of the trellis, graph G
72
s0 s1 s2 s3 sn+m-3 sn+m-2
Symbol nodes
Channel nodes
0y 1y 2y 3y 2−+mny 1−+mny
Figure 4.3: A graph that represents constraints imposed by the channel on thenoiseless channel output sequences.
is a generic cycle-free representation of a PR channel, irrespective of its memory
length. However, the size of the set represented by the channel nodes increases
exponentially with increase in memory length. If memory length is m, the size of
this set is 2(m+2).
4.1.2 Message-Passing Symbol Detector
Now, we describe a MP detection algorithm that operates on graph G and produces
APPs of output symbols, from which APPs of channel inputs are derived. If xk and
yk are the channel input and noiseless output at time k, then
yk = f (xk, xk−1, . . . , xk−m) , (4.1)
where the function f() is determined by the channel response h(D). Let A =
{a0, a1, . . . , a2m−1} be the set of possible noiseless channel output symbols when the
73
current channel input xk = 0, and let B = {b0, b1, . . . , b2m−1} be the set of possible
noiseless channel output symbols when the channel input xk = 1. Then, APPs of
the channel input xk are given as,
p (xk = 0|r) =2m−1∑
i=0
p(yk = ai|r). (4.2)
Though elements of set A and B may not be unique, we emphasize that they
correspond to a unique state transition in the corresponding channel trellis. Since
the MP detection algorithm is not operated explicitly on the trellis, we simply refer
the quantities p(yi) as symbol probabilities rather than state transition probabilities.
Let ϕi denote the set of symbol pairs represented by channel node si of the
graph shown in Fig. 4.3. Therefore, (yi, yi+1) ∈ ϕi, ∀ i. The constraints imposed on
the output sequences by the channel can be viewed as a code, where each channel
node si corresponds to a code of length 2 and the set ϕi corresponds to its set of
codewords. With the knowledge of ϕi for all channel nodes, the output sequence can
thus be decoded without the use of state variables. Since the underlying channel
graph G is a tree, APPs can be estimated optimally using the SPA. Now, the MP
detection algorithm is given as follows.
MP Detection Algorithm:
1. Initialization: Since, channel inputs are i.i.d, all state transitions are initially
equally likely. Therefore, p(yi) = 1/2m+1, ∀ i = 0, 1, . . . , n + m − 1.
74
2. Message from symbol nodes to channel nodes during the tth iteration:
M (t)yk→sk−1
= M (t−1)sk→yk
M (t)yk→sk
= M (t−1)sk−1→yk
, ∀ k = 1, . . . , n + m − 2
(4.3)
where, M(t)yk→sk−1
denotes the message sent from symbol node yk to channel
node sk−1 during the tth iteration. Other terms are defined similarly.
3. Message from channel nodes to symbol nodes during the tth iteration:
M (t)sk→yk
=p(yk|rk, rk+1, ϕk)
p(rk|yk)p(yk), ∀ yk
M (t)sk→yk+1
=p(yk+1|rk, rk+1, ϕk)
p(rk+1|yk+1)p(yk+1), ∀ yk+1.
(4.4)
Messages received from the symbol nodes during the current iteration serve
as the a priori symbol probabilities for the channel node operation.
4. APPs of channel output symbols: After repeating the above steps for a fixed
number of iterations, APPs of channel output symbols are calculated as,
M (t)yk
= M (t)sk→yk
· M (t)sk−1→yk
· p(rk|yk) · p(yk) (4.5)
where, p(yk) is the initial a priori probability.
5. APPs of channel input bits: The algorithm halts after calculating channel
input APPs using Eqn. 4.2.
Remark : p(yk|rk, rk+1, ϕk) and p(yk+1|rk, rk+1, ϕk) in Eqn. 4.4 can be com-
puted in a straightforward way, since |ϕk| ≤ 2m+2, and m is small. In the context
75
of a trellis, this is same as operating BCJR only on the two sections of the trellis
represented by the channel node sk.
When the number of iterations equal the length of the sequence, the APPs
obtained using the above algorithm are same as that obtained from the BCJR algo-
rithm. Usually, only a small number of iterations are required to obtain performance
close to optimal. However, as observed in [12], all schedulings of the BCJR algo-
rithm, except the fully-serial scheduling, result in an error floor if enough iterations
are not run. Like other MP detection algorithms proposed in the literature, this
algorithm is more complex than BCJR, primarily due to the parallel scheduling of
the algorithm, but it can potentially reduce latency time and is suitable for high-
speed applications. Its most important advantage is that it can be combined with
a LDPC MP decoder to obtain a joint MP decoder.
4.2 Joint Message-Passing Symbol-Decoding Algorithm
In this section, we extend the graphical model described earlier to include con-
straints imposed by the parity checks of the LDPC code. The tripartite graph
shown in Fig. 4.4 is obtained by including the parity check nodes to the channel
graph of Fig. 4.3. Connections between the parity check nodes and the symbol
nodes are defined in the same way as the connections between the parity check
76
s0 s1 s2 s3 sn+m-3 sn+m-2
Check nodes
Channel nodes
c0 c1 c2 c3 cq
0y1y 2y 3y 1−+mny
Figure 4.4: A graph that represents constraints imposed by the channel and theparity checks of the LDPC code on the noiseless channel output sequences. Paritychecks imposes certain constraints on the channel output sequences by imposingconstraints on the channel input sequences.
nodes and the variable nodes. Using this combined graph, a joint decoding algo-
rithm is developed that estimates the symbol APPs using both the channel and the
code information simultaneously. The joint decoding algorithm is outlined below.
Joint MP Decoding Algorithm:
1. Initialization: Symbol a priori probabilities p(yi) = 1/2m+1.
2. Message from symbol nodes to channel nodes during the tth iteration: If H =
77
{hij} is the parity check matrix of the LDPC code, then for every k, compute,
M (t)yk→sk−1
= M (t−1)sk→yk
·∏
j|hjk=1
M (t−1)cj→yk
M (t)yk→sk
= M (t−1)sk−1→yk
·∏
j|hjk=1
M (t−1)cj→yk
.
(4.6)
3. Message from symbol nodes to check nodes during the tth iteration: For every
i and k such that hik = 1, compute,
M (t)yk→ci
= M (t−1)sk→yk
· M (t−1)sk−1→yk
·∏
j|hjk=1,j 6=i
M (t−1)cj→yk
. (4.7)
4. Message from channel nodes to symbol nodes during the tth iteration: This is
same as Eqn. 4.4.
5. Message from check nodes to symbol nodes during the tth iteration: For every
i and k such that hik = 1, compute,
M (t)ci→yk
=p(yk|ri, Ci)
p(rk|yk)p(yk), ∀ yk (4.8)
where, Ci denotes the event that the check ci is satisfied and ri denotes the set
of received samples at the locations of the variable nodes connected to check
ci.
6. APPs of channel symbols: For every k, compute,
M (t)yk
=M (t)sk→yk
· M (t)sk−1→yk
·∏
j|hjk=1
M (t)cj→yk
·
p(rk|yk) · p(yk)
(4.9)
where, p(yk) is the initial a priori probability.
78
0 0
1
0
1
00
11
1
1
0
0
0ix jx kx
Figure 4.5: Trellis diagram of a single parity check code of length 3. The statesindicate the parity of the incoming sequences.
7. APPs of channel input bits: After every iteration, the channel input APPs
are calculated using Eqn. 4.2. All the above steps are repeated until either
the decoded sequence satisfies all channel and code constraints or a preset
maximum number of iterations is reached.
Remark : Observe that the check node operation of the LDPC MP decoder is
modified to provide symbol information for use in joint decoding. In the traditional
decoder, the check node operation is efficiently computed by the tanh function,
whereas the new check node operation is more complex. We describe next the best
method to compute this new operation.
For ease of exposition, consider a degree 3 check node cl = xi ⊕ xj ⊕ xk.
The check node cl implies that the three variable nodes form a single parity check
code, denoted by CX . This is compactly represented by the code trellis shown in
79
Fig. 4.5. All paths beginning and ending at state 0 correspond to codewords of
the code CX . Input bit APPs conditioned on the event Cl can be determined by
operating the BCJR algorithm on this trellis, which simply turns out be the tanh
check node operation.
The check node cl is connected to symbol nodes yi, yj and yk in the combined
graph. By imposing constraints on the variable nodes xi, xj and xk, the check cl
also imposes certain constraints on the corresponding output symbols. In other
words, the check node cl implies that the three symbol nodes also form a code,
which we denote by CY . In order to obtain symbol APPs, a trellis for the code
CY , referred to as the expanded code trellis, is constructed by expanding the edges
of the code trellis shown in Fig. 3.6(b). The expanded code trellis is shown in
Fig. 4.6. An edge representing a state transition in code trellis is now replaced
by 2m edges representing all possible noiseless channel outputs generated during
the corresponding state transition. For example, if xi = 0 is transmitted, the
corresponding noiseless channel output yi ∈ A. Further, the edge labels are elements
of set A or B depending on whether the corresponding edge label in code trellis is
0 or 1. If the three symbol nodes are independent, i.e. they have at least m other
symbol nodes between them, then all paths beginning and ending at state 0 of
the expanded code trellis correspond to codewords of the code CY . Therefore, the
symbol APPs of yi, yj and yk conditioned on the event Cl is determined by operating
80
0 0
1
0
1
0
iy jy kyA A A
A
BB
BB
Figure 4.6: Expanded code trellis of Fig. 3.6(b). The state transition edge labelsare either elements of set A or B.
the BCJR algorithm on the expanded code trellis.
If the symbol nodes connected to a check node are not independent, the
above method can still be used for an approximate calculation of Eqn. 4.8. In
principle, the joint symbol decoder can be applied for channels with any memory
length, although the number of check nodes that violates the independence property
will be lower for channels with small memory.
The biggest drawback of the joint decoder is perhaps the large increase in
complexity of the check node operation. If wr is the row weight of the code, then
every check node operation approximately requires memory to store (1 + 2m−1)4wr
numbers and requires (1 + 5 · 2m−2)8wr multiplications and (1 + 2m)4wr additions.
81
4.3 Simulation Results
We illustrate the performance of the joint symbol-decoding algorithm by simulating
an LDPC coded PR system, where the channel is given by the impulse response
[1, 0,−1] (PR4), and the LDPC code is of length 1908 and rate 0.89 [33]. The
channel output sequences are decoded by using both turbo-equalizer and the joint
symbol-decoder. When turbo-equalizer is used, the number of global iterations is
restricted to 5 and the number of internal LDPC decoder iterations is restricted to
3. These settings give the best bit error rate (BER) performance. Increasing the
number of global iterations beyond 5 does not improve the performance significantly.
When the joint symbol-decoder is used, the number of iterations is restricted to 60.
The performance comparison is shown in Fig. 4.7. At a signal-to-noise ratio (SNR)
of 5.4 dB the BER obtained by the joint decoder is almost an order of magnitude
better than the turbo-equalizer. Also, the figure suggests that the gain increases
with increasing SNR.
Among the 212 parity checks of this code, 100 parity checks contain at least
one pair of variable nodes (or symbol nodes) that are not independent. Overall,
apart from the cycles in the bipartite code graph, this resulted in 134 four-cycles
and 136 six-cycles in the combined graph. However, the joint decoder was applied
to the code without any modification, implying that the computation of Eqn. 4.8
82
3.5 4 4.5 5 5.5 610
−6
10−5
10−4
10−3
10−2
10−1
SNR(dB)
BE
R
Turbo−equalizer (BER after every iteration)Joint decoder
Figure 4.7: Bit error rate comparison of (1908,212) random LDPC code on a PR4channel when decoded using turbo-equalizer and the joint decoder.
was approximate. In spite of this, the decoder was able to achieve significant gain
over the turbo-equalizer.
To make the computation of Eqn. 4.8 exact, it is necessary to ensure that
all parity checks satisfy the independence assumption. If a given LDPC code does
not satisfy this assumption, then the columns of their parity check matrix can be
suitably permuted to satisfy this assumption. Permutation of checks preserves the
code structure and their properties. Although, existence of such a permutation
cannot be guaranteed, it is usually possible to find such a permutation by trial
83
3.5 4 4.5 5 5.5 610
−6
10−5
10−4
10−3
10−2
10−1
SNR(dB)
BE
R
Turbo−equalizer (BER after every iteration)Joint decoderJoint decoder, no 4−cycles
Figure 4.8: Performance of column-permuted (1908,212) random LDPC code thatdoes not have any four or six-cycles in the combined graph (excluding cycles in thebipartite code graph).
and error for channels with small memory. For the LDPC code discussed above
and for the PR4 channel, such a permutation exists that removes all four and six-
cycles introduced in the tripartite graph (excluding cycles in the bipartite code
graph). The performance obtained by the joint decoder for this modified code is
shown in Fig. 4.8. It is interesting to note that there is no significant change in its
performance.
Fig. 4.8 suggests that the small cycles introduced in the tripartite graph
84
does not have a detrimental effect. However, as in the case of AWGN channel, we
believe small cycles will have an effect in the error floor region. Consider the random
LDPC code of length 273 and rate 0.7 [33]. There are 22 four-cycles and 20 six-cycles
in the tripartite graph, excluding cycles in the code graph. The performance of this
code on the [1, 1,−1,−1] (EPR4) channel for both the turbo-equalizer and joint
decoder is shown in Fig. 4.9. The turbo-equalizer is run for 10 internal iterations
and 4 global iterations, whereas the joint decoder is run for 40 iterations. Clearly,
both bit error and frame error performance of joint decoder suffers from error floor.
Two suitable permutations were obtained for the (273,82) random LDPC
code, that removes the small cycles. The first permutation removes all four-cycles
of this code, but increases the number of six-cycles to 26. The second permutation
removes all four and six-cycles. Performance for both these column-permuted codes
is shown in Fig. 4.10. Removal of just the four-cycles does not improve the perfor-
mance. However, removal of both the four and six-cycles from the code results in
the decoder overcoming the error floor. As a result, there is about 1 dB gain at a
BER of 10−5. In this case too, it is evident that the gain increases with increase in
SNR.
85
3 3.5 4 4.5 5 5.5 6 6.5 710
−6
10−5
10−4
10−3
10−2
10−1
100
SNR (dB)
BE
R/F
ER
FER BER
Turbo−Equalization: 4 Global iterations, 10 MP iterations
Joint Decoding:40 Iterations
Turbo−equalizer (BER)Turbo−equalizer (FER)Joint decoder (BER)Joint decoder (FER)
Figure 4.9: Performance of (273,82) random LDPC code on a EPR4 channel whendecoded using turbo-equalizer and joint decoder.
4.4 Summary
The problem of joint detection and decoding of LDPC coded signals over partial
response channels is considered. In order to jointly use both the channel and code in-
formation, the LDPC decoder is modified to produce information on channel output
symbols rather than on channel inputs. This is combined with a message-passing
detector to develop a joint decoder that estimates channel input APPs by first es-
timating channel output symbol APPs. The performance of this decoder is shown
86
3 3.5 4 4.5 5 5.5 6 6.5 710
−6
10−5
10−4
10−3
10−2
10−1
100
SNR (dB)
BE
R/F
ER
TE (BER)TE (FER)JD (BER), no permutationJD (FER), no permutationJD (BER), no 4−cyclesJD (FER), no 4−cyclesJD (BER), no 6−cyclesJD (FER), no 6−cycles
Figure 4.10: Performance of column-permuted (273,82) random LDPC code thatdoes not have any four or six-cycles in the combined graph (excluding cycles in thebipartite code graph).
to significantly outperform that of the turbo-equalizer for random LDPC codes on
channels with memory 1 and more. At the time of writing this dissertation, this
remained the only decoder to provide significant improvement over turbo-equalizer
for LDPC coded ideal partial response systems (memory more than 1) with AWGN.
The small cycles introduced in the combined graph involving both channel
and code constraints is shown to result in high error floor region. But, it is shown
that by some simple column permutation of the parity check matrix of the code,
87
these cycles can be removed and consequently error floor can be lowered.
88
CHAPTER 5
TERNARY MAGNETIC RECORDING SYSTEM
In Chapter 1, many novel magnetic recording architectures that have been proposed
to increase areal densities to 1 Tb/in2 and beyond were briefly discussed. These
architectures are radically different from the conventional systems. However, all
magnetic recording systems have always been analyzed as a binary inter-symbol
interference (ISI) communication system. Binary symbols are represented by the
presence or absence of a magnetic transition. While designing a magnetic recording
system, it is desirable to minimize the width of the channel transition response.
It is well-known that the gradient of a magnetic transition determines the width
of its corresponding readback response. Sharper the transition, smaller the width
of the response. Once the system parameters are optimized and fixed to obtain
the sharpest magnetic transitions, the channel response is fixed and thus can be
analyzed as any other binary communication system. Appropriate signal processing
algorithms can be designed for this channel in order to achieve the required error
rate performance.
89
In this chapter, a novel magnetic recording system is introduced and studied
from a read channel perspective. Specifically, this work explores the advantages of
a magnetic recording channel that is capable of storing three symbols in a unit area
of the medium; absence of a transition and the presence of a transition with one of
two predetermined gradients. Transitions with different gradients lead to different
readback responses, resulting in the need to develop appropriate receivers, equalizers
and detectors for this channel. Such a system has some natural advantages over
the binary system. For the same writable area and for storing the same amount
of information, the symbol area in ternary system will be higher than the bit area
in binary system. Consequently, it reduces ISI and inter-track interference (ITI).
However, the use of the third symbol reduces the signal minimum distance of the
ternary system. In general, ternary systems will be useful whenever the gain due to
higher symbol area overcomes the loss due to lower minimum distance. We show
that, under certain conditions, one such case is systems with high radial density.
Depending on the magnetic recording technology, there are different ways
the transition gradient can be varied. For example, in conventional systems, the
gradient of the transition can be varied by modulating the write current amplitude
and in heat-assisted magnetic recording (HAMR) [37], by modulating the tempera-
ture to which the medium is heated. There are many aspects of magnetic recording
90
system that should be considered before feasibility and benefits of the ternary sys-
tem can be conclusively determined. Our work provides a detailed analysis from
the signal processing perspective. The rest of the chapter is organized as follows. In
Section 5.1, we describe longitudinal ternary recording system and derive optimal
receivers, generalized partial response (GPR) targets and equalizers for this system
in Section 5.2 and 5.3. We compare the performance of this system with high-radial
density binary systems in Section 5.4 and derive some conclusions in Section 5.6.
5.1 Channel Model
The response of a longitudinal magnetic recording system to an isolated transition
is given by the Lorentzian function,
h(t) =V
1 + (2t/w)2 (5.1)
where, V is the peak voltage and w is the width of the response at half the peak
voltage (PW50). In this work, all symbols for both binary and ternary systems are
represented in the NRZI format. If a = {ak} denote the binary input sequence,
then output r(t) is given as,
r(t) = a ∗ h(t) (5.2)
=∑
k
akh(t − kT ) + z(t) (5.3)
91
where, z(t) denotes additive white Gaussian noise (AWGN) in the channel.
The three possible input symbols to the ternary channel, shown in Fig. 5.1
are represented by 0,±1 and ±2 where, 1 and 2 both represent transitions but with
different gradients. The response to these two transitions are both characterized by
Lorentzian functions, but with different widths and peak voltages,
h(t, 1) =V1
1 + (2t/w1)2
h(t, 2) =V2
1 + (2t/w2)2 .
(5.4)
Therefore, the channel response for a ternary system is dependent on the input
symbol and consequently, the output r(t) can be expressed as,
r(t) =∑
k
akh(t − kT, ak) + z(t). (5.5)
The relationship between the two responses is given as,
V2 =
(
w1
w2
)
V1. (5.6)
If T is the time period of the system, then the normalized density is calculated as
w/T and therefore, the relationship between the two responses can also be expressed
as,
V2 =
(
ND1
ND2
)
V1 (5.7)
where, ND1 and ND2 are the normalized densities for symbols 1 and 2 respectively.
92
{ }2,1,0 ±±=k
a),( kath
0N
+
Figure 5.1: Ternary channel: the channel transition response is dependent on theinput symbol.
A sequence of binary information can be converted to an equivalent ternary
sequence of smaller length using some binary-to-ternary mapping. Consequently,
for the same writable area, the symbol area of the ternary system will be higher than
that of the bit area in binary system that stores the same amount of information.
For systems with high linear density, larger symbol area results in a smaller ISI
and for systems with high radial density, larger symbol area results in a smaller
ITI. These are the two main motivations for considering such a system. However,
since the two transition responses are highly correlated, the minimum distance of
the ternary system will be lower. In general, ternary systems will be useful over the
corresponding binary systems, whenever the benefits of lower interference overcomes
the loss due to lower minimum distance. Before such a case can be discussed, it is
necessary to first develop appropriate read channel signal processing algorithms for
the ternary system.
93
T
T
'ka
''ka
)1,(th )1,( th −
)2,(th )2,( th −
'ky
''ky
0N
+
Figure 5.2: Optimal receiver for the ternary magnetic recording channel.
5.2 Optimal Receiver
In this section, the optimal receiver in the maximum likelihood (ML) sense for
the ternary system is derived. The ML receiver is the one that selects the input
sequence (a) that maximizes the probability of receiving r(t) given that a was sent.
It is well-known that [38, chapter 10], calculating this probability is proportional to
calculating the metric
PM(a) = −∫ ∞
−∞
∣
∣
∣
∣
∣
r(t) −∑
k
akh (t − kT, ak)
∣
∣
∣
∣
∣
2
dt. (5.8)
By expanding this equation, we get
PM(a) = −∫ ∞
−∞
|r(t)|2 dt
+∑
k
2ak
∫ ∞
−∞
r(t)h(t − kT, ak)dt
−∑
k
∑
m
∫ ∞
−∞
h(t − kT, ak)h(t − mT, am)dt.
(5.9)
Let the input sequence a = {ak} be split into two sequences a′
= {a′
k} and
94
a′′
= {a′′
k} of the same length such that,
a′
k =
ak if ak = ±1
0 otherwise
and,
a′′
k =
ak if ak = ±2
0 otherwise.
Therefore, a = a′
+ a′′
. Using these sequences in Eqn. 5.9, it can be shown that
the ML receiver is the one that maximizes the correlation metric,
CM(a) = 2∑
n
[
a′
ny′
n + a′′
ny′′
n
]
−∑
n
∑
m
anamxanam
n−m (5.10)
where,
y′
k =∑
n
a′
nx11n−k +
∑
n
a′′
nx21n−k + z
′
k
y′′
k =∑
n
a′
nx12n−k +
∑
n
a′′
nx22n−k + z
′′
k
xpqk =
∫
h(t, p)h(t + kT, q)dt
z′
k =
∫
z(t)h(t − kT, 1)dt
z′′
k =
∫
z(t)h(t − kT, 2)dt.
Eqn. 5.10 is similar to the correlation metric of the conventional binary system [38].
From the equations above, it is apparent that xpqk is the output of a filter matched
to h(t, q) whose input is h(t, p) where p, q ∈ {1, 2}. Consequently, the sufficient
95
statistics y′
k and y′′
k can be obtained at the output of two filters that are matched
to the two responses respectively. Therefore, the optimal receiver for the ternary
system is as shown in Fig. 5.2.
5.3 Equalization and Detection
5.3.1 Equivalent Discrete Time System
In this section, generalized partial response (GPR) targets and equalizers are derived
and appropriate detectors are discussed. Using Eqn. 5.10, an equivalent discrete
time system for the ternary channel can be developed, which is shown in Fig. 5.3.
Interestingly, this shows the equivalence of the ternary system to the two-track/two-
head system with inter-track interference, which was first characterized and studied
by Barbosa [39]. Detection of ternary signals can therefore be considered as the
detection of data in two neighboring tracks with inter-track interference when read
by two heads simultaneously. However, unlike the two-track/two-head system, the
data in the two tracks are correlated, since the binary input to the two channels are
dependent and the inter-track interference are of comparable energy to the primary
channel. As in conventional magnetic recording systems, it is necessary to shorten
the length of the channel response in order to lower the complexity of the detector,
and is known as equalization. This topic has been extensively investigated for binary
96
'kz
''kz
'ky
''ky
'ka 11X
12X
21X
22X''ka +
+
Figure 5.3: Discrete time equivalent system of the ternary magnetic recording sys-tem.
systems, but are not directly applicable to our case. We describe next a multi-input
multi-output (MIMO) generalized partial response (GPR) equalization scheme for
ternary systems.
5.3.2 MIMO Equalizer
In this scheme, the equalizer is a set of FIR filters that equalizes all the four channels
(including the cross-track interference channels) simultaneously to four different
targets. This set of filters is termed as a MIMO equalizer with two-inputs and two-
outputs, as shown in Fig. 5.4. If the channel output and equalizer are expressed as
97
JointDetector
'ky
''ky
ka
=
2221
1211qq
qq����Figure 5.4: MIMO equalizer: a set of four FIR filters that equalizes all four channels.
matrix of polynomials,
Y =
[
y′
(D) y′′
(D)
]
Q =
q11(D) q12(D)
q21(D) q22(D)
then, the ideal equalizer output is given as,
Y · Q =
a′
(D)p11(D) + a′′
(D)p21(D)
a′
(D)p12(D) + a′′
(D)p22(D)
T
where, p11(D), p22(D), p12(D) and p21(D) are the predetermined target polynomials
for channels X11, X22, X12 and X21 respectively.
5.3.3 GPR Targets
In [40], Moon proposed a method to simultaneously obtain GPR targets and equal-
izers that minimizes the equalization error in the mean square sense. He showed
that these targets with real coefficients significantly perform better than targets
with integer coefficients. In this section, GPR targets and equalizers for the MIMO
98
equalization scheme of the ternary system are derived in a similar way. Consider
the system shown in Fig. 5.5, where the channel X, equalizer C and the desired
target F are expressed as 2 × 2 polynomial coefficient matrices,
X =
x11 x12
x21 x22
C =
c11 c12
c21 c22
=
[
C1 C2
]
F =
f11 f12
f21 f22
=
[
F1 F2
]
.
Therefore, the mean of the squared errors between the output of C and F are
expressed as,
E[
e′
k
2]= CT
1 RY C1 + F T1 RAF1 − 2CT
1 RY AF1
E[
e′′
k
2]= CT
2 RY C2 + F T2 RAF2 − 2CT
2 RY AF2,
(5.11)
where, RA, RY are the input and output auto-correlation matrices respectively,
and RY A is the cross-correlation matrix between the input and the output. The
idea of the following derivation is to simultaneously obtain targets and equalizers
that minimizes the sum of the mean squared errors given in Eqn. 5.11. Note that
if the input sequences to the two channels are independent, then the equalization
errors at the two outputs will be independent. If J = E[
e′
k
2+ e
′′
k
2], then F and
C are obtained by evaluating and equating to 0, the gradients ∂J/∂C1, ∂J/∂C2,
99
X
Channel Equalizers
Targets
'ka''
ka
F
'ky
''ky
'kd''
kd
C'ks''
ks
Figure 5.5: GPR calculation for ternary magnetic recording system with MIMOequalizer.
∂J/∂F1 and ∂J/∂F2. However, in order to avoid the trivial all-zero solution, certain
constraint is imposed on the solution space. For binary systems, use of monic
constraint was long known to produce the best result and was formally shown to be
optimal in [41]. For MIMO systems, the corresponding optimal constraint is given
as,
det
f11(0) f12(0)
f21(0) f22(0)
= 0 (5.12)
where, fij(0) is the first coefficient of the equalizer filter fij, for i, j ∈ {1, 2}. This
constraint is the equivalent monic constraint for the MIMO system. Using the
Lagrange technique, the function to minimize is modified as J = E[
e′
k
2+ e
′′
k
2] −
2m [f11(0)f22(0) − f12(0)f21(0)], where m is the Lagrange multiplier. Evaluating
various gradients to 0 and after some simplifications, the resulting equation is given
100
as,
Z
f11 f12
f21 f22
= m
f22(0) f21(0)
0 0
......
−f12(0) f11(0)
0 0
......
(5.13)
where,
Z =
z11 z12
z21 z22
= RA − RTY AR−1
Y RY A.
If
Z−1 =
z11 z12
z21 z22
then, from Eqn. 5.13,
md =
f11(0) f12(0)
f21(0) f22(0)
f11(0) f12(0)
f21(0) f22(0)
T
(5.14)
where,
d =
z11(1, 1) z12(1, 1)
z21(1, 1) z22(1, 1)
.
These equations are solved by first finding m such that det(md) = 1. Then, the
monic coefficients of the four FIR equalizer filters are determined from Eqn. 5.14
and the other coefficients from Eqn. 5.13. Finally, the targets are determined by
solving the equations ∂J/∂C1 = 0 and ∂J/∂C2 = 0.
101
For the two-track/two-head system with interference, it is well-known that
the optimal detector can be implemented by the Viterbi algorithm that jointly
detects data from both channels. But, in this case, due to the dependence between
the two input sequences, the joint detector simply reduces to a ternary Viterbi
detector.
5.4 Application to High-Radial Density Systems
One of the methods used to increase areal density of a magnetic recording system is
to increase its radial density by reducing the track width. Since the read heads do
not scale down in size proportionately, these systems usually suffer from interference
from data stored in adjacent tracks during the readback process. The amount of
interference from an adjacent track is characterized by the fraction of its width that
is directly under the read head. If wt and wh > wt are the track width and the
read head width respectively, then assuming no guard band between tracks and
assuming the head is centered around the main track, the interference from each
adjacent track is given as λ = (wh − wt) / (2wt). Therefore, the readback signal
consists of all the signal energy from the main track data and a λ fraction of the
signal energy from each of the adjacent track data [42] [43].
Systems with inter-track interference has been studied extensively and
102
LPF C JointViterbi
Main Track
Adjacent Track
T
Equalizes bothmain and
adjacent trackdata
Adjacent Track
)(th
)(thλ
)(thλ
Figure 5.6: PRML binary magnetic recording system with ITI.
many ways to improve their performance have been proposed. Most commonly,
the use of more than one head to read several adjacent tracks simultaneously has
been suggested. It has been shown that by taking advantage of the correlation be-
tween the readback signals, the minimum distance of the system can be increased
[42]. In this chapter, systems with only one read head is considered, which is more
practical. Such a system is shown in Fig. 5.6. Since the channel response of all the
tracks are same, the single FIR equalizer simultaneously equalizes both the main
track data and the adjacent track data. But, the adjacent track response is equal-
ized to λ times the partial response target. The optimal detector for this system is
implemented by the Viterbi algorithm that jointly detects the main track and the
adjacent track data. After the detection of every sector, the adjacent data detected
is simply discarded as in [43].
As described earlier, use of ternary signaling can be useful in systems with
103
high radial density. Consider a binary system with ITI with a certain areal density
that is used to store some amount of binary information. Now, the binary informa-
tion can be converted to ternary sequences using some binary-to-ternary mapping
and can be stored using a ternary system with a higher symbol area. Consequently,
the ternary system can afford to store the same amount of information with lesser
number of tracks, but with larger width. If the width of the track can be increased
by more than wh−wt, then the interference λ becomes 0, as the read head no longer
overlaps with adjacent tracks. The increase in track width is dependent on the rate
of the mapping. If R denotes the mapping code rate, then the track width in the
ternary system can be increased to w′
t = R×wt. Note that R is greater than 1 and
is upper-bounded as,
R ≤ log(3)
log(2)≈ 1.585. (5.15)
Therefore, the maximum interference that the ternary system can alleviate is
λmax =(1 − Rmax)
2≈ 0.3. (5.16)
Use of ternary signaling will be useful when the gain in performance due to allevia-
tion of interference is greater than the loss due to decrease in minimum distance of
the ternary system. These depend on a number of factors like normalized density,
amount of interference, signal-to-noise ratio (SNR) etc. In the following example,
one such case where the ternary system merits some consideration is presented.
104
In order for a meaningful comparison between the binary and the ternary
systems, the definition of SNR needs to be carefully considered. The SNR definition
used in this work is given as [44],
SNR = 10 · log10
Ei
N0
(5.17)
where, N0 is the power spectral density height of the electronics (white Gaussian)
noise and Ei is the energy of the normalized impulse response (refer to [44] for further
details). Usually, Ei is set to 1 for the binary system (without loss of generality,
this is assumed to be the maximum energy that can be extracted from the system).
For the equivalent ternary system, the electronics noise power remains the same,
since the bandwidth of the system is unchanged. But, due to the introduction
of a new symbol with lesser energy, ternary sequences normally have less energy
than the corresponding binary sequences. Nevertheless, the noise power is not
proportionately reduced for the same SNR, since the introduction of a symbol with
lesser energy is a deliberate deviation from optimality.
5.5 Simulation Results
Consider a binary system with channel transition response width w1 and peak v1
and operating at normalized density 2 with λ = 0.25. The performance of this
system is compared with its corresponding ternary system that does not suffer from
105
inter-track interference, but stores the same amount of information. In order for
the track width to be equal to the head width, a mapping code of rate 1.5 should
be used. The performance of the ternary system depends on the new symbol that is
introduced. Width of the response for this symbol w2 is always greater than w1 (and
v2 < v1). Choosing w2 is same as choosing the slope of the transition, since there
is a one-to-one correspondence between the transition gradient and the peak and
width of its response. Choosing a much higher w2 will increase the distance between
the two responses but at the same time also increases ISI. Similarly, choosing a w2
that is closer to w1 will decrease the minimum distance between the two responses
but will also decrease ISI. Therefore, the new symbol needs to be carefully chosen
that minimizes the bit error rate. By simulation, it was determined that w2 =
1.5 × w1 gave good performance for the ternary system. Therefore, the ternary
system is operated at a normalized density of 2 with respect to symbol 1 (ND1)
and normalized density 3 with respect to symbol 2 (ND2). A rate 3/2 mapping code
is used and is shown in Tab. 5.1. Optimal GPR targets of length 5 and optimal
detectors were used for both systems. The bit error performance comparison
between the two systems is shown in Fig. 5.7, which shows that the ternary system
gives more than 1 dB performance improvement over the binary system at a bit
error rate (BER) of 10−5. Moreover, the optimal detector for the ternary system
requires only 162 states, whereas the optimal joint detector for the binary system
106
Table 5.1: Binary-to-Ternary Mapping Code
Binary Ternary000 00001 01010 02011 12100 11101 21110 22111 20
requires 212 states. Since, the mapping codeword length is only 3, it does not result
in any error propagation as is confirmed by Fig. 5.7, which shows that the ternary
symbol error rate (SyER) is not significantly different from the BER.
Fig. 5.8 shows a similar performance comparison for the case when λ = 0.2
for the binary system. The corresponding ternary system requires a mapping code
of rate 1.4 to nullify the interference. But, the mapping code of rate 1.5 used above
is retained, which allows the system to have a track width equal to the head width
and also increase the time period of the system by a factor of 1.4/1.5 (increases
SNR). Therefore, the ternary system operates at normalized densities 1.87 and 2.8
corresponding to the two symbols. In this case, the binary system performs better
at low SNRs, where as the ternary system performs better at high SNRs. In general,
the performance of the binary systems will improve as the interference reduces and
after a certain threshold, it will no longer be beneficial to store information using
the ternary system. This threshold will vary depending on the various parameters
107
23 24 25 26 27 28 29 30 3110
−6
10−5
10−4
10−3
10−2
10−1
SNR (dB)
Bit
Err
or R
ate/
Sym
bol E
rror
Rat
e
Ternary (SyER), ND = 2 & 3Ternary (BER), ND 2 & 3Binary (BER), λ = 0.25
f11
: 1.43 −2.72 −5.63 −1.77 0.74 f12
: 0.08 −2.05 −3.44 −1.66 −0.06 f21
: 0.00 4.94 2.89 0.36 −0.21 f22
: 0.69 2.78 2.14 0.63 0.03
Figure 5.7: BER comparison between a binary system with ITI (λ = 0.25) andternary system with no ITI (both systems store the same amount of information).
of the system.
An additional benefit of the ternary system is that the binary-to-ternary
mapping code can be exploited to impose certain characteristics on the input se-
quences. For example, they can be designed to avoid certain error events or min-
imize number of transitions and hence reduce nonlinear effects etc. Particularly,
they can be used to impose certain channel constraints, like the (d, k) run-length
limited (RLL) constraints (see [45] and references therein), without any additional
code rate penalty. For example, in the rate 3/2 mapping code used above, 8 binary
108
22 23 24 25 26 27 28 29 3010
−6
10−5
10−4
10−3
10−2
10−1
SNR (dB)
Bit
Err
or R
ate
(BE
R)
Ternary (BER), ND 1.87 & 2.8Binary, λ = 0.2
f11
: 1.35 −2.86 −6.15 −1.69 0.73f12
: 0.08 −2.24 −3.82 −1.72 −0.02f21
: 0.00 5.44 2.01 0.06 −0.12f22
: 0.74 3.01 1.77 0.36 0.025
Figure 5.8: BER comparison between a binary system with ITI (λ = 0.2) andternary system with no ITI (both systems store the same amount of information).
message words of length 3 are mapped to 8 out of 9 ternary codewords of length
2. If the unused ternary vector is chosen to be the all-zero vector, then the map-
ping code can impose the (0,2) constraint. A binary constraint code imposing the
(0,2) constraint has a capacity of 0.88. Various other RLL constraints can also be
imposed by increasing the codeword length or decreasing the mapping code rate
wherever possible. If a code of rate 6/4 is used instead of 3/2, then the mapping
can be designed to impose the (4,4) constraint. Although the d-constraint do not
represent the same property in ternary and binary systems, they may serve the
109
same purpose.
5.6 Summary
In this chapter, we explored the advantages of a magnetic recording channel that
is capable of making transitions with two different but predetermined gradients.
By developing appropriate signal processing algorithms for this channel, we showed
that this system performs better than a binary system with ITI. Its advantage
stems from the fact that the ternary systems are equivalent to two-track/two-head
systems, where data from both the heads are used for detection. Moreover, using
the binary-to-ternary mapping used in the ternary system, we showed that some
RLL constraints can be imposed, that were previously possible only with low-rate
constraint codes. Overall, our work provides a favorable view of ternary magnetic
recording systems from a signal processing perspective. It also encourages us to
consider further, more complicated systems involving non-linear noise and other
effects that may be associated with the ternary channel.
110
CHAPTER 6
HEAT-ASSISTED MAGNETIC RECORDING SYSTEM
Heat-Assisted Magnetic Recording (HAMR) was recently proposed as a viable next-
generation storage technology [46] that can achieve densities of 1 Tb/in2 or more.
As discussed briefly in Chapter 1, in this technique, a high anisotropy medium
is heated by a laser at the location of transition during the write process. In
order to exploit the potential of HAMR, there are many challenges that need to be
overcome. It is important to study and analyze the system, so as to identify and
understand its characteristics. In this chapter, we introduce HAMR, explain the
numerical models used to analyze the system and illustrate some of its important
characteristics. The rest of the chapter is organized as follows. Section 6.1 describes
the principle of heat-assisted magnetic recording. Section 6.2 introduces the thermal
Williams-Comstock model, which has been shown to represent a HAMR system
appropriately by incorporating all thermal and magnetic properties. Using this
model, along with the microtrack model described in Section 6.3, we determine
the unique characteristics of the HAMR system in Section 6.4, with a particular
111
focus on the effects of thermal profile on transition characteristics and on non-
linear transition shift (NLTS) in HAMR. Finally, we summarize the key findings in
Section 6.5.
6.1 Principle of Heat-Assisted Magnetic Recording
In heat-assisted magnetic recording, during the write process the magnetic media
is heated by a laser attached to the write-head assembly. After magnetizing, the
medium is rapidly cooled down to room temperature. The resultant magnetization
of the medium is determined by its hysteresis loop (MH loop), which shows the
relationship between the applied field (Ha) and the resultant magnetization (M).
An example of an MH-loop is shown in Fig. 6.1. For stable magnetic transitions,
a certain field is applied so that even after removing the field, the medium retains
a certain level of magnetization. This is called remnant magnetization (Mr). The
ease of making a transition in a medium is measured by its coercivity (Hc), which
is the field required to reverse the direction of magnetization. The coercivity and
remnant magnetization are known to vary approximately linearly with temperature.
Therefore, if a transition is written at a temperature higher than room temperature,
a smaller field is required to make a transition. But, almost instantaneous cooling
ensures that the medium saturates at Mr corresponding to the room temperature.
112
H a
M
H c
+ M r
- M r
Room temperature High temperature
Figure 6.1: Example of an MH loop, illustrating its dependence on temperature
The process of heating while writing enables sharper magnetic transitions
to be made on high-coercive materials, thereby achieving higher stability and areal
density. The read process in HAMR is identical to that of conventional magnetic
recording systems. Though the fundamental idea behind HAMR is simple, the
process of rapid heating and cooling brings about a number of practical challenges.
More information on these can be obtained from [47, 48, 49].
113
M
1
+ M r (T)
- M r (T)
) , ( T x h
M
1
+ (T)
- (T)
) , ( T x h
Figure 6.2: MH loop with effective field
6.2 Thermal Williams-Comstock Model
The William-Comstock model is a well-known approximate analytical model that
describes the transition characteristics in a conventional magnetic recording system.
In [50], this model was extended by incorporating thermal gradients in order to de-
termine the transition characteristics of a longitudinal HAMR system. Specifically,
this model, known as the thermal Williams-Comstock model, captures the effects
of heating through the thermal gradients of coercivity and remnant magnetization.
Application of this model to perpendicular HAMR system is also discussed in this
section.
114
As mentioned earlier, the MH loop of a magnetic material is dependent on
the temperature. In order to account for this behavior a new quantity called effective
field (h) is introduced. It is defined as the ratio of applied field and coercivity,
h(x, T ) =Ha(x)
|Hc(T (x))| (6.1)
where T (x) is the temperature profile along the direction of the movement of head
(x). Note that the dependence of coercivity on temperature is shown explicitly.
The new MH loop defined using h is shown in Fig. 6.2, where irrespective of the
temperature, h is 1 at the point of reversal of magnetization. The magnetization
gradient with respect to position is defined as,
dM
dx=
dM
dh
dh
dx. (6.2)
Following a similar approach as in the Williams-Comstock model, the resultant
equation, known as the slope equation is derived as,
dM(x)
dx=
dM(x)
dHa
[
dHh(x)
dx+
dHd(x)
dx− dHc(T )
dT
dT
dx
]
(6.3)
where, M(x) is magnetization, Hh, Hd are the head and demagnetizing field respec-
tively. Demagnetizing field is defined as the field from the magnetized medium,
which opposes the field from the head. Note that the magnetization gradient de-
pends on the coercivity thermal gradient.
Two quantities that completely characterize a magnetic transition are its
transition center (or location) and length. Transition center is defined as the point
115
where the magnetization of the medium reverses its direction. It occurs when the
net applied field equals the coercivity of the medium. The net applied field is defined
as Ha = Hh + Hd. Therefore, the transition center is the solution to the equation
Hc(x) = Hh(x) + Hd(x) (6.4)
.
Traditionally, magnetic transitions are described using an arctangent pro-
file. Sometimes, hyperbolic tangent is argued to be better suited [51]. An arctangent
transition is described as
M(x) =2Mr(T (x))
πtan−1
(
x − x0
a
)
(6.5)
where, x0 is the transition center, a is known as the transition parameter and the
expression πa is known as the transition length. The goal of the thermal Williams-
Comstock model is to determine a and x0 of a transition for a given HAMR system
set-up. As mentioned before, transition center can be evaluated using Eqn. 6.4.
The transition parameter is evaluated using the slope equation. See [50] for the
derivation of each quantity of the slope equation.
Demagnetizing field is known to depend on both transition center and pa-
rameter. Therefore from Eqn. 6.3 and Eqn. 6.4, it can be seen that the two un-
knowns, transition center and parameter are dependent on each other and cannot
be solved analytically. But, if the thermal spot size of the laser used for heating is
116
assumed to be large, then the equations can be simplified and solved analytically.
For large spot size, since the thermal gradients are small, demagnetizing field can
be ignored in the calculation of transition center, thus making x0 independent of a.
Also, the demagnetizing field gradient equation simplifies, which in turn simplifies
the slope equation.
If the spot size is not large, then the demagnetizing field cannot be ignored.
Therefore, the system of equations (Eqn. 6.3 and Eqn. 6.4) can only be solved
iteratively [52]. In the first iteration, for a random value of a, x0 is calculated using
Eqn. 6.4. Then using this value of x0, Eqn. 6.3 is evaluated for a. In the second
iteration, this new value of a is used to determine the new x0 and so on. The
iteration continues until both x0 and a converges to some required accuracy.
Thermal Williams Comstock Model can be applied for both longitudinal
and perpendicular HAMR. Irrespective of the direction of magnetization, the tran-
sition can be considered to follow an arctangent profile. In order to solve the slope
equation, the head and demagnetizing field needs to be evaluated. The field expres-
sions for both longitudinal and perpendicular recording is explained in the following
sections.
117
Medium
Head
y
x
Figure 6.3: Longitudinal head and field
6.2.1 Longitudinal HAMR
In longitudinal recording, the medium is magnetized in the longitudinal direction
(along the media). The set-up of a longitudinal system is shown in Fig. 6.3,
which shows the poles of the head and the medium beneath it. The longitudinal
field between the two poles magnetizes the media in the longitudinal direction.
Assuming an infinitely long and wide head with a finite gap width, Karlqvist derived
an analytical expression for the field intensity in the medium [53]. The expression
for the longitudinal component of the field is given as
Hx(x, y) =H0
π
[
tan−1
(
x + g/2
y
)
− tan−1
(
x − g/2
y
)]
(6.6)
where g is the gap width and H0 is the deep gap field. For the purpose of thermal
Williams-Comstock model, the field is evaluated at the center of the medium. If
t is the thickness of the medium and d the distance from the bottom of the head
to the medium surface, then Hx is evaluated at y = d + t/2. The effect of the Hy
118
component of the field is ignored in longitudinal recording. The demagnetizing field
from a transition can be expressed as [54]
Hd(x) = −dM
dx∗ Hstep
x (x) (6.7)
where the field from a sharp transition at the center of the medium is given as
Hstepx (x) =
1
πtan−1
(
t
2x
)
. (6.8)
Therefore, assuming an arctangent transition, the demagnetizing field can be calcu-
lated as below. If the laser spot size is large, then the demagnetizing field equation
can be simplified, details of which are given in [50].
Hd(x) =−2
π2
∫ +∞
−∞
[
Mr(x′)a
a2 + (x0 − x′)2+ tan−1
(
x′ − x0
a
)
dMr(T (x′))
dx′
]
tan−1
(
t
2(x − x′)
)
dx′
(6.9)
6.2.2 Perpendicular HAMR
In perpendicular recording, the perpendicular field of the head magnetizes the
medium in the vertical direction. The actual set-up of a perpendicular system
is shown in the top of Fig 6.4. A highly permeable layer (or keeper) is deposited be-
neath the medium through which the field is conducted from one pole to the other.
Note that the field is in the vertical direction in the medium. The field intensity,
derived by Westmijze [55], is given as a complex function that needs to be solved
numerically. But, by considering the equivalent system set-up (bottom of Fig. 6.4),
119
Medium
Keeper
Pole Head
Medium
Pole Head
y
x
Image
Figure 6.4: Perpendicular head and field: actual head configuration and equivalentset-up
an analytical head field expression can be derived [56, 57]. An image of the pole
head can be considered to be symmetrically placed beneath the medium. When
this set-up is turned sideways, the similarity with the longitudinal set-up can be
readily seen. Therefore, to a good approximation the field in a perpendicular head
can be evaluated using the Karlqvist expression, taking into account the change in
coordinate system for the perpendicular recording. Therefore, the perpendicular
component of the head field is given as
Hy(x, y) =H0
π
[
tan−1
(
y + g/2
x
)
− tan−1
(
y − g/2
x
)]
(6.10)
where g is the gap width between the pole head and its image. If d is the head-
medium distance and t is the medium thickness, then g = 2d + 2t. For the ther-
mal Williams-Comstock model, the field is evaluated at the center of the medium
(y = t/2). The effect of the Hx component of the field is ignored in perpendicu-
lar recording. As mentioned before, the Karlqvist expression is not a reasonable
120
approximation for fields close to the head gap. Therefore, Eqn. 6.10 is valid only
for x > 0. Since, in general, the transition occurs away from the pole edge this
expression can be faithfully used in the thermal Williams-Comstock model for eval-
uation of transition characteristics. The demagnetizing field can be calculated using
Eqn. 6.7, but with the perpendicular field component of a sharp transition as shown
below.
Hd(x) = −dM
dx∗ Hstep
y (x) (6.11)
where the field from a sharp transition at the center of the medium is given as [58]
Hstepy (x) =
1
πtan−1
(
2x
t
)
(6.12)
Therefore, considering an arctangent transition the demagnetizing field can be cal-
culated as,
Hd(x) =−2
π2
∫ +∞
−∞
[
Mr(x′)a
a2 + (x0 − x′)2+ tan−1
(
x′ − x0
a
)
dMr(T (x′))
dx′
]
tan−1
(
2(x − x′)
t
)
dx′
(6.13)
For a large laser spot size, the second term becomes insignificant. Consequently,
the demagnetization field can be reduced to,
Hd(x) ≈− 2
π2
∫ ∞
−∞
Mr(x′)a
a2 + (x′ − x0)2tan−1
(
2(x − x′)
t
)
dx′
=2Mr(T (x))
πtan−1
(
x − x0
t/2 + a
)
(6.14)
121
and the gradient of the demagnetizing field can be simplified if we ignore the thermal
gradient of the remnant magnetization, i.e.,
dHd(x)
dx≈ −2Mr(T (x))
π(a + t/2)(6.15)
Furthermore, similar to the derivation for longitudinal recording in [50], we can
show that the analytical expression for the transition parameter in perpendicular
recording is,
a = −γ
2+
1
2
√
γ2 +4Hc(1 − S)t
∆π
∣
∣
∣
∣
x0
, (6.16)
where,
∆ =dHh
dx− dHc
dT
dT
dx
∣
∣
∣
∣
x0
=Hgg
π(x20 + (g/2)2)
− dHc
dT
dT
dx
∣
∣
∣
∣
x0
,
γ =2Mr
∆π− t
2+
2Hc(1 − S)
∆π.
(6.17)
6.3 Microtrack Model
The process of heating and magnetization of a medium is two-dimensional. But,
the Williams-Comstock model provides only a one-dimensional solution to the prob-
lem of determining the transition characteristics. Therefore, they do not accurately
model the magnetic transition of a track with finite width. In this section, we ex-
plain the microtrack model that closely captures the effects of this two-dimensional
process. In this model, a magnetic track is divided into several sub-tracks of equal
122
Track
Apply thermal Williams - Comstock model to every sub - track.
x 0
a
Sub - Track
Apply thermal Williams - Comstock model to every sub - track.
x 0
a
Sub - Track
Figure 6.5: Microtrack modeling of HAMR channel
width as shown in Fig. 6.5. The temperature profile resulting from the heating is de-
noted by T (x, y) where, x and y represent the directions along and across the track
respectively. If N is the number of sub-tracks and ∆z their width, then for each
sub-track, the temperature profile is approximated by the one-dimensional function
T (x, y = i · ∆z), −N/2 ≤ i ≤ N/2. The more sub-tracks, the better the approx-
imation. For a given system set-up, thermal Williams-Comstock model is applied
independently to each of these sub-tracks to determine the corresponding transi-
tion parameter and center. If the readback response for each of these sub-tracks is
h(t, a), then the total response of the transition is given as,
p(t) =1
N
N∑
i=1
h(t − τi, ai) (6.18)
123
where, τi and ai are the transition center and parameter of the ith sub-track. The
isolated transition responses for longitudinal and perpendicular recording are ex-
pressed using Lorentzian and error functions respectively [44, Chapter 6].
6.4 Unique Characteristics of HAMR
Designing a HAMR system for optimal performance is a multi-dimensional problem.
The system performance is dependent not only on magnetic properties like coercivity
and remnant magnetization, but also on their variation with temperature, heating
profile (peak temperature, width at half of peak temperature) [52], and also on
the position of peak temperature [50]. The position of the laser spot needs to be
properly chosen in order to obtain the optimal performance. Since the laser is
attached to the head assembly, it is not known whether it is possible to align the
peak temperature with the gap center or whether it is the best possible set-up. In
this section, we report on some of the unique characteristics of longitudinal HAMR
induced by thermal spot.
For any system parameters, using the thermal Williams-Comstock model
and the microtrack model, the transition length, location and magnetization can be
determined across a track of finite width. Fig. 6.6 shows an example of a typical
magnetization pattern of a transition in the medium for a longitudinal recording
124
system. This figure illustrates several features of a transition that are induced
by heating and are unique to HAMR. Firstly, we note that the transition occurs
at different location across the track and secondly, the transition is sharp at the
center and decreases progressively toward the edges of the track. Since, the thermal
profile is Gaussian in both dimensions, maximum temperature occurs at the center
of the track and minimum at the edges of the track. As coercivity varies linearly
with temperature, solution to Eqn. 6.4 varies for different sub-tracks, inducing
the transition curvature across the track. The solution to transition location is
a function of y, since temperature T is a function of both x and y. Thermal
Williams-Comstock model suggests that high coercivity gradient induces sharper
transitions. Since, coercivity gradient along the track is small at the edges, we
notice a significant increase in transition length near the edges. From Eqn. 6.4,
we note that the transition location also depends on the head field. Fig. 6.7
shows the shift in transition location with respect to the gap center at various
deep gap fields for a system with Hc(x, y) = −500 · T (x, y) + 5 · 105 A/m and
Mr(x, y) = −300·T (x, y)+3·105 A/m. It also clearly shows the transition curvature
across the track.
One of the most important quantity that determines the storage capacity
of HAMR is the width of the readback signal of its transitions. The width when
measured at half their peak value is denoted by PW50 and is commonly used for
125
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
−400 −300 −200 −100 0 100 200 300
100
200
300
400
500
600
700
800
900
Down Track Position (nm)
Cro
ss T
rack
Pos
ition
(nm
)
Figure 6.6: Magnetization of the medium after a transition is made
purpose of comparison. Smaller the PW50, greater the storage capacity of the disk.
Therefore, it is necessary to investigate the effects of thermal profile on readback
signal width. In the following analysis we assume a high coercive material, where
Hc(x, y) = −2000 · T (x, y) + 1.6 · 106 A/m and Mr(x, y) = −1200 · T (x, y) + 1.2 ·
106 A/m. Other system parameters are shown in Table 6.1. Fig. 6.8 shows the
PW50 of the readback signal at various peak temperatures for different deep gap
fields. In general, higher temperatures lead to smaller PW50, but show signs of
saturating or even increasing at very high temperatures. Though not shown here,
this phenomenon is more pronounced with low coercive materials. For a given head
126
−600 −400 −200 0 200 400 600120
125
130
135
140
145
150
155
160
165
170
Cross Track Position (nm)
Tra
nsiti
on C
entr
e (n
m)
Hg: 550 kA/m
Hg: 650 kA/m
Hg: 750 kA/m
Figure 6.7: Transition center curvature across the track for various deepgap fields. For this system, Hc(x, y) = −500 · T (x, y) + 5 · 105 A/m andMr(x, y) = −300 · T (x, y) + 3 · 105 A/m
field, there is an optimal temperature that achieves minimum PW50. Since, higher
temperatures result in lower coercivity, it is natural to expect PW50 to be better
at high temperatures, but the reason for their saturation or increase at very high
temperatures is not obvious.
6.4.1 Effects of Laser-Spot Position
The readback width not only depends on the peak temperature, but also on the
position of the laser (or consequently the position of the peak temperature) with
respect to the head gap. Along the track, the laser can be positioned either in
the direction of the head movement (up-track or +X) or opposite to it (down-track
or -X). For all cases, the laser is assumed to be at the center of the track in the
127
Table 6.1: System Parameters
Full-width half-max of laser spot 70 nmWrite head gap 100 nmDistance from pole to medium 20 nmRead head gap 5 nmWidth of the track 160 nmNumber of microtracks 23
cross-track direction. Positioning anywhere else is undesirable and is not discussed
in this paper. Fig. 6.9 shows the PW50 for the same system as before with peak
temperature 350 ◦C, at different laser spot positions around the head gap center
(0 nm). Minimum PW50 occurs when the laser is positioned just to the right of
the gap center. Moving the laser away from the gap in either direction, greatly
increases the readback signal width, though it seems to decrease after reaching a
peak for down-track laser spot positions. To explain such a behavior needs careful
investigation. Without a proper understanding of the source of these changes, it will
be difficult to generalize the behavior of any HAMR system, which has many degrees
of freedom for optimization. In the following analysis, we identify the reasons for
such changes and argue that it is indeed a general behavior of HAMR with high
coercive materials.
The two parameters that determine the PW50 of the readback signal of a
transition are the transition curvature of the track and parameter of each sub-track.
Changing the position of the laser not only changes the temperature profile but also
128
0 50 100 150 200 250 300 350 400 450 50050
60
70
80
90
100
110
Peak Temperature ( oC)
PW
50
Hg = 17⋅ 105 A/m
Hg = 14⋅ 105 A/m
Hg = 20⋅ 105 A/m
Figure 6.8: Readback signal width at various peak temperatures for different gapfields
the coercivity of the medium. Consequently, it alters both the transition center and
parameter. Fig. 6.10 shows an example of how the head field and the coercivity
look at different laser spot positions along the down-track direction. Observe that
the point of intersection of coercivity and head field is different at different laser
spot positions. This figure is a one-dimensional illustration of how the coercivity
changes with laser spot, i.e. it is an illustration for one of the sub-tracks.
Fig. 6.11 shows the transition centers across the track at different laser spot
positions in the down-track direction. It reveals two general trends. As the laser is
129
−150 −100 −50 0 50 100 15050
51
52
53
54
55
56
57
Alignment of laser with respect to gap center (nm)
PW
50
Figure 6.9: PW50 at various laser spot positions in Longitudinal HAMR
moved down-track from the gap center, transition location initially move away from
the gap center until they occur at the positive coercivity gradient region. Thereafter,
moving the laser further down-track, moves the location back closer to the gap
center. As the transition location moves away from the gap center, the curvature
deteriorates, since the center now occurs closer to the peak temperature, where
the variation in thermal gradient across the track is the highest. Consequently,
in this example, the transition curvature is at its worst when the laser is aligned
at around -96 nm (location closest to peak temperature) and improves on either
side of this position. Fig. 6.12 shows the cross-track transition parameter profile
130
−200 −150 −100 −50 0 50 100 150
2
4
6
8
10
12
14
x 105
Down−Track Location (nm)
A/m
Coercivity at Various Laser Spot Positions
Hc
Hh
0 −32 −64 −96 −128 Alignment
Figure 6.10: Coercivity for various laser spot positions
for the same down-track laser spot positions. As before, two trends are identified.
The parameter generally increases when laser is moved down-track toward the lower
gradient region. In the figure, at an alignment of -96 nm there is a huge increase
largely contributed by almost-zero coercivity gradient and low head field. If the
transition location occurs when both coercivity gradient and head field gradient
are positive, then higher coercivity gradient would result in a higher transition
parameter. Since, coercivity gradient decreases toward track edges, we observe that
the parameter at the center of the track increases much faster than the parameter
131
−80 −60 −40 −20 0 20 40 60 80−75
−70
−65
−60
−55
−50
Cross−Track Location (nm)
Tra
nsiti
on C
ente
r (n
m)
0 nm−32 nm−64 nm−96 nm−128 nm
Figure 6.11: Transition centers across the track at various laser spot positions inthe down-track direction
at the edges of the track, when laser is moved down-track. At an alignment of -128
nm, though the parameter at the center remains higher than at the edges, its values
have improved on account of better head field gradient at the transition location.
When laser is moved up-track, the coercivity gradient is always negative at
the transition location. Also, farther the laser is from the gap center, farther the
transition location is from the peak temperature. Therefore, as shown in Fig. 6.13,
there is almost no curvature for positions far to the right of the gap center. For
132
−80 −60 −40 −20 0 20 40 60 805.5
6
6.5
7
7.5
8
8.5
9
Cross−Track Location (nm)
Tra
nsiti
on P
aram
eter
(nm
)
0 nm−32 nm−64 nm−96 nm−128 nm
Figure 6.12: Transition parameter across the track at various laser spot positionsin the down-track direction
the same reason, as shown in Fig. 6.14, the change in transition parameter is al-
most identical throughout every sub-track. However, since the transition location is
pushed to the lower temperature regions, where the coercivity gradients are smaller,
transition parameter increases. PW50 is directly proportional to the transition pa-
rameter and is inversely proportional to the extent of curvature. If the transition
location of the sub-tracks of a track are misaligned, then their combined readback
response will be much wider than the response of a track whose sub-track locations
are aligned with each other. As the laser is positioned along the down-track direc-
tion, PW50 will increase on account of deteriorating curvature though the transition
133
−80 −60 −40 −20 0 20 40 60 80−65
−60
−55
−50
−45
−40
Cross−Track Location (nm)
Tra
nsiti
on C
ente
r (n
m)
0 nm32 nm64 nm96 nm128 nm
Figure 6.13: Transition centers across the track at various laser spot positions inthe up-track direction
parameter will be less. It will peak when the transition location occurs, where the
temperature of the medium is maximum. As the laser is positioned along the up-
track direction, PW50 will increase on account of increasing transition parameter,
though the curvature will improve. The minimum PW50 can be obtained for laser
spot alignments close to the gap center. This explains the behavior observed in Fig.
6.9.
The effect of position of laser was also investigated in perpendicular HAMR.
The system parameters were same as the one chosen for longitudinal HAMR except
134
−80 −60 −40 −20 0 20 40 60 805.5
6
6.5
7
7.5
8
8.5
9
9.5
Cross−Track Location (nm)
Tra
nsiti
on P
aram
eter
(nm
)
0 nm32 nm64 nm96 nm128 nm
Figure 6.14: Transition parameters across the track at various laser spot positionsin the up-track direction
for the coercivity, which is Hc(T ) = −2000T +2.1·106A/m. The change in transition
curvature and parameter for various laser positions to the left of the gap centre is
shown in Fig 6.15 and Fig 6.16 respectively. The effects are very similar to the ones
observed with longitudinal HAMR.
135
−100 −80 −60 −40 −20 0 20 40 60 80 100−60
−55
−50
−45
−40
−35
−30
−25
−20
−15
Cross−Track Location (nm)
Tra
nsiti
on C
entr
e (n
m)
Effects of Laser Position in Perpendicular HAMR
0 nm−32 nm−64 nm−96 nm−128 nm
Figure 6.15: Transition centres at various position of laser to the left of gap centre
6.4.2 Non-Linear Transition Shift (NLTS)
Non-linear transition shift is the phenomenon in which a transition that is currently
written is pulled closer to the previously written transitions due to their demagne-
tizing fields. Fig 6.17 illustrates the principle behind NLTS. It shows the head field
of the current transition, demagnetizing field of the previous transition located at
0 nm and their sum, applied field. Because of the net increase in the applied field
the transition centre is shifted to the left. This shift, denoted as ∆, is known as the
non-linear transition shift. In order to combat this shift, transitions are deliberately
136
−100 −80 −60 −40 −20 0 20 40 60 80 10020
25
30
35
40
45
50
Cross−Track Location (nm)
Tra
nsiti
on P
aram
eter
(nm
)
Effect of Laser Position on Perpendicular HAMR
0 nm−32 nm−64 nm−96 nm−128 nm
Figure 6.16: Transition parameter at various position of laser to the left of gapcentre
delayed (precompensation) so that after NLTS, they are located at the intended po-
sition. The amount of NLTS depends on, among other factors, the distance between
transitions, transition length and the demagnetizing field of the past transitions. In
conventional recording systems, demagnetizing and head field is the same across
the track. Therefore, NLTS experienced by the transition across the track is the
same. But that is not the case in HAMR. The demagnetizing field of a transition
137
−50 0 50 100 150−2
0
2
4
6
8
10x 10
5 Principle of NLTS
Downtrack Location (nm)
Fie
ld S
tren
gth
(A/m
)
Hdemag
Hh
Hh + H
demagH
c∆
Medium Thickness δ = 20 nm
Figure 6.17: Effect of demagnetizing field of past transitions
in longitudinal HAMR is given as ([59])
Hd(x) = − 2
π2
∫ ∞
−∞
[
Mr(x′)a
a2 + (x − x0)2− tan−1
(
x′ − x0
a
)
dMr(T )
dT
dT (x′)
dx′
]
tan−1 δ
2(x − x′)dx′
(6.19)
Note that the demagnetizing field is dependent on Mr, its gradient with respect
to position and the transition parameter. Heating the medium at the location
of current transition also heats up adjacent bits. Since the transition is curved
and the thermal profile is Gaussian in both dimensions, the temperature at the
transition location of the previous transition across the track is different. This
difference in temperature induces different Mr across the track as it varies linearly
with temperature. Variation in Mr along with the variation in transition parameter
results in varying demagnetizing field intensities exhibited by previously written
138
transition(s) across the track. Therefore, in HAMR the pull experienced by the
current transition will vary across the track leading to uneven shifts of the transition
centres. The new location of transition centre after NLTS is the solution to the Eqn
6.20. Note that the large spot approximation is used here, i.e. the demagnetizing
field of the current transition is neglected.
Hc(x) = Hh(x) +∑
i
H ′di(x) (6.20)
Where, Hdi is the demagnetizing field of the ith past transition. In general, effect
of Hdi is small for i > 2. Unlike in conventional recording, change in transition
centre changes the corresponding transition parameter because of change in ther-
mal gradients. Again, using the large spot approximation equations the transition
parameters can be recalculated.
NLTS in an Isolated Dibit
This section describes in detail the various effects of NLTS in an isolated dibit. The
first transition in a dibit is not affected by NLTS but the second transition is pulled
closer to the first one. The system parameters considered to study the NLTS were
similar to the one given in Tab 6.1, except that the medium thickness (δ) and deep
gap field (H0) were 20 nm and 14e05 A/m respectively. Also the laser was assumed
to be aligned to the gap centre for all cases considered. Fig 6.18 shows the NLTS
139
variation across track at different normalized densities, i.e at various bit periods.
NLTS values are shown as a % of bit period. As the normalized density increases,
apart from the expected increase in NLTS, the unevenness of NLTS across the track
increases too. Therefore, the transition curvature of the second transition of the
dibit changes after the shift. The transition centres at the centre of the track move
much closer to the first transition than the ones at the edges of the track. Such
a behavior is more prominent at high densities. As the density increases, the first
transition moves closer to the peak temperature region of the second transition, i.e
towards a region of high temperature gradient variation across track. Therefore, the
demagnetizing fields of the first transition (and thus NLTS) across the tracks are
now more uneven. In contrast, the NLTS in a dibit for the conventional system as
shown in Fig 6.19 for normalized density 3 does not vary across the track. Therefore,
the transition curvature remains unchanged even after the shift.
Since, the transition curvature and parameter after NLTS varies in HAMR,
the amount of average NLTS or the shift in peak readback signal cannot be pre-
dicted. Fig 6.20 shows the average amount of shift at various bit lengths. It follows
a similar pattern as in the conventional recording system [60]. Since, the demag-
netizing fields start to reduce as we get closer to the transition centre the average
NLTS reduces at very high densities.
Change in transition centre and parameter also brings about a change in
140
−80 −60 −40 −20 0 20 40 60 8015
20
25
30
35
40
Cross track Location (nm)
% N
LTS
Variation in NLTS across the track at various normalized densities
ND 2.0ND 2.5ND 3.0ND 3.5
PW50 = 65 nm
Figure 6.18: NLTS variation across track at various normalized densities for HAMR
PW50. In conventional recording systems, NLTS brings about an increase in PW50
[54] because of the increase in variance of the transition centres around the mean.
In HAMR, variance of the transition centre reduces because of the reduction in cur-
vature due to uneven NLTS. But, since NLTS moves the transition centres towards
lower temperature regions PW50, in general, would increase. Fig 6.21 shows this
change in PW50 at various bit lengths.
NLTS in an Isolated Series of Transitions
In the previous section the effects of NLTS in a dibit was studied. In this section,
the effect of NLTS for an isolated series of consecutive transitions will be studied.
With the same system set-up as before, a series of 10 consecutive transitions were
141
−80 −60 −40 −20 0 20 40 60 8019.5
20
20.5
21
21.5
22
Cross−Track Location (nm)
% N
LTS
Figure 6.19: NLTS variation across track at ND 3.0 for conventional longitudinalrecording system
written. Transition 2 is pulled closer by transition 1 and transition 3 is affected by
both transition 1 and 2. Since transition 1 is of the same polarity as 3, it will repel
transition 3. All other transitions are affected in a similar way. Fig 6.22 shows NLTS
as a % of bit length for all transitions at normalized density 3. Note that alternate
transitions experience an uneven NLTS across the track. This can be explained
using Fig 6.23. Transition curvatures of three consecutive transitions, both before
and after NLTS is shown. Dotted lines indicate transition centres before NLTS and
solid lines indicate transition centres after NLTS. For the first transition, both are
the same. NLTS experienced at the centre of the track for transition 2 is more than
at the edges. Therefore, the transition locations at the centre of the track shifts more
to the left than the transition locations at the edges, thereby straightening out the
142
10 20 30 40 50 60 703
3.5
4
4.5
5
5.5
6
6.5
7
Distance between transitions (nm)
Shi
ft in
Pos
ition
of P
eak
Rea
dbac
k S
igna
l (nm
)
Average NLTS variation with bit length
Figure 6.20: Average NLTS at various bit lengths
curvature. When the third transition is written, the distance between transitions is
now different all across the track. Specifically, at the centre of the track the distance
between transitions is more than at the edges. Since NLTS decreases exponentially
with distance, the increased distance between the transitions compensates for the
increase in NLTS at the centre of the track. Therefore, transition 3 experiences a
more even NLTS across the track. When a fourth transition is written, it behaves
in a similar way as transition 2. Therefore, we see an alternating pattern of uneven
NLTS. This effect is even more prominent at high densities as shown in Fig 6.24 (at
normalized density 4).
Eventually, after a number of consecutive transitions has been made, the
NLTS would saturate. Fig 6.25 indicates the saturation of average NLTS at various
143
10 20 30 40 50 60 7066
66.5
67
67.5
68
68.5
Distance between transitions (nm)
PW
50 (
nm)
PW50
variation (due to NLTS) with bit length
PW50
without NLTS= 65 nm
Figure 6.21: PW50 variation due to NLTS at various bit lengths
normalized densities. As expected, NLTS saturates more quickly for low normalized
densities. Note that in conventional systems since the transition curvature does
not change after NLTS, the shift is purely determined by the distance between
transitions. Therefore, every alternate transition experiences high NLTS like in
HAMR but is same across the track. In this case, average NLTS is same as the
NLTS experienced by every track, which is shown in Fig 6.26 for normalized density
3.
144
−80 −60 −40 −20 0 20 40 60 805
10
15
20
25
30
35
Cross−Track Location (nm)
% N
LTS
Cross−Track NLTS Variation in a Series of Transition
2345678910
ND = 3.0 T = 22 nm
Figure 6.22: NLTS variation for a series of transitions at ND 3
6.5 Summary
In this chapter we introduced the thermal Williams-Comstock model for both longi-
tudinal and perpendicular heat-assisted magnetic recording. Using this model and
the microtrack model, several characteristics of HAMR were illustrated. Specif-
ically, the effects of peak temperature and laser spot alignment on the readback
response were discussed. By analyzing the underlying effects it had on the transi-
tion length and location, we identified the general system behavior to changes in
145
1 2 3
Before NLTS
After NLTS
Figure 6.23: Depiction of transition curvatures of 3 consecutive transitions beforeand after NLTS
thermal profile.
Further, the effects of NLTS was studied in the longitudinal HAMR system.
It was determined that, unlike the conventional system, the NLTS experienced by
a transition is not only dependent on its nearest transitions, but also on other
transitions as well. This is a consequence of a significant change in curvature of a
transition due to NLTS. The resultant effect on the width of the readback response
was also studied.
146
−80 −60 −40 −20 0 20 40 60 800
5
10
15
20
25
30
35
40
45Cross−Track NLTS Variation in a Series of Transition
Cross−Track Location (nm)
% N
LTS
2345678910
ND = 4.0 T = 16.5 nm
Figure 6.24: NLTS variation for a series of transitions at ND 4
147
1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
30
35
40Average NLTS Variation in a Series of Transitions
Transitions
% A
vera
ge N
LTS
ND = 2ND = 3ND = 4
Figure 6.25: Average NLTS variation in a series of transitions at various normalizeddensities for HAMR
148
1 2 3 4 5 6 7 8 9 100
5
10
15
20
25
Transitions
% A
vera
ge N
LTS
Figure 6.26: Average NLTS variation in a series of transition at various normalizeddensities for conventional systems
149
CHAPTER 7
TWO-DIMENSIONAL MAGNETIC RECORDING SYSTEM
From the previous chapter, it is clear that HAMR relies on sophisticated write-
process for achieving ultra-high densities. Two-dimensional magnetic recordin
(TDMR) is unique because it uses conventional media, relying instead on sophisti-
cated signal processing and coding algorithms.
In TDMR, high storage density is achieved by reducing the number of
grains used for storing one bit of information. This could be as low as 1 grain/bit.
TDMR is seen as a potential candidate for achieving 10 Tb/in2. There are significant
challenges that need to be overcome before the potential of TDMR can be harnessed.
The challenges arise mainly because of two factors. By reducing the density to 1
grain/bit, the SNR reduces to as low as 0 dB. Further, the severe inter-symbol
interference (ISI) due to high density is of two-dimensional (2D) nature, requiring
the development of sophisticated 2D detectors and possibly 2D error correction
codes that can operate at such low SNR’s. The overall idea behind TDMR is to
store information at very high densities and use advanced 2D detectors and low-rate
150
(around 0.5) error correction codes to effectively achieve a user density in the range
of 1-10 Tb/in2. Theoretical achievable densities is determined by the capacity of
the channel, which is difficult to determine for the TDMR channel. Some work on
capacity bounds has been done on very simplified TDMR channels [18].
In this chapter, we provide some experimental lower bounds on capacity
for the TDMR system with a density of 1 bit/grain. This is done by first showing
that the TDMR system can be considered as a binary error and erasure channel
(BEEC) and subsequently calculating the capacity of the corresponding BEEC. We
do this analysis for two suitable TDMR read-channel models that have received
much attention recently; Voronoi-grain model [23] and the random-grain model
[18]. They both provide a good trade-off between complexity and accuracy, but
differ primarily in the possibility of write-errors. Write-errors occur when some
input bits do not get stored in any grain.
The rest of the chapter is organized as follows. The system model is de-
scribed in Section 7.1 and the method to view TDMR as BEEC is described in
Section 7.2. Using this method some error-erasure rates and the corresponding ca-
pacity bounds are derived in Section 7.3. Finally, some conclusions are drawn in
Section 7.4.
151
Figure 7.1: Two-dimensional magnetic recording (TDMR) system model.
7.1 System Model
Fig. 7.1 shows the TDMR system model. Two dimensional binary input x is written
onto a medium consisting of irregular grains resulting in a magnetized pattern m.
The primary source of noise in TDMR is due to the irregular grain boundaries of the
medium. Electronic noise is negligible compared to the medium noise and therefore
not considered in this model. The sampled readback signal from the read-head is
denoted as y. The impulse response of the read-head is wider than one bit in both
dimensions resulting in 2D ISI. Using y, the detector makes some decision x on the
input x.
TDMR read-channel model consists mainly of three components; model
for the medium, model for the write-process and model for the read-process. We
describe below two TDMR read-channel models.
152
7.1.1 Voronoi-grain Model
In this model, all instances of the media are assumed to be generated by a per-
turbation of the ideal medium. An ideal medium is defined as the one where the
grains are of the same size and shape and are equi-spaced. Geometrically, they are
represented by a set of points S on a square lattice as shown in Fig. 7.2(a). Voronoi
regions of these points correspond to grains of the ideal medium. These ideal grains
are referred to as cells and the points are referred to as cell-centers.
An instance of a conventional medium is generated by randomly shifting
the cell-centers and generating their corresponding Voronoi regions. The shift is
based on some known probability distribution and it is assumed that the shifted
centers remain within their corresponding cell boundaries. An example of such an
instance of the medium of size 3 × 3 cells is shown in Fig. 7.2(b), where ’∗’ marks
the positions of ideal cell-centers and ’·’ marks the shifted position of the cell-
centers. The Voronoi tiling corresponding to the points marked by ’·’ correspond to
the grains of the conventional medium. The probability distribution governing the
shifts is given by the class of distributions known as Tikhonov distribution, whose
probability density function (pdf) is given as,
pγ(∆x) =exp (γ cos(∆x))
2πI0(γ)0 ≤ ∆x ≤ 2π, 0 ≤ γ ≤ ∞,
where, I0 is the zeroth order modified Bessel function of the second kind. When
153
γ = 0, the pdf turns out to be uniform and when γ = ∞, the pdf turns out to be a
delta function. It can be thought of as corresponding to conventional medium and
patterned medium respectively. Pdf for values of γ between 0 and ∞ turns out to
be some bell-shaped curves of different peaks and width. Therefore, the regularity
of the grain structure can be controlled by varying γ. In other words, by varying
γ, the variance of the grain area can be varied. It also serves as a parameter to
compare with other medium generating models. For γ = 0, the variance of grain
area is about 6.3%.
The write-head does not have the knowledge of the grain structure of the
medium and therefore assumes the medium to be ideal. Consequently, it attempts
to write at the ideal cell-centers. The write-model is a rule that determines which
grains get magnetized when the head attempts to write at cell-centers. In this
model, we say that the magnetization of a grain is determined by the cell in which
its center (shifted cell-centers) is contained. From our earlier assumption that the
shifted cell-centers lie within their corresponding cells, it can be concluded that
there will be no write-errors in this model. There is a one-to-one correspondence
between the input bits (corresponding to each cell) and the grains. In reality, write-
errors are expected to be significant in TDMR systems. However, this model is
sufficient for the design of detectors, which can never correct write-errors. In this
work, it also serves to determine the effects of write-errors on capacity.
154
(a) (b)
Figure 7.2: Voronoi-grain model: all media are assumed to be a perturbation of theideal medium. (a) ideal medium and (b) an instance of a non-ideal medium, whereevery Voronoi region corresponds to a grain.
The readback signal is obtained by convolving the magnetized medium
with the system impulse response. The impulse response of the TDMR system is
a truncated unit-energy 2D Gaussian function spanning 3 cells in both dimensions
with a PW50 of 1 cell width.
7.1.2 Random-grain Model
The primary difference between the random-grain model and the Voronoi-grain
model is that the media are not considered to be perturbed versions of the ideal
medium. In this model, as many points as the number of grains required are ran-
domly placed in a 2D plane. Voronoi regions of these points correspond to the
grains of the medium. In general, this will result in a medium with very high vari-
ance of the grain area. In order to reduce the variance, some q% of the grains with
155
the largest area is divided into two and q% of the grains with the smallest area
are removed. As a result, the variance of grain area is reduced while maintaining
the same number of grains. Variance obtained for q = 7 is similar to the variance
obtained by the voronoi-grain model for γ = 0, which is 6%.
The magnetization of a grain is determined by the cell in which its centroid
(not same as the grain center) is contained. In this model, a cell may not contain
any centroid or may contain more than one centroid. Therefore, there is no one-to-
one correspondence between the inputs and the grains and consequently will lead
to write-errors. The readback signal is obtained in the same way as described in
the last section. An example of the random-grain model’s write-process is shown in
Fig. 7.3(a) and Fig. 7.3(b).
7.2 TDMR as BEEC
Let the binary input of the TDMR system be denoted as X = {xi,j} , (i, j) ∈
[0, 1, · · · , n−1]. Let the corresponding magnetization of the medium be denoted as
156
(a) (b)
Figure 7.3: Random-grain model: (a) input bits, and (b) the resultant magnetizedmedium after writing the input bits onto the medium (with write-errors).
m(t1, t2). Then the readback output Y = {yi,j} is given as,
yi,j =
∫∫
h(τ1, τ2)m(i − τ1, j − τ2) dτ1 dτ2
=+1∑
k1=−1
+1∑
k2=−1∫∫
Ak1,k2
h(τ1, τ2)m(i − τ1, j − τ2) dτ1 dτ2.
Where, h(t1, t2) is the impulse response of the system and Ai,j is the region spanning
the (i, j)th cell. The second part of the equation is obtained because the impulse
response spans 3× 3 cells. In an ideal medium, the cell boundaries are same as the
grain boundaries and as a result m(t1, t2) can simply be determined by the inputs.
Therefore, for the ideal medium, the sample readback output can simply be given
157
as,
yi,j =+1∑
k1=−1
+1∑
k2=−1
hk1,k2xi−k1,j−k2
,
where,
hk1,k2=
∫∫
Ak1,k2
h(τ1, τ2) dτ1 dτ2.
Therefore, for an ideal medium the output can be exactly determined by simply per-
forming the above discrete-convolution. Consequently, the inputs can be determined
by simply using a zero-forcing equalizer. Observe that for the TDMR system all of
the noise is due to the noise in the medium. That is, xi,j = xi,j = (h−1 ∗ y)i,j, where
h−1 corresponds to the inverse of the channel response and it can be pre-calculated.
For TDMR systems with non-ideal medium, we ignore the irregular grain regions
and simply use the same zero-forcing equalizer. Obviously, in this case xi,j 6= xi,j
and in fact, xi,j ∈ R. We refer to this quantity as effective magnetization, as it is
proportional to the difference in the area of the region with opposite polarities in
the (i, j)th cell.
Decision on xi,j can be made based on the pdf of xi,j. For a given system
model, the pdf of xi,j can be determined by simulations. Fig. 7.4 and Fig. 7.5 shows
the pdf of effective magnetization for the Voronoi-grain model and the random-grain
model, respectively. The two bell-shaped curves correspond to the probabilities
p(xi,j/xi,j = −1) and p(xi,j/xi,j = +1), respectively. The pdf corresponding to
158
−4 −3 −2 −1 0 1 2 3 40
0.01
0.02
0.03
0.04
0.05
0.06
x
p(x|x
)
x = −1x = +1
Figure 7.4: Pdf of effective magnetization determined by simulations using theVoronoi-grain model.
the random-grain model is wider than the one corresponding to the Voronoi-grain
model, because of the presence of write-errors. The knowledge of these pdf’s is used
to make decision on the inputs. Specifically, some threshold t is chosen and the
decision on the input is made based on the following rule,
xi,j =
1 if xi,j ≥ t
−1 if xi,j ≤ −t
e if − t < xi,j < t.
Where, e indicates erasure. With the knowledge of the pdf of effective magnetization
and the above rule, we can determine the error and erasure rate for a given threshold.
Observe that the errors and erasures may be correlated in both dimensions, but for
further analysis we assume them to be independent.
159
−4 −3 −2 −1 0 1 2 3 40
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
x
p(x|x
)
Figure 7.5: Pdf of effective magnetization determined by simulations using theRandom-grain model.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Error Probability
Era
sure
Pro
babi
lity
0.20.40.60.80
Figure 7.6: Error and erasure rates for various thresholds and variances of the grainarea, when TDMR is modeled using Voronoi-grain model.
7.3 Capacity of TMDR
When TDMR is modeled using the Voronoi-grain model, the error and erasure rates
are as shown in Fig. 7.6 for various thresholds. As the threshold t increases, the
160
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
0.4
0.5
0.6
0.7
0.8
0.9
1
Error Rate
Cap
acity
0.20.40.60.80
Figure 7.7: Capacity for BEEC corresponding to the error and erasure rates shownin Fig. 7.6.
error rate decreases and the erasure rate increases. When t = 0, error rate is about
6.5% and erasure rate is 0 for γ = 0. Note that the error and erasure rate drop
significantly for even small increase in γ. Since, there are no write errors in this
model, all errors are entirely readback errors. Assuming that the errors and erasures
occur independently, the capacity of the corresponding BEEC can be calculated as,
C = (1 − ǫ) + (1 − p − ǫ) log2(1 − p − ǫ) +
p log2(p) − (1 − ǫ) log2(1 − ǫ), (7.1)
where, ǫ and p denote the erasure and error rate, respectively. Fig. 7.7 shows the
capacity for the BEEC corresponding to the error and erasure rates shown in Fig.
7.6. For a given system, the threshold can be chosen such that the capacity is max-
imized. For γ = 0, the maximum capacity is about 0.73 and increases significantly
161
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Error Probability
Era
sure
Pro
babi
lity
Figure 7.8: Error and erasure rates for various thresholds, when TDMR is modeledusing the random-grain model. The variance of grain area is same as that obtainedusing the Voronoi-grain model with γ = 0.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Error probability
Cap
acity
Figure 7.9: Capacity for BEEC corresponding to the error and erasure rates shownin Fig. 7.8.
for small increase in γ. Also, note that the threshold that maximizes the capacity
depends on γ.
Fig. 7.8 shows the corresponding error and erasure rates for random-grain
162
model (with q = 7%). The error rates are much higher compared to the Voronoi-
grain model primarily due to the presence of write-errors, which is about 8%. Ne-
glecting the write-errors, the error rate is 7.5% (for t = 0), similar to the error rate
obtained for the Voronoi-grain model with γ = 0 (which results in almost the same
variance of the grain area as the random-grain model). The effect of write-errors
significantly reduces the capacity as shown in Fig. 7.9. In this case, the maximum
capacity is 0.4. This capacity acts as a lower bound for the TDMR capacity.
7.4 Summary
In this chapter, we described an experimental technique to calculate lower bounds
on capacity of two-dimensional magnetic recording system. Using the Voronoi-grain
model, the lower bound on the capacity was shown to be about 0.73 ignoring write-
errors. Using the Random-grain model, which incorporates write-errors, we showed
that the lower bound on capacity reduces to 0.4.
163
CHAPTER 8
CONCLUSIONS
Next-generation magnetic recording systems that attempt to achieve areal densities
beyond 1 Tb/in2 will require a number of innovations in various components of
the system. This dissertation contributes toward the advancement of various read-
channel algorithms.
In Chapter 3 and 4, we proposed two methods to improve upon the per-
formance of turbo-equalizer. We showed that the joint use of both the channel
and code structure information to decode is an effective way to improve error rate
performance beyond that achieved by turbo-equalizer. In one method, we showed
that this can partially be achieved by incorporating some LDPC parity checks in
the channel trellis. Our simulation results indicate that even the inclusion of a
small number of parity checks yield significant performance improvement either in
terms of the error rate or the number of iterations required for decoding conver-
gence. Due to complexity constraints, this method has a practical limitation to
the number of checks that can be incorporated in the trellis. This limitation was
164
partially mitigated by developing a method to incorporate the maximum number
of checks for a given complexity. Although a very practical method when the num-
ber of checks included is small, it is difficult to extend this algorithm without an
unacceptable increase in complexity. However, in the second method, a more ad-
vanced decoding algorithm was developed, which while already providing a better
performance than any other joint decoder available today, also has the potential for
further improvements. The method of exchanging information on channel output
symbols rather than on channel binary inputs is fundamentally different from other
decoders. Although, more complex than the turbo-equalizer, it appears that this
method provides a natural way to combine information from both the channel and
the LDPC code for channels with memory. More importantly, unlike the current
methodologies, it does not pose a particular difficulty for channels with memory
more than 1. Consequently, the superior performance of our joint symbol decoder
over turbo-equalizer for channels with memory more than 1 was observed.
Further improvements in areal density requires radically new architectures.
In Chapter 5, we proposed a novel ternary magnetic recording architecture from a
read-channel perspective. With the development of corresponding optimal signal
processing algorithms, we showed that this system is equivalent to a two-track/two-
head magnetic recording system with ITI. Use of parallel multiple heads is known
to be the best way to mitigate interference from adjacent tracks. Unfortunately, the
165
prohibitive cost and complexity involved in using more than one head has always
discouraged any further development in this direction. The simulation results for
the ternary system suggests that it may be possible to derive, at least partially, the
benefits of having multiple heads without actually having one. As with other new
recording technologies proposed, it would be premature to conclude on the merits
and demerits of this system without considering the numerous other factors in a
practical magnetic disk drives.
Research into HAMR and TDMR systems are in a relatively more advanced
stage. At least in theory, their potential is known, although there is a lot of uncer-
tainty with respect to harnessing it. In Chapter 6, using a sophisticated numerical
model, we provided a detailed analysis of the effects of thermal profile on the transi-
tion characteristics of the HAMR channel. In particular, it was determined that the
position of laser in the head assembly is an important consideration in optimizing
the channel response. Further, we showed that the process of heating while writing
can significantly alter the non-linear transition shift experienced by the transitions.
In Chapter 7, we discussed some channel modeling techniques for TDMR system,
that have proved to be useful for the evaluation of read-channel algorithms. More-
over, using this model we determined experimentally some bounds on the capacity
of such systems. These results are encouraging enough to continue research in this
area, although it was determined that the write-errors can be a prohibitive factor
166
in the TDMR system.
167
REFERENCES
[1] R. G. Gallager, “Low-density parity-checks codes,” IEEE Trans. Inform. The-ory, vol. 8, no. 1, pp. 21–28, Jan. 1962.
[2] D. J. C. Mackay and R. M. Neal, “Near Shannon limit performance of low-density parity-check codes,” Electronics Lett., vol. 32, no. 18, pp. 1645–1646,Aug. 1996.
[3] J. L. Fan, A. Friedmann, E. Kurtas, and S. W. McLaughlin, “Low density par-ity check codes for magnetic recording,” in Proc. 37th Annual Allerton Confer-ence on Communication, Control and Computing, Sept. 1999, pp. 1314–1323.
[4] B. Vasic, B. Djordjevic, and R. Kostuk, “Low-density parity check codes anditerative decoding for long haul optical communication systems,” J. LightwaveTechnol., vol. 21, no. 2, pp. 438–446, Feb. 2003.
[5] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes,” in Proc. IEEE InternationalConference on Communications (ICC), vol. 2, Geneva, Switzerland, May 1993,pp. 1064–1070.
[6] J. Hagenauer and P. Hoeher, “A Viterbi algorithm with soft-decision outputsand its applications,” in Proc. IEEE Global Telecommunications Conference(GLOBECOM), vol. 3, Dallas, Texas, Nov. 1989, pp. 1680–1686.
[7] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linearcodes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. 20,no. 2, pp. 284–287, Mar. 1974.
[8] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inform. Theory, vol. 47, no. 2, pp. 498–519,Feb. 2001.
[9] R. Radhakrishnan, S. Sankaranarayanan, F. Erden, and B. Vasic, “Detectors
168
with auxiliary ldpc parity checks,” Seagate Technology, Pittsburgh, PA, Tech.Rep., Aug. 2008.
[10] J. S. Yedidia, W. Freeman, and Y. Weiss, “Constructing free-energy approxi-mations and generalized belief propagation algorithms,” IEEE Trans. Inform.Theory, vol. 51, no. 7, pp. 2282–2312, July 2005.
[11] P. Pakzad and V. Anantharam, “Kikuchi approximation method for joint de-coding of LDPC codes and partial-response channels,” IEEE Trans. Commun.,vol. 54, no. 7, pp. 1149–1153, July 2006.
[12] B. M. Kurkoski, P. H. Siegel, and J. K. Wolf, “Joint message-passing decodingof LDPC codes and partial-response channels,” IEEE Trans. Inform. Theory,vol. 48, no. 6, pp. 1410–1422, June 2002.
[13] G. Colavolpe and G. Germi, “On the application of factor graphs and the sumproduct algorithm to ISI channels,” IEEE Trans. Commun., vol. 53, no. 5, pp.818–825, May 2005.
[14] G. Colavolpe, “On LDPC codes over channels with memory,” IEEE Trans.Wireless Commun., vol. 5, no. 7, pp. 1757–1765, July 2006.
[15] R. Radhakrishnan and B. Vasic, “Joint message-passing symbol-decoding ofLDPC coded signals over partial-response channels,” in Proc. IEEE Interna-tional Conference on Communications (ICC), Dresden, Germany, June 2009.
[16] ——, “Ternary magnetic recording system,” IEEE Trans. Magn., submittedfor publication.
[17] R. E. Rottmayer et al., “Heat-assisted magnetic recording,” IEEE Trans.Magn., vol. 42, no. 10, pp. 2417–2421, Oct. 2006.
[18] R. Wood, M. Williams, A. Kavcic, and J. Miles, “The feasibility of magneticrecording at 10 tera bits per square inch on conventional media,” in Proc.of The Magnetics Recording Conference (TMRC), New York, NY, USA, July2008.
[19] B. Terris et al., “Patterned media for future magnetic data storage,” Microsys-tem Technologies, vol. 13, no. 2, pp. 189–196, Nov. 2006.
169
[20] R. Radhakrishnan, F. Erden, C. He, and B. Vasic, “Transition response charac-teristics of heat-assisted magnetic recording and their performance with MTRcodes,” IEEE Trans. Magn., vol. 43, no. 6, pp. 2298–2300, June 2007.
[21] R. Radhakrishnan, B. Vasic, F. Erden, and C. He, Characterization of heat-assisted magnetic recording, ser. DIMACS Series in Discrete Mathematics andTheoretical Computer Science, Feb. 2008, vol. 73, pp. 23–40.
[22] R. Radhakrishnan, F. Erden, C. He, and B. Vasic, “Transition responsecharacteristics of heat-assisted magnetic recording,” in Proc. of 10th JointMMM/Intermag Conference, Baltimore, USA, Jan. 2007.
[23] A. R. Krishnan, R. Radhakrishnan, and B. Vasic, “Read-channel modeling fordetector design of two-dimensional magnetic recording systems,” IEEE Trans.Magn., accepted for publication.
[24] A. R. Krishnan, R. Radhakrishnan, A. Kavcic, B. Vasic, W. Ryan, and F. Er-den, “Two-dimensional magnetic recording: read channel modeling and detec-tion,” IEEE Trans. Magn., accepted for publication.
[25] R. Radhakrishnan, A. R. Krishnan, and B. Vasic, “Error and erasure ratefor two-dimensional magnetic recording systems,” in International Conferenceon Telecommunications in Modern Satellite, Cable and Broadcasting Services(TELSIKS), Nis, Serbia, Oct. 2009, accepted for publication.
[26] G. D. Forney, “Maximum-likelihood sequence estimation of digital sequences inthe presence of intersymbol interference,” IEEE Trans. Inform. Theory, vol. 18,no. 3, pp. 363–378, May 1972.
[27] J. L. Fan and J. M. Cioffi, “Constraint coding techniques for soft-iterativedecoders,” in Proc. IEEE Global Telecommunications Conference (GLOBE-COM), vol. 1B, Rio de Janeireo, Brazil, Dec. 1999, pp. 723–727.
[28] J. Moon and B. Brickner, “Maximum transition run codes for data storagesystems,” IEEE Trans. Magn., vol. 32, no. 5, pp. 3992–3994, Sept. 1996.
[29] R. D. Cideciyan, J. D. Coker, E. Eleftheriou, and R. L. Galbraith, “Noise basedmaximum likelihood detection combined with parity-based post-processing,”IEEE Trans. Magn., vol. 37, no. 2, pp. 714–720, Mar. 2001.
170
[30] T. Conway, “A new target response with parity coding for high density mag-netic recording channels,” IEEE Trans. Magn., vol. 34, no. 4, pp. 2382–2386,July 1998.
[31] R. Lynch, E. M. Kurtas, A. Kuznetsov, E. Yeo, and B. Nikolic, “The searchfor a practical iterative detector for magnetic recording,” IEEE Trans. Magn.,vol. 40, no. 1, pp. 213–218, Jan. 2004.
[32] Y. Kou, S. Lin, and M. P. C. Fossorier, “Low-density parity-check codes basedon finite geometries: A rediscovery and new results,” IEEE Trans. Inform.Theory, vol. 47, no. 7, pp. 2711–2736, Nov. 2001.
[33] D. Mackay. Encyclopedia of sparse graph codes. [Online]. Available:http://www.inference.phy.cam.ac.uk/mackay/codes/data
[34] G. D. Forney, “Coset codes - part II: Binary lattices and related codes,” IEEETrans. Inform. Theory, vol. 34, pp. 1152–1187, Sept. 1988.
[35] F. R. Kschischang and V. Sorokine, “On the trellis structure of block codes,”IEEE Trans. Inform. Theory, vol. 41, pp. 1924–1937, Nov. 1995.
[36] R. J. McEliece, “On the BCJR trellis for linear block codes,” IEEE Trans.Inform. Theory, vol. 42, no. 4, pp. 1072–1092, July 1996.
[37] R. E. Rottmayer et al., “Heat-assisted magnetic recording,” IEEE Trans.Magn., vol. 42, no. 10, pp. 2417–2421, Oct. 2006.
[38] J. G. Proakis, Digital communications, 4th ed. McGrawHill, 2001.
[39] L. C. Barbosa, “Simultaneous detection of readback signals from interferingmagnetic recording tracks using array heads,” IEEE Trans. Magn., vol. 26,no. 5, Sept. 1990.
[40] J. Moon and W. Zeng, “Equalization for maximum likelihood detectors,” IEEETrans. Magn., vol. 31, no. 2, Mar. 1995.
[41] R. Venkataramani and F. Erden, “A posteriori equivalence: a newperspective for design of optimal channel shortening equalizers,” IEEETrans. Inform. Theory, submitted for publication. [Online]. Available:http://arxiv.org/abs/0710.3802
171
[42] E. Soljanin and C. N. Georghiades, “Multihead detection for multitrack record-ing channels,” IEEE Trans. Inform. Theory, vol. 44, no. 7, pp. 2988–2997, Nov.1998.
[43] B. G. Roh, S. U. Lee, and J. Moon, “Single-head/single-track detection ininterfering tracks,” IEEE Trans. Magn., vol. 38, no. 4, pp. 1830–1838, July2002.
[44] B. Vasic and E. Kurtas, Coding and signal processing for magnetic recordingsystems. CRC Press, 2004.
[45] K. A. S. Immink, P. H. Siegel, and J. K. Wolf, “Codes for digital recorders,”IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2260–2299, Oct. 1998.
[46] M. H. Kryder, “Future trends in magnetic storage technology,” in Proc. ofNorth American Perpendicular Magnetic Recording Conference (NAPMRC),Jan. 2003, pp. 68–68.
[47] T. W. McDaniel and W. A. Challener, “Issues in design of media for hybridrecording (invited paper),” Trans. of Magn. Soc. of Japan, vol. 2, no. 4, pp.316–321, June 2002.
[48] T. W. McDaniel, W. A. Challener, and K. Sendur, “Issues in heat-assistedperpendicular recording,” IEEE Trans. Magn., vol. 39, no. 4, pp. 1972–1979,July 2003.
[49] W. A. Challener, T. W. McDaniel, C. Mihalcea, K. Mountfield, K.Pelhos, andK. Sendur, “Light delivery techniques for heat-assisted magnetic recording,”Jpn. J. Appl. Phys., vol. 42, no. 2B, pp. 981–988, 2003.
[50] T. Rausch, J. A. Bain, D. D. Stancil, and T. E. Schlesinger, “Thermal williams-comstock model for predicting transition length in a heat-assisted magneticrecording system,” IEEE Trans. Magn., vol. 40, no. 1, pp. 137–147, Jan. 2004.
[51] Y. Zhang and H. N. Bertram, “A theoretical study of nonlinear transitionshift,” IEEE Trans. Magn., vol. 34, no. 4, pp. 1955–1957, July 1998.
[52] M. F. Erden, T. Rausch, and W. Challener, “Cross-track location and tran-sition parameter effects in heat-assisted magnetic recording,” IEEE Trans.Magn., vol. 41, no. 6, pp. 2189–2194, June 2005.
172
[53] O. Karlqvist, “Calculation of magnetic field in the ferromagnetic layer of amagnetic drum,” Trans. Roy. Inst. Tech., no. 86, 1954.
[54] W. X. Shan and T. M. Alexander, Magnetic information storage technology.Academic Press, 1999.
[55] W. Westmijze, “Studies on magnetic recording,” Phillips Res. Rep. Part II,vol. 8, no. 3, 1953.
[56] J. C. Mallinson and H. N. Bertram, “On the characteristics of pole-keeper headfields,” IEEE Trans. Magn., vol. 20, no. 5, pp. 721–723, Sept. 1984.
[57] ——, “A theoretical and experimental comparison of longitudinal and verticalmodes of magnetic recording,” IEEE Trans. Magn., vol. 20, no. 3, pp. 461–467,May 1984.
[58] H. N. Bertram, Theory of magnetic recording. University Press, 2003.
[59] T. Rausch, “Experimental and theoretical investigation of HAMR,” Ph.D. dis-sertation, Carnegie Mellon University, Pittsburgh, PA, USA, 2003.
[60] J. Goldberg and J. Wolf, “Implementation and analysis of nonlinear effectsin the microtrack model,” IEEE Trans. Magn., vol. 35, no. 5, pp. 2256–2258,Sept. 1999.
