In patterned media, track pitch may be small causing adjacent tracks to have significant interference on the replay waveform from the main data track. To mitigate ...
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
Two-Dimensional Generalized Partial Response Equalizer for Bit-Patterned Media Sheida Nabavi and B. V. K. Vijaya Kumar Data Storage Systems Center (DSSC), Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
Abstract- The use of bit-patterned media is one of the approaches being investigated to extend magnetic recording densities to 1 Tbit/in2 and beyond. In patterned media, track pitch may be small causing adjacent tracks to have significant interference on the replay waveform from the main data track. To mitigate the effect of such inter-track interference (ITI), we propose the use of a two-dimensional (2D) generalized partial response (GPR) equalizer. We select both the equalizer and the partial response target using the minimum mean squared error (MMSE) criterion. However, we avoid the need for a 2D Viterbi algorithm by imposing a constraint on the 2D target that forces the adjacent track contributions (in the ideal case) to zero. Simulation results show that this 2D equalizer significantly improves the bit error rate (BER). In this work, the effect of a 2D GPR equalizer on the performance of a patterned media system in the presence of track mis-registration (TMR) is also investigated. Based on the simulation results, the 2D equalization method appears to be more tolerant to TMR than the conventional GPR. The main drawback of the proposed method is the need for simultaneously acquiring the signals from three adjacent tracks. Keywords- bit-patterned media; Intertrack interference; twodimentional equalization;
I.
INTRODUCTION
Due to the rapid growth of internet and digital communications, the need for higher storage density continues to increase. Conventional magnetic recording technologies are expected to reach their physical and engineering limits in the near future. In a magnetic recording system, the signal-tonoise ratio (SNR) in the replay or readback signal is related to the number of magnetic grains per recorded bit. To increase the density, the bit size should be reduced while maintaining satisfactory SNR. That requires a reduction in the grain size. Grain size, however, cannot be shrunk significantly without the magnetization of the grains becoming thermally unstable (at room temperatures) or super-paramagnetic [1]. Among the most promising approaches proposed to achieve magnetic recording density of 1 Tbit/in2 and beyond is the use of bitpatterned media [2], which we will simply refer to as patterned media. In patterned media each bit is stored in a single-domain magnetic island. Because of the nonmagnetic regions between magnetic islands, patterned media offer the advantages of reduced transition and track edge noise, reduced non-linear bit shift and simplified tracking [3], [4]. While much attention has been paid to the fabrication of patterned media, little investigation has been focused on the processing of readback signals from such media. Replay process for longitudinal bit-patterned media was investigated
in [4]-[7]. Patterned media channels for perpendicular media, based on the conventional two-dimensional (2D) approaches of calculating reciprocity integral, have been discussed by Hughes [6], [7]. In order to accurately model the readback signal, Nutter et al. [8] used a three-dimensional (3D) approach to investigate the effect of island geometry on the readback signal. The effects of geometry and distribution of the patterned islands on read channel performance were examined in [9] in terms of bit error rate (BER) vs. SNR. In [9], [10] the performance of the perpendicular bit-patterned media was evaluated for different equalizers. The effect of the read head offset or track mis-registeration (TMR) on the performance of the read channel was investigated in [11], [12]. In high-density patterned media, islands are likely distributed closely, both down-track and cross-track. Therefore, in addition to inter-symbol interference (ISI) there will also be inter-track interference (ITI). This inevitable ITI can degrade the channel performance significantly. Also, TMR can further increase ITI due to the read head sensing the islands in the adjacent tracks [11]. Therefore for patterned media, both ISI and ITI have to be considered. In this work, we use the 1 Tbit/in2 operating point and we try to mitigate the effect of ITI and ISI by using an equalizer. In patterned media, the writing frequency and track positions are fixed by the patterning of the islands. Fixed distribution of the islands and the knowledge of the along-track pulse response and the across-track profile, enable us to use a 2D approach. To combat ITI and ISI, we propose a 2D equalizer with a 2D partial response (PR) target. In this work, instead of choosing a fixed PR target, we optimize both the equalizer coefficients and the PR target based on minimum mean square error (MMSE) criterion. However, we avoid the need for a 2D Viterbi algorithm (VA) by imposing a constraint on the 2D PR target that forces ITI to zero and keeps only the controlled ISI. Then the conventional VA can handle this known ISI. This method can be used in other 2D data storage systems such as holographic data storage systems [13] and 2D optical storage system (TwoDOS) [14] as well. The rest of this paper is organized as follows. In Section II, we describe the channel model that we have used for the patterned media channel simulator. In Section III, we describe a system with conventional GPR equalizer and we show the effect of ITI on the performance of that system. Design of the optimal 2D equalizer and target are presented in Section IV along with the results of applying the 2D GPR equalizer followed by the conventional Viterbi detector. In Section V, the effect of read head offset on the channel performance for
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
both 1D and 2D GPR targets is investigated. Section VI provides our conclusions.
w(k) a(k)
II.
CHANNEL MODEL
To model the channel, we used the replay response to an isolated magnetic island given in [6], [8] and [9]. Based on [6], we model an island pulse response for perpendicular patterned media with soft underlayer (SUL) and without SUL, as the sum of two Lorentzian pulses: one corresponding to the island upper surface charge and the other corresponding to the island bottom surface charge that is smaller and broader with opposite polarity [6]. For the case of media with SUL, The Lorentzian pulse corresponding to the island bottom surface charge is smaller and wider than that of for the media without SUL. As a result the pulse response for media without SUL is a triplet pulse; but the pulse response for media with SUL is not a triplet pulse, as shown in Fig.1. The across-track profile is modeled as a Lorentzian pulse. In patterned media, the size and shape of the recorded domains are constrained by the geometry of the magnetic islands. Therefore Nutter et al. considered the magnetization variation both along the track and across the track direction in the replay model to obtain a more accurate model [8], [9]. In [8] and [9] the replay response has been simulated by extending the reciprocity integral over 3D space. In this work the pulse responses and track profiles have been synthesized to have the similar characteristics as those in [9]. The synthesized readback pulse for an isolated square island with length of 12.5 nm has a PW50 of 18.8 nm for the case of medium without SUL and a PW50 of 21.2 nm for the case of medium with SUL [9], as shown in Fig 1. The track profile PW50 is 29.5 nm in the case of medium without SUL and track profile PW50 is 31.2 nm in the case of medium with SUL [9]. The square islands of length 12.5 nm and period of 25 nm correspond to 1 Tbit/in2 areal density.
Fig. 1. Replay pulse response for isolated square island of length 12.5 nm for the cases of media with SUL (solid curve) and without SUL (dashed curve)
Equalizer
x(k) Channel
y(k) +
Target G(D)
z(k) F(D) d(k)
Viterbi Detector
â(k)
e(k) +
Fig. 2. Schematic of the system
III.
PATTERNED MEDIA WITH
ITI
To investigate the effect of ITI on the performance of the channel, we simulate two cases: single track (no ITI present) and multi-track (ITI present). In this simulator, we use the GPR equalizer [15], [16] because it gives the best channel performance in terms of BER vs. SNR [9]. The schematic of the simulated system is shown in Fig. 2. A. 1D Simulator In Fig. 2, a(k) is the random, uncoded binary input data. In the case of single track, the channel output is generated as the linear superposition of island pulse responses. In the case of multi-track, signal interferences due to adjacent tracks are calculated in a similar way, but with the pulse response evaluated with the read head displaced off track by a track pitch. For the case of the main track and two adjacent interfering tracks, the channel output (including ITI), is the sum of the three readback signals: readback signal of the main track and the two interference signals from the adjacent tracks [11]. This channel output signal is then sampled at sampling instants commensurate with the period of the islands. The channel output is represented by x(k) in Fig. 2. For this work we assumed that the patterned medium is ideal, i.e., no jitter noise in the positions of the islands is assumed and the only noise included is additive white Gaussian noise (AWGN). Here y(k) is the output of the patterned media channel that is corrupted by AWGN w(k) and z(k) is the output of the equalizer that is input to the Viterbi detector. Equalizer F(D) is a finite impulse response (FIR) filter that reshapes the output of the channel close to the known and desired signal. This desired signal, d(k), is generated by a target polynomial G(D), and e(k) is the difference value between desired value and real value of the equalizer output. The error response e(k) is used to optimize the equalizer and the partial response target based on the MMSE approach [15], [16]. The dashed lines in Fig. 2 indicate the filter optimization path. The output of the Viterbi detector is â(k); and it is the estimated value of a(k). In the simulator we set the length of the equalizer to 11 and the length of the target polynomial to 3. Longer equalizer and target do not improve the performance of the channel significantly. To have a target with a good match to the pulse response, we considered the target G(D) = [g-1 g0 g1], and imposed the monic constraint, i.e., g0=1. In this analysis the SNR is defined as follows.
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(1)
SNR = 20 log(V p / σ ) ,
where Vp =1 is the peak signal of the readback waveform and σ is the standard deviation of the noise. B. Results The results of the simulator for the cases of medium with and without SUL; and for the cases of single track and multitrack are shown in Fig. 4 and Fig. 5, respectively. The results are very close to the results in [9]. For the case of single track, G(D) = [0.1492 1 0.1492] for media with SUL and G(D) = [-0.1368 1 -0.1368] for media without SUL. For the case of multi-track, G(D) = [0.0986 1 0.0984] for media with SUL and G(D) = [-0.0734 1 -0.0738] for media without SUL. As is evident in Figs. 4 and 5, ITI significantly degrades the performance of the system. For both media with SUL and without SUL, for a target BER of 10-4, ITI causes more than 7 db SNR loss. To mitigate the effect of the ITI on the performance of the channel, we propose the use of 2D GPR. IV. 2D GPR EQUALIZER To design and implement a 2D GPR equalizer, we assume that we have the readback signals of three adjacent tracks. Reading three tracks of data for cross talk cancellation (XTC) was investigated for optical recording [17], [18]. In theory, reading three tracks may be possible by having multi-head read elements or buffering synchronized (synchronization is much easier in patterned media than in continuous media because the island positions are well known a priori) data from three adjacent tracks. Although such read architectures for magnetic recording may be difficult to implement currently, we hope that the benefits of using 2D GPR will provide some incentive for work towards such read-head designs. Also in [17] and [18] it was shown that reading three tracks and using them in equalization is beneficial. A. Designing 2D equalizer and 2D target In this method, the equalizer and the target polynomial are in 2D. The schematic of 2D representation of the system is shown in Fig. 3. Because the distributions of islands are fixed, we can mention bits or islands based on their position rather than time. Here the k and j indices correspond to the alongtrack and the across-track position of an island, respectively. The target polynomial, G(D1,D2), and equalizer polynomial F(D1 , D2), can be written as (2) and (3). w(j,k) a(j,k)
Equalizer
x(j,k)
Channel
y(j,k)
+ Target G(D1 ,D2)
z(j,k)
F(D1 ,D2) d(j,k)
Viterbi Detector
â(j,k)
e(j,k)
+
Fig. 3. Schematic of the 2D channel that processes data corresponding to multiple adjacent tracks in parallel
G ( D1 , D2 ) =
L
L
∑ ∑g
D1m D2n
(2)
f m ,n D1m D2n
(3)
m ,n
m =− L n =− L
F ( D1 , D2 ) =
M
N
∑ ∑
m =− M n =− N
where D1 and D2 correspond to unit shifts in the across-track and along-track directions, respectively. The target polynomial is of size (2L+1) by (2L+1) whereas the equalizer coefficient matrix is of size (2M+1) by (2N+1). In this work we set L=1, M=1 and N=3. Therefore G and F can be represented equivalently by the following matrices. g −1,−1 G = g0,−1 g1,−1
f −1,−3 F = f 0,−3 f1,−3
g −1,1 g 0,1 g1,1
g −1,0 g 0,0 g1,0
f −1,−2 … f0,−2
…
f1,−2
…
f −1,3 f0,3 f1,3
(4)
(5)
The output of the equalizer z(j,k) is the 2D convolution of the equalizer and the channel output y(j,k). The readback signal from each track is obtained as discussed in previous section for the case of multi-track. Then all readback signals have contributions of their adjacent tracks. The ideal output of the target polynomial d(j,k) is also computed as the 2D convolution of the target and the input data. The goal is to detect the main track data. The main track corresponds to j=0. The main track equalizer output and the main track desired output are represented by z(0,k) and d(0,k) respectively and these can be expressed as follows. M
z k = z ( 0, k ) =
N
∑ ∑
f (m , n) y (− m , k − n)
(6)
g (m , n )a (− m , k − n )
(7)
m=− M n=− N
d k = d (0, k ) =
L
L
∑ ∑
m=− L n=− L
For the case of M=1, we need three adjacent tracks of readback signal (y(-1,k), y(0,k) and y(1,k)). To synthesize three adjacent track readback signals containing ITI, we use five tracks of their corresponding input data. The error signal e(k ) used to adjust the equalizer coefficients and the GPR target, is given as follows. e( k ) = zk − d k
(8)
We represent the data sequence, the equalizer coefficients and the target polynomial coefficients by vectors as shown in the following.
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f = f − M ,− N
f − M , − N +1 …
g = g − L ,− L y k = yM , k + N
a k = aL ,k + L
f 0,0 …
f M , N
f M , N −1
g L , L
g − L ,− L +1 … g0,0 … g L ,L −1
T
(9) (10)
T
y− M , k − N (11)
yM , k + N −1 … y0, k … y− M , k −( N −1)
aL, k + L −1 … a0,k
T
y− L ,k − L (12)
… y− L ,k − ( L −1)
e( k ) = f T y k - gT ak
(13)
The mean square error can then be obtained as follows.
{
2
} = E {( f y
- gT a k )( f T y k - gT a k )
T
T
k
}
= f T Rf − 2f T Tg + g T Ag
(14)
where R is the auto-correlation matrix of the channel output data, T is the cross-correlation of the input data and the channel output data and A is the auto-correlation of the input data, i.e., R = E {y k yTk } , T = E {y k ak T } and A = E {ak aTk } , where
E denotes the expectation. When minimizing the mean squared error (MSE) in (14), to avoid the trivial answer of f=g=0, we should impose some constraints on g. In addition, we also want to avoid the use of 2D Viterbi algorithm because of its high complexity. Therefore we propose a constraint on g that forces the adjacent tracks contributions (in the ideal case) to zero. An example of a target of size 3×3, i.e., the length of ISI and ITI are 3, with the constraint is shown in (15) and in vector form in (16). 0 G = g 0, −1 0
0 0 1 g 0,1 0 0
(15)
g = 0 0 0 g 0, −1 1 g0,1 0 0 0
T
(16)
The above constraint can be expressed as follows. (17)
ET g = c
where
c = [1 0 0 0 0 0 0 ] , T
(18)
and,
0 1 0 T E = 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 . 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
J = f T Rf − 2f T Tg + gT Ag − 2λ T (ET g - c),
(20)
T
Based on the vector representation, we can express the error as follows.
E ( e( k ) )
With this constraint, we can minimize the following Lagrange functional to achieve constrained minimization of the MSE.
(19)
where λ, is a vector containing the Lagrange multipliers. By setting the gradients of J with respect to f, g, λ to zero vectors, we obtain the optimized target and equalizer coefficient vectors as follows. λ = (ET (A - TT R -1T)-1 E)-1 c
(21)
g = (A - TT R -1T)-1 Eλ
(22)
f = R -1Tg
(23)
B. 2D simulator In this work, we obtain the 2D equalizer and the “effectively 1D” generalized partial response target using (21), (22) and (23). This 2D equalizer is applied to three parallel readback signals from the three adjacent tracks, as shown in Fig. 3. However, a conventional 1D Viterbi detector (designed for the “effectively 1D” GPR target) is used for the 2Dequalized main track data. We tried different lengths of equalizers and targets. To avoid unnecessary complexity we consider the size of 3×3 for the target and the size of 3×7 for the equalizer. Longer target and equalizer do not improve the channel performance significantly; but increase the complexity. The targets for media with SUL and without SUL are given in (24) and (25), respectively. 0 0 0 G = 0.1302 1 0.1302 0 0 0 0 0 0 G = −0.0862 1 −0.0862 0 0 0
(24)
(25)
C. Results The results of the simulator for the cases of medium with and without SUL for 2D equalizers are shown in Fig. 4 and Fig. 5, respectively. SNR was defined in (1). As shown, the use of the 2D GPR equalizer improves the bit error rate (BER) performance of the channel. For target BER at 10-4, we gain more than 4 dB in SNR for media with SUL and about 4 dB in SNR for media without SUL. The BER performance of the multi-track channel with 2D GPR equalizer is not as good as the single-track channel (i.e., with no ITI) with conventional GPR equalizer as can be seen in Fig. 4 and Fig. 5. This is probably due to noise enhancement caused by forcing adjacent track interference to zero. The improved BER performance comes at the cost of using a 2D equalizer and the need to read data from three adjacent tracks.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
V. EFFECT OF READ HEAD OFFSET Read-head offset (also known as track mis-registration or TMR) can increase ITI due to the read head sensing the islands in the adjacent tracks; and as a result can degrade the performance of the channel. In the real system the amount of TMR is unknown and the presence of small amount of TMR is inevitable. Therefore using equalizers that are more tolerant to TMR is essential. In this work we also investigated the effect of TMR on the BER performance of the patterned media for the cases of using the conventional GPR equalizer and using the 2D GPR equalizer. A. Simulating TMR In the presence of TMR, the readback signal is modeled as described in section III; but we include the read head offset. Therefore the readback signal from the main track is modeled as the linear superposition of pulse response evaluated at the off-track position; and interferences due to the immediately adjacent track signals are computed in a similar way, but with the pulse response evaluated with the read head displaced from track center by a track pitch ± off-track. The readback signal is the summation of these three signals.
Fig. 6. BER vs SNR for 10% and 20% TMR, solid lines are for 1D equalizer and dashed lines are for 2D equalizer, media with SUL
Fig. 7. BER vs SNR for 10% and 20% TMR, solid lines are for 1D equalizer and dashed lines are for 2D equalizer, media without SUL Fig. 4. BER vs SNR using 2D GPR equalizer, medium with SUL
B. Results For different TMR values, we compared the BER performance of channel for the cases of using the conventional 1D GPR equalizer and using 2D GPR equalizer followed by the conventional Viterbi detector. To investigate the performance of these equalizers in the presence of unknown TMR, a 1D GPR equalizer and a 2D equalizer are designed for 0% TMR and are used for different values of TMR. In this work the value of TMR is defined as (26). TMR=
head offset × 100 island period
(26)
The results of applying the 1D GPR equalizer and the 2D GPR equalizer (designed for 0% TMR) to the channel with Fig. 5. BER vs SNR using 2D GPR equalizer, medium without SUL
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
10% and 20% TMR are shown in Fig. 6 and Fig. 7, for media with and without SUL respectively. It is evident from Fig. 6 and Fig. 7 that the 2D equalizer is more tolerant to TMR. For 10% off-track, at a target BER of 10-4, for both media with SUL and without SUL, the loss of SNR for conventional equalizer is about 1 dB, whereas it is less than 0.5 dB for the 2D equalizer and conventional Viterbi detector. For the 20% off-track case, a target BER of 10-4 and media with SUL, SNR loss is about 5dB, whereas it is less than 3 dB for the 2D equalizer. Also for the 20% off track case at target BER of 10-4 and for media with no SUL, we lose about 4 dB SNR, whereas the SNR loss is about 3 dB when using the 2D equalizer. Therefore we can gain SNR by using a 2D GPR followed by 1D Viterbi. VI. CONCLUSIONS In this work we proposed the use of a zero-ITI forcing 2D GPR equalizer followed by conventional 1D Viterbi detector for processing signals read from 3 adjacent tracks of bit patterned media. Simulation results show that this method can significantly improve the performance of the channel in terms of BER versus SNR. Also we simulated the effect of TMR and observed that the 2D GPR equalizer outperforms the 1D GPR equalizer. The main drawback of the proposed method is the need to acquire signals from three adjacent tracks simultaneously. While this may be beyond the reach of current read-head technology, our proposed 2D equalization method offers sufficient benefits to spur research in such read-head technology. REFERENCES [1] [2] [3]
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