Editing Partial-response maximum-likelihood

{{Short description|Method for interpreting data in digital storage systems}}

In [[computer data storage]], '''partial-response maximum-likelihood''' ('''PRML''') is a method for recovering the [[Digital signal (electronics)|digital data]] from the weak analog read-back signal picked up by the [[Disk read-and-write head|head]] of a magnetic [[Hard disk drive|disk drive]] or [[tape drive]]. PRML was introduced to recover data more reliably or at a greater [[areal density (computer storage)|areal-density]] than earlier simpler schemes such as peak-detection.<ref>G. Fisher, W. Abbott, J. Sonntag, R. Nesin, "[https://ieeexplore.ieee.org/document/542278 PRML detection boosts hard-disk drive capacity]", IEEE Spectrum, Vol. 33, No. 11, pp. 70-76, Nov. 1996</ref> These advances are important because most of the digital data in the world is stored using [[magnetic storage]] on [[hard disk drive|hard disk]] or [[tape drive]]s.

Ampex introduced PRML in a tape drive in 1984. IBM introduced PRML in a disk drive in 1990 and also coined the acronym PRML. Many advances have taken place since the initial introduction. Recent [[read/write channel]]s operate at much higher data-rates, are fully adaptive, and, in particular, include the ability to handle nonlinear signal distortion and non-stationary, colored, data-dependent noise ([[noise-predictive maximum-likelihood detection|PDNP or NPML]]).

''Partial response'' refers to the fact that part of the response to an individual bit may occur at one sample instant while other parts fall in other sample instants. ''Maximum-likelihood'' refers to the detector finding the bit-pattern most likely to have been responsible for the read-back waveform.

== Theoretical development ==
[[File:Class 4 Partial-Response Eye Diagram.jpg|thumb|Continuous-time Partial-Response (class 4) and corresponding 'eye pattern']]
'''Partial-response''' was first proposed by Adam Lender in 1963.<ref>A. Lender, "[https://ieeexplore.ieee.org/abstract/document/6373379 The duobinary technique for high-speed data transmission]", Trans. AIEE, Part I: Communication and Electronics, Vol. 82, No. 2, pp. 214-218, May 1963</ref> The method was generalized by Kretzmer in 1966. Kretzmer also classified the several different possible responses,<ref>E. Kretzmer, "[https://ieeexplore.ieee.org/document/1089288 Generalization of a Technique for Binary Data Communication]", IEEE Trans. Comm., Vol. 14, No. 1, pp. 67-68 Feb. 1966</ref> for example, PR1 is duobinary and PR4 is the response used in the classical PRML. In 1970, Kobayashi and Tang recognized the value of PR4 for the [[magnetic recording]] channel.<ref>H. Kobayashi and D. Tang, "[https://ieeexplore.ieee.org/document/5391640 Application of Partial-response Channel Coding to Magnetic Recording Systems]", IBM J. Res. Dev., Vol, 14, No. 4, pp. 368-375, July 1970</ref>

'''[[Maximum-likelihood]]''' decoding using the eponymous [[Viterbi algorithm]] was proposed in 1967 by [[Andrew Viterbi]] as a means of decoding [[convolutional codes]].<ref>A. Viterbi, "[https://ieeexplore.ieee.org/document/1054010 Error bounds for convolutional codes and an asymptotically optimum decoding algorithm]", IEEE Trans. Info. Theory, Vol. 13, No. 2, pp. 260-269, Apr. 1967</ref>

By 1971, [[Hisashi Kobayashi]] at [[IBM]] had recognized that the Viterbi algorithm could be applied to analog channels with inter-symbol interference and particularly to the use of PR4 in the context of Magnetic Recording<ref>H. Kobayashi, "[https://ieeexplore.ieee.org/document/1054689 Correlative level coding and maximum-likelihood decoding]", IEEE Trans. Inform. Theory, vol. IT-17, PP. 586-594, Sept. 1971</ref> (later called PRML). (The wide range of applications of the Viterbi algorithm is well described in a review paper by [[Dave Forney]].<ref>D. Forney, “[https://www2.isye.gatech.edu/~yxie77/ece587/viterbi_algorithm.pdf The Viterbi Algorithm]”, Proc. IEEE, Vol. 61, No. 3, pp. 268-278, Mar. 1973</ref>) A simplified algorithm, based upon a difference metric, was used in the early implementations. This is due to Ferguson at [[Bell Labs]].<ref>M. Ferguson, ”[https://ieeexplore.ieee.org/document/6774130 Optimal reception for binary partial response channels]” Bell Syst. Tech. J., vol. 51, pp. 493-505, Feb. 1972</ref>

== Implementation in products ==
[[File:PRML chronology circa 1994 (scanned Nov 1, 2019).pdf|thumb|Early PRML chronology (created around 1994)]]
The first two implementations were in Tape (Ampex - 1984) and then in hard disk drives (IBM - 1990). Both are significant milestones with the [[Ampex]] implementation focused on very high data-rate for a digital instrumentation recorder and [[IBM]] focused on a high level of integration and low power consumption for a mass-market HDD. In both cases, the initial equalization to PR4 response was done with analog circuitry but the Viterbi algorithm was performed with digital logic. In the tape application, PRML superseded 'flat equalization'. In the HDD application, PRML superseded [[run-length limited|RLL]] codes with 'peak detection'.

=== Tape recording ===
The first implementation of PRML was shipped in 1984 in the Ampex Digital Cassette Recording System (DCRS). The chief engineer on DCRS was [[Charles Coleman (engineer)|Charles Coleman]]. The machine evolved from a 6-head, transverse-scan, digital [[video tape recorder]]. DCRS was a cassette-based, digital, instrumentation recorder capable of extended play times at very high data-rate.<ref>T. Wood, "[http://www.thic.org/pdf/Oct96/ampex.twood.pdf Ampex Digital Cassette Recording System (DCRS)]", THIC meeting, Ellicott City, MD, 16 Oct., 1996 (PDF)</ref> It became Ampex' most successful digital product.<ref>R. Wood, K. Hallamasek, "[https://www.computerhistory.org/collections/catalog/102788145 Overview of the prototype of the first commercial PRML channel]", Computer History Museum, #102788145, Mar. 26, 2009</ref>

The heads and the read/write channel ran at the (then) remarkably high data-rate of 117&nbsp;Mbit/s.<ref>C. Coleman, D. Lindholm, D. Petersen, and R. Wood, "[https://web.archive.org/web/20191007051735/https://ieeexplore.ieee.org/document/5261308 High Data Rate Magnetic Recording in a Single Channel]", J. IERE, Vol., 55, No. 6, pp. 229-236, June 1985. (invited) (Charles Babbage Award for Best Paper)</ref> The PRML electronics were implemented with four 4-bit, [[Plessey]] [[analog-to-digital converter]]s (A/D) and [https://en.wikichip.org/wiki/fairchild/100k 100k ECL logic].<ref>Computer History Museum, #102741157, "[https://www.computerhistory.org/collections/catalog/102741157 Ampex PRML Prototype Circuit]", circa 1982</ref> The PRML channel outperformed a competing implementation based on "Null-Zone Detection".<ref>J. Smith, "[https://ieeexplore.ieee.org/document/1089924 Error Control in Duobinary Data Systems by Means of Null Zone Detection]", IEEE Trans. Comm., Vil 16, No. 6, pp. 825-830, Dec., 1968</ref> A prototype PRML channel was implemented earlier at 20&nbsp;Mbit/s on a prototype 8-inch HDD,<ref name=8inch>R. Wood, S. Ahlgrim, K. Hallamasek, R. Stenerson, "[https://ieeexplore.ieee.org/document/1063460 An Experimental Eight-inch Disc Drive with One-hundred Megabytes Per Surface]", IEEE Trans. Mag., vol. MAG-20, No. 5, pp. 698-702, Sept. 1984. (invited)</ref> but Ampex exited the HDD business in 1985. These implementations and their mode of operation are best described in a paper by Wood and Petersen.<ref>R. Wood and D. Petersen, "[https://ieeexplore.ieee.org/document/1096563 Viterbi Detection of Class IV Partial Response on a Magnetic Recording Channel]", IEEE Trans. Comm., Vol., COM-34, No. 5, pp. 454-461, May 1986 (invited)</ref> Petersen was granted a patent on the PRML channel but it was never leveraged by Ampex.<ref>D. Petersen, "[https://patents.google.com/patent/US4504872A/en Digital maximum likelihood detector for class IV partial response]", US Patent 4504872, filed Feb. 8, 1983</ref>

=== Hard disk drives ===
In 1990, IBM shipped the first PRML channel in an HDD in the [[History of IBM magnetic disk drives#IBM 0681|IBM 0681]] It was full-height 5¼-inch form-factor with up to 12 of 130&nbsp;mm disks and had a maximum capacity of 857&nbsp;MB.

The PRML channel for the IBM 0681 was developed in [[IBM Rochester]] lab. in Minnesota<ref>J. Coker, R. Galbraith, G. Kerwin, J. Rae, P. Ziperovich, "[https://ieeexplore.ieee.org/document/278677 Implementation of PRML in a rigid disk drive]", IEEE Trans. Magn., Vol. 27, No. 6, pp. 4538-43, Nov. 1991</ref> with support from the [[IBM Zurich]] Research lab. in [[Switzerland]].<ref>R.Cidecyan, F.Dolvio, R. Hermann, W.Hirt, W. Schott "[https://ieeexplore.ieee.org/document/124468 A PRML System for Digital Magnetic Recording]", IEEE Journal on Selected Areas in Comms, vol.10, No.1, pp.38-56, Jan 1992</ref> A parallel R&D effort at IBM San Jose did not lead directly to a product.<ref>T. Howell, et al. "[https://ieeexplore.ieee.org/document/104703 Error Rate Performance of Experimental Gigabit per Square Inch Recording Components]", IEEE Trans. Magn., Vol. 26, No. 5, pp. 2298-2302, 1990</ref> A competing technology at the time was 17ML<ref>A. Patel, "[https://www.researchgate.net/publication/224663211 Performance Data for a Six-Sample Look-Ahead 17ML Detection Channel]", IEEE Trans. Magn., Vol. 29, No. 6, pp. 4012-4014, Dec. 1993</ref> an example of Finite-Depth Tree-Search (FDTS).<ref>R. Carley, J. Moon, "[https://patents.google.com/patent/US5136593A/en Apparatus and method for fixed delay tree search]", filed Oct. 30th, 1989</ref><ref>R. Wood, "[https://ieeexplore.ieee.org/document/42527 New Detector for 1,k Codes Equalized to Class II Partial Response]", IEEE Trans. Magn., Vol. MAG-25, No. 5, pp. 4075-4077, Sept. 1989</ref>

The IBM 0681 read/write channel ran at a data-rate of 24&nbsp;Mbit/s but was more highly integrated with the entire channel contained in a single 68-pin [[Plastic leaded chip carrier|PLCC]] [[integrated circuit]] operating off a 5 volt supply. As well as the fixed analog equalizer, the channel boasted a simple adaptive digital ''cosine equalizer''<ref>T. Kameyama, S. Takanami, R. Arai, "[https://ieeexplore.ieee.org/document/1059216 Improvement of recording density by means of cosine equalizer]", IEEE Trans. Magn., Vol. 12, No. 6, pp. 746-748, Nov. 1976</ref> after the A/D to compensate for changes in radius and/or changes in the magnetic components.

=== Write precompensation ===
The presence of nonlinear transition-shift (NLTS) distortion on [[Non-return-to-zero|NRZ]] recording at high density and/or high data-rate was recognized in 1979.<ref>R. Wood, R. Donaldson, "[https://ieeexplore.ieee.org/document/1060300 The Helical-Scan Magnetic Tape Recorder as a Digital Communication Channel]", IEEE Trans. Mag. vol. MAG-15, no. 2, pp. 935-943, March 1979</ref> The magnitude and sources of NLTS can be identified using the 'extracted dipulse' technique.<ref>D. Palmer, P. Ziperovich, R. Wood, T. Howell, "[https://ieeexplore.ieee.org/document/1065310 Identification of Nonlinear Write Effects Using Pseudo-Random Sequences]", IEEE Trans. Magn., Vol. MAG-23, no. 5, pp. 2377-2379, Sept. 1987</ref><ref>D. Palmer, J. Hong, D. Stanek, R. Wood, "[https://ieeexplore.ieee.org/document/5680698/ Characterization of the Read/Write Process for Magnetic Recording]", IEEE Trans. Magn., Vol. MAG-31, No. 2, pp. 1071-1076, Mar. 1995 (invited)</ref>

Ampex was the first to recognize the impact of NLTS on PR4.<ref>P. Newby, R. Wood, "[https://ieeexplore.ieee.org/document/1064566 The Effects of Nonlinear Distortion on Class IV Partial Response]", IEEE Trans. Magn., Vol. MAG-22, No. 5, pp. 1203-1205, Sept. 1986</ref> and was first to implement [[Write precompensation]] for PRML NRZ recording. 'Precomp.' largely cancels the effect of NLTS.<ref name=8inch /> Precompensation is viewed as a necessity for a PRML system and is important enough to appear in the [[BIOS]] HDD setup<ref>{{Cite web |url=http://www.kva.kursk.ru/bios1/HTML1/standard.html |title=Kursk: BIOS Settings - Standard CMOS Setup, Feb 12, 2000 |access-date=October 8, 2019 |archive-date=October 4, 2018 |archive-url=https://web.archive.org/web/20181004103524/http://www.kva.kursk.ru/bios1/HTML1/standard.html |url-status=dead }}</ref> although it is now handled automatically by the HDD.

== Further developments ==
=== Generalized PRML ===
PR4 is characterized by an equalization target (+1, 0, -1) in bit-response sample values or (1-D)(1+D) in polynomial notation (here, D is the delay operator referring to a one sample delay). The target (+1, +1, -1, -1) or (1-D)(1+D)^2 is called Extended PRML (or EPRML). The entire family, (1-D)(1+D)^n, was investigated by Thapar and Patel.<ref>H.Thapar, A.Patel, "[https://ieeexplore.ieee.org/document/1065230 A Class of Partial Response Systems for Increasing Storage Density in Magnetic Recording]", IEEE Trans. Magn., vol. 23, No. 5, pp.3666-3668 Sept. 1987</ref> The targets with larger n value tend to be more suited to channels with poor high-frequency response. This series of targets all have integer sample values and form an open [[Eye pattern|eye-pattern]] (e.g. PR4 forms a ternary eye). In general, however, the target can just as readily have non-integer values. The classical approach to maximum-likelihood detection on a channel with intersymbol interference (ISI) is to equalize to a minimum-phase, whitened, matched-filter target.<ref>D. Forney, "[https://ieeexplore.ieee.org/document/1054829 Maximum Likelihood Sequence Estimation of Digital Sequences in the Presence of Intersymbol Interference]", IEEE Trans. Info. Theory, vol. IT-18, pp. 363-378, May 1972.</ref> The complexity of the subsequent Viterbi detector increases exponentially with the target length - the number of states doubling for each 1-sample increase in target length.

=== Post-processor architecture ===
Given the rapid increase in complexity with longer targets, a post-processor architecture was proposed, firstly for EPRML.<ref>R. Wood, "[https://ieeexplore.ieee.org/document/281375 Turbo-PRML, A Compromise EPRML Detector]", IEEE Trans. Magn., Vol. MAG-29, No. 6, pp. 4018-4020, Nov. 1993</ref> With this approach a relatively simple detector (e.g. PRML) is followed by a post-processor which examines the residual waveform error and looks for the occurrence of likely bit pattern errors. This approach was found to be valuable when it was extended to systems employing a simple parity check<ref>{{Cite journal|last=Conway|first=T.|date=July 1998|title=A new target response with parity coding for high density magnetic recording channels|url=https://ieeexplore.ieee.org/document/703887|journal=IEEE Transactions on Magnetics|volume=34|issue=4|pages=2382–2386|doi=10.1109/20.703887|url-access=subscription}}</ref><ref>R. Cideciyan, J. Coker; E. Eleftheriou; R. Galbraith, "[https://ieeexplore.ieee.org/document/917606 NPML Detection Combined with Parity-Based Postprocessing]", IEEE Trans. Magn. Vol. 37, No. 2, pp. 714–720, March 2001</ref><ref>M. Despotovic, V. Senk, "Data Detection", Chapter 32 in ''[https://www.researchgate.net/publication/328870436 Coding and Signal Processing for Magnetic Recording Systems]'' edited by B. Vasic, E. Kurtas, CRC Press 2004</ref>

=== PRML with nonlinearities and signal-dependent noise ===
As data detectors became more sophisticated, it was found important to deal with any residual signal nonlinearities as well as pattern-dependent noise (noise tends to be largest when there is a magnetic transition between bits) including changes in noise-spectrum with data-pattern. To this end, the Viterbi detector was modified such that it recognized the expected signal-level and expected noise variance associated with each bit-pattern. As a final step, the detectors were modified to include a 'noise predictor filter' thus allowing each pattern to have a different noise-spectrum. Such detectors are referred to as Pattern-Dependent Noise-Prediction (PDNP) detectors<ref>J. Moon, J. Park, "[https://ieeexplore.ieee.org/abstract/document/920181 Pattern-dependent noise prediction in signal dependent noise]" IEEE J. Sel. Areas Commun., vol. 19, no. 4, pp. 730–743, Apr. 2001</ref> or [[noise-predictive maximum-likelihood detection|noise-predictive maximum-likelihood detectors]] (NPML).<ref>E. Eleftheriou, W. Hirt, "[https://ieeexplore.ieee.org/document/539233 Improving Performance of PRML/EPRML through Noise Prediction]". IEEE Trans. Magn. Vol. 32, No. 5, pp. 3968–3970, Sept. 1996</ref> Such techniques have been more recently applied to digital tape recorders.<ref>E. Eleftheriou, S. Ölçer, R. Hutchins, "[https://ieeexplore.ieee.org/document/5438946 Adaptive Noise-Predictive Maximum-Likelihood (NPML) Data Detection for Magnetic Tape Storage Systems]", IBM J. Res. Dev. Vol. 54, No. 2, pp. 7.1-7.10, March 2010</ref>

== Modern electronics ==
Although the PRML acronym is still occasionally used, advanced detectors are more complex than PRML and operate at higher data rates. The analog front-end typically includes [[Automatic gain control|AGC]], correction for the nonlinear read-element response, and a low-pass filter with control over the high-frequency boost or cut. Equalization is done after the ADC with a digital [[FIR filter]]. ([[Two-dimensional magnetic recording|TDMR]] uses a 2-input, 1-output equalizer.) The detector uses the PDNP/NPML approach but the hard-decision Viterbi algorithm is replaced with a detector providing soft-outputs (additional information about the reliability of each bit). Such detectors using a soft Viterbi algorithm or [[BCJR algorithm]] are essential in iteratively decoding the [[low-density parity-check code]] used in modern HDDs. A single integrated circuit contains the entire read and write channels (including the iterative decoder) as well as all the disk control and interface functions. There are currently two suppliers: [[Broadcom]] and [[Marvell Technology Group|Marvell]].<ref>{{cite web |url=https://www.marvell.com/storage/assets/Marvell_88i9422_Soleil_pb_FINAL.pdf |title=Marvell 88i9422 Soleil SATA HDD Controller |date=September 2015 |access-date=2019-10-09 |archive-url=https://web.archive.org/web/20161213104211/http://www.marvell.com/storage/assets/Marvell_88i9422_Soleil_pb_FINAL.pdf |archive-date=2016-12-13 |url-status=dead }}</ref>

== See also ==
* [[Maximum likelihood]]
* [[Viterbi algorithm]]

== References ==
{{Reflist}}

== Further reading ==
* [http://pcguide.com/ref/hdd/geom/dataPRML-c.html The PC Guide: PRML]
* [http://www.guzik.com/solutions_chapter9.asp Online Chapter "Introduction to PRML"], from Alex Taratorin's book ''Characterization of Magnetic Recording Systems: A Practical Approach''

[[Category:Computer storage devices]]