:..ECTROMAGNETISM
`
`:s and engineering which still plays
`1ew technology. Electromagnetism
`ers, material scientists, and applied
`,e aspects of applied and theoretical
`ly important in modern and rapidly
`the needs of researchers, students,
`
`Magnetic Information Storage
`Technology
`
`Shan X. Wang
`Department of Materials Science and Engineering
`(and of Electrical Engineering)
`Stanford University
`Stanford, California
`
`Alexander M. Taratorin
`IBM
`Almaden Research Center
`San Jose, California
`
`~
`TSM
`
`sEDITOR
`E PARK, MARYLAND
`
`@
`
`. \•,
`
`ACADEMIC PRESS
`San Diego London Boston
`New York Sydney Tokyo Toronto
`
`UMN EXHIBIT 2024
`LSI Corp. et al. v. Regents of Univ. of Minn.
`IPR2017-01068
`
`
`Page 1 of 95
`
`

`

`The cover page illustrates a hard disk drive used in modem computers. Hard
`disk drives are the most widely used information storage devices. The block
`diagram shows the principle of the state-of-the-art disk drives. Any information
`such as text files or images are first translated into binary user data, which are
`then encoded into binary channel data. The binary data are recorded in magnetic
`disks by a write head. For example, the magnetization pointing to the right (or
`left) represents "l" (or "O"). The magnetization pattern generates a voltage
`waveform when passing underneath a read head, which could be integrated with
`the write head and located near the tip of the stainless suspension. The voltage
`waveform is then equalized, i.e., reshaped into a proper form. This equalization
`step, along with a so-called maximum likelihood detection algorithm, allows us
`to detect the binary channel data with a high reliability. The trellis diagram shown
`with 16 circles and 16 arrows is the foundation of maximum likelihood detection.
`The detected binary channel data are then decoded back to the binary user data,
`which are finally translated back to the original information.
`
`This book is printed on acid-free paper. 0
`
`Copyright © 1999 by Academic Press
`
`All rights reserved.
`No part of this publication may be reproduced or transmitted in any form or by
`any means, electronic or mechanical, including photocopy, recording, or any
`information retrieval system, without permission in writing from the publisher.
`
`ACADEMIC PRESS
`a division of Harcourt Brace & Company
`525 B Street, Suite 1900, San Diego, California 92101-4495, USA
`http:/ /www.apnet.com
`
`Academic Press
`24-28 Oval Road, London NWl 7DX, UK
`http://www.hbuk.co.uk/ap/
`
`Library of Congress Cataloging-in-Publication Data
`
`Wang, Shan X.
`Magnetic information storage technology / Shan
`X. Wang, Alex Taratorin.
`p. cm. -
`(Electromagnetism)
`Includes index.
`ISBN 0-12-734570-1
`I. Wang, Shan X.
`1. Magnetic recorders and recording.
`II. Taratorin, A. M.
`III. Title.
`IV. Series.
`TK7881.6.W26
`1998
`621.39'7-dc21
`
`98-24797
`CIP
`
`Printed in the United States of America
`99 00 01 02 03 MB
`9 8 7 6 5 4 3 2 1
`
`I
`
`
`Page 2 of 95
`
`

`

`Contents
`
`Foreword
`Preface
`List of Acronyms
`List of Symbols
`
`1. Introduction
`1.1 Overview of Digital Magnetic Information Storage
`1.1.1 Information Storage Hierarchy
`1.1.2 Magnetic and Optical Storage Devices
`1.1.3 Overview of Magnetic Hard Disk Drives
`1.1.4 History and Trends in Digital Magnetic Recording
`1.2 Review of Basic Magnetics
`1.2.1 Magnetic Field
`1.2.2 Unit System
`1.2.3 Demagnetizing Field
`1.2.4 Ferromagnetism and Magnetic Hysteresis
`1.2.5 Magnetic Anisotropy
`1.2.6 Magnetic Domains
`1.2.7 Magnetization Process
`
`2. Fundamentals of Inductive Recording Head and Magnetic
`Medium
`2.1 Fourier Transforms of Magnetic Fields in Free Space
`2.2 Inductive Recording Heads
`2.2.1 Karlqvist Head Equation
`2.2.2 Head Magnetic Circuit
`2.3 Magnetic Recording Medium
`2.3.1 Magnetic Fields of Step Transition
`2.3.2 Magnetic Fields of Finite Transition
`
`3. Read Process in Magnetic Recording
`3.1 Reciprocity Principle
`3.2 Readback from Single Transition
`3.3 Readback from Multiple Transitions
`3 .3 .1 Linear Bit Shift
`3.3.2 Square-wave Recording and Roll-off
`
`xv
`xvii
`xxi
`xxiv
`
`1
`1
`1
`3
`5
`10
`12
`12
`14
`15
`17
`20
`23
`24
`
`31
`31
`35
`35
`40
`42
`43
`44
`. ;_'-.
`49
`49
`52
`56
`57
`58
`
`ix
`
`
`Page 3 of 95
`
`

`

`X
`
`Contents
`
`4. Write Process in Magnetic Recording
`4.1 Models of Write Process
`4.1.1 Williams-Comstock Model
`4.1.2 Head Imaging and Relaxation Effects
`4.1.3 Demagnetization Limit
`4.2 Nonlinearities in Write Process
`
`5. Inductive Magnetic Heads
`5.1 Modeling of Inductive Heads
`5.1.1 Analytical Models
`5.1.2 Transmission Line Models
`5.1.3 Finite Element and Boundary Element Modeling
`5.1.4 Micromagnetic Models
`5.2 Fabrication of Thin-Film Inductive Heads
`5.2.1 Wafer Fabrication Process
`5.2.2 Head-Slider Form Factor
`5.3 Inductive Head as Reader
`5.3.1 Undershoot
`5.3.2 Readback Noises of Inductive Readers
`5.4 Inductive Head as Writer
`5.4.1 Write Head Design Considerations
`5.4.2 Head Field Rise Time
`5.5 Side Read, Side Write, and Side Erase
`5.5.1 Side Fringing Field
`5.5.2 Side Read/Write/Erase
`5.5.3 Track Profile
`5.6 High-Saturation Inductive Head Materials
`5.6.1 Requirements of Inductive Head Materials
`5.6.2 Survey of New Inductive Head Materials
`
`6. Magnetoresistive Heads
`6.1 Anisotropic Magnetoresistive (AMR) Effect
`6.2 Magnetoresistive Read Head and Bias Schemes
`6.2.1 Transverse Bias of Magnetoresistive Heads
`6.2.2 Longitudinal Bias of Magnetoresistive Heads
`6.3 Magnetoresistive Head Readback Analysis
`6.3.1 Reciprocity Principle for Magnetoresistive Heads
`6.3.2 Small Gap and Thin Medium Limit
`6.3.3 Advantage of Magnetoresistive Over Inductive Read Head
`6.3.4 Fourier Transform of Magnetoresistive Head Read Signal
`6.3.5 Transmission Line Model of Magnetoresistive Heads
`6.4 Practical Aspects of Magnetoresistive Heads
`6.4.1 Side Reading Asymmetry
`6.4.2 Readback Nonlinearity
`6.4.3 Baseline Shift
`
`.. \ ..
`
`65
`65
`67
`73
`75
`76
`
`81
`82
`82
`85
`88
`90
`90
`90
`96
`99
`99
`100
`103
`104
`107
`110
`111
`112
`113
`114
`115
`117
`
`123
`123
`125
`128
`133
`13$
`139
`144
`146
`147
`148
`150
`151
`154
`155
`
`
`Page 4 of 95
`
`

`

`Contents
`
`Contents
`
`ing
`
`ffects
`
`ement Modeling
`
`Heads
`
`rders
`
`IS
`
`;e
`
`terials
`-Aaterials
`:aterials
`
`Effect
`.s Schemes
`1e Heads
`five Heads
`alysis
`·esistive Heads
`it
`er Inductive Read Head
`five Head Read Signal
`~toresistive Heads
`Heads
`
`65
`65
`67
`73
`75
`76
`
`81
`82
`82
`85
`88
`90
`90
`90
`96
`99
`99
`100
`103
`104
`107
`110
`111
`112
`113
`114
`115
`117
`
`123
`123
`125
`128
`133
`138
`139
`144
`146
`147
`148
`150
`151
`154
`155
`
`6.4.4 Thermal Asperity
`6.4.5 Electrostatic Discharge Sensitivity
`6.5 Fabrication of MR Heads
`6.5.1 Magnetoresistive Head Wafer Process
`6.5.2 Materials Used in Magnetoresistive Head Fabrication
`6.6 Giant Magnetoresistive (GMR) Heads
`6.6.1 Giant Magnetoresistance Effect
`6.6.2 GMR Read Heads
`6.6.3 Tunneling Magnetoresistive Effect
`
`7. Magnetic Recording Media
`7.1 Magnetic Medium Requirements
`7.2 Particulate Media
`7.3 Thin-Film Media
`7.3.1 Magnetic Recording Layer
`7.3.2 Microstructures of the Under/ayer and Magnetic Layer
`7.3.3 Tribological Requirements
`7.3.4 Fabrication of Low-Noise Thin-Film Disks
`7.3.5 Thermal Stability of Bits
`7.4 Keepered Media
`7.4.1 Effect of Keeper on Spacing Loss
`7.4.2 Modulated Head Efficiency Model
`7.4.3 Extension to Magnetoresistive Heads
`7.5 Patterned Media
`7.5.1 Single-Domain Particles and Superparamagnetic Limit
`7.5.2 Read/Write Processes and Noises in Patterned Media
`7.5.3 Challenges in Patterned Media
`
`8. Channel Coding and Error Correction
`8.1 Encoding for Magnetic Recording Channels
`8.2 Channel Bit Representation by NRZ and NRZI Data
`8.3 Run-length-limited (RLL) Codes
`8.4 User Data Rate, Channel Data Rate, and Flux Frequency
`8.5 Principles of Error Detection and Correction
`8.5.1 Error Detection by Parity Checking
`8.5.2 Hamming Distance and Error Detection/Correction Theorems
`8.5.3 Cyclic Codes
`8.5.4 Matrix Formulation of ECC
`
`9. Noises
`9.1 Noise Formulation
`9.1.1 Power Spectral Density
`9 .1.2 Autocorrelation
`9.1.3 Power Spectral Density of Noise and Signal
`9.2 Medium Noise Mechanisms
`
`xi
`
`157
`158
`159
`159
`162
`163
`163
`168
`171
`
`177
`178
`179
`180
`180
`182
`187
`189
`191
`192
`193
`194
`197
`198
`199
`200
`202
`
`207
`207
`209
`212
`215
`217
`219
`221
`223
`228
`
`233
`234
`235
`'237
`238
`241
`
`
`Page 5 of 95
`
`

`

`xii
`
`Contents
`
`9.2.1 Transition Noise
`9.2.2 Particulate Medium Noise
`9.2.3 Modulation Noise
`9.3 Head and Electronics Noises
`9.4 SNR Considerations for Magnetic Recording
`9 .5 Experimental Measurements of SNR
`9.5.1 Peak-Signal-to-Erased-Medium-Noise Measurement
`9.5.2 Spectrum Analyzer Measurement of SNR
`9.5.3 Cross-Correlation Measurement of SNR
`
`10. Nonlinear Distortions
`10.1 Hard Transition Shift and Overwrite
`10.1.1 Hard Transition Shift
`10.1.2 Overwrite
`10.1.3 Overwrite at a Frequency Ratio of 2
`10.1.4 Overwrite at Different Frequency Ratios
`10.1.5 Spectrum of the Overwrite Signal
`10.2 Nonlinear Transition Shift (NLTS)
`10.2.1 NLTS in Dibit
`10.2.2 Precompensation of NLTS
`10.2.3 Interactions of NLTS with HTS
`10.2.4 NLTS in a Series of Transitions
`10.2.5 Data Rate Effects and Timing NLTS
`10.3 Measurement of Nonlinear Transition Shift
`10.3.1 Method of Spectral Elimination
`10.3.2 Methods Based on Pseudo-random Sequences
`10.3.3 Comparison of Spectral Elimination and Pseudo-random
`Methods
`10.4 Partial Erasure
`10.5 Magnetoresistive Read Head Nonlinearity
`
`11. Peak Detection Channel
`11.1 Peak Detection Channel Model
`11.2 BER at the Threshold Detector
`11.3 BER at the Zero-Crossing Detector
`11.4 Window Margin and Bit-Shift Distribution
`
`242
`247
`253
`256
`257
`260
`261
`262
`264
`
`267
`267
`267
`273
`275
`278
`' 284
`287
`287
`291
`293
`299
`307
`314
`314
`323
`
`328
`331
`335
`
`345
`345
`347
`349
`353
`
`12. PRML Channels
`12.1 Principle of Partial Response and Maximum Likelihood
`12.1.1 PR4 Channel
`12.1.2 Maximum Likelihood Detector
`12.2 Partial Response
`12.2.1 PR4 Channel Bandwidth and Frequency Response
`12.2.2 Partial-response Polynomial and Classification
`12.2.3 Channel and User Densities of PR4, EPR4, and E2PR4
`Channels
`
`· °''
`
`361
`361
`363
`367 ·
`369
`369
`372
`
`380
`
`
`Page 6 of 95
`
`

`

`Contents
`
`Contents
`
`Recording
`\JR
`:-Noise Measurement
`mt of SNR
`t of SNR
`
`write
`
`"?.tltio of 2
`ruency Ratios
`Signal
`'S)
`
`HTS
`ions
`ng NLTS
`tsition Shift
`tion
`·andom Sequences
`nination and Pseudo-random
`
`mlinearity
`
`or
`istribution
`
`:i Maximum Likelihood
`
`'.or
`
`d Frequency Response
`and Classification
`of PR4, EPR4, and E2PR4
`
`242
`247
`253
`256
`257
`260
`261
`262
`264
`
`267
`267
`267
`273
`275
`278
`284
`287
`287
`291
`293
`299
`307
`314
`314
`323
`
`328
`331
`335
`
`345
`345
`347
`349
`353
`
`361
`361
`363
`367
`369
`369
`372
`
`380
`
`12.2.4 Principles of Equalization
`12.3 Clock and Gain Recovery
`12.4 Maximum Likelihood Detection
`12.4.1 PRML State Diagram and Trellis
`12.4.2 Maximum Likelihood or Viterbi Detection Algorithm
`12.4.3 Interleave and Sliding Threshold in PR4 Channel
`12.4.4 Error Events in Maximum Likelihood Detection
`12.5 PRML Error Margin Analysis
`12.6 Performance of PRML Channels
`
`13. Decision Feedback Channels
`13.1 Principle of Decision Feedback
`13.2 Fixed-Depth Tree Search with Decision Feedback and (1,7) Code
`13.3 Equalization for MDFE Channel
`13.4 Error Rate of MDFE Channel and Error Propagation
`13.5 Timing and Gain Recovery in MDFE Channel
`
`14. Off-Track Performance
`14.1 Structure of Magnetic Track
`14.2 Track Misregistration and Off-Track Performance
`14.3 Off-Track Capability and 747 Curve
`
`15. Head-Disk Assembly Servo
`15.1 Head Servomechanism
`15.2 Sector Servo Disk Format
`15.3 Position Error Signal
`
`16. Fundamental Limitations of Magnetic Recording
`16.1 Superparamagnetism and Time-Dependent Coercivity
`16.1.1 Superparamagnetism
`16.1.2 Magnetic Viscosity or Aftereffect
`16.1.3 Time-Dependent Coercivity
`16.2 Dynamic Effects: Medium, Head, and Electronics
`16.2.1 Medium Switching Time
`16.2.2 Write Head and Electronics
`
`17. Alternative Information Storage Technologies
`17.1 Optical Disk Recording
`17.1.1 CD-ROM
`17.1.2 DVD
`17.1.3 Phase-Change Optical Recording
`17.1.4 Magneto-Optic Recording
`17.1.5 Optical Head
`17.1.6 RLL Codes in Optical Disk Recording
`17.1.7 Servomechanism of Optical Head
`
`xiii
`
`381
`388
`393
`393
`399
`403
`408
`415
`421
`
`433
`433
`437
`442
`444
`447
`
`451
`451
`453
`456
`
`467
`467
`469
`472
`
`479
`480
`480
`482
`486
`488
`489
`491
`
`. \,
`
`495
`495
`496
`499
`501
`503
`507
`511
`513
`
`
`Page 7 of 95
`
`

`

`xiv
`
`17.2 Holographic Recording
`17.2.1 Basic Principle of Holography
`17.2.2 Photorefractive Effect
`17.2.3 Holographic Recording System
`17.2.4 Theoretical capacity and practical limitations of holographic
`recording
`17.3 Flash Memory
`17.3.1 Why Flash?
`17.3.2 Program/Erase Endurance
`17.3.3 Flash Memory and Its Future
`17.4 Magnetic RAM
`
`Index
`
`Contents
`
`516
`516
`518
`520
`
`521
`522
`522
`521:
`524
`526
`
`531
`
`
`Page 8 of 95
`
`

`

`CHAPTHR 11
`
`Peak Detection Channel
`
`Peak detection channel is a simple and reliable method of data detection.
`Peak detection was the first detection channel utilized in magnetic disk
`drives. It was extensively used for several decades and is still found in
`many disk drive products. In this chapter we will describe the main
`principles of peal5 detection channel operation and consider the error
`rates of this channel.
`
`11.1 PEAK DETECTION CHANNEL MODEL
`
`;i
`
`A block diagram of a typical peak detection channel is shown in Fig. 11.1.
`It is based on the assumption that each transition results in a relatively
`sharp peak of voltage. The goal of the peak detection channel is to detect
`each individual voltage peak.
`The input analog signal is passed through two paths. One path quali(cid:173)
`fies a peak of voltage by rectification and threshold detection. When a
`voltage level exceeds some threshold, a comparator is turned on and a
`rectangular pulse appears· at the output of the threshold detector.1
`2 The
`'
`other path consists of a differentiator and a zero-crossing detector. A
`voltage peak will correspond to a zero-crossing after differentiation. The
`zero-crossing detector generates a short rectangular pulse for each zero(cid:173)
`crossing. If a zero-crossing is detected and it is located within the region
`where signal amplitude exceeds a specific threshold, a transition is de(cid:173)
`tected and a "qualified" pulse appears at the peak detector output.
`The "coincidence" scheme used in the peak detection channel makes . -,,
`it robust to find the positions of magnetic transitions. If only the threshold
`detector is used, the pulse generated by the rectifier is relatively wide,
`which cannot give the exact location of the transition. On the other hand,
`if only the zero-crossing detector is used, a lot of extra zero-crossings
`
`
`Page 9 of 95
`
`

`

`· 346
`
`CHAPTER 11 Peak Detection Channel
`
`Readback
`Signal
`
`V
`
`Differentiator
`
`Zero-Crossing
`Detector
`
`K
`.._--•I Rectifier. ~ - -~(cid:141)
`
`_
`
`_
`
`~
`L:_I
`
`Detected
`Transitions
`
`Threshold
`Detector
`
`(cid:143).
`
`FIGURE 11.1. Block diagram of peak detection channel.
`
`caused by noises will be mistaken as magnetic transitions. Only when
`· both a zero-crossing and a rectified pulse are detected simultaneously, a
`magnetic transition is found reliably.
`To distinguish between adjacent transitions and to combat instabilities
`of the disk rotational speed, each pulse of voltage is detected inside an
`appropriate detection window, also called a timing window and should· be
`,equal to the channel bit period: A special phase-locked loop (PLL) system
`is used to provide a detection window for each channel bit. The PLL
`updates its frequency based on detected pulses. Each incoming transition
`or voltage pulse is searched inside its detection window. As shown in
`Fig. 11.2, each pulse should be detected after the previous channel bit and
`before the next channel bit, so the timing window is equal to a channel
`bit period or bit cell. If a peak detection channel uses (1,7) modulation
`. encoding, the detection window is equal to 50% of the minimum timing
`distance between two transitions that are written in the magnetic medium.
`The performance of a detection channel is often characterized by
`channel bit rate as well as bit error rate (BER). Bit error rate Pe is the
`probability of mistaking a "O" as a "l", or mistaking a "l" as a "O" due
`to the noises, distortions, or interferences in the channel. The reciprocal
`of Pe means 1 error per 1/P e bits transferred in the channel. Obviously,
`
`
`Page 10 of 95
`
`

`

`I.APTER 11 Peak Detection Channel
`
`11.2 BER AT THE THRESHOLD DETECTOR
`
`347
`
`_L
`
`Zero-Crossing
`Detector
`
`~ L:_J '
`
`"hreshold
`>elector
`
`Detected
`Transitions
`
`Channel Clock
`
`I
`11
`
`I
`QI
`I
`
`I
`I
`11
`
`.
`I
`etect,on
`D
`Window
`(Bitcell)
`
`Det~ction
`Window
`(Bitcell)
`
`Detebtion
`Window
`(Bitcell)
`
`tion channel.
`
`FIGURE 11.2. Clock and detection window in peak detection channel.
`
`nagnetic transitions. Only when
`;e are detected simultaneously, a
`
`sitions and to combat instabilities
`of voltage is detected inside an
`l a timing window and should be
`phase-locked loop (PLL) system
`r for each channel bit. The PLL
`pulses. Each incoming transiti~n
`:letection window. As shown m
`1fter the previous channel bit and
`tg window is equal to a channel
`1 channel uses (1,7) modulation
`1 to 50% of the minimum timing
`written in the magnetic medium.
`annel is often characterized by
`? (BER). Bit error rate Pe is the
`or mistaking a "l" as a "0" due
`~s in the channel. The reciprocal
`~rred in the channel. Obviously,
`
`we would like Pe to be as small (ideally zero) as possible. The BER can
`be reduced through error detection and correction. At present, the corrected
`BER is usually in 10- 12-10-11 range, while the raw (uncorrected) BER is
`typically 10-9-10-7.
`The.error rate at the threshold detector is determined by the probabil(cid:173)
`ity of drop-outs when the pulse amplitude falls below the specified thresh(cid:173)
`old or the probability of strong noise outbursts when total media and
`electronic noise exceeds the specified signal level. The error rate at the
`zero-crossing detector is determined by random shifts of the zero-crossing
`position from the correct peak location. Random noises cause fluctuations
`of the zero-crossing position in the differentiated readback signal. An
`error occurs when the zero-crossing position falls beyond the detection
`window, i.e., when zero-crossing is detected earµ.er or later than the cur(cid:173)
`rent bit cell. Next we will examine how to calculate BER.
`
`11.2 BER AT THE THRESHOLD' DETECTOR
`
`The error rate of the threshold detector may be calculated from the channel
`SNR. If the zero-to-peak signal voltage amplitude is V0 _P and the rms
`
`
`Page 11 of 95
`
`

`

`11.3 BEE
`
`Sim:
`burst ex,
`
`Therefon
`is stable;
`of the cl,
`channel E
`at the tlu
`for by tb
`medium
`above.
`
`11.3 BEE
`
`Zero-crosE
`tive of vo]
`both ofwl
`is equal tc
`signal are
`These zero
`An err
`shifts out c
`hasaGaus1
`ity of a bit
`
`Theem
`for the all
`readback v.,
`~ecording ru
`IS:
`
`348.
`
`CHAPTER 11 Peak Detection Channel
`
`noise voltage is V rms,nt then the SNR at the threshold detector is defined
`as:
`
`SNR(dB) = 20 log : (cid:143) -p .
`rms, n
`
`(11.1)
`
`To estimate the BER of the threshold detector in the peak detection channel,
`we assume that the noise voltage n in the recording channel is approxi(cid:173)
`mately Gaussian with a zero mean value and a standard deviation of
`<T = V rms,n- This means that the probability density of the noise voltage
`n is given by the Gaussian distribution:
`
`(11.2)
`
`Now we will consider the probability of errors in the threshold detec(cid:173)
`tor with a threshold fixed at 50% of the zero-to-peak signal amplitude. If
`there is no peak of voltage in the channel, we may detect a false transition
`when the noise outburst exceeds one-half of zero-to-peak signal voltage.
`The probability of this error event, i.e., mistaking a "O" for "l", is given
`by the following integral of the Gaussian distribution function:
`
`P011 = r . ~ 1---w exp(-t~)dn.
`
`v0_p/2 V 2'TT<T
`
`(11.3)
`
`Define the following integral of the Gaussian distribution as the comple(cid:173)
`mentary error function:
`
`erfc(x) = ,J;. r exp( -y2 )ay.
`
`Alternately, one can define a Q(x) function:
`
`f""
`1
`Q(x) = ~ rn--
`v 27T x
`
`1
`~ 1n
`2
`e-y 12dy = -
`erfc(x/ v 2),
`2
`
`which represents the probability that a unit-variance zero-mean Gaussi
`noise exceeds x. Then we can easily obtain that the BER is.
`
`(VSNR)
`(VSNR)
`1
`( Va-p )
`1
`2V2 = Q -
`2V2<r = 2 erfc
`P011 = 2erfc
`
`2-
`
`•
`
`
`Page 12 of 95
`
`

`

`[ER 11 Peak Detection Channel
`
`11.3 BER AT THE ZERO-CROSSING DETECTOR
`
`349
`
`threshold detector is defined
`
`Similarly, a "1" signal will be mistaken as "O" if a negative noise out(cid:173)
`burst exceeding a value of -V0_P/2 occurs. Such a BER can be derived as:
`
`Vo-p.
`T
`rms, n
`
`(11.1)
`
`in the peak detection channel,
`·ecording channel is approxi(cid:173)
`md a standard deviation of
`density of the noise voltage
`
`.!t__)
`2a2-
`.
`
`(11.2)
`
`=rrors in the threshold detec(cid:173)
`·to-peak signal amplitude. If
`may detect a false transition
`zero-to-peak signal voltage.
`tking a "O" for "1", is given
`;tribution function:
`
`2
`n
`- - dn
`)
`2a2-
`.
`
`(11.3)
`
`distribution as the comple-
`
`(11.4)
`
`n2 }
`
`00
`
`(11.6)
`
`I-Vo-p/2 1
`{
`V2'T{(T exp - 2a2 dn
`P110 = _
`=.!.erfc(VSNR) = Q(VSNR).
`
`2
`
`2V2
`
`2
`
`Therefore, the error rate at any bit is a constant if the signal amplitude
`is stable and the channel noise is Gaussian. The BER is a strong function
`of the channel SNR. The expected BER at the threshold detector for a
`channel SNR of 20 dB is 3 X 10-7, while that for a channel SNR of 24 dB
`at the threshold detector is <10- 15. However, the factors not accounted
`for by the above model, such as the signal amplitude instability and
`medium defects, may greatly increase the threshold errors predicted
`above.
`
`11.3 BER AT THE ZERO-CROSSING DETECTOR
`
`Zero-crossing detector locates the transition by looking at the time deriva(cid:173)
`tive of voltage signal. The readback signal V(t) is mixed with noise n(t),
`both of which are differentiated, so the output signal of the differentiator
`is equal to V'(t) + n'(t). As a result, the zero-crossing locations in the
`signal are shifted from the transition locations, as shown in Fig. 11.3.
`2
`These zero-crossing shifts are also called the peak-shifts or bit-shifts.1
`'
`An error will occur at the zero-crossing detector if the zero-crossing
`shifts out of the detection window. Assume that the zero-crossing shift t8
`has a Gaussian distribution with a standard deviation rTt, then the probabil(cid:173)
`ity of a bit error due to the zero-crossing shift can be calculated as:
`00 ~ exp{-2
`t~2 }dt8 = erfc( ~ ~ ) = 2Q(Tw/
`2 V 2rTt
`
`Pe= 2J
`Tw/2
`
`2'1rrTt
`
`°t
`
`2
`).
`
`CTt
`
`(11.7)
`
`rfc(x/V2),
`
`riance zero-mean Gaussian
`1t the BER is.
`
`~) = Q(~). (11.5)
`
`The error rate at the output of the zero-crossing detector can be derived
`for the all "1" s NRZI pattern, whieh has an approximately sinusoidal
`readback waveform: V(t) = V 0 sin(wt), where w = 21Tf = 7rlTw is the
`recording angular frequency. The signal at the output of the differentiator
`is:
`
`.. '-..
`
`dV(t)
`V*(t) = V'(t) = - - =wV0 cos(wt)
`.
`dt
`
`(11.8)
`
`
`Page 13 of 95
`
`

`

`350
`
`CHAPTER 11 Peak Detection Channel
`
`dV(t)!dt
`
`dn(t)ldt
`
`dV(t)ldt+dn(t)!dt
`
`Differentiated Signal
`
`Differentiated Noise
`
`Differentiated
`Signal + Noise
`
`FIGURE 11.3. Zero-crossing and noise (after differentiation) in peak detection
`channel. Tw is the detection window or channel bit period.
`
`Therefore, the zero-crossings occur when
`V*(tn) =O, tn = (2n + l)Tw/2, n = 0, ±1, ±2,
`If the random noise after the differentiator is n*, then the zero-crossing
`is shifted by t5, and they are related by the following:
`
`V* (t + t) + n* = V*(t ) + dV*I t + n* = 0
`dt
`_
`,
`n
`s
`n
`s
`n* = -tsddV*I = -t/d2;/ .
`In
`tn
`
`f
`
`I
`n
`
`f
`
`Calculating the rms value of noise at the differentiator output, we obtain
`that ·
`
`Vrms,n = \i«n*)2) = l~zt;/tnlTt.
`
`Combining Equations (11.8), (11.9), and (11.7), the BER atthe zero-crossing
`detector becomes
`
`
`Page 14 of 95
`
`

`

`R 11 Peak Detection Channel
`
`11.3 BER AT THE ZERO-CROSSING DETECTOR
`
`35l
`
`dV(t)/dt+dn(t)/dt
`
`Differentiated
`Signal + Noise
`
`irentiation) in peak detection
`period.
`
`J, ±1, ±2, ...
`
`n*, then the zero-crossing
`lowing:
`
`ts + n* = 0,
`
`!ntiator output, we obtain
`
`(11.9)
`
`e BER at the zero-crossing
`
`( VSNR*)
`\/2
`.
`
`'. wTw
`
`2
`
`(11.10)
`
`where 'Va-p = wV0 is the zero-to-peak signal amplitude after differentia(cid:173)
`tion, and SNR* = <"Va-p!Vrms,n)2 is the SNR at the output of the differentia- ·
`tor.
`
`Assume that the noise power spectral density from the readback head
`is approximately constant (white noise):
`'T}(w) = 'TJ,
`
`which is valid within the system bandwidth as determined by the cutoff
`anguiar frequency %· The ideal differentiator has the following :frequency
`response:
`
`H(iw) = iw,
`
`Therefore, the noise power after the differentiation is given by the integral:
`
`)2 = J(JJC uJ'T}dW = (JJ~'TJ = ~N
`N* = (V*
`rms, n
`3
`3
`,
`O
`where N = % 'TJ is the noise power before differentiation. Since the differen(cid:173)
`tiated signal has an amplitude of wV0, the SNR after differentiation is
`w2"6
`al
`SNR* = dN/3 = 3 d SNR,
`where SNR = V5/N is the SNR before differentiation. Most of the signal
`energy is concentrated in the frequency range where w < we, so SNR*
`(after differentiation) may be smaller than SNR (before differentiation).
`For example, if w = %12.25, then the SNR is reduced by 2.3 dB due to
`differentiation. For the all 'T's pattern without modulation encoding, w
`= 7rlT w, so the bit-shift error rate becomes
`
`(11.11)
`
`(11.12)
`
`C
`
`C
`
`Pe = erfc 7T
`
`( VsNR*)
`( VSNR)
`\/2 = erfc 2.42
`\/2
`.
`2
`
`2
`
`The argument of the complimentary function is 2.42 times that in Equation
`(11.6). Therefore, in this case, the bit-shift error rate at the zero-crossing
`detector is much smaller than the threshold error rate at the threshold
`detector. In contrast, for the all ''l'' s pattern with (d, k) encoding, w =
`7T!Tw(d + 1), i.e., the detection window is now smaller than transition
`period. As a result, the bit-shift error rate will dominate if d 2c: 2. In
`general, the total BER in a peak detection channel can be expressed as
`
`. ;_ ...
`
`
`Page 15 of 95
`
`

`

`352
`
`CHAPTER 11 Peak Detection Channel
`
`where Pe,bs is the bit-shift error rate at the zero-crossing detector, and
`Pe,th is the threshold error rate at the threshold detector, both of which
`must be much smaller than 1.
`It must be cautioned that SNR is not the only factor that affects the
`error rate at the zero-crossing detector. Both liner intersymbol interference
`(ISI) and nonlinear transition shift (NLTS) can cause peak-shifts. In this
`case, the peaks and the zero-crossings in the readback signal are shifted
`from the desired transition locations, which should be at the center of the
`detection window. Furthermore, noise is mixed to the distorted signal, so
`the zero-crossings may be shifted even more from the center of the detec(cid:173)
`tion window, as illustrated in Fig. 11.4. Consequently, the BER at the zero(cid:173)
`crossing detector increases. Based on Equation (11.7), the bit-shift BER
`taking ISI and NLTS into consideration can be expressed as follows:
`(Tw/2-LI)
`(Tw/2+LI)
`1
`_ 1
`,
`Pe,bs - 2 erfc V2lTt
`+2 erfc V2lTt
`where ±LI is the net bit-shift, and we assumed that the peak has equal
`probabilities to shift early or late. Note that peak detection channels are
`usually used with (1,7) or (2,7) RLL codes, so the transition separations
`are relatively long. In other words, for peak detection channels linear ISI
`is more significant than NLTS. The latter is a critical factor in PRML
`channels (Chapter 12).
`
`dV(t)ldt .
`
`dn(t)!dt
`
`dV(t)!dt+dn(t}!dt
`
`ISi
`
`Differentiated Signal
`
`Differentiated Noise
`
`Tw/2
`. :,_,
`
`ISl+/5
`
`Differentiated
`Signal + Noise
`
`FIGURE 11.4. Bit-shifts caused by intersymbol interference (ISi) and noise ·
`peak detection channel.
`
`
`Page 16 of 95
`
`

`

`1..PTER 11 Peak Detection Channel
`
`the zero-crossing detector, and
`ireshold detector, both of which
`
`,t the only factor that affects the
`,th liner intersymbol interference
`'S) can cause peak-shifts. In this
`t the readback signal are shifted
`ch should be at the center of the
`mixed to the distorted signal, so
`tore from the center of the detec(cid:173)
`:msequently, the BER at the zero(cid:173)
`quation (11.7), the bit-shift BER
`can be expressed as follows:
`(Twf2+L1)
`1
`·-erfc
`~ ~ ,
`2
`V 2(Tt
`
`ssumed that the peak has equal
`that peak detection channels_ are
`~es, so the transition separations
`eak detection channels linear ISI
`ter is a critical factor in PRML
`
`(t
`
`dV(t)I dt+dn(t)I dt
`
`~ t
`,/2
`
`e
`
`Tw/2
`
`181+15
`
`Differentiated
`Signal + Noise
`
`1bol interference GSI) and noise in
`
`11.4 WINDOW MARGIN AND BIT-SHIFT DISTRIBUTION
`
`353
`
`11.4 WINDOW MARGIN AND BIT-SHIFT DISTRIBUTION
`
`The output of the zero-crossing detector of a peak detection channel
`produces sharp pulses at the locations where the zero-crossings are de(cid:173)
`tected, as shown in Fig. 11.5. As we have discussed in the previous section,
`
`Channel Clock
`
`I a,
`
`I
`I
`
`I
`
`I a,
`
`I
`I
`
`I
`
`I
`
`,'ji,
`
`1
`,-;-
`'" f 111
`'" I 111
`'" I 111
`'" I 111
`'" I Jli
`'" I 111
`'" I 11:
`
`1
`,7ii 1hil1
`11 111
`I Ill I
`11 11•
`r 1111
`,,,,. 1 •,11
`1,111
`I Ill I
`,,,,,
`,,,,,
`,,,,.
`" I
`
`I 1111
`t 1,11
`I
`luJ
`I
`
`Detection
`Window
`(Bitcell)
`
`Detection
`Window
`(Bitcell)
`
`Detection
`Window
`(Bitcell)
`
`FIGURE 11.5. The distribution of the readback pulses (top) and the output of
`zero-crossing detector (bottom).
`
`
`Page 17 of 95
`
`

`

`354
`
`CHAPTER 11 Peak Detection Channel
`
`an error occurs when the corresponding pulse falls outside the detection
`window. The error rate at the output of the zero-crossing detector can be
`measured by counting the pulses falling outside the prescribed detection
`window. However, the error rate of an actual magnetic recording channel
`tends to be very low, so the pulse counting measurement may take a long
`time. In fact, the raw BER of a magnetic disk drive is typically -10-9• At
`such a low probability level, more than 1010 bits of data should be collected
`and analyzed to obtain reliable statistics. This would require > 104 disk
`revolutions and at least several minutes of measurement time for each
`experimental condition, just to capture the data bits.
`An effective and fast method for evaluating error performance is
`based on the window margin analysis.3 To understand the principle of
`window margin, let us imagine that we are able to measure the exact
`position of each pulse at the output of the zero-crossing detector and to
`accumulate the histogram of such positions, as shown in Fig. 11.6. The
`height of the histogram H(tk) at each timing position t = tk corresponds
`to the total number of pulses having a timing shift of tk. The sum of H(tk)
`equals to the total number of detected pulses.
`
`I H(t)
`I
`I
`T7
`
`1
`
`\
`
`1
`
`-J
`
`~
`
`.
`
`r
`
`I
`I
`I ,
`I
`I
`
`_;
`
`-Tw/2
`
`~
`
`~ ....
`
`r
`
`t
`
`'
`
`FIGURE 11.6. Histogram of peak shifts at the output of zero-crossing detector.
`
`
`Page 18 of 95
`
`

`

`TER 11 Peak Detection Channel
`
`11lse falls outside the detection
`~ zero-crossing detector can be
`1tside the prescribed detection
`tal magnetic recording channel
`measurement may take a long
`sk drive is typically - 10-9
`• At
`bits of data should be collected
`This would require > 104 disk
`of measurement time for each
`e data bits.
`aluating error performance is
`ro understand the principle of
`are able to measure the exact
`e zero-crossing detector and to
`ms, as shown in Fig. 11.6. The
`.ng position t = tk corresponds
`ting shift of tk. The sum of H(tk)
`uses.
`
`'t)
`
`11.4 WINDOW MARGIN AND BIT-SHIFT DISTRlBUTION
`
`355
`
`Once we know the histogram, the number of pulses with a bit-shift
`value larger than ts, N(ts), can be expressed as the sum of H(tk) for Jtkl >
`ts;
`
`N(ts) = ~ H(tk) + ~ H(tk),
`tk>ts
`tk< -ts
`Obviously, the total number of pulses in the histogram is N = N(t =
`tot
`s
`0). Th~ larger th: ts ~alue, the smaller the N(t5 ) value. If we plot N(ts)/Ntot
`vs. ts m a logarithmic scale, we get a bit-shift plot as shown in Fig. 11.7.
`The horizontal axis of this plot represents the bit-shift value (in nanosec(cid:173)
`onds), and the half detection window (or half timing window) equals to 5 ns.
`When the pulse from the output of the zero-crossing dete