Tabu Search Aided Multi-Track Detection Scheme for Bit-Patterned Media Recording

Abstract

A tabu search (TS) aided multi-track detection scheme for bit patterned magnetic recording is proposed. The proposed detection scheme adopts the simplified multi-track detector with a two-dimensional generalized partial response (GPR) target, uses a priori information, and employs a reliability test and a TS detector. The proposed detection scheme processes signals in three stages. First, the single-track detector with a one-dimensional GPR target detects the data for the main-track and side-tracks. Second, the reliability test is performed to detect bits with low reliability in the side-tracks by comparing the distance between the readback signal and the estimated and reconstructed readback signal. If the distance is greater than a specific value, the TS detector is conducted to improve the reliability of bits with low reliability in the side-tracks. Lastly, the simplified multi-track detector detects the main-track data using a priori information from the outputs of the TS or single-track detectors for the side-track data. The proposed and conventional detection schemes are compared in terms of the bit error rate performances. Simulation results show that the proposed detection scheme with slightly additional complexity has more than 3 dB gains compared to the conventional detection schemes.

1. Introduction

Bit-patterned media recording (BPMR) is one of the promising technologies that is considered to attain an areal density of up to 4 Tb/in² [1,2]. In BPMR, high areal density is enabled by reducing the inter-symbol and inter-track distances. Therefore, inter-track interference (ITI) in the cross-track also occurs in addition to inter-symbol interference (ISI) in the along-track. As the recording density increases, the performance degradation caused by ITI becomes more severe. Therefore, robust signal processing techniques to mitigate ITI are essential [1].

Various detection schemes were introduced for high density BPMR channels [3,4,5]. The single-track detection using a one-dimensional (1D) equalizer with a 1D generalized partial response (GPR) target was proposed in [3]. By using the 1D equalizer and 1D GPR target, the performance of the single-track detection operating on four states and eight branches is severely degraded. To employ a two-dimensional (2D) GPR target, the joint-track detection was proposed in [4] by applying the 1D equalizer with the 2D GPR target. For the complexity reduction in the detectors using the 2D GPR target, the joint-track detection used a reduced-state trellis diagram operating on four states and eight branches. Despite considering the 2D GPR target, the performance of the joint-track detection is also severely degraded due to the reduced-state trellis diagram [4] where ITI is strong. To overcome the performance degradation, the simplified multi-track detection was proposed in [5]. The simplified multi-track detection with the reduced-state trellis diagram achieves a substantial gain by using a priori information of the side-tracks from the single-track detection, since a priori information contributes to the reliability improvement of the branch metric. However, as the recording density increases, the performance of the simplified multitrack detection is gradually degraded, since the reliability of a priori information from the single-track detection is seriously reduced. Therefore, an efficient detection scheme for improving the reliability of side-track information should be studied for high density BPMR systems.

In this paper, we propose a tabu search (TS) aided multi-track detection scheme for BPMR systems to enhance the performance of the simplified multi-track detector by improving the reliability of side-track information. The proposed detection scheme using the 2D equalizer with the 2D GPR target adopts the simplified multi-track detector using the simplified detection algorithm in [4], uses a priori information as in [5], and employs a reliability test and a TS detector. That is, the proposed detection scheme additionally considers the reliability test and TS detector to the simplified multi-track detection in [5]. To improve the reliability of side-track information, the TS detector is conducted for bits with low reliability in the side-tracks in the proposed detection scheme, where bits with low reliability are selected by the reliability test. For minimizing the increase in complexity, the TS algorithm [6,7], which is a low-complexity local search procedure, is considered for the additional detector, and the TS detector works for a very small number of bits with low reliability. This improved reliability of side-track information from the TS detector is employed as a priori information in the simplified multi-track detection in [5] for the main-track data. For the reliable bits, the a posteriori probabilities (APPs) computed by the single-track detection are delivered to the simplified multi-track detector. Then, the simplified multi-track detector detects the main track data by using a priori information from the outputs of the TS or single-track detectors for the side-tracks. By using a priori information with improved reliability, the proposed detection scheme with slightly additional complexity has the significantly enhanced performance for high density BPMR systems.

2. Proposed Detection Scheme

The proposed detection scheme consists of a 1D equalizer, a 2D equalizer, a single-track detector, a simplified multi-track detector, a reliability test, and a TS detector as shown in Figure 1. That is, the reliability test and TS detector are additionally considered in the proposed detection scheme to the conventional simplified multi-track detection proposed in [5].

The proposed detection scheme operates in three stages. First, the single-track detector detects the data by using the 1D equalizer with the 1D GPR target for the main-track {0} and side-tracks {−2, −1, +1, +2}. Second, the reliability test is taken to detect bits with low reliability in the side-track {−1, +1} by comparing the distance between the readback signal and the estimated and reconstructed readback signal from the single-track detector in the first stage. If the distance is greater than the specific value, the TS detector is conducted to improve the reliability of bits with low reliability. Finally, the simplified multi-track detector detects the main-track data by using the outputs of the TS or single-track detectors on {−1, +1} tracks as a priori information.

2.1. Single-Track Detector

In the single-track detector, the APPs of the data on tracks {−2, −1, 0, +1, +2} are calculated by the Bahl–Cocke–Jelinek–Raviv algorithm [8] based on the output of the 1D equalizer. The detected data on {−2, −1, 0, +1, +2} tracks are used for the second stage. The APPs on the side-tracks {−1, +1} are used as a priori information for the main-track data in the simplified multi-track detector in the third stage.

2.2. Reliability Test

The proposed detection scheme performs a reliability test, and then conducts the TS detector for the unreliable bits on {−1, +1} tracks to reduce the complexity of the detector for a priori information of the simplified multi-track detector. If the distance between the readback signal $r_{i,k}$ with the reconstructed readback signal ${\overline{r}}_{i,k}$ on {−1, +1} tracks is larger than the specific value $\delta$, we define the corresponding bit $a_{i,k}$ as an unreliable bit, where $i$ is the track index and $k$ is time index. The reconstructed readback signal is calculated by the convolution operation of a three-by-three channel response and the detected data on {−2, −1, 0, +1, +2} tracks from the single-track detector. For example, the calculation of the reconstructed readback signals ${\overline{r}}_{-1,k}$ and ${\overline{r}}_{+1,k}$ uses the detected data ${\widehat{a}}_{i,k}$ from the single-track detector, where $i$ = {−2, −1, 0} and $k$ = {k − 1, k, k + 1}, and $i$ = {0, +1, +2} and $k$ = {k − 1, k, k + 1}, respectively. As $\delta$ increases, an increase in the number of bits performing the TS detector improves the reliability of overall bits, while it accompanies lots of complexities. It is important to properly set the $\delta$ in consideration of the complexity of detector and the reliability of bits. Based on the reliability test results, we determine that a priori information is set to the APP of the single-track detector or the output of the TS detector.

2.3. TS Detector

The low-complexity TS detector is applied in the proposed detection scheme to improve the reliability for bits with low reliability on {−1, +1} tracks. Assuming $a_{+1,k}$ is the bit with low reliability and the received intensities are $r_{+1,k-1}$, $r_{+1,k}$, and $r_{+1,k+1}$, the detector for $a_{+1,k}$ should consider 15 bits, $a_{i,k}$, where $i$ = {0, +1, +2} and $k$ = {$k$ − 2, $k$ − 1, $k$, $k$ + 1, $k$ + 2}. Therefore, the total number of candidate vectors to be considered in the exhaustive search is $2^{15}$ = 32,768, which is too complicated. To reduce the complexity with high reliability, we employ the TS algorithm [6,7] for the detection of the unreliable bits, which is a local neighborhood search strategy with a tabu list recording the visited solution vector in an iterative manner. In the multi-input multi-output detection schemes [9,10] for wireless communication systems, the TS detector has a similar bit error rate (BER) performance to the exhaustive search, even though it only searches for less than 0.5% vectors compared to the exhaustive search.

The TS algorithm performs four steps, initialization, neighbor vector enumeration, tabu move, and tabu list updating in an iterative manner, as shown in Figure 2. The search algorithm starts from the initial solution vector ${\mathit{x}}^{\left(0\right)}.$ At $l$ iteration, the neighbor vectors are generated based on the solution vector ${\mathit{x}}^{\left(l\right)}$, and the best vector is selected as a new solution vector ${\mathit{x}}^{\left(l+1\right)}$, among the neighbor vectors excluding the solution vectors in the tabu list in terms of their metric, which is the solution vector for the next iteration. This process is terminated if the maximum number of iterations M is reached, and the best solution vector is chosen from the solution vectors in all iterations as the final solution vector. To escape from local minima, the algorithm prohibits the moves to the solution vectors of the past few iterations referred to as the tabu period P by using the tabu list. The iterative process of the TS algorithm is as follows:

Step (1) 

Initialization: An initial solution vector ${\mathit{x}}^{\left(0\right)}$ is set to a vector having the detected data ${\widehat{a}}_{i,k}$, where $i$ = {0, +1, +2} and $k$ = {$k$ − 2, $k$ − 1, $k$, $k$ + 1, $k$ + 2} from the single-track detector. In addition, the tabu list is initialized while the iteration number and the tabu period are set to M and P, respectively.

Step (2) 

Neighbor Vector Enumeration: The N neighbor vectors at each iteration are derived from ${\mathit{x}}^{\left(l\right)}$ with the length of N by changing the opposite bit for each bit position. We take a simple example with the initial solution vector ${\mathit{x}}^{\left(0\right)}$ = [−1, +1, −1, +1]. According to the definition of neighborhood, the four neighbor vectors (${\mathit{n}}_1$ = [+1, +1, −1, +1], ${\mathit{n}}_2$ = [−1, −1, −1, +1], ${\mathit{n}}_3$ = [−1, +1, +1, +1], and ${\mathit{n}}_4$ = [−1, +1, −1, −1]) are ${\mathit{x}}^{\left(0\right)}$’s neighbors.

Step (3) 

Tabu Move: A new solution vector ${\mathit{x}}^{\left(l+1\right)}$ with the lowest metric is selected among the neighbor vectors excluding the solution vectors in the tabu list. The TS measures the distance metric between $r_{i,k}$ and ${\overline{r}}_{i,k}^j$, where $k$ = {$k$ − 1, $k$, $k$ + 1} and $j$ = {1 · · · M}, where ${\overline{r}}_{i,k}^j$ is the reconstructed signal with the j-th neighbor vector. The selected solution vector ${\mathit{x}}^{\left(l+1\right)}$ is used as the initial solution vector in the next iteration.

Step (4) 

Tabu List Updating: The new solution vector obtained in Step 3 is recorded in order on the tabu list. When the tabu list with a length of P is full, the new solution vector will push the first solution vector out of the list, and the new solution vector is added in the tabu list.

Then, the detected bit $a_{+1,k}$ from the final solution vector of the TS detector is transferred to the simplified multi-track detector.

2.4. Simplified Multi-Track Detector

In the simplified multi-track detector [4], the APP of the main-track data is calculated based on a priori information of the side-track, P($a_{-1,k-1}$) and P($a_{+1,k-1}$). This a priori information of the side-track is obtained from the single-track detector or the TS detector. By using a priori information with the improved reliability from the TS detector, the performance of the simplified multi-track detector is significantly enhanced.

3. Simulation Results

Nabavi’s channel model is considered in this paper as described in [11]. The 2D Gaussian pulse response is described by

$$h\left(\mathit{x},y\right)=A\exp \left(-\frac{1}{2c^2}\left[{\left(\frac{\mathit{x}}{{\mathrm{PW}}_{50\_\mathrm{along}}}\right)}^2+{\left(\frac{y+{\Delta }_{TMR}}{{\mathrm{PW}}_{50\_\mathrm{cross}}}\right)}^2\right]\right),$$

(1)

where A = 1 is the peak amplitude and $c=1/2.3548$ is a constant. As in [11], the replay response of a square island with length a = 11 nm has ${\mathrm{PW}}_{50\_\mathrm{along}}$ = 19.5 nm and ${\mathrm{PW}}_{50\_\mathrm{cross}}$ = 24.7 nm. For achieving different areal densities, the islands on rectangular grids are arranged by varying the island periods $T_{\mathit{x}}$ on the along-track and $T_y$ on the cross-track. In this simulation, the arrangements of islands with $T_{\mathit{x}}$ = 14 nm and $T_y$ = 15 nm, and $T_{\mathit{x}}$ = 12.7 nm and $T_y$ = 14 nm are considered, which give areal densities round 3.0 Tb/in² and 3.5 Tb/in², respectively. The signal-to-noise ratio (SNR) is defined as ${10\log }_{10}\left(A/{\sigma }^2\right)$, where ${\sigma }^2$ is the variance of the additive white Gaussian noise. The value of track misregistration (TMR) is defined as

$${\Delta }_{TMR}=\left(TM{R}_y\times T_y \right)/100,$$

(2)

where $TM{R}_y$ is the percentage of TMR. For the equalizer, the 1D equalizer with 11 coefficients and the 2D equalizer with 11 × 11 array coefficients are used. We set $\delta$ so that the ratio of the number of bits entering the TS detector to total bits $\gamma$ is 0.2%, 0.5%, 1%, and 2%, respectively. For the TS detector, we chose the parameters with M = 20 and P = 10.

Figure 3 shows the BER performances for the various detection schemes with 0% TMR when an areal density is 3.0 Tb/in². The joint-track detection [4] has no performance gain compared to the single-track detection [3]. Despite the use of the 2D GPR target, the joint-track detection suffers from the performance degradation due to the simplified detection algorithm [4] in high density BPMR channels. By using a priori information of the side-tracks, the simplified multi-track detection [5] has more than 2 dB gains compared to the single-track detection. To further improve the BER performances, the proposed detection detects the main-track data by using more reliable a priori information of two side-tracks from the TS detector. As $\gamma$ increases, the BER performances of the proposed detection are gradually improved. However, when $\gamma$ is more than 0.5%, the BER performances of the proposed detection become saturated. When $\gamma$ is 0.5%, the proposed detection has more than 3 dB gains compared to the simplified multi-track detection at the BER of 10⁻⁵.

Figure 4 shows the BER performances for the various detection schemes with 0% TMR when an areal density is 3.5 Tb/in². The simplified multi-track detection has approximately 1 dB gains compared to the single-track detection at the BER of 10⁻⁴, while the simplified multi-track detection has more than 2 dB gains than the single-track detection when an areal density is 3.0 Tb/in². As the recording density increases, the BER performances of the simplified multi-track detection are gradually decreased. When an areal density is 3.5 Tb/in², the BER performances of the proposed detection are saturated when $\gamma$ is more than 1.0%. When $\gamma$ is 1.0%, the proposed detection has more than 4 dB gains compared to the simplified multi-track detection at the BER of 10⁻⁴.

Figure 5 show the BER performances for the various detection scheme with 10% TMR when an areal density is 3.5 Tb/in². Due to the influence of TMR, the BER performances for all detection schemes deteriorated. However, the single-track and joint-track detections show particularly the serious BER performance degradation. The simplified multi-track detection has the more severe BER performance degradation than the proposed detection. When $\gamma$ is 2.0%, the proposed detection has more than 6 dB gains compared to the simplified multi-track detection at the BER of 2 × 10⁻⁴. The BER performances of the proposed detection are saturated when $\gamma$ is more than 1.0% below the BER of 2 × 10⁻⁵.

4. Conclusions

In this paper, we proposed a TS aided multi-track detection scheme using a priori information for high density BPMR channels. By applying a low-complexity local search algorithm for the additional detector, the proposed detection scheme showed significant performance improvement even with low complexity. Simulation results showed that the proposed detection scheme with slightly additional complexity has more than 3 dB and 4 dB gains compared to the conventional detection schemes in 3.0 Tb/in² and 3.5 Tb/in², respectively. In addition, at 10% TMR, the proposed technique has more than 6 dB gains than the conventional detection schemes in 3.5 Tb/in². Therefore, the proposed detection scheme is suitable for high density BPMR channels. In the future, we will research techniques to improve the detection accuracy of unreliable bits for high density BPMR channels.