Effective Generalized Partial Response Target and Serial Detector for Two-Dimensional Bit-Patterned Media Recording Channel Including Track ...

applied
 sciences
Article
Effective Generalized Partial Response Target and
Serial Detector for Two-Dimensional Bit-Patterned
Media Recording Channel Including
Track Mis-Registration
Thien An Nguyen and Jaejin Lee *
 Department of Information Communication Convergence Technology, Soongsil University, Seoul 06978, Korea;
 anthienng1995@soongsil.ac.kr
 * Correspondence: zlee@ssu.ac.kr; Tel.: +82-2-820-0901
 



 Received: 23 July 2020; Accepted: 14 August 2020; Published: 19 August 2020 


 Abstract: With the development of 5G technology, programs are gradually moving to cloud services.
 This leads to an increasing demand for storage. In the field of high-density data storage, bit-pattern
 media recording (BPMR) is considered a promising approach, as it can expand the data density to
 4 Tb/in2 . However, in high-density BPMR, bits or magnetic islands are very close to each other, leading
 to significant intertrack interference (ITI) from the cross-track direction and intersymbol interference
 (ISI) from the down-track direction. To minimize two-dimensional interference, including ITI and
 ISI, the serial detector method has been highly effective. However, in this method, the signal at the
 output of the first decoder is still a hard output. Therefore, we suggest methods to convert the output
 of the first detector into a soft output. Additionally, we have developed a new form of generalized
 partial response target to overcome the track mis-registration. The results show that our proposed
 methods apparently improve bit error rate performance.

 Keywords: information storage; bit-patterned media recording; intersymbol interference; partial
 response maximum likelihood; soft-output viterbi algorithm

1. Introduction
 In the field of data storage, conventional magnetic recording systems have an area density limit
of ~1 Tb/in2 because of the superparamagnetic effect [1]. Therefore, bit-patterned media recording
(BPMR) has been proposed as a promising candidate to solve this problem. In BPMR, one bit is stored
in a single-domain island surrounded by a nonmagnetic region, and the areal density (AD) can reach
up to 4 Tb/in2 [2]. Moreover, BPMR technology can decrease the nonlinear transition shift, offer easier
tracking, and exhibit good thermal stability [2,3]. For increasing the storage capacity, the distance
between magnetic islands must be closer, which produces intersymbol interference (ISI) and intertrack
interference (ITI) from the down-track (horizontal) and cross-track (vertical) directions, respectively.
These types of interference are referred to as two-dimensional (2D) interference in BPMR systems.
 Many methods are used to reduce 2D interference. We can use modulation methods like those of
Nguyen [4], with an error-correcting 5/6 code to avoid fatal interference as much as possible and gain
error correction; of Buajong [5], with a combination of a rate-3/4 modulation code and ITI subtraction
to reduce ITI; and of Kanjanakunchorn [6], with a rate-5/6 constructive ITI code to handle ITI. We can
use detection in combination with a partial response (PR) target and an equalizer. To reduce ITI,
Nabavi [7] proposed a modified trellis for BPMR. In [8], Nabavi suggested a 2D equalizer with a PR
target and optimized equalizer coefficients to eliminate ISI and ITI. Based on Nabavi [7,8], a hybrid 2D
equalizer was proposed by Wang [9] to improve channel estimation when ITI is known. Additionally,

Appl. Sci. 2020, 10, 5738; doi:10.3390/app10175738 www.mdpi.com/journal/applsci

Appl. Sci. 2020, 10, 5738 2 of 10

to mitigate 2D interference, Kim [10] proposed a 2D soft-output Viterbi algorithm (2D SOVA), which is
also known as a parallel detection model, for holographic data storage, then developed an iterative 2D
SOVA for bit-patterned media [11]. Jeong, based on the parallel detection model, proposed a multipath
ISI structure that fits the staggered BPMR structure [12]. In addition to the idea of parallel detection,
recently the idea of serial detection appeared [13]. In serial detection, the 2D interference is separated
into two serial interference components. This helps reduce the complexity of 2D Viterbi detection of
the BPMR channel with 2D interference. However, in serial detection, the inner detection still has an
output that is referred to as a hard output. This is a disadvantage of serial detection compared to
parallel detection, which has a soft output in inner detection.
 In this study, we propose two methods to create a soft output for the inner detector. The first
method is to use an equalizer to create a soft output. The second method is based on the Viterbi
algorithm combined with channel interference to create a soft output. This method is highly effective,
because it can reduce errors compared to a six-level signal and preserve the interference information of
the signal. Moreover, when the channel experiences track mis-registration (TMR), which degrades
system performance [14,15], the previous PR target is no longer effective in overcoming TMR. Therefore,
we have proposed a new form of PR target for the 2D BPMR channel with TMR. Finally, we investigated
the ability of serial detection in the presence of media noise.
 The rest of this paper is organized as follows. In Section 2, we explain our algorithm and show
how we design the soft output for serial detection. Section 3 briefly presents the BPMR channel model
and illustrates the simulation results. Finally, the conclusion is drawn in Section 4.

2. New PR Target and Soft Output for Serial Detection

2.1. New PR Target for TMR
 To implement serial detection, the PR target with polynomial G has the following matrix form [13].
  
  rp p rp 
  
 G =  r 1 r ,
  (1)
 
 rp p rp
 

where r and p are the interference coefficients from horizontal and vertical directions, respectively. This
form has a coefficient at four corners, which is the product of horizontal and vertical coefficients, and
the ITI coefficients are symmetric. Consequently, the equalizer output can be analyzed as given below:

 z[i, k] ≈ a[ j, k] ∗ G
    
  rp p rp   p  h
   i (2)
 = a[ j, k] ∗  r 1 r  = a[ j, k] ∗  1  ∗ r 1 r ,
  
    
 rp p rp p

where * is the convolution operator.
 With the above analysis, the 2D interference of the channel includes two serial one-dimensional
(1D) interference components. In other words, the original signal is distorted by vertical interference
then horizontal interference. Therefore, the detection composed of a 1D horizontal detector in series
with a 1D vertical detector, as shown in Figure 1.
 Since the channel is no longer symmetric in the vertical direction when the system has TMR,
the upper interference coefficient (g-1.0 ) and the lower interference coefficient (g1,0 ) are estimated with
different values. Therefore, a new form of PR target such as is required.
  
  rp1 p1 rp1 
  
 G =  r 1 r 
  (3)
 
 rp2 p2 rp2
 

 , Detector

 , 
 , Channel , Equalizer , Horizontal Vertical , 
 Modulation Demodulation
 BPMR detector detector
 Target
 , , 
Appl. Sci. 2020, 10, 5738 1 1 MMSE 3 of 10
 down-track
 cross-track
 
With this asymmetric form, aGsignal 1z[j,k]
 like (2) is given as and applied to the serial detector, as shown
 
in Figure 1.
 z[ j, k] ≈ a[ j, k] ∗Figure G 1. Serial
   detection scheme.  
  rp1 p1 rp1   p1  h
     i (4)
 Since the channel = isa[no ] ∗  r symmetric
 j, klonger 1 r in  =
  thea[vertical
 j, k] ∗  direction
 1  ∗ r when
 1 rthe system has TMR, the
    
 upper interference coefficient (grp -1.02) and
 p2 the
 rp2lower interference p2 coefficient (g1,0) are estimated with
 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 11
 different values. Therefore, a new form of PR target such as is required.

 ,  rp1 p1 rp1  Detector
 G =  r 1 r  , (3)
 
 Modulation
 , Channel ,  rp2 , p2
 Equalizer 
 rpHorizontal
 2 Vertical , 
 Demodulation
 
 BPMR detector detector
 Target
 With this asymmetric form, a signal z[j,k] like (2) is given as and applied to the serial detector, as shown
 , , 
 in Figure 1. 1 1 MMSE
 
 [
 z j, k
 cross-track ] ≈down-track
 a [ j, k ] ∗ G
  rp1 p1 rp1   p1 
 1 
 ] ∗  r  = a [ j , k ] ∗  1  ∗ [ r 1 r ]
 G (4)
 = a [
 j, k r 1
  rp2 p2 rp2   p2 
 Figure
 Figure 1.1. Serial
 Serial detection
 detection scheme.
 scheme.

2.2. Soft Output Using an Equalizer
 2.2.Since
 Soft Output Usingisanno
 the channel Equalizer
 longer symmetric in the vertical direction when the system has TMR, the
 upper In interference
 In the serial detector forthe
 the serial detector for
 coefficient (gPR
 the -1.0)target
 and the
 PR target without
 lowerTMR
 without as
 as shown
 interference
 TMR in Figure
 Figure 1,
 showncoefficient
 in 1,(g
 the
 the1,0) first
 first (inner)
 are(inner)
 estimateddetector
 with
 detector
converts
 different values. Therefore,equalizer
 a new output
 form of signal
 PR z[j,
 target k] into
 such the
 as is six-level
 required. signal s[j,k]
 converts the real-valued equalizer output signal z[j, k] into the six-level signal s[j,k] (they are -2p-1, -1,
 the real-valued (they are -2p-1, -
2p-1, -2p+1,
 1, 2p-1, 1, and
 -2p+1, 2p+1.)
 1, and and
 2p+1.) andthen
 then thethe
 second
 second (outer)
 (outer) detector
 detector converts
 converts the
 the signal into
 signal intothe
 theoriginal
 original
  rp1 p1 rp1 
signal a[j,k], which is is
 the 2-level
 2-levelsignal
 signal[13] since the signal s[j,k] is a six-level output. This loses the
 signal a[j,k], which the G =  r the
 [13] since r  s[j,k] is a six-level output. This loses the
 1 signal (3)
interference information and reduces system
 interference information and reduces systemperformance. performance. To solve
 To this problem,
 solve this problem,we wepropose
 proposetwotwo
  rp2 p2 rp2 
methods.
 methods.First,First,we
 wedesign
 designan anequalizer
 equalizerfor forreplacing
 replacing the the inner
 inner detector, because the
 detector, because theequalizer
 equalizercancan
estimate
 With thisaasymmetric
 estimate signal
 a signalsimilar
 form,
 similar toto
 athetheoutput
 signal z[j,k]oflike
 output ofthe(2)
 the inner detector
 is given
 inner as andand
 detector create
 applied
 and toathesoft output,
 serial
 soft which
 detector,
 output, which helps
 as helps
 shown
preserve
 Figurethe
 in preserve interference
 1. the interference information.
 information.The Thesystem
 systemwith withthe
 theequalizer
 equalizer is presented in in Figure
 Figure2.2.
 z [ j, k ] ≈ a [ j, k ] ∗ G
 , Detector
  rp1 p1 rp1   p1 
   , 
 , k ] ∗  1  ∗ [ r 1Vertical
 (4)
 , Channel
 = a [ j , k ] ,∗  r Equalizer
 1 r , = a [ jHorizontal r] , 
 Modulation Demodulation
 BPMR  rp2 p2 rp2   p2 
 equalizer detector
 Target

 , , 
 1
2.2. Soft Output Using an Equalizer 1 MMSE
 down-track
 In the serial detector cross-track
 for the PR target without TMR as shown in Figure 1, the first (inner) detector
 
 converts the real-valued equalizer G output
 1 signal z[j, k] into the six-level signal s[j,k] (they are -2p-1, -
 cross-track target
 1, 2p-1, -2p+1, 1, and 2p+1.) and then the second (outer) detector converts the signal into the original
 , 
 , 
 signal a[j,k], which is the 2-level signal [13] since 1 the signal s[j,k] is a LMS output. This loses the
 six-level
 
 interference information and reduces system performance. To solve this problem, we propose two
 methods. First,Figure we2. design an equalizer
 System with a horizontal forequalizer
 replacing the inner
 replacing detector, (inner)
 the horizontal because the equalizer can
 detector.
 estimate aFigure
 signal2.similar
 System to
 withthea output
 horizontal of the inner replacing
 equalizer detector and create a soft
 the horizontal output,
 (inner) which helps
 detector.
 In the
 preserve the model, the coefficient
 interference information.of target G is estimated
 The system with theby using aisminimum
 equalizer presentedmean square
 in Figure 2. error
(MMSE) method. After finding the coefficients of target G, we have the horizontal (down-track)
and vertical (cross-track) target coefficients,
 , 
 respectively. BasedDetector on the coefficients of cross-track,
the horizontal equalizer can be designed for replacing the inner detector. , 
 The coefficients of the
horizontal equalizer, which replaces ,
 the inner detector, are estimated by using , 
 , Channel Equalizer , Horizontal Vertical the least mean square 
 Modulation Demodulation
 BPMR equalizer detector
(LMS) algorithm.
 Target

 , , 
 1 1 MMSE
 down-track
 cross-track
 
 G 1 
 cross-track target
 
 , , 
 1 LMS
 

In the model, the coefficient of target G is estimated by using a minimum mean square error
(MMSE) method. After finding the coefficients of target G, we have the horizontal (down-track) and
vertical (cross-track) target coefficients, respectively. Based on the coefficients of cross-track, the
horizontal equalizer can be designed for replacing the inner detector. The coefficients of the
horizontal equalizer,
Appl. Sci. 2020, 10, 5738 which replaces the inner detector, are estimated by using the least mean square
 4 of 10
(LMS) algorithm.

2.3. Soft
2.3. Soft Output
 Output Utilizing
 Utilizing aa Horizontal
 Horizontal Detector
 Detector and
 and Interference
 Interference
 With the
 With the above
 above method,
 method, the
 the equalizer
 equalizer substituting for the
 substituting for the inner
 inner detector
 detector produces
 produces aa soft
 soft output
 output
for the outer detector. However, this method produces signals that have more errors than
for the outer detector. However, this method produces signals that have more errors than a six-level a six-level
signal. To
signal. To remove
 remove this
 this error,
 error, we
 we have
 have toto use
 use the
 the Viterbi
 Viterbi algorithm
 algorithm atat the
 the inner
 inner detector
 detector to
 to restore
 restore aa
six-level signal like a conventional serial detector. We then design a feedback unit to ascertain
six-level signal like a conventional serial detector. We then design a feedback unit to ascertain the the
interference information and add the interference information with the six-level signal. The
interference information and add the interference information with the six-level signal. The model of model of
the inner
the inner detector
 detector with
 with soft
 soft output
 output is
 is presented
 presented in
 in Figure
 Figure 3.
 3.

 Detector

 , 
 , ̂ , 
 1 

 , , , 
 , Channel Equalizer , Horizontal Vertical
 Modulation Demodulation
 BPMR detector , detector
 Target
 soft-output inner detector
 , , , 
 1 1 MMSE
 down-track
 cross-track

 G 1 
 
 Figure
 Figure 3.
 3. System
 System with
 with aa soft-output
 soft-output horizontal
 horizontal (inner)
 (inner) detector.
 detector.

 In Figure
 Figure3,3,the thesignal
 signal
 z[j,z[j,
 k], k], which
 which is approximately
 is approximately a 2D aconvolution
 2D convolution of theand
 of the target target
 the and the
 original
original
data a[j, data
 k], is a[j, k], is presented
 presented as follows:as follows:
 z [ j , k ] ≈ a [ j , k ] ∗ G + w[ j , k ]
 z[ j, k] ≈ a[ j, k] ∗ G + w[ j, k]
  p1  ph1 
  
 (5)
 = a[ j, k=] ∗a[ j,1k ] ∗∗ 1 r* [ r1 1 r r ] ++ww[ [j,j,k k] ,],
   i (5)
  
 p2  p2 
 

where w[j,k] is an additive distortion component, i.e., colored noise plus residual interference. For ease
where w[j,k] is an additive distortion component, i.e., colored noise plus residual interference. For ease
  p1  
  p1 
of signal analysis, we set b [ j , k ] = a [ j , k ] ∗  1  . We can represent (5) as follows
of signal analysis, we set b[ j, k] = a[ j, k] ∗  1 . We can represent (5) as follows
  p2  
 p2
 z [ j , k ] ≈ b [ j , k h] ∗ [ r 1 r ] +i w [ j , k ]. (6)
 z[ j, k] ≈ b[ j, k] ∗ r 1 r + w[ j, k]. (6)
 The signal z[j,k] is passed through the inner detector. In an ideal condition, the inner detector
convertsThe the signal
 signal z[j,k]
 z[j,k] into thethrough
 is passed signal b[j,k] ∈ {-2p-1,
 the inner -1, 2p-1,
 detector. In -2p+1,
 an ideal 1, 2p+1}, in which
 condition, the horizontal
 the inner detector
interference
converts the has been
 signal eliminated.
 z[j,k] However,
 into the signal b[j,k] in fact, because
 ∈ {-2p-1, -1, 2p-1,of-2p+1,
 the noise signalinw[j,k],
 1, 2p+1}, whichitthe
 only restores
 horizontal
the approximation of b[j,k]. We refer to that signal as [j,k]= b[j,k]+e [j,k].
interference has been eliminated. However, in fact, because of the noise signal w[j,k], it only restores
 b When continuing to use the
 [j,k]approximation
the signal for the outer detection,
 of b[j,k]. We refer thetoperformance
 that signal as is degraded.
 b̂[j,k]= b[j,k]+eTo solve
 b [j,k].this issue,
 When we estimate
 continuing the
 to use
noise w[j,k].
the b̂[j,k] signalWith [j,k],
 for the wedetection,
 outer predict the the noise
 performance signal w[j,k] and setToitsolve
 is degraded. to this
 [j,k] issue,
 according to the
 we estimate
following equations:
the noise w[j,k]. With b̂[j,k], we predict the noise signal w[j,k] and set it to ŵ[j,k] according to the
following equations: 
 z [ j, k ] = b [ j, k ] ∗ [ r 1 r ] = b [ j, k ] *[r 1 r ] + eb [ j, k ] *[r 1 r ], (7)
 h i
 ẑ[ j, k] = b̂[ j, k] ∗ r 1 r = b[ j, k] ∗ [ r 1 r ] + eb [ j, k] ∗ [ r 1 r ], (7)
  [ j, k ] = z [ j, k ] − z [ j, k ] = w[ j, k ] − e [ j, k ] * [ r 1 r ].
 w (8)
 b
 h i
 ŵ[ j, k] = z[ j, k] − ẑ[ j, k] = w[ j, k] − eb [ j, k] ∗ r 1 r . (8)

 The signal ŵ[j,k] is the noise information that is added to the output signal of the inner detector.
Thus, the final output of the inner detector is calculated as follows:
 h i
 s[ j, k] = b̂[ j, k] + ŵ[ j, k] = b[ j, k] + eb [ j, k] − eb [ j, k] ∗ r 1 r + w[ j, k]. (9)

Appl. Sci. 2020, 10, 5738 5 of 10

 Meanwhile, the r coefficient is usually quite small. So eb [j,k]*[r 1 r] is close to eb
 h i
 eb [ j, k] ≈ eb [ j, k] ∗ r 1 r . (10)

and eb [j,k]- eb [j,k]*[r 1 r] is close to zero. Finally, the soft value s[j,k] is fed to the outer detector.

 s[ j, k] ≈ b[ j, k] + w[ j, k]. (11)

3. Simulation Results

3.1. BPMR Channel Model
 With the original data u[k] e {0,1}, it is modulated into the 2D data array a[j,k] e {−1,1}. The input
data a[j,k] are taken into the BPMR channel, in which the signals are interfered by ISI and ITI. At the
output of the channel, additive white Gaussian noise is added to present the noise model. In this
research, we apply a 2D Gaussian function to represent the 2D island response of the BPMR channel as
follows [12]:
 1 x + ∆x 2 z + ∆z 2
 "    #!
 P(x, z) = A exp − 2 + , (12)
 2c PWx PWz
where x and z are the down- and cross-track directions, respectively; ∆x and ∆z are the down- and
cross-track bit location fluctuations, respectively; c is 1/2.3548, which represents the relationship
between the standard deviation of a Gaussian function and PW50 , which is a parameter of the pulse
width at half of the peak amplitude; and PWx and PWz are the PW50 components of the down- and
cross-track pulses, respectively.
 The BPMR channel pulse response is expressed as
  
 h[ j, k] = P jTx , kTz − ∆o f f , (13)

where j and k are the discrete indices in the down- and cross-track directions, respectively; Tx and Tz
are the bit period and track pitch, respectively, and ∆o f f is the read-head offsets for the cross-track.
TMR is defined as the ratio between the head offset size and the magnetic-island period, as follows:

 ∆o f f
 TMR(%) = . (14)
 Tz

 The readback signal y[j,k] for BPMR is given by

 y[ j, k] = a[ j, k] ∗ h[ j, k] + n[ j, k], (15)

where a[j,k], h[j,k], and n[j,k] are the 2D discrete input data, 2D channel response, and electronic noise
modeled as additive white Gaussian noise (AWGN) with variance σ2 and zero mean, respectively.

3.2. Simulation Results
 In this experiment, the first method is shown in Figure 2, which uses an equalizer instead of the
inner detector. The channel output y[j,k] was inputted to 2D equalizer. This equalizer has a size of 5 × 5.
At the same time, the coefficients of the equalizer and the GPR target were determined by calculating
the error values e[j,k] and minimizing this error with the MMSE algorithm [13]. Then, the output
of the equalizer z[j,k] was passed to the horizontal equalizer, which had a size of 5 × 5, and the
coefficients were updated by finding the error values ei [j,k] and using the LMS algorithm to reduce
ISI. Finally, the outputs of the horizontal equalizer, s[j,k], were transferred to the vertical detector to
reduce ITI and restore the original input data â[j,k]. The channel signal-to-noise ratio (SNR) is defined
as 10log10 (1/σ2 ), where σ2 is AWGN power. To simulate the second method, we built a scheme as
shown in Figure 3. The system with the six-level output signal for serial detection was built in the same

Appl. Sci. 2020, 10, 5738 6 of 10

Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 11
manner as Figure 1. In our experiment, we refer to the system in Figure 3 as soft-output horizontal
(inner)
defineddetection,
 as 10log10the(1/ system
 ), wherein Figure
 is AWGN 2 as horizontal
 power. (inner) equalizer,
 To simulate and system
 the second in Figure
 method, 1 asa
 we built
conventional
scheme as shown serialindetection.
 Figure 3. The system with the six-level output signal for serial detection was built
 First, we compare
in the same manner as Figure the bit 1.error rateexperiment,
 In our (BER) performance of the
 we refer to three systems
 system as shown
 in Figure 3 as in Figure 4.
 soft-output
We simulated 10 pages with a size of 1200 × 1200 bits and an AD of 3 Tb/in 2 (T = T = 14.5 nm) [16].
horizontal (inner) detection, the system in Figure 2 as horizontal (inner) equalizer, x z and system in
To simplify the problem, the coefficients
Figure 1 as conventional serial detection. of the channel without TMR effect and media noise used in
the simulation were given as
 First, we compare the bit error rate  (BER) performance of three  systems as shown in Figure 4.
We simulated 10 pages with a size of 1200  0.0824
 × 12000.3876 0.0824
 bits and an AD  of 3 Tb/in2 (Tx = Tz = 14.5 nm) [16].
 H =  0.2125 1 0.2125 . (16)
  
To simplify the problem, the coefficients  of the channel without TMR  effect and media noise used in
the simulation were given as 0.0824 0.3876 0.0824

 10-1
 Soft output horizontal (inner) detection
 Soft output horizontal (inner) detection (optimal)
 Horizontal (inner) equalizer
 10-2 Conventional serial detection

 10-3

 10-4

 10-5

 10-6

 10 11 12 13 14 15 16 17 18 19 20
 SNR(dB)

 Figure 4. Bit error rate (BER) performance of the proposed models without track mis-registration (TMR).
 Figure 4. Bit error rate (BER) performance of the proposed models without track mis-registration
 (TMR).
 At SNR = 20 dB, the PR target was calculated as
  
  0.0503 0.4348 0.0503
   0.0824 0.3876 0.0824  
 
 G =H= 0.1158 1 0.1158  . . (17)
   0.2125 0.2125   (16)
  0.0824
 0.0503 0.3876 0.0824 
 0.4348 0.0503

 The results
 At SNR = 20showed
 dB, the that the system
 PR target with soft as
 was calculated output of the inner detector achieves better BER
performance than the system with hard output of the inner detector. At a BER of 10−6 , the gain of the
  0.0503 0.4348 0.0503
soft output horizontal detector is ~1.6 dB higher than that of the conventional serial detection and
 G =  0.1158 1 0.1158  . (17)
0.5 dB higher than that of the horizontal equalizer. Because soft output preserves the interference
  0.0503 0.4348 0.0503
information, the outer detector decides more accurately. In Figure 3, the soft-output inner detector has
two functions.
 The resultsOne is to reduce
 showed that theerrors
 systemcompared
 with softtooutput
 a six-level
 of thesignal,
 innerwhich improves
 detector thebetter
 achieves accuracy
 BER
performance than the system with hard output of the inner detector. At a BER of 10−6, the gain of the
soft output horizontal detector is ∼1.6 dB higher than that of the conventional serial detection and 0.5
dB higher than that of the horizontal equalizer. Because soft output preserves the interference

Appl. Sci. 2020, 10, 5738 7 of 10

 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 11

of the inner detector. The other is the preservation of interference information. The resulting noise
 information, the outer detector decides more accurately. In Figure 3, the soft-output inner detector
information is then provided to the outer detector, which is the second detector in the serial detector,
 has two functions. One is to reduce errors compared to a six-level signal, which improves the
thus improving the performance of the entire serial detector. For GPR target to reach the optimal
 accuracy of the inner detector. The other is the preservation of interference information. The resulting
coefficient, which is the same as the channel coefficients, we choose the matrix G = H. The result
 noise information is then provided to the outer detector, which is the second detector in the serial
of this case is represented by the dashed line in Figure 4, and the proposed scheme is close to the
 detector, thus improving the performance of the entire serial detector. For GPR target to reach the
optimal performance.
 optimal coefficient, which is the same as the channel coefficients, we choose the matrix G = H. The
 In the next simulation, we include the effect of TMR and the channel matrix H is changed according
 result of this case is represented by the dashed line in Figure 4, and the proposed scheme is close to
to the level of TMR (%). We show the examples of 10 and 20% TMRs. The system is simulated when
 the optimal performance.
AD is 3 Tb/in2 . The BER performance of this simulation with 10% and 20% TMR is shown in Figures 5
 In the next simulation, we include the effect of TMR and the channel matrix H is changed
and 6, respectively. The coefficients of channels with 10% and 20% TMR were shown as
 according to the level of TMR (%). We show the examples of 10 and 20% TMRs. The system is
 simulated when AD is 3 Tb/in2. The BER  performance of this simulation
  with 10% and 20% TMR is
  0.0675 0.3176 0.0675 
 shown in Figures 5 and 6, respectively.  The coefficients of channels
 H10% =  0.2105 0.9906 0.2105 
  with 10% and 20% TMR were
 (18)
 shown as 
 0.0986 0.4641 0.0986
 

  0.0675 0.3176 0.0675 
and
 H10% = 0.2105 0.9906 0.2105  (18)
 
 
  0.0543 0.2554 0.0543
  0.0986 0.4641 0.0986 
 
 
 
 H20% =  0.2046 0.9628 0.2046 , (19)
  
 0.1159 0.5452 0.1159

 10-1
 Soft output horizontal (inner) detection (asymmetric GPR)
 Soft output horizontal (inner) detection (symmetric GPR)
 Horizontal (inner) equalizer (asymmetric GPR)
 10-2 Horizontal (inner) equalizer (symmetric GPR)
 Conventional serial detection (asymmetric GPR)
 Conventional serial detection (symmetric GPR)

 10-3

 10-4

 10-5

 10-6

 10 11 12 13 14 15 16 17 18 19 20
 SNR(dB)
 Figure 5. BER performance of the proposed models with 10% TMR.
 Figure 5. BER performance of the proposed models with 10% TMR.
 Meanwhile, the targets estimated for the channels with 10% and 20% TMR were
 and  
  0.0428 0.4391 0.0428 
   0.0543 0.2554 0.0543 
 G10% =  0.0974 1 0.0974  (20)
 H 20% =  0.2046 0.9628 0.2046  ,  (19)
 0.0445 0.4567 0.0445
  0.1159 0.5452 0.1159 

Appl. Sci. 2020, 10, 5738 8 of 10

and  
  0.0288 0.4673 0.0288 
  
 G20% =  0.0617 1 0.0617 ,
  (21)
 
 0.0298 0.4833 0.0298
 

respectively. The target was calculated at SNR = 20 dB. Because of the effect of TMR, the channel
coefficients were asymmetric. To estimate this asymmetry, an asymmetric GPR like (3) was applied
to the BPMR systems. The results show that the asymmetric GPR target performs better than the
symmetric GPR
 Appl. Sci. 2020, 10, xtarget when
 FOR PEER faced with TMR. In addition, we also experiment with the symmetric
 REVIEW 8 of 11
GPR like (1) by assigning the ITI coefficients to the average of p1 and p2 .

 10-1
 Soft output horizontal (inner) detection (asymmetric GPR)
 Soft output horizontal (inner) detection (symmetric GPR)
 Horizontal (inner) equalizer (asymmetric GPR)
 10-2 Horizontal (inner) equalizer (symmetric GPR)
 Conventional serial detection (asymmetric GPR)
 Conventional serial detection (symmetric GPR)

 10-3

 10-4

 10-5

 10-6

 10 11 12 13 14 15 16 17 18 19 20
 SNR(dB)

 Figure 6. BER performance of the proposed models with 20% TMR, respectively.
 Figure 6. BER performance of the proposed models with 20% TMR, respectively.
 Figure 7 shows the effect of TMR on serial detection with soft output for the inner detector. In fact,
 Meanwhile, the targets estimated for the channels with 10% and 20% TMR were
we do not know exactly what level TMR appears. Therefore, we simulated with SNR = 15 dB and TMR
from 10% to 30%. Two models proposed for0.0428 the soft0.4391
 output0.0428  inner detector outperformed the
 for the
 
 G10% = 0.0974 1 0.0974 
conventional serial detection model. However, the proposed models  are more sensitive to TMR. This is
 (20)
because the soft output for the inner detector0.0445
 preserves0.4567 0.0445 information including information
 interference
about TMR. Therefore, changes in TMR values also change the performance of the proposed model.
 and
 Finally, we simulated our proposed scheme with a 6% fluctuation in the position [17]. In this
case, the coefficients are changed according 0.0288 0.4673 0.0288
 to the position  bit island. The results in Figure 8
 of the
 
 G 20% =  0.0617 1 , −5
show that soft output for the inner detector still achieved a0.0617
 gain of
  ~2 dB at a BER of 10 . However,(21)
two methods of soft-output achieve almost 0.0298 the same0.4833
 result0.0298 
 because position fluctuation causes the
target coefficients to lose the formats given in (1) and (3).
 respectively. The target was calculated at SNR = 20 dB. Because of the effect of TMR, the channel
 coefficients were asymmetric. To estimate this asymmetry, an asymmetric GPR like (3) was applied
 to the BPMR systems. The results show that the asymmetric GPR target performs better than the
 symmetric GPR target when faced with TMR. In addition, we also experiment with the symmetric
 GPR like (1) by assigning the ITI coefficients to the average of p1 and p2.
 Figure 7 shows the effect of TMR on serial detection with soft output for the inner detector. In
 fact, we do not know exactly what level TMR appears. Therefore, we simulated with SNR = 15 dB
 and TMR from 10% to 30%. Two models proposed for the soft output for the inner detector

Appl. Sci. 2020, 10, 5738 9 of 10
 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 11

 10-1
 Soft output horizontal (inner) detection (asymmetric GPR)
 Soft output horizontal (inner) detection (symmetric GPR)
 Horizontal (inner) equalizer (asymmetric GPR)
 10-2 Horizontal (inner) equalizer (symmetric GPR)
 Conventional serial detection (asymmetric GPR)
 Conventional serial detection (symmetric GPR)

 10-3

 10-4

 10-5

 10-6

 10 12 14 16 18 20 22 24 26 28 30
 TMR %
 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 11
 Figure 7. BER7.performance
 Figure ofofthe
 BER performance theproposed models
 proposed models according
 according to TMR.
 to TMR.

 Finally, we simulated our proposed scheme with a 6% fluctuation in the position [17]. In this
 10-1
 case, the coefficients are changed according to the position of the bit island. The results in Figure 8
 Soft output horizontal (inner) detection
 show that soft output for the inner detector still achieved a gain of ∼2 dB at a BER of 10−5. However,
 Horizontal (inner) equalizer
 two methods of soft-output achieve almost the sameConventional
 result because position
 serial fluctuation causes the
 detection
 target coefficients
 10-2 to lose the formats given in (1) and (3).

 10-3

 10-4

 10-5

 10-6

 10 11 12 13 14 15 16 17 18 19 20
 SNR(dB)

 BER8.performance
 Figure 8.Figure of of
 BER performance thetheproposed models
 proposed models with
 with 6% position
 6% position fluctuation.
 fluctuation.

 4. Conclusions
4. Conclusions
 We have proposed soft output schemes for the inner detector in serial detection. Based on an
 We have proposed soft output schemes for the inner detector in serial detection. Based on an
 equalizer and the Viterbi algorithm, our proposed model shows that the inner detector with soft-
equalizer and theachieves
 output Viterbibetter
 algorithm, our proposed
 BER performance model
 than that of theshows that the
 inner detector inner
 with detectorsignal.
 the six-level withInsoft-output
achieves better BERthe
 particular, performance
 proposed modelthan
 alsothat of high
 brings the performance
 inner detector with the
 to channels withsix-level
 TMR. signal. In particular,
the proposed model
 Author also brings
 Contributions: high performance
 Conceptualization, T.A.N. and to
 J.L.;channels with
 methodology, TMR.
 T.A.N. and J.L.; software, T.A.N.;
 validation, T.A.N. and J.L.; formal analysis, T.A.N.; investigation, T.A.N. and J.L.; writing—original draft
 preparation, T.A.N.; writing—review and editing, T.A.N. and J.L.; supervision, J.L.; project administration, J.L.;
 funding acquisition, J.L. All authors have read and agreed to the published version of the manuscript.

 Funding: This work was supported by a National Research Foundation of Korea (NRF) grant funded by the
 Korea government (MSIT) (No. NRF-2019R1FA1046899).

 Conflicts of Interest: The authors declare no conflict of interest.

Appl. Sci. 2020, 10, 5738 10 of 10

Author Contributions: Conceptualization, T.A.N. and J.L.; methodology, T.A.N. and J.L.; software, T.A.N.;
validation, T.A.N. and J.L.; formal analysis, T.A.N.; investigation, T.A.N. and J.L.; writing—original draft
preparation, T.A.N.; writing—review and editing, T.A.N. and J.L.; supervision, J.L.; project administration, J.L.;
funding acquisition, J.L. All authors have read and agreed to the published version of the manuscript.
Funding: This work was supported by a National Research Foundation of Korea (NRF) grant funded by the
Korea government (MSIT) (No. NRF-2019R1FA1046899).
Conflicts of Interest: The authors declare no conflict of interest.

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