IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001 9

A Decision-Feedback Equalizer with Pattern-Dependent Feedback forMagnetic Recording Channels

Yungsoo Kim and Hwang-Soo Lee, Member, IEEE

Abstract—A new nonlinear equalizer for high-density magneticrecording channels is presented. It has a structure of the decision-feedback equalizer (DFE) with a nonlinear model at the feedbacksection and a dynamic threshold detector. The feedback nonlinearmodel is a sequence of look-up tables (LUTs) indexed by time, andeach table is addressed by transition pattern formed by one futureand past transitions. We call this new nonlinear equalizer the pat-tern-dependent DFE (PDFE). The feedback nonlinear model can-cels the trailing nonlinear intersymbol interference (ISI), and thenthe data decision is made by considering the precursor nonlinearISI caused by one future symbol. We propose a tap optimizationcriterion SNR for the PDFE which in effect tries to maximize theoutput signal to noise ratio, and derive a closed-form solution forthe tap values. We compare the detection performance of PDFEwith that of the DFE and the RAM-DFE on experimental chan-nels. The RAM-DFE is a DFE with one large LUT at its feedbacksection. The results show that the PDFE yields a significant per-formance improvement over the DFE and the RAM-DFE. Also thePDFE derived in this paper achieves a superior performance com-pared with the PDFE derived by the minimum mean-square-errorcriterion.

Index Terms—Detection, DFE, magnetic recording, nonlinearequalizer.

I. INTRODUCTION

D IGITAL magnetic storage channels experience severenonlinear distortions at high linear densities [1]–[4].

During the writing process, transitions written previously causethe next transition to shift in position and adjacent transitionspartially erase one another, resulting in an amplitude reductionof of tje read back signal.

Many models have been proposed to accurately represent thechannel characteristics and to design more reliable receivers.The finite-state machine (FSM) model in [5] is a simple look-up-table of channel responses to a group of past, current, and futuresymbols. It is very accurate when the number of past and futurebits considered in the model is sufficiently large, but it gives uslittle insight into the channel characteristics. In RAM-DFE [6],the FSM model is used in place of the feedback filter of a DFE,to cancel the postcursor linear and nonlinear intersymbol inter-ference (ISI). However, the nonlinear distortions caused by thefuture transitions are not completely removed in the RAM-DFE

Paper approved by M. Z. Win, the Editor for Equalization and Diversity of theIEEE Communications Society. Manuscript received August 29, 1998; revisedJuly 14, 1999 and January 31, 2000. This paper was presented in part at theInternational Conference on Communications (ICC), Atlanta, GA, 1998.

Y. Kim is with Samsung Advanced Institute of Technology, Suwon 440-600,Korea (e-mail: ysk@sait.samsung.co.kr).

H.-S. Lee is with the Department of Electrical Engineering, Korea Ad-vanced Institute of Science and Technology, Taejon 305-701, Korea (e-mail:hslee@ee.kaist.ac.kr).

Publisher Item Identifier S 0090-6778(01)00252-5.

and degrades the equalizer performance at a high recording den-sity.

A simple model proposed in [7] is based on a superposition ofthreedifferent pulse shapes. Each pulse shape is empirically ob-tained by actually measuring the channel response to one of thethree data transition patterns; an isolated transition, two transi-tions separated by one bit period, and two transitions separatedby two bit periods. This model is based on an intuitive under-standing of the storage channel, but it lacks flexibility in that ithas only three pulse shapes. In the DFE-NC [8], this model isused as the feedback section of a DFE structure. After an in-tuitive consideration of this model, the DFE-NC is modified tothe DFE-DT [8] that takes into account the precursor nonlinearISI in data detection. This modification leads to a significantperformance improvement. However, the performance can notbe improved any further, since we cannot have more than threepulse shapes for the model.

In [9], a new nonlinear model is proposed that is also basedon a superposition of pulse shapes. But this model haspulse shapes and selects the pulse shape value based on the datatransition pattern formed bypast and future symbols. Equiv-alently, it can be viewed as a sequence of tables in which eachtable is addressed by transition pattern. The modeling accuracyimproves as the model considers more symbols in the pattern,and is flexibly traded off against the model complexity. Thismodel is likewise implemented as the feedback section of a DFEstructure in [9], but with the pattern constrained to one futureand past transitions. The data decision is made by consideringthe nonlinear distortion caused by one symbol in the future.

A basic receiver design criterion is to maximize the ratio, where is the minimum distance between points in

signal constellations and is the output error power. But eventhough for the equalizer in [9] is affected by the equalizertap values, it is designed with the simple minimum mean squareerror (MMSE) criterion.

In this work, we re-derive the nonlinear equalizer of [9] ina more rigorous manner. We describe more fully how a treesearch detection into one future symbol is reduced to a dynamicthreshold detection. We also propose a new tap optimizationcriterion SNR which tries to maximize , and derive aclosed-form solution for the tap values. In Section II, the equal-izer derivation is presented. In Section III, simulation results arepresented. Finally, this paper is concluded in Section IV.

II. NONLINEAR EQUALIZER

Given a sequence of received signals ,the new nonlinear equalizer decides on the value of the binarysymbol as the value of for which the output

0090–6778/01$10.00 © 2001 IEEE

10 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001

error power is the minimum. The output erroris obtained byfirst filtering the received signal by a finite impulse response(FIR) filter and then subtracting by theoutput of the nonlinear model, as below:

(1)

(2)

where is an estimate of ,is the data transition, is

the data transition bit. denotes the state of transitionpattern defined by a sequence of data transition bits

. denotes a vector ofsize from to , inclusive.

The nonlinear model has a sequence of functions or tablesindexed by time , . The term

represents the channel response at current timedue to adata transition which occurred at time , and ittakes into account the nonlinear interaction of with itsneighboring transitions of one future andpast transitions,

. The model output is then generatedby a superposition of the table output values.

The table value can be expressed as a product of two vectorsas below

(3)

where and. is the total

number of patterns. denotes a delta function withvalue 1 when and 0 otherwise. The column vectorhas 0’s for all its elements except at one position where ,and acts as an address vector for. denotes the transpose of.

In (1), the model requires values for the past, the current andone future symbols; the past estimated values ,are assumed to be correct, and are fed back and used as themodel input. The terms that require values for the current andone future symbols in (1) are separated from the terms that arealready determined by the past decisions, and they are denotedas and , respectively. Defining and substituting(3) into (1) gives

(4)

(5)

(6)

where

(7)

(8)

(9)

Fig. 1. Decision trees into one future bit.

......

...

...

For the given past data , the unknown termcan take four values depending on the sequence ,or equivalently . However, when ,

, independent of , which shows that actuallyhas three different values. This is depicted in Fig. 1. The valueof or will be decided as the one for which is the min-imum

if and

otherwise(10)

where is the value of when , that is, when thereare no transitions at time. and are the values of

with a transition at time when and, respectively.

The decision rule of (10) can be simplified if we make rea-sonable assumptions based upon our intuitive understanding ofthe magnetic recording channel. The following useful terms aredefined below

(11)

(12)

(13)

(14)

Using the above definitions, (10) can be restated as below

if and

otherwise.(15)

If , is either or 0. Then, and ,which correspond to thepositivetransition , are morepositivethan , and thus, and areboth positive.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001 11

In this case, (15) is further simplified to a threshold decisionusing a worst-case decision boundary

ifotherwise.

(16)

Similarly, when , is either or 0. andthat correspond to thenegativetransition , are

morenegativethan , and and areboth negative.Then, the decision rule (15) is also simplified to (16) with theworst-case decision boundary . There-fore, if and always have the same sign, and the signis opposite to that of , we have a threshold detector (16).

The worst-case decision boundary for (16) is triviallyfound if we recall the nonlinear distortion effect in datarecording, in which any two adjacent transitions partially eraseeach other and cause an amplitude reduction in the read backsignal [1], [4]. This signal reduction by adjacent transitionsmakes sure that value is always smaller than , andthus is always the worst-case decision boundary

(17)

The detector will make fewer decision errors if the distancesand are kept large in magnitude while making the

equalizer output error as small as possible. A reasonable opti-mization criterion for the equalizer is to maximize SNRde-fined as

(18)

where denotes an ensemble average. This is a simpleand effective criterion for this equalizer, compared with, forexample, the one maximizing .Another simple measure of optimization criterion that worksalmost equally well is .

In order to proceed with the equalizer optimization, we needto express SNRin terms of and . Referring back to (9),

and are restated in terms of as below

(19)

(20)

where

and is the value of when . and arethe values when and

, respectively. Recalling (5), the mean square error (MSE) is

(21)

Since and are independent of, maximizing SNRwith respect to is equivalent to minimizing the MSE. The op-timum that minimizes the MSE is found as

(22)

Fig. 2. Proposed nonlinear equalizer with� = 1; M = 4; N = 6.

Substituting (22) into (21) gives us an another expression for theMSE in terms of

(23)

where

(24)

Now, differentiating (18) with respect to , and setting it to0 gives

(25)

Substituting (19), (20), and (23) into (25), and solving forgives

(26)

From (26), we conclude that is the eigenvector of thematrix and correspondsto the eigenvalue. For the optimum tap values, we choose theeigenvector corresponding to the maximum eigenvalue of (26).Once is found, can be obtained from (22).

The way the equalizer is derived in [9] is different from theabove method in that one reference patternis selected among

patterns and is fixed to 1. Then, the rest of the equal-izer tap values are optimized to minimize the MSE. The se-lection of the reference pattern significantly affects perfor-mance, and the tap values with the best performance are chosenby simulation.

Fig. 2 depicts as an example the nonlinear equalizer with, , and . It is a DFE with the pattern-

dependent nonlinear model at the feedback section, which wesimply call the PDFE. The data transition pattern for the non-linear model, , consists of one fu-ture and one past transition bits, and spansin time. The figure represents the data detection of (16) where

andwith when

, respectively. The column number of the boxesat the feedback section corresponds to the time index; theright most column is for and the left most column isfor . The row number corresponds to the pattern number

; the top row is for and the bottom row isfor . In this figure, the past transition sequence is

12 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001

Fig. 3. Feedback tap values for PDFE(1).

TABLE ID AND T FOR PDFE(1)WHEN a = �1

and the pattern number se-quence is . Therefore,

and .In Fig. 3, the feedback tap values are plotted as an examplewhich are obtained using the nonlinear signal of Section III.Table I lists and values for the PDFE of Fig. 3 when

. It can be noted that and are both pos-itive and is always smaller than in magnitude. If

, all the values in the table reverse their signs.

III. SIMULATION

The performance of the PDFE is tested with an actual readback signal and compared with those of the DFE and theRAM-DFE [6]. A 127-bit maximal-length pseudorandom bi-nary sequence (PRBS) was generated by polynomialand was recorded repeatedly on the disk. Each frame of the readback waveforms, containing one period of the PRBS sequence,was aligned and averaged over each bit position, so that theadditive noise would be removed from the signal.

Since the averaged measured signal has only 127 differentvalues, the signal is perfectly represented by an FSM model ofsize 128. This FSM model of size 128 is an accurate model ofthe channel if both the linear and nonlinear intersymbol inter-ferences extend to a period of 7 bits and negligible outside ofthat period [5], [6]. The signal-to-distortion ratio for this signal,which is defined as the signal power divided by the modelingerror power, is 12.7 dB for the linear model [9].

Figs. 4 and 5 show the detection bit-error rate (BER) per-formance of the PDFE, the DFE, and the RAM-DFE. All theequalizers use the actual estimated bits in the feedback loop inthe simulation. The equalizer input signalis sum of the noise-less channel output and the additive white Gaussian noise.The signal is generated by the FSM model and the noiseis added with the input signal-to-noise ratio (SNR) defined as

Fig. 4. BER performance for (0, 4/4) RLL coded data.

Fig. 5. BER performance for MTR 3 coded data.

as in [6]. The nonlinear signal for the equalizertraining (or estimation) is generated from a random, uncodedbinary sequence, while the signal for data detection is gener-ated from a coded binary sequence. This ensures that all the datapatterns occur in the training data that are required by the feed-back sections of the RAM-DFE and the PDFE. If the equalizersare trained using the coded data, many of the feedback contentsof the nonlinear equalizers are left undefined after the training.This results in a serious performance degradation. The additivenoise power is computed with respect to the training data, andheld fixed throughout the training and the data detection periods.

PDFE(2) denotes the PDFE with in which the pat-tern considers two past transitions and one future transition,while PDFE(1) is the PDFE with one past and one future tran-sitions. PDFE-SNRd represents the PDFE optimized to maxi-mize SNR (18). PDFE-mse represents the PDFE optimized bythe MMSE criterion [9]. The tap values for the DFE and theRAM-DFE are obtained by the MMSE criterion [6]. The en-semble averages are taken 16 384 times. For all cases, the feed-forward filter has 11 taps. In the feedback section, we use aseven-tap FIR filter for the DFE, a RAM of size 128 for theRAM-DFE, and the nonlinear model with for the PDFE.

In Fig. 4, the data is encoded by the (0,4/4) run length-limited(RLL) code [10]. PDFE(2)-SNRd achieves the best perfor-mance while the DFE is the worst for BER less than 1E-5. Theperformance gain achieved by PDFE-SNRd over PDFE-mseis significant; PDFE(1)-SNRd outperforms PDFE(1)-mseby 3 dB at BER = 1E-5, while PDFE(2)-SNRd outperforms

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 1, JANUARY 2001 13

Fig. 6. BER performance for causal channel.

PDFE(2)-mse by about 0.8 dB at 1E-5. Also PDFE(2)-SNRdoutperforms the RAM-DFE by more than 2 dB at 1E-6.The performance of the PDFE improves as we increase thecomplexity ( ) of the feedback model.

In Fig. 5, the data is encoded by the maximum transitionrun (MTR) 3 code [11], [12]. PDFE(1)-mse has the worst per-formance, and PDFE(2)-SNRd outperforms the RAM-DFE byabout 2.9 dB at 1E-6.

In Fig. 6, a nonlinear channel model is derived from the orig-inal channel model such that the new channel is causal and hasonly trailing ISI, by averaging the FSM model values over thefuture symbols for each sequence of the current and past sym-bols. For this channel, the RAM-DFE outperforms the PDFE.

These results clearly show that the PDFE is more effectivethan the RAM-DFE when the channel contains the nonlinearISI caused by future symbols. The performance gain achievedby the PDFE over the RAM-DFE in Figs. 4 and 5 is attributed tothe data decision that takes into account the precursor nonlinearISI caused by one future symbol. When the channel is causaland has only trailing ISI, there will be no gain in looking intofuture symbols for detection.

IV. CONCLUSIONS

We conclude that for a real magnetic recording channel,the new nonlinear equalizer, the PDFE, derived in this paper

is more effective than the DFE and the RAM-DFE. While theRAM-DFE cancels the trailing nonlinear ISI only, the PDFEcancels the trailing nonlinear ISI and makes the data decisionconsidering the precursor nonlinear ISI caused by one futuresymbol. This accounts for the large performance improvementachieved by the PDFE over the RAM-DFE when the channelcontains precursor nonlinear ISI. The performance of the PDFEimproves as we increase the complexity of the nonlinear model.Also the tap optimization method by maximizing SNRpro-posed in this paper achieves a large improvement in equalizerperformance over the method based on the MMSE criterion.

ACKNOWLEDGMENT

The authors wish to thank the anonymous reviewers for theirhelpful comments and their thorough review. Their remarksgreatly improved the quality of the paper.

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