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91

Maximum-Likelihood Sequence Detector forDynamic Mode High Density Probe Storage

Naveen Kumar∗, Pranav Agarwal†, Aditya Ramamoorthy∗ and Murti Salapaka†∗Dept. of Electrical and Computer Engineering

Iowa State University, Ames, IA 50010Email: nk3,adityar@iastate.edu

†Dept. of Electrical and Computer EngineeringUniversity of Minnesota, Minneapolis, MN 55455

Email: agar0108,murtis@umn.edu

Abstract— There is an ever increasing need for storing data insmaller and smaller form factors driven by the ubiquitous useand increased demands of consumer electronics. A new approachof achieving a few Tb per in2 areal densities, utilizes a cantileverprobe with a sharp tip that can be used to deform and assess thetopography of the material. The information may be encodedby means of topographic profiles on a polymer medium. Theprevalent mode of using the cantilever probe is the static modethat is known to be harsh on the probe and the media. Inthis paper, the high quality factor dynamic mode operation,which is known to be less harsh on the media and the probe,is analyzed for probe based high density data storage purposes.It is demonstrated that an appropriate level of abstraction ispossible that obviates the need for an involved physical model.The read operation is modeled as a communication channelwhich incorporates the inherent system memory due to the inter-symbol interference and the cantilever state that can be identifiedusing training data. Using the identified model, a solution to themaximum likelihood sequence detection problem based on theViterbi algorithm is devised. Experimental and simulation resultsdemonstrate that the performance of this detector is severalorders of magnitude better than the other existing schemes andconfirm performance gains that can render the dynamic modeoperation feasible for high density data storage purposes.

I. I NTRODUCTION

The rapid growth in the personal computer industry andthe Internet is a major factor underlying the demand forultra-high capacity storage devices. Demands of a few Tbper in.2 are predicted in the near future. Currently storagesystems are mainly based on magnetic, optical and solid statetechnologies. However all these technologies are reachingfundamental limits on their achievable areal densities.

A promising high density storage methodology, that is thefocus of this paper utilizes a sharp tip at the end of a microcantilever probe to create (or remove) and read indentations(see [15] for an example). Recently, experimentally achievedtip radii near 5 nm on a micro-cantilever were used to createareal densities close to 4 Tb/in2 [7], [1]. The presence/absenceof an indentation represents a bit of information. The mainadvantage of this method is the significantly higher arealdensities that are possible. The work of [15], [8] uses thestatic mode operation of the atomic force microscope (AFM)

This research was supported in part by NSF grants ECCS-0802019 andECCS-0802117

where the cantilever’s operation is in contact mode whichdegrades the medium and the probe. In the dynamic modeoperation given in this paper, the cantilever with a high qualityfactor is forced sinusoidally using a dither piezo and it gentlytaps on the medium intermittently which reduces the mediumand the probe wear [16]. High quality factor cantilevers aredesirable for a dynamic mode operation because the effectof a topographic change on the medium, on the oscillatingcantilever, lasts much longer (approximatelyQ cycles whereQ is the quality factor) and the signal to noise ratio increaseswith higher quality factors (improves as

√Q [12]). However,

the advantages of high quality factor with respect to resolutionbecome disadvantages with respect to bandwidth which isapparent as the effect of a topographic change lasts longerand therefore information on the medium has to be tempo-rally spaced to reduce inter-symbol interference. Thus thechallenges of designing good detection schemes are twofold;the first is that of modeling that remains tractable for a datastorage purpose which might not encapsulate the entiretyof the behavior inherent in the dynamic mode but remainsuseful for detecting information stored on the medium andthe second is to use the model to exploit the advantages ofhigh quality factor cantilevers without sacrificing bandwidth.Both these challenges are met in this paper and the results arecorroborated with experimental results.

The cantilever dynamics is complex with a number of phys-ical intricacies that can render the modeling of the dynamicmode intractable for high performance data storage purposes.In this paper, it is demonstrated that an appropriate level ofabstraction is possible that obviates the need for an involvedphysical model. The dynamic mode read operation is modeledas a communication channel model which incorporates theinherent system memory due to the inter-symbol interferenceand the cantilever state that can be identified using trainingdata. Using the identified model, a solution to the maximumlikelihood sequence detection problem based on the Viterbialgorithm is devised. Experimental and simulation resultscorroborate the analysis of the detector, demonstrate thattheperformance of this detector is several orders of magnitudebetter than the performance of other existing schemes andconfirm performance gains that can render the dynamic modeoperation feasible for high density data storage purposes.

(a) (b)Fig. 1. (a) Shows the main components of a probe based storagedevice.(b) Shows a block diagram representation of the cantilever systemG beingforced by white noise (η), tip-media forceh and the dither forcingg. Theoutput of the blockG, the deflectionp is corrupted by measurement noiseυthat results in the measurementy. Tip media forceh = φ(p).

The paper is organized as follows. In Section II, backgroundand related work of the probe based data storage system ispresented. Section III deals with the problem of designing andanalyzing the data storage unit as a communication system andfinding efficient detectors for the channel model. Section IVand Section V report results from simulation and experimentsrespectively. Section VI summarizes the main findings of thispaper and provides the conclusions and future work.

II. BACKGROUND AND RELATED WORK.

In the standard atomic force microscope setup, which hasformed the basis of probe based data storage, the cantileverdeflection is measured by a beam-bounce method where alaser is incident on the back of the cantilever surface andthe laser is reflected from the cantilever surface into a splitphotodiode (see Figure 1(a)). The advantage of the beam-bounce method is the high resolution (low measurement noise)and high bandwidth (in the 2-3 MHz) range. The disadvantageis that it is more cumbersome for integrating this methodinto a parallel operation where multiple cantilevers operatein parallel. There are attractive measurement mechanisms thatintegrate the cantilever motion sensing onto the cantileveritself. These include piezo-resistive sensing [4] and thermalsensing [6]. For the dynamic mode operation there are variousschemes to actuate the cantilever that include electrostatic [5],mechanical by means of a dither piezo that actuates the supportof the cantilever base and magnetic [9]. In this paper, it is as-sumed that the cantilever is actuated by a dither piezo and thesensing mechanism employed is the beam bounce method (seeFigure 1(a)). Even though the experimental setup reported inthis paper uses a particular scheme of detection and actuationof the cantilever, the paradigm developed for data detection islargely applicable in principle to other modes of detectionandactuation of the cantilever. For channel modeling purposes,the AFM dynamics is viewed in the systems perspective as aninterconnection of a linear cantilever system with the nonlineartip-media interaction forces (φ) in feedback (See Figure 1(b)).The interaction forcesφ depend on the tip deflectionp [13].

A. Cantilever-Observer Architecture

As mentioned earlier, the highQ cantiever gives high signalto noise ratio and less tip-media wear but it is limited in band-width. The quality factor of the cantilever can be reduced tofasten the decay of the transient resulting in higher bandwidth.However, low Q operation results in lower resolution andhigher forces. This tradeoff between resolution and bandwidthcan be well tackled by using the cantilever-observer frame-work shown in Figure 2. The cantilever-observer frameworkdecouples the bandwidth fromQ allowing one to use a high

Q system giving better resolution and high bandwidth at thesame time [12]. It gives the flexibility to shorten the impulseresponse of the channel and cancels the effect of the ditherat the output. In this framework, the error process is denotedby e which is the actual deflection signal minus the estimateddeflection signal. With the dither input (g), thermal noise (η),measurement noise (v) and the observer with Kalman gain, theerror processe approaches a zero mean wide sense stationarystochastic process whenφ = 0 i.e. no tip-media interaction ispresent [12]. Ifφ is neglected from the system, the systemfrom the dither input (g) to error process (e) can be modeledas a linear system whose impulse response is denoted byΓ(t)with zero mean white Gaussian noisen(t) with varianceVwhich is characterized by thermal and measurement noise (SeeFigure 3).

Fig. 2. An observer architecture for the system in Figure 1(b)

III. C HANNEL MODEL AND DETECTORS

The cantilever based data storage system can be modeledas a communication channel as shown in Figure 3. Thecomponents of this model are explained below in detail.Shaping Filter (b(t)): The model takes as input the bitsequence(a0, a1 . . . aN−1) whereak, k = 1, . . . , N − 1 isequally likely to be 0 or 1. In the probe storage context, ‘0’refers to the topographic profile beinglow and ‘1’ refers tothe topographic profile beinghigh. Each bit has a duration ofT seconds. This duration can be found based on the length ofthe topographic profile specifying a single bit and the speedofthe scanner. The height of the high bit is denoted byA. Thecantilever interacts with the media by gently tapping it when itis high. When the media is low, typically no interaction takesplace. We model the effect of the medium height using a filterwith impulse responseb(t) (shown in Figure 3) that takes asinput, the input bit impulse traina(t) =

∑N−1k=0 akδ(t− kT ).

The output of the filter is given bya(t) =∑N−1

k=0 akb(t−kT )which is high/low when the corresponding bit is ‘1’/‘0’.Nonlinearity Block (φ): The cantilever oscillates at frequencyfc which means that in each cantilever cycle of durationTc(= 1/fc), the cantilever hits the media at most once if themedia is high during a timeTc. In each high bit durationT , the cantilever hits the mediaq times (i.e.T = qTc) withvarying magnitudes. Therefore, forN bits, the output of thenonlinearity block is given by,a(t) =

∑Nq−1k=0 νk(a)δ(t −

kTc), where νk denotes the magnitude of thekth impactof the cantilever on the medium. We are approximating thenonlinearity block output as a sequence of impulsive forceinputs to the cantilever. The strength of the impulsive forceinputs are dependent on previous impulsive hits; preciselybecause the previous interactions affect the amplitude of theoscillations that in turn affect how hard the hit is at a particularinstant of time. The exact dependence is very hard to model

2

Fig. 3. Continuous time channel model of probe based data storage system

deterministically and therefore we chose a Markov model forthe sequence of impact magnitudes for a single bit duration,

νi = G(ai, ai−1, . . . , ai−m) + bi (1)

whereνi = [νiq νiq+1 . . . ν(i+1)q−1]T , G(ai, ai−1, . . . , ai−m)

is a function of current and lastm bits,m denotes the inherentmemory of the system andbi is a zero mean i.i.d. Gaussianvector of lengthq.Channel Response (Γ(t)): The Markovian modeling of theoutput of the nonlinearity block as discussed above allowsus to break the feedback loop in Figure 2. The rest of thesystem can then be modeled by treating it as a linear systemwith impulse responseΓ(t). The net system also includes thelinear observer that is implemented as part of the detector.Theimpulse responseΓ(t) can be found in closed form for a givenset of parameters of cantilever-observer system [11].Channel Noise (n(t)): The measurement noise (from theimprecision in measuring the cantilever position) and thermalnoise (from modeling mismatches) can be modeled by a singlezero mean white Gaussian noise process (n(t)) with powerspectral density equal toV .

The continuous time innovation outpute(t) becomes,

e(t) =

Nq−1∑

k=0

νk(a)Γ(t− kTc) + n(t) = s(t, ν(a)) + n(t),

where s(t, ν(a)) =∑Nq−1

k=0 νk(a)Γ(t − kTc) and ν(a) =(ν0(a), ν1(a) . . . νNq−1(a)). The sequence of impact valuesνi is assumed to follow a Markovian model as explainedabove,Γ(t) is the channel impulse response andn(t) is azero mean white Gaussian noise process.A. Sufficient Statistics for Channel model

A set of sufficient statistics for the channel model isderived in this section. Given the probabilistic model onνand finite bit sequence (a), an orthogonal information losslessdiscretization ofe(t) by expansion over an orthonormal finite-dimensional basis with dimensionN can be achieved whereNorthonormal basis functions span the signal space formed bys(t, ν(a)). The discretized innovation by expandinge(t) overN orthonormal basis functions is given by,e = s(ν(a)) + n,where e = (e0, e1 . . . eN ), s(ν(a)) = (s0, s1 . . . sN ),n = (n0, n1 . . . nN ) and n ∼ N(0, V IN×N ) whereIN×N stands for N × N identity matrix. The maximumlikelihood estimate of the bit sequence can be found asˆa =argmaxa∈{0,1}N f(e|a) where ˆa = (a0, a1 . . . aN−1) is theestimated bit sequence andf denotes a pdf. The termf(e|a)can be further simplified as,

f(e|a) =∫

ν

f(e|a, ν)f(ν|a)dν =

∫

ν

1

(2πV )N

2

× exp−||e− s(ν(a))||2

2Vf(ν|a)dν =

1

(2πV )N

2

exp−||e||22V

×∫

ν

exp−(||s(ν(a))||2 − 2eT s(ν(a)))

2Vf(ν|a)dν

Fig. 4. Discretized channel model with whitened matched filter

where ||.||2 denotes Euclidean norm,f(e|a, ν) andf(ν|a) denote the respective conditional pdf’s andν = (ν0, ν1 . . . νNq−1). The correlation betweeneand s(ν(a)) can be equivalently expressed as an integralover time because of the discretization procedure. We have,eT s(ν(a)) = νT z′ where z′k =

∫∞

−∞e(t)Γ(t − kTc)dt for

0 ≤ k ≤ Nq− 1 is the output of a matched filterΓ(−t) withinput e(t) sampled att = kTc, ν = (ν0, ν1 . . . νNq−1) andz′ = (z′0, z′1 . . . z′Nq−1). f(e|a) can now be written as,

f(e|a) =1

(2πV )N

2

exp−||e||22V

︸ ︷︷ ︸

h(e)

×∫

ν

exp−||s(ν(a))||2

2Vexp

νT z′

Vf(ν|a)dν

︸ ︷︷ ︸

F(z′|a)

Thus f(e|a) can be factorized intoh(e) (dependent onlyon e) and F(z′|a) (for a given a dependent only onz′).Using the Fisher-Neyman factorization theorem [3], it followsthat z′ is a vector of sufficient statistics for the detectionprocess. From the definition of sufficient statistics, we canclaim,f(e|a)

f(z′|a)= C, where C is a constant independent of

a. Thus the detection problem can be reformulated as,ˆa =argmaxa∈{0,1}N f(z′|a). The bit detection problem dependsonly on the matched filter outputs (z′). These matched filteroutputsz′k for 0 ≤ k ≤ Nq − 1 can be further simplified as,z′k =

∑Nq−1k1=0 νk1

(a)h′k−k1

+ n′k, whereh′

k−k1=

∫∞

−∞Γ(t −

kTc)Γ(t − k1Tc)dt and n′k =

∫∞

−∞n(t)Γ(t − kTc)dt. A

whitening matched filter can be determined to whiten outputnoise n′

k [10]. We shall denote the discretized output ofwhitened matched filter shown in Figure 4 aszk, such thatzk =

∑I

k1=0 νk−k1(a)hk1

+nk, where the filter{hk}k=0,1,...,I

denotes the effect of the whitened matched filter and thesequence{nk} represents Gaussian noise with varianceV .

B. Viterbi Detector Design

Note that the outputs of the whitened matched filterz, con-tinue to remain sufficient statistics for the detection problem.Therefore, we can reformulate the detection strategy as,

ˆa = arg maxa∈{0,1}N

f(z|a)

= arg maxa∈{0,1}N

ΠN−1i=0 f(zi|a, zi−1

0 ) (2)

where z = [z0 z1 . . . zNq−1]T . Let zi be the received

output vector corresponding to theith input bit i.e. zi =[ziq ziq+1 . . . z(i+1)q−1]

T andzi−10 = [zT0 zT1 . . . zTi−1]

T . In ourmodel, the channel is characterized by finite impulse responseof length I i.e. hi = 0 for i < 0 and i > I. In this workwe assume thatI ≤ mIq i.e. the inter-symbol-interference(ISI) length in terms ofq hits is equal tomI . Let m be theinherent memory of the system (see (1)). The length of channelresponse is known which meansmI is known but the value

3

Fig. 5. Dependency graph for the model withI = mIq and m is theinherent memory of the system

of m can’t be found becausem is the inherent memory ofthe system and depends on the experimental parameters ofthe system. In the experimental results section, we describehow we estimate the value ofm from experimental data. Thereceived output vectorzi can now be written as,

zi =

hI . . h0 0 . . 00 hI . . h0 0 . 0. . . . . . . . . . . . . .0 . . 0 hI . . h0

νiq−I

ν1+iq−I

...ν(i+1)q−1

+ ni

= Hνii−mI+ ni

where νi = [νiq νiq+1 . . . ν(i+1)q−1]T , νii−mI

=[νTi−mI

. . . νTi ]T , ni = [niq n1+iq . . . n(i+1)q−1]

T .

We now simplify the factorization in (2) so that decodingcan be made tractable. We construct the dependency graph ofthe concerned quantities which is shown in Figure 5. Usingthe Bayes ball algorithm [14], we conclude that

f(zi|νii−mI, a, zi−1

0 ) = f(zi|νii−mI), (3)

f(νi−mI|a, zi−1

0 ) = f(νi−mI|ai−1

0 , zi−10 ), (4)

f(νi−k|νi−k−1i−mI

, a, zi−10 ) = f(νi−k|νi−k−1

i−mI, ai−mI−1

0 ,

ai−1i−m−k, z

i−10 ), ∀ 1 ≤ k ≤ mI − 1 (5)

f(νi|νi−1i−mI

, a, zi−10 ) = f(νi|aii−m) (6)

where ai−10 = [a0 a1 . . . ai−1]. Although the conditional

pdf f(νi−mI|a, zi−1

0 ) andf(νi−k|νi−k−1i−mI

, a, zi−10 ) depend on

the entire past, we assume that the dependence is rapidlydecreasing. This is observed in simulation and experimentaldata as well. We make the following assumptions on thisdependence,

f(νi−mI|ai−1

0 , zi−10 ) ≈ f(νi−mI

|ai−1i−m−mI

, zi−1i−mI

) (7)

f(νi−k|νi−k−1i−mI

, ai−mI−10 , ai−1

i−m−k, zi−10 )

≈ f(νi−k|νi−k−1i−mI

, ai−1i−k−m, zi−1

i−k), ∀ 1 ≤ k ≤ mI − 1 (8)

i.e. the dependence is restricted to only the immediate neigh-bors in the dependency graph. Using the above assumptionsand dependency graph results,f(zi|a, zi−1

0 ) can be further

simplified as,

f(zi|a, zi−10 ) =

∫

f(zi|νii−mI, a, zi−1

0 )f(νii−mI|a, zi−1

0 )dνii−mI

=

∫

f(zi|νii−mI, a, zi−1

0 )f(νi−mI|a, zi−1

0 )

· ΠmI−1k=1 f(νi−k|νi−k−1

i−mI, a, zi−1

0 )f(νi|νi−1i−mI

, a, zi−10 )dνii−mI

=

∫

f(zi|νii−mI)f(νi−mI

|ai−10 , zi−1

0 )ΠmI−1k=1 f(νi−k|νi−k−1

i−mI,

ai−mI−10 , ai−1

i−m−k, zi−10 )f(νi|aii−m)dνii−mI

(Using (3),(4),(5),(6))

=

∫

f(zi|νii−mI)f(νi−mI

|ai−1i−m−mI

, zi−1i−mI

)ΠmI−1k=1 f(νi−k|

νi−k−1i−mI

, ai−1i−k−m, zi−1

i−k)f(νi|aii−m)dνii−mI(Using (7),(8))

= f(zi|aii−m−mI, zi−1

i−mI).

By defining a stateSi = aii−m−mI+1, this can be furtherexpressed asf(zi|Si, Si−1, z

i−1i−mI

). Again using Bayes ballalgorithm, we conclude that

f(zii−mI|νii−2mI

, aii−m−mI) = f(zii−mI

|νii−2mI), (9)

Π2mI−1k=1 f(νi−2mI+k|νi−2mI+k−1

i−2mI, aii−m−mI

)

= ΠmI−1k=1 f(νi−2mI+k|νi−2mI+k−1

i−2mI, aii−m−mI

)

· Π2mI−1k=mI

f(νi−2mI+k|ai−2mI+ki−2mI−m+k), (10)

f(νi|νi−1i−2mI

, aii−m−mI) = f(νi|aii−m). (11)

The pdf of zii−mI= [zTi−mI

. . . zTi ]T given current stateSi

and previous stateSi−1 is given by,

f(zii−mI|Si, Si−1) = f(zii−mI

|aii−m−mI)

=

∫

f(zii−mI|νii−2mI

, aii−m−mI)f(νii−2mI

|aii−m−mI)dνii−2mI

=

∫

f(zii−mI|νii−2mI

, aii−m−mI)f(νi−2mI

|aii−m−mI)

· Π2mI−1k=1 f(νi−2mI+k|νi−2mI+k−1

i−2mI, aii−m−mI

)

· f(νi|νi−1i−2mI

, aii−m−mI)dνii−2mI

=

∫

f(zii−mI|νii−2mI

)f(νi−2mI|aii−m−mI

)ΠmI−1k=1

· f(νi−2mI+k|νi−2mI+k−1i−2mI

, aii−m−mI)Π2mI−1

k=mIf(νi−2mI+k|

ai−2mI+ki−2mI−m+k)f(νi|aii−m)dνii−2mI

(Using (9),(10),(11))

where the last step is obtained using results fromdependency graph and all the terms in the last stepexcept ΠmI−1

k=1 f(νi−2mI+k|νi−2mI+k−1i−2mI

, aii−m−mI) and

f(νi−2mI|aii−m−mI

) are Gaussian distributed. This impliesthat the pdf ofzii−mI

given(Si, Si−1) is not exactly Gaussiandistributed. If the number of states in the detector is increasedit can be modeled as a Gaussian, but this increases thecomplexity. In order to keep the decoding tractable we makethe assumption thatf(zii−mI

|Si, Si−1) is Gaussian i.e.

f(zii−mI|Si, Si−1) ∼ N(Y(Si, Si−1), C(Si, Si−1))

where Y(Si, Si−1) is the mean andC(Si, Si−1) is the co-variance. With our state definition, we can reformulate thedetection problem as a maximum likelihood state sequence

4

Fig. 6. Comparison of various detectors for simulation data. The Bayes curveis not visible in the graph as it coincides with the LMP curve.

detection problem [2],

ˆS = argmaxall S

f(z|S) = argmaxall S

ΠN−1i=0 f(zi|S, z0 . . . zi−1)

= argmaxall S

ΠN−1i=0 f(zi|Si, Si−1, z

i−1i−mI

)

= argmaxall S

ΠN−1i=0

f(zii−mI|Si, Si−1)

f(zi−1i−mI

|Si, Si−1)

= arg minall S

N−1∑

i=0

log|C(Si, Si−1)||c(Si, Si−1)|

+ (zii−mI− Y(Si, Si−1))

T

· C(Si, Si−1)−1(zii−mI

− Y(Si, Si−1))− (zi−1i−mI

− y(Si, Si−1))T c(Si, Si−1)

−1(zi−1

i−mI− y(Si, Si−1))

where ˆS is the estimated state sequence,c(Si, Si−1) is the up-permIq×mIq principal minor ofC(Si, Si−1) andy(Si, Si−1)collects the firstmIq elements ofY(Si, Si−1). It is assumedthat the first state is known. With the metric given above,Viterbi decoding can be applied to get the maximum likelihoodstate sequence.

C. LMP, GLRT and Bayes Detector

In [12], the hit detection algorithm is proposed whichdoesn’t consider the modeling of channel memory and workswell only when the hits are sufficiently apart. In [11], variousdetectors for hit detection like locally most powerful (LMP),generalized likelihood ratio test (GLRT) and Bayes detectorare presented. These detectors also ignore the inherent memoryin the channel and perform detection of single hits. Subse-quently a majority type rule is used for bit detection.

IV. SIMULATION RESULTS

We performed simulations with the following parameters.The first resonant frequency of the cantileverf0 = 63.15KHz, quality factorQ = 206, the number of hits in highbit duration is equal to13 i.e. q = 13, the cantilever forcingis chosen such that the cantilever amplitude is24 nm (withoutany interaction with media), tip-media separation is 28 nm,discretized thermal and measurement noise variance are0.1and0.001 respectively. With these thermal and measurementnoise values, a Kalman observer is designed and the lengthof the channel impulse response (I) is approximately24 fordesigned observer which meansmI is equal to2. Bit error rate(BER) will improve by taking higher values of the inherentmemory of the system (m) but it will increase complexity of

detector. Considering the tradeoff between BER andm, weare usingm = 1 for simulation data. We used a topographicprofile where high and low regions denote the bit ‘1’ and ‘0’respectively and the bit sequence is generated randomly. Thereare 13 hits when the bit is ‘1’. The simulation was performedwith the above parameters using the Simulink model thatmimics the experimental station that provides a qualitative aswell as a quantitative match to the experimental data. Theinnovations were obtained at the output of the observer fordifferent tip-media interaction. We define the system SNR astip media interaction (nm) divided by total noise variance.

In Figure 6, we compare the results of four differentdetectors. The locally most powerful (LMP), generalized like-lihood ratio test (GLRT) and Bayes detector proposed in [11]perform hit detection, as against bit detection. The minimumprobability of error for all detectors decreases as the tip-mediainteraction increases because hits become harder on mediaif tip-media interaction is increased which makes detectioneasier. The Viterbi detector gives best performance among alldetectors because it incorporates the Markovian property of νin the metric used for detection. At an SNR of 10.4 dB theViterbi detector has a BER of3 × 10−6 as against the LMPdetector that has7× 10−3.

V. EXPERIMENTAL RESULTS

In experiments, a cantilever with resonant frequencyf0 =71.78 KHz and quality factorQ = 67.55 is oscillated nearits resonant frequency. A freshly cleaved mica sheet is placedon top of a high bandwidth piezo. This piezo can position themedia (mica sheet) in z-direction with respect to cantilever tip.A random sequence of bits is generated through a FPGA boardand applied to the z-piezo. High level is equivalent to1000 mVand represents bit ‘1’ and low level is0 V and represents bit‘0’ thus creating a pseudo media profile of6 nm height. Thebit width can be changed using FPGA controller. Thus we canchange the bit width from350 µs till 60 µs. The tip is engagedwith the media at a single point and its instantaneous amplitudein response to its interaction with z piezo is monitored. Thecontroller gain is kept sufficiently low such that the operationis effectively in open loop. The gain is sufficient to cancelpiezo drift and maintain a certain level of tip-media interaction.An observer is implemented in another FPGA board which isbased on the plant’s free air model and takes plant dither anddeflection signals as its input and gives out innovation signalat the output. The innovation signal is used to detect bits bycomparing various bit detection algorithms. The experimentswere performed on Multimode AFM, from Veeco Instruments.Considering a bit width of40 nm and the amount of time forscanning of60 µs gives a tip velocity equal to2/3 × 10−3

m/sec. The total scan size is 100 micron which means thecantilever will take0.15 seconds to complete one full scan.Read scan speed for this operation is6.66 Hz. The read scanspeed for different bit width can be found in a similar manner.

With above setup, experiments are performed for differentbit widths varying from60 µs to 300 µs. For 300 µs bitwidth, there are around21 hits in high bit duration andViterbi decoding is applied on the innovation signal obtainedfrom experiment. For experimental model,I is approximately

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Fig. 7. BER for Viterbi, LMP and GLRT for different bit widthsvarying from60 µs to 300 µs for experimental data. There is a very marginal differencebetween LMP and GLRT curve which is not visible in the graph but LMPdoes perform better than GLRT.

24 which meansmI is equal to2. It is hard to estimatethe inherent memory of the system (m) from experimentalparameters. We have variedm from 0 to 2 and found dif-ferent BER using these values ofm. The total number ofstates in the Viterbi detector is2m+mI . For m ≥ 1, theimprovement in BER is quite marginal as compared to theincreased complexity of Viterbi decoding. Considering thetradeoff between the complexity of Viterbi decoding andm,we usem = 1 for which BER from Viterbi decoding is equalto 1× 10−5 whereas BER from locally most powerful (LMP)test is 0.26. BER in the case of Viterbi decoding is quiteless as compared to BER for usual thresholding detectors. Ifthe bit width is decreased to60 µs which means there arearound4 hits in high bit duration, BER for Viterbi decodingis 7.56 × 10−2 whereas BER for LMP is0.49 which meansthat LMP is doing almost no bit detection. As the bit widthis decreased, there is more ISI between adjacent bits whichincreases BER. BER for all detectors and bit widths is shownin Figure 7. It is evident that Viterbi decoding gives remarkableresults on experimental data as compared to LMP detector.Viterbi detector captures cantilever dynamics behaviour quiteprecisely in terms of mean and covariance matrix for differentstate transitions. Thresholding detectors like LMP and GLRTgiven in [11] perform very badly on experimental data. Forbit sequence like ‘000011111’, cantilever gets enough timetogo into steady state in the beginning and hits quite hard onmedia when bit ‘1’ comes after a long sequence of ‘0’ bits.The likelihood ratio for LMP and GLRT rise significantly forthe bit ‘1’ which can be easily detected through thresholding.However a sequence of continuous ‘1’ bits keeps the cantileverin steady state and the cantilever starts hitting mildly on mediawhich means the likelihood ratio remains small for these bits.It is very likely that long sequence of ‘1’ bits will not getdetected by threshold detectors.

VI. CONCLUSIONS AND FUTURE WORK

We presented the dynamic mode operation of a cantileverprobe with a high quality factor and demonstrated its appli-cability to a high-density probe storage system. The systemis modeled as a communication system by modeling thecantilever interaction with media. The bit detection problemis solved by posing it as a maximum likelihood sequencedetection followed by Viterbi decoding. Simulation and exper-

imental results show that Viterbi detector outperforms allthedetectors and gives remarkably low BER. For dynamic modeoperation with high quality factors to be practical for datastorage applications, it is evident that the modeling paradigmdeveloped is crucial as the performance of other methods doesnot meet the stringent requirements of data storage metrics.

Constrained codes are used in almost all forms of storageincluding magnetic hard drives. It is quite certain that con-strained codes will help the detection scheme with some lossof rate. In future work, the trade-off between the rate lossand detector performance will be researched. In experimentaldata, some amount of jitter is inevitably present which is wellhandled by our algorithm as the jitter is smaller than the bitwidth chosen. But if the jitter is in the order of bit width, whichmay be the case at very high densities we need to apply moreadvanced modeling and detection techniques which will bepart of future work.

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