Performance and implementation of adaptive partial response maximum likelihood detection

3806 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 5, SEPTEMBER 1998

Performance and Implementation of AdaptivePartial Response Maximum Likelihood Detection

Ke Han and Richard R. Spencer,Senior Member, IEEE

Abstract—Motivated by previous comparison work, a configu-ration for partial response maximum likelihood detection usingthe Viterbi algorithm (PRML/VA) detectors with adaptive targetpolynomials is examined. In this configuration, a mean-quarederror decision feedback equalization (MSE-DFE) is used to adaptboth the forward equalizer and the target channel for the Viterbidetector. The performance of this adaptive PRML/VA is analyzedand compared with other detection techniques. The issue ofconvergence speed is also studied.

Index Terms—Adaptive target polynomials, analytical perfor-mance comparison, partial response maximum likelihood detec-tion.

I. INTRODUCTION

PARTIAL response maximum likelihood detection usingthe Viterbi algorithm (PRML/VA) [9], decision feedback

equalization (DFE) [10], [13], and fixed-delay tree search withdecision feedback (FDTS/DF) [2] are the three sampling de-tection techniques most often considered for digital magneticrecording. In a previous paper, detailed performance analysesfor each of these different detection techniques were providedand their relative performances in different situations werecompared [11]. The work presented here was motivated by theresults of that comparison. In this paper, a configuration foradaptive PRML is analyzed. There have been many researchactivities to optimize the target polynomials for maximumlikelihood sequence detection (MLSD) while reducing thecomplexity of the Viterbi detector [14]–[16]. In this paper,an effort has been made to relate the adaptive PRML detectorwith the well understood DFE detector and complete someperformance comparisons in a wider spectrum of detectiontechniques.

The paper is organized as follows. In Section II, the designand performance analysis of each detector is briefly reviewed,and the comparison results of our previous work are summa-rized [11]. Then in Section III, the motivation and derivationof this adaptive PRML configuration are discussed. Theoreticalcalculations of the performance are also given and comparedwith those of other detectors in this section. In Section IV,some implementation issues, such as the adaptation algorithm

Manuscript received October 7, 1997; revised May 1, 1998.K. Han is with Quantum Corporation, Milpitas, CA 95035 USA (e-mail:

[email protected]).R. R. Spencer is with the Department of Electrical and Computer

Engineering, University of California, Davis, CA USA 95616 (e-mail:[email protected]).

Publisher Item Identifier S 0018-9464(98)06135-4.

Fig. 1. Block diagram of the magnetic recording read channel.

and the complexity of the implementation are considered. Thenin Section V, the issue of convergence of the adaptation andthe tradeoff between the final performance and the conver-gence speed are discussed. Finally, some conclusions are givenin Section VI.

II. OPTIMAL DETECTOR DESIGNS

A block diagram of a disk drive read channel with asampling detector is shown in Fig. 1.

In this block diagram, the incoming binary data sequenceis assumed to be written on the disk in a nonreturn-

to-zero (NRZ) format to create the magnetization . Theterms and are media noise and electronic noise,respectively. is the transfer function of the magneticrecording channel. is a continuous-time receive filter.The output of this filter is sampled at a rate of .is a discrete time equalizer and its output is fed into a detectorthat will yield , the estimate of the input sequence .

A. PRML/VA Detection

A PRML read channel with a VA detector can be modeledas shown in Fig. 2.

In Fig. 2, equalizes the channel to a target poly-nomial . An error signal is defined to be thedifference between the outputs of the target channel and theequalized channel. An optimal mean-squared error (MSE)design is to choose the equalizer so that the powerof the error signal will be minimized.

Usually, the error signal in front of the VA detector is notwhite, and the effect of error correlation should be dealt withseparately for each different possible error event. Suppose thelength of an error event is and the target polynomial is

(1)

0018–9464/98$10.00 1998 IEEE

HAN AND SPENCER: PERFORMANCE AND IMPLEMENTATION OF ADAPTIVE PRML 3807

Fig. 2. A read channel with a PRML/VA detector.

Fig. 3. A read channel with a DFE detector.

then it can be derived [11], [14] that the probability of thiserror event is

Prob

(2)

where is the distance between the correctand incorrect branches in the trellis, and

are the auto-correlation of the errorsignal.

Knowing the probability of each error event, the bit errorrate (BER) can be approximately given by [4]

BER

(3)

where is a path that can have a minimum-distance errorevent, is the set of all such paths, is given by(2), and is the number of errors in error event .

B. DFE Detection

A read channel with a DFE detector can be modeled asshown in Fig. 3.

Here, removes the precursor intersymbol interfer-ence (ISI) and is a feedback equalizer, a strictlycausal filter that removes all the post-cursor ISI. An errorsignal is defined to be the difference between the inputand the output of the slicer. An optimal design is to choose

Fig. 4. A read channel with a FDTS/DF detector.

both and so that the power of the errorsignal is minimized.

To deal with the effect of error propagation in the DFEdetector, an -element vector can be defined that consistsof the past decision errors, which will affect the currentdecision

(4)

Given the present error state vector and the probabil-ity distribution of the error signal (consisting of noise andmis-equalization), the conditional probability of a particulardecision error term can be calculated

Prob Prob Prob (5)

where Prob can be evaluated given the probabilitydistribution of the error signal.

These conditional probabilities also define the transitionprobabilities from one error state to the others. Assumingthe successive error signal samples are independent, this isa Markov process [5]. With all the transition probabilitiesavailable, the following transition equation can be defined( )

......

. . ....

... (6)

where is the probability of a particular error state attime and is the probabilityof the next error state being given the current error statebeing . Here the time subscripts are dropped becauseisnot a function of time. Since each error state can only evolveinto one of three error states, there are only three nonzeroelements in each row or column. The fact that this matrix issparse is critical and greatly reduces the memory required to


implement a solution. The steady-state probabilities of all theerror states can be derived iteratively. Then the BER of theDFE is given by [11]

BER Prob (7)

where is the steady-state probability of error state.

C. FDTS/DF Detection

A read channel with a FDTS/DF detector can be modeledas shown in Fig. 4.

The forward equalizer equalizes the channel to acertain target. The feedback equalizer will cancel the“tail” of the target response. The FDTS decision unit, whichis a depth-limited exhaustive tree search algorithm, will see atruncated version of the target response.

The design of a FDTS/DF detector can be divided intotwo steps. First, the optimal forward equalizer and optimaltarget polynomial are chosen so that the power of thedifference signal between the target channel and the equalizedchannel is minimized. The second step is to divide the optimaltarget polynomial into two parts. Assuming the orderof is , if the depth of the tree structure in the FDTSblock is , then the first terms of will constitutethe target polynomial that the FDTS sees, and the laterterms of will constitute the feedback filter .

Because of the finite depth of the tree structure and the de-cision feedback, error propagation is introduced. Fortunately,in order to evaluate the performance of the FDTS/DF detectorwith error propagation included, a method that is similar tothe evaluation of the performance of the DFE can be used.

Since the past decisions and the future inputsaffect the current decision, an-element error state vector isdefined, which consists of the past decisions and the

future inputs (note that the time is regarded here asthe current time)

(8)

Approximating the successive error samples as independent,the progression of this error vector with time is a Markovprocess, and the transition probability can be calculated.

Suppose the current error state is given by (8) and the correctinput at time is , then the transition probabilityfrom this state to the state

is

Prob

Prob

Prob Prob

Prob metrics of paths starting with

metrics of paths starting with (9)

Similarly, all other transition probabilities can be calculated.The transition probability matrix is again sparse. Based on thisidea, an algorithm was developed to calculate the BER of theFDTS/DF detector with error propagation included [11].

Based on the above algorithms, a theoretical comparison ofthe three detection techniques has been performed. The overallperformance comparison results are given in Fig. 5. In thiscomparison, a Lorentzian model is used for the channel, and anideal lowpass filter (LPF) with a bandwidth equal to half of thebaud rate is used as the front-end filter. For the VA detectors,the target polynomials have the form .The media noise and the electronic noise terms make equalcontributions to the noise power at the output of the LPF. Inall cases, enough taps have been used in the equalizers thatadding taps does not improve the performance noticeably.

III. PRML/VA WITH ADAPTIVE TARGET CHANNELS

From the above comparison, it can be seen that FDTS/DFwith complexity outperforms every other detectorshown. The main reason for this impressive performance isthat the overall target polynomial of the FDTS/DF detectoris optimized at each different linear density. Since the treestructure of the decision unit is a sub-optimal form of a Viterbidetector, it seems reasonable to suggest that a PRML/VAdetector with a target polynomial optimized at each lineardensity should perform even better.

In fact, for a given system, the best performance is givenby a whitened matched filter (WMF) followed by a PRML/VAdetector, as is shown in Fig. 6 [4].

In this figure, is a continuous-time matched filterand is a discrete-time whitening filter. To simplify thefollowing discussion, only the electronic noise is consideredunless stated otherwise.

This design of the optimal PRML/VA is identical to thedesign of a ZF-DFE for the same system. In fact it can beproven [4] that for a ZF-DFE designed for the same system,the forward equalizer is given by

(10)

and the feedback equalizer is given by

(11)

where is the target polynomial of the optimalPRML/VA detector.

Therefore, the PRML/VA detector with optimal target poly-nomial can be related to the zero forcing DFE (ZF-DFE) asshown in Fig. 7.

In a real system, the ZF criterion cannot be implementedadaptively, because noise and mis-equalization error cannotbe separated. So a more practical criterion, the MSE criterion,has to be used. The MSE design of an optimal PRML/VAdetector can also be modeled by the block diagram in Fig. 2.Remember that is now an unknown target polynomialthat is to be optimized.

Assuming that there is only electronic noise, the powerspectrum of the error signal , as defined in Fig. 2, is givenby

(12)

where is the power spectrum of the binary data.


Fig. 5. Comparison for PRML/VA, DFE, and FDTS/DF with 50/50 noise and fixed BER of1E � 8. N = 8 for FDTS/DF.

Fig. 6. The optimal configuration of a PRML detector.

The optimal design is to choose both , which shouldbe strictly causal, and the forward equalizer , so thatthe power spectrum of the error signal is minimized for allfrequencies.

Again, this MSE design of an optimal PRML/VA detectoris identical to the design of a MSE-DFE for the same system[4]. In fact, the forward equalizer of the MSE-DFE is given by

mse (13)

and the feedback equalizer of the MSE-DFE is given by

mse (14)

So this MSE-designed optimal PRML/VA detector can berelated to the MSE-DFE as shown in the Fig. 8. Since aMSE-DFE detector can be implemented adaptively, in thesame way a MSE-designed optimal PRML/VA detector canbe implemented adaptively. Fig. 8 is the adaptive PRML/VAdetector configuration that is proposed in this paper.

This configuration is optimal in the following sense. First,the power of the error signal at the input to the VA detectoris minimized given that the target polynomial is monic;second, the adaptation of the MSE-DFE tends to whiten theerror signal in front of the detector, so the performance ofthe PRML/VA detector is not significantly reduced by noisecorrelation. Thus, it can be expected that when the SNR isreasonably high, and the numbers of taps of both the forwardand feedback equalizers are reasonably large, the performanceof the adaptive PRML/VA should be very close to that of theoptimal one given in Fig. 6.

Based on the configuration given in Fig. 8, the performanceof the adaptive PRML/VA detector (APRML/VA) has beenanalyzed and compared with other detectors. Again in thiscalculation, a Lorentzian channel model is used and it isassumed that the media noise and the electronic noise makeequal contributions to the noise power at the output of thefront end receive filter, for which an ideal lowpass filter isused. The complexity of the VA detector is limited by limitingthe number of feedback taps in the MSE-DFE. Because ofthe finite order of the target polynomials, the error signal atthe input of the detector is not completely whitened. So inthe performance calculation, the effect of the error correlationis included. Some comparison results are given in Fig. 9. Ascan be seen, for the same order of target polynomial, (EPR4and APRML/third, EEPR4 and APRML/fourth), the adaptivePRML/VA detector shows significant improvement over the


Fig. 7. Implementation of the optimal PRML using ZF-DFE.

Fig. 8. Implementation of adaptive PRML using MSE-DFE.

Fig. 9. Comparison between adaptive PRML and other detectors with 50/50 noise and fixed BER of1E � 8. N = 8 for FDTS/SD.


Fig. 10. Configuration of a fully adaptive DFE detector.

performance of the VA detector with standard PR polynomials.When the recording density is relatively high, the APRML/VAwith the fourth-order target polynomial will outperform theFDTS/DF detector with and . Note that at lowlinear densities, the FDTS/ detector performs better thanthe APRML/VA detectors shown. This result is due to the factthat the FDTS/DF detector has an eighth-order overall targetpolynomial, which is divided between the FDTS detector andthe DF filter, and therefore has a better chance to whiten theerror signal and to maximize the detection SNR. On the otherhand, the APRML/VA with the fourth-order target does nothave the added benefit of the feedback and the error signalat the input of the VA detector is, therefore, not as close tobeing white. So the performance of the adaptive PRML/VAdetector still suffers to a larger extent from error correlationthan a FDTS/DF detector with a comparable complexity inthe tree structure.

IV. I MPLEMENTATION

A. Stochastic Gradient (SG) Algorithm

From Fig. 8, it can be seen that if an adaptive DFE isimplemented, the forward equalizer of the adaptive DFE canbe used as the forward equalizer for the PRML detector, whilethe coefficients of the feedback equalizer can be used as thoseof the target polynomial for the PRML detector. An adaptiveDFE is illustrated in Fig. 10.

The forward and feedback equalizers must be FIR filterswhere

(15)

and

(16)

A common algorithm for adapting the DFE is the stochasticgradient algorithm (SG), which minimizes the power of theerror signal at every sample time by adjusting the coefficientsof both the forward and the feedback equalizers. The SGalgorithm can be described by

(17)

where

(18)

(19)

and

(20)

B. Complexity

In our previous comparison, enough forward and/or feed-back taps are used to achieve essentially the best possibleperformance. In fact, for the PRML detectors, the number offorward taps does not have to be very large. In Fig. 11, theperformance of the following seven detectors are compared:a DFE detector, two PRML detectors with EPR4 and EEPR4targets, two adaptive PRML detectors with third- and fourth-order targets, and two FDTS/DF detectors with and

where, for each of the seven detectors, only seven tapshave been used for the forward equalizer. The same channeland noise models are used as in the previous comparison.Comparing Fig. 11 with Fig. 5, it can be seen that by re-ducing the number of forward taps from infinity to seven,the performance degradation of every PRML detector is lessthan 0.45 dB in terms of channel SNR for the practical rangeof recording densities ( ). Actually theperformance degradation is less than 0.25 dB for most ofthe PRML detectors over most of the practical density range.The performance of the DFE detector is, on the other hand,more sensitive to the number of forward and feedback taps. Itcan be seen that a performance degradation ranging from 0.5dB at lower densities to 1.5 dB at higher densities has beenintroduced by reducing the complexity of the DFE to sevenforward taps and four feedback taps. The performances of theFDTS/DF detectors with and are also givenin Fig. 11, but the curves shown are only approximations ofthe real performances. Because of the finite number of feedforward taps, the error signal at the input of the FDTS decisionunit is not completely white, but correlated. In the calculationof the performance of the FDTS/DF detectors, the effect oferror correlation was not considered, since a good methodto deal with this effect has not been found. Based on theperformance degradation due to noise correlation observedin the APRML detectors, a rough estimate is that the realperformance of the FDTS/DF detectors should be 0.4–0.6 dBworse than the curves given in Fig. 11, which would makethe performance of the APRML/fourth approximately equal tothat of the FDTS/DF with .

V. CONVERGENCE AND PERFORMANCE

It has been proven in [4] that the speed of convergence ofthe adaptation process is decided by the eigenvalue spread ofthe auto-correlation matrix given by

(21)


Fig. 11. Comparison between PRML/VA, DFE, and adaptive PRML with complexity constraints.

Fig. 12. Performance comparison between DFE and APRML detectors with and withoutL = 0 constraints. (50/50 noise, BER= 1:0e � 8, 7forward taps, no ITI).


Fig. 13. Eigenvalue spreads of the auto-correlation matrices with different configurations of the forward equalizers at different linear densitiesand afixed channel SNR of 16 dB.

where

(22)

and

(23)

The smaller the eigenvalue spread is, the faster the conver-gence will be.

The diagonal submatrices of the auto-correlation matrix,, and are usually diagonally dominant,

meaning that in each row, the amplitude of the element on thediagonal of the matrix is bigger than the other terms. In orderto make the eigenvalue spread of the matrixsmall, it is desir-able to make the amplitudes of the elements of the nondiagonalmatrices and as small as possible. FromFig. 10 it can be seen that ’s are the binary inputs to thechannel, and ’s are the sampled outputs of the channel. Intu-itively, it is expected that current inputs and current outputs aremore correlated, while the past inputs and future outputs areless correlated. So, if is set in (23), there will be less cor-relation between vector and vector . Thus the amplitudesof the elements of the matrices and willbe smaller, and the eigenvalue spread of the auto-correlation

matrix will be smaller. Because of the constraint, theoverall performance of the detector after convergence mighthave to suffer to some extent. So there is a tradeoff involvedbetween the final performance and the convergence speed.

A theoretical comparison of the performances of severaldetectors with general forward equalizers and with anti-causalforward equalizers is shown in Fig. 12. In this comparison, it isassumed that the adaptation process for each different detectorhas fully converged. For all the detectors, seven taps were usedfor the forward equalizers. But there are two different kindsconfigurations for the forward equalizers. For the first group ofDFE, APRML/third, and APRML/fourth detectors, the generalconfiguration was used for the forward equalizer, i.e.,and . Their performances are represented by the DFE,APRML_third, and APRML_fourth curves in the plot. Forthe second group of DFE, APRML/third, and APRML/fourthdetectors, the anti-causal configuration was used for the for-ward equalizer, i.e., and . Their performancesare represented by the DFE_AC, APRML_third_AC, andAPRML_fourth_AC curves in the plot. In this figure, it canbe seen that because of the constraint, the performanceof each different detector has been degraded by 0.6–2.0 dBdepending on recording density.


Fig. 14. Eigenvalue spreads of the auto-correlation matrices versus numberof causal taps in the forward equalizer for a particular detector APRML/fourthat PW50 = 2:5 and channel SNR of 16 dB.

In Fig. 13, the eigenvalue spreads of the auto-correlationmatrices are compared for APRML/third and APRML/fourthdetectors with different configurations of forward equalizers ata fixed channel SNR of 16 dB. From this figure, it can be seenthat because of the constraint, the eigenvalue spreadsof the auto-correlation matrices of the detectors are greatlyreduced. Since the convergence speed of the adaptation isinversely exponentially proportional to the eigenvalue spread,the difference between the eigenvalue spreads in Fig. 13represents a much greater difference in convergence speed.For example, at recording density , the eigenvaluespreads of APRML/third and APRML/fourth are reduced by afactor of 8 because of the constraint. But in simulations,it takes a quarter million training samples for the APRML/thirdand APRML/fourth with general forward equalizers to fullyconverge while it takes only a thousand training samples forthe APRML/third and APRML/fourth with anti-causal forwardequalizers to fully converge.

Fig. 14 shows the eigenvalue spreads of the auto-correlationmatrix versus the number of causal taps in the forwardequalizer for a particular detector, APRML/fourth, at a lineardensity of T, and a channel SNR of 16 dB.From this figure, it can be seen that a single causal tapin the forward equalizer can cause a significant increase inthe eigenvalue spread and, therefore, a great decrease in theconvergence speed. This is because the eigenvalue spread isdecided by the minimum eigenvalue (the maximum eigenvaluedoes not change very much with the number of causal taps). Asingle causal tap can produce at least one element with largeamplitude in each of the two nondiagonal matricesand , and therefore reduce the amplitude of at least

one eiginvalue dramatically. Further increasing the number ofcausal taps beyond one does not increase the eigenvalue spreadvery much since it does not further reduce the amplitude ofthe minimum eigenvalue significantly.

From the above discussion, it can be seen that a tradeoffexists between the performance and convergence for theadaptive PRML/VA detectors. Which configuration of theforward equalizer to use depends on what kind of applicationis considered. If the system parameters are changing rapidlyand on-line adaptation is needed, then it may be suitable tochoose the anti-causal forward equalizer and sacrifice someperformance to quickly follow the change of the parameters.In some other applications, the hard disk can be dividedinto many small zones according to different radii, and foreach small zone known, sets of parameters for the forwardequalizer and target polynomial are used. When the read headis moving into this zone, the known sets of parameters areloaded in to achieve the best possible performance. In thiscase, the convergence speed is not a critical issue, and it maybe desirable to choose the general configuration of the forwardequalizer to guarantee the best performance.

VI. CONCLUSIONS

A configuration for adaptive PRML detection has beenpresented. It uses an adaptive MSE-DFE to provide the op-timal forward equalizer and optimal target polynomial forthe PRML/VA detector at each different linear density. Theimplementation is practical and optimal in the sense thatthe power of the error signal in front of the detector isminimized and the error signal is whitened. Performanceanalyses and comparisons with other detectors indicate thatthis is a promising technique for the future.

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Ke Han received the B.S. degree in electrical engineering from the Universityof Science and Technology of China in 1987, the M.S. degree from theInstitute of Automation, Chinese Academy of Science, China, in 1990, andthe Ph.D. degree from the University of California, Davis, in 1996.

Since 1995, he has been a Channel Design Engineer at Quantum Corpora-tion, Milpitas, CA. His fields of interests are detection, equalization, timingrecovery, and continuous-time filtering for digital magnetic read channels.

Richard R. Spencer(S’82–M’86–SM’97) received the B.S.E.E. degree fromSan Jose State University, CA, in 1978, and the M.S. and Ph.D. degreesin electrical engineering from Stanford University, CA, in 1982 and 1987,respectively.

While completing the B.S. degree, he worked in the Radio ProductsLaboratory at Aydin Energy Division, Palo Alto, CA, designing circuits forthe IF section of microwave transceivers, and at Electro Magnetic FilterCompany, Mountain View, CA, conducting electromagnetic compatibilitytesting. In 1978, he joined Memorex Corporation, Santa Clara, CA, wherehe first worked on disk drive read/write electronics and later was the ProjectManager for an integrated circuit process monitor chip and test system. AtStanford University, his research concentrated on circuit design for integratedsensors. Since leaving Stanford, he has been with the Department of Electricaland Computer Engineering at the University of California, Davis, where heis currently an Associate Professor. His research interests include analogsignal processing circuitry, disk/drive read/write electronics, circuits for digitalcommunication, and integrated sensors.

Dr. Spencer received the UCD-IEEE Outstanding Undergraduate TeachingAward in 1991–1992 and 1992–1993. He was a co-organizer of the IEEESolid-State Circuits & Technology Committee Workshop on Integrated Sen-sors in 1988, and has been a Session Chair for the International Solid-StateCircuits Conference and the Symposium on VLSI Circuits. He was a GuestCo-Editor of the IEEE JOURNAL OF SOLID-STATE CIRCUITS in December 1992.He was a member of the Program Committee for the ISSCC from 1987–1993,and was the Chair of the Sensors, Imagers, and Displays subcommittee from1995–1997.

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Performance and implementation of adaptive partial response maximum likelihood detection