Editing Partial-response maximum-likelihood (section)

== 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>