Estimating Interference with a Two-Dimensional Viterbi Algorithm for Bit-Patterned Media Recording

Abstract

Bit-patterned media recording (BPMR) is proposed as a candidate for future magnetic data storage to overcome superparamagnetism. The distance between magnetic islands in BPMR must be reduced to increase the areal density (AD). As magnetic islands become closer, two-dimensional (2D) interference is increased, including intersymbol interference (ISI) based on the down-track direction and intertrack interference (ITI) from the cross-track direction. We propose an estimator to predict interference from neighboring (upper and lower) tracks. This estimator exploits the 2D Viterbi algorithm (VA) with reduced states. We removed the interference from the neighboring track and applied a simple 1D VA to detect the received signal. The simulation results show that our model performs better than previously proposed models.

1. Introduction

Mechanisms such as bit-patterned media recording (BPMR) [1], heat-assisted magnetic recording (HAMR) [2], and 2D magnetic recording (TDMR) [3,4] were developed to increase the areal density (AD) of magnetic data storage devices. In BPMR, the distance between magnetic islands must be reduced to increase the AD. As the distance between magnetic islands becomes closer, interference between them increases. This is two-dimensional (2D) interference, which comprises intertrack interference (ITI) from the cross-track direction and intersymbol interference (ISI) from the down-track direction. In addition, the received signal is disturbed by track misregistration (TMR) and media noise [5,6]. To combat this interference, error-correcting or detection algorithms are required. As a result, we designed an interference-estimation scheme to improve detection.

For error-correcting codes, Nguyen and Lee proposed a 9/12 two-dimensional modulation code to avoid the isolated patterns [7]. To reduce the ITI, a rate-3/4 modulation code was proposed by Buajong and Warisarn [8]. To avoid the patterns causing 2D interference, a rate-8/9 2-D modulation code was designed by Kovintavewat, Arayangkool, and Warisarn [9]. To combat 2D interference, a modified 2D Viterbi algorithm (VA) using 2D modulation-encoding constraints was proposed by Sokjabok, Warisarn, Koonkarnkhai, and Lee [10]. For the structure of a staggered BPMR, an error-correcting 5/6 modulation code was introduced by Nguyen and Lee [11], which helps to correct errors and reduce isolated patterns in the staggered BPMR. With the same code rate, a rate-5/6 constructive ITI code was designed by Kanjanakunchorn and Warisarn to mitigate ITI [12]. With the flexibility of a neural network, Jeong and Lee proposed a decoding scheme based on a multilayer perceptron for BPMR [13].

For detection, Cideciyan et al. introduced a partial-response maximum-likelihood (PRML) method to convert 2D interference into 1D interference [14], which was subsequently developed into a general partial response (GPR) to improve accuracy [15]. To apply the Viterbi algorithm (VA) for 2D interference, Nabavi et al. proposed a modified VA, which mitigates the effect of ITI [16]. GPR and MVA were used by Wang and Kumar to design a hybrid 2D equalizer [17]. Nguyen and Lee developed a feedback scheme for MVA to improve ITI prediction [18]. To combat ITI with a low implementation cost, Sadeghian and Barry proposed an effective detection technique in [19]. In addition, Shi and Barry [20] also proposed a multitrack detector with 2D pattern-dependent noise prediction, which significantly outperformed a conventional 2D Viterbi detector when the channel noise was pattern-dependent. Kim and Lee introduced an iterative 2D soft-output VA (SOVA) for BPMR systems [21]. This scheme was inspired by the 2D SOVA for holographic data storage systems and designed as a parallel structure of two 1D VA detectors along the horizontal and vertical directions, respectively [22]. Nguyen and Lee proposed a serial detection scheme for two 1D VA detectors along the horizontal and vertical directions [23]. In serial detection, the signal is detected by the horizontal detector, and then the output signal is detected by the vertical detector. Furthermore, a soft output between horizontal and vertical detection was introduced to improve the performance of serial detection [24].

Because the VA is used to remove 1D interference, the above detection algorithms are modifications of the VA to handle 2D interference. Thus, we can use the estimator to convert 2D interference into 1D interference and apply the conventional VA. For ITI estimation, Buajong and Warisarn used a GPR target to create feedback for a multitrack, multihead system to estimate ITI [25]. To remove the ITI effect from the desired track, an ITI cancellation model using the feedback of sidetrack information was proposed by Koonkarnkhai, Warisarn, and Kovintavewat [26]. In [18], owing to the asymmetrical target, the authors were able to extract the ITI information. To improve detection on the center track, Chang and Cruz designed a multitrack detection technique to estimate the interference from the sidetrack [27]. Recently, Jeong and Lee proposed an ITI estimation scheme based on a neural network [28] to achieve interference with high accuracy.

In this paper, we propose an ITI estimator that exploits 2D VA. First, we considered the sum of product between the signal and the interference as the symbols, which can be detected by 2D VA. After detecting these symbols, we summed the suitable symbols to estimate the interference from the sidetrack. Then, we removed these interferences from the received signal to convert 2D interference into 1D interference. The simulation results show that the ITI information improves the quality of the equalizer output signal and the performance of the BPMR systems.

The remainder of this study is organized as follows. Section 2 explains the 2D VA for interference estimation. Section 3 presents the proposed detection model. Section 4 presents and discusses the simulations and results. Finally, Section 5 concludes the study.

2. Estimating Interference with 2D VA

2.1. GPR Target

First, had to determine the GPR target of the BPMR channel, H, to design an appropriate VA detector, which is the main idea of PRML. Figure 1 illustrates the estimation method for the GPR target during the training process.

We followed the procedures mentioned in [23,24] to estimate coefficients of the target and equalizer matrices, G and F, respectively, which can be written as:

$$G=\left[\begin{array}{ccc}{g}_{-1,-1}& g_{-1,0}& g_{-1,1}\\ g_{0,-1}& g_{0,0}& g_{0,1}\\ g_{1,-1}& g_{1,0}& g_{1,1}\end{array}\right], \mathrm{and}$$

(1)

$$F=\left[\begin{array}{ccccc}{f}_{-2,-2}& f_{-2,-1}& f_{-2,0}& f_{-2,1}& f_{-2,2}\\ f_{-1,-2}& f_{-1,-1}& f_{-1,0}& f_{-1,1}& f_{-1,2}\\ f_{0,-2}& f_{0,-1}& f_{0,0}& f_{0,1}& f_{0,2}\\ f_{1,-2}& f_{1,-1}& f_{1,0}& f_{1,1}& f_{1,2}\\ f_{2,-2}& f_{2,-1}& f_{2,0}& f_{2,1}& f_{2,2}\end{array}\right]\hspace{0.33em},$$

(2)

From Equations (1) and (2), the signals d[j,k] and z[j,k] can be achieved as:

$$d\left[j,k\right]={\displaystyle \sum _{m=-1}^1{\displaystyle \sum _{n=-1}^{1}a\left[j-m,k-n\right]g_{m,n}}}, \mathrm{and}$$

(3)

$$z\left[j,k\right]={\displaystyle \sum _{m=-2}^2{\displaystyle \sum _{n=-2}^{2}y\left[j-m,k-n\right]f_{m,n}}}.$$

(4)

We defined the vectors as:

$$g={\left[\begin{array}{ccccccccc}{g}_{-1,-1}& g_{-1,0}& g_{-1,1}& g_{0,-1}& g_{0,0}& g_{0,1}& g_{1,-1}& g_{1,0}& g_{1,1}\end{array}\right]}^T,$$

(5)

$$f={\left[\begin{array}{ccccccccc}{f}_{-2,-2}& f_{-2,-1}& f_{-2,0}& f_{-2,1}& \dots & f_{2,-1}& f_{2,0}& f_{2,1}& f_{2,2}\end{array}\right]}^T,$$

(6)

$$a={\left[\begin{array}{ccccccc}a\left[j-1,k-1\right]& a\left[j-1,k\right]& a\left[j-1,k+1\right]& \dots & a\left[j+1,k-1\right]& a\left[j+1,k\right]& a\left[j+1,k+1\right]\end{array}\right]}^T, \mathrm{and}$$

(7)

$$y={\left[\begin{array}{ccccccc}y\left[j-2,k-2\right]& y\left[j-2,k-1\right]& y\left[j-2,k\right]& \dots & y\left[j+2,k\right]& y\left[j+2,k+1\right]& y\left[j+2,k+2\right]\end{array}\right]}^T,$$

(8)

where ^T is transpose operator. Based on Equations (5)–(8), Equations (3) and (4) can be rewritten as:

$$d\left[j,k\right]=g^{T}a, \mathrm{and}$$

(9)

$$z\left[j,k\right]=f^{T}y.$$

(10)

To find the parameters of the GPR target G and the equalizer F, we solved the following optimization problem.

$$\begin{array}{l}\arg \min E\left\{{\left(f^{T}y-g^{T}a\right)}^2\right\},\\ s.t.E^{T}g=c\end{array}$$

(11)

where

$$E^T=\left[\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right], \mathrm{and}$$

(12)

$$c={\left[\begin{array}{ccccc}0& 0& 1& 0& 0\end{array}\right]}^T.$$

(13)

The solutions to Equation (11) are presented as:

$$\lambda ={\left(E^T{\left(A-T^T{R}^{-1}T\right)}^{-1}E\right)}^{-1}c,$$

(14)

$$g={\left(A-T^T{R}^{-1}T\right)}^{-1}E\lambda ,$$

(15)

$$g_{-1,-1}=g_{-1,1}=g_{1,-1}=g_{1,1}=g_{0,-1}{g}_{-1,0}=g_{0,-1}{g}_{1,0},$$

(16)

$$f=R^{-1}Tg,$$

(17)

where $\lambda$ is a vector containing the Lagrange multipliers, A = E{aa^T}, R = E{yy^T}; and T = E{ya^T}. With Equations (14)–(17), the GPR target can be achieved as:

$$G=\left[\begin{array}{ccc}t& p& t\\ r& 1& r\\ t& p& t\end{array}\right],$$

(18)

where r and p are the horizontal and vertical interferences, respectively; and t = rp.

2.2. 2D VA

In BPMR systems, the received signal is disturbed by 2D interference. Therefore, we needed to remove or mitigate 2D interference into 1D interference to apply 1D VA. In this section, we introduce 2D VA. Based on the 2D VA, we propose an interference estimator for converting 2D interference into 1D interference in the next section (Section 2.3).

As shown in Figure 1, the output of the equalizer z[j,k] can be written as:

$$\begin{array}{lll}z\left[j,k\right]& =& d\left[j,k\right]+w_F\left[j,k\right]\\ & =& {\displaystyle \sum _{n=-1}^1{\displaystyle \sum _{m=-1}^{1}a\left[j-n,k-m\right]g_{n,m}}}+w_F\left[j,k\right].\end{array}$$

(19)

where w_F [j,k] denotes the colored noise. By minimizing the mean square error (MSE), we w_F [j,k] can be ignored while analyzing the estimated signal. The interfered signal can be rewritten as:

$$\begin{array}{ll}{\displaystyle \sum _{n=-1}^1{\displaystyle \sum _{m=-1}^{1}a\left[j-n,k-m\right]g_{n,m}}}& =\\ & +ta\left[j-1,k-1\right]+pa\left[j-1,k\right]+ta\left[j-1,k+1\right]\\ & +ra\left[j,k-1\right]+a\left[j,k\right]+ra\left[j,k+1\right]\\ & +ta\left[j+1,k-1\right]+pa\left[j+1,k\right]+ta\left[j+1,k+1\right].\end{array}$$

(20)

When we compare the target, G, and equation (20), interference [r 1 r] is from the main track and interference [t p t] is from the upper and lower tracks. In this study, we defined [t p t] as ITI, [t r t]^T as ISI, [p 1 p]^T as vertical interference (VI), and [r 1 r] as horizontal interference (HI) in G. ITI affects the current symbol based on the symbols in neighboring (upper and lower) tracks, and ISI affects the current symbol based on the symbols in neighboring (previous and next) sample times.

Moreover, we defined a vector, v, as follows:

$$v=\left[\begin{array}{ccc}v\left[j,k-1\right],& v\left[j,k\right],& v\left[j,k+1\right]\end{array}\right],$$

(21)

where

$$v\left[j,k-1\right]=ta\left[j-1,k-1\right]+ra\left[j,k-1\right]+ta\left[j+1,k-1\right],$$

(22)

$$v\left[j,k\right]=pa\left[j-1,k\right]+a\left[j,k\right]+pa\left[j+1,k\right], \mathrm{and}$$

(23)

$$v\left[j,k+1\right]=ta\left[j-1,k+1\right]+ra\left[j,k+1\right]+ta\left[j+1,k+1\right].$$

(24)

With the assignments in Equations (22)–(24), the ISI estimator can be designed. Similarly, if v is assigned along the horizontal direction, as in Equations (25)–(27), the ITI estimator can be designed.

$$v\left[j-1,k\right]=ta\left[j-1,k-1\right]+pa\left[j-1,k\right]+ta\left[j-1,k+1\right].$$

(25)

$$v\left[j,k\right]=ra\left[j,k-1\right]+a\left[j,k\right]+ra\left[j,k+1\right].$$

(26)

$$v\left[j+1,k\right]=ta\left[j+1,k-1\right]+pa\left[j+1,k\right]+ta\left[j+1,k+1\right].$$

(27)

Based on Equations (21)–(24), a trellis can be designed to detect v using VA. Because the input, a[j,k], is 1 or −1, the values of v can be calculated, as listed in

Table 1. Results of v, depending on a[j,k], from Equations (22)–(24).

a[j − 1,k − 1]/ a[j,k − 1]/ a[j + 1,k − 1]	a[j − 1,k]/ a[j,k]/ a[j + 1,k]	a[j − 1,k + 1]/ a[j,k + 1]/ a[j + 1,k + 1]	v[j,k − 1]	v[j,k]	v[j,k + 1]
−1	−1	−1	−r − 2t	−1 − 2p	–r − 2t
−1	−1	1	−r	−1	−r
1	−1	−1
−1	1	−1	r − 2t	1 − 2p	r − 2t
1	−1	1	−r + 2t	−1 + 2p	−r + 2t
−1	1	1	r	1	r
1	1	−1
1	1	1	r + 2t	1 + 2p	r + 2t

. Thus, the trellis has 36 states for [v[j,k − 1], v[j,k]], and each state has six output branches for v[j,k + 1].

2.3. Estimating Interference

After calculating and determining the survivor path (in Section 2.2), the state and input branch at each step of the survivor path can be identified. The state contains the information of v[j,k − 1] and v[j,k], and the output branch contains the information of v[j,k + 1]. Therefore, ISI can be achieved using v[j,k − 1] and v[j,k + 1], and it can be written as:

$$ISI[j,k]=v\left[j,k-1\right]+v\left[j,k+1\right].$$

(28)

Considering ISI[j,k] in Equation (28), we subtracted it from the signal, z[j,k], to create a signal that is only distorted by 1D VI. Then, the VI signal was detected by the 1D VA. Using Equation (19), the VI signal was derived as follows:

$$\begin{array}{lll}{s}_v\left[j,k\right]& =& z\left[j,k\right]-ISI\left[j,k\right]\\ & =& {\displaystyle \sum _{n=-1}^{1}a\left[j-n,k\right]g_{n,0}}+w_F\left[j,k\right].\end{array}$$

(29)

On the other hand, if v is assigned along the horizontal direction to Equations (25)–(27), the ITI estimator can be created. Thus, the signal ITI[j,k] can be achieved, and the signal s_h[j,k] can be acquired as follows:

$$ITI[j,k]=v\left[j-1,k\right]+v\left[j+1,k\right].$$

(30)

$$s_h\left[j,k\right]={\displaystyle \sum _{m=-1}^{1}a\left[j,k-m\right]g_{0,m}}+w_F\left[j,k\right].$$

(31)

In the proposed model (in Section 3), the signals ISI[j,k] or ITI[j,k] are the output of the estimator. If 1D VA is applied according to the vertical direction, the estimator is used with Equations (28) and (29) to find the ISI[j,k] and remove the horizontal interference. If 1D VA is applied according to the horizontal direction, the estimator is used with Equations (30) and (31) to find the ITI[j,k] and remove the vertical interference. ISI[j,k] or ITI[j,k] are selected according to circumstances of the channel. This is mentioned in the simulation section (Section 4).

3. Proposed Model

Figure 2 shows the proposed detection model. The original data, u[k] $\in$ $\left\{0/1\right\}$, are modulated into the signal, a[j,k]$\in \left\{-1/1\right\}$, and stored in the BPMR channel, H. This channel includes 2D interference and Gaussian noise (w[j,k]). The output of the channel y[j,k] goes through an equalizer, F, to adjust it close to the desired signal, d[j,k]. The parameters of the equalizer, F, and the target, G, were estimated using the MMSE algorithm during the training period. The output of the equalizer, F, is z[j,k], which is supplied to the estimator (using 2D VA) to achieve the ISI[j,k] or ITI[j,k] (depending on the direction mentioned in Section 2.2).

3.1. BPMR Channel

The readback signal from the BPMR channel suffers from 2D interference and Gaussian noise and can be written as:

$$y\left[j,k\right]=a\left[j,k\right]*h\left[j,k\right]+w\left[j,k\right],$$

(32)

where * is the convolution operation; j and k are the discrete indices along the down- and cross-track directions, respectively; y[j,k] is the readback signal; w[j,k] is additive white Gaussian noise (AWGN) with zero mean and variance ${\sigma }^2$; and h[j,k] is the BPMR channel pulse response, as follows:

$$h\left[j,k\right]=P\left(j{T}_x,k{T}_q-{\Delta }_{off}\right),$$

(33)

where P(x, q) is a 2D Gaussian function used to represent the 2D island response of the BPMR channel, as in [29].

$$P\left(x,q\right)=A\exp \left(-\frac{1}{2c^2}\left[{\left(\frac{x+{\Delta }_x}{P{W}_x}\right)}^2+{\left(\frac{q+{\Delta }_q}{P{W}_q}\right)}^2\right]\right),$$

(34)

where A is the peak amplitude (in this study, A = 1); x and q are the down- and cross-track directions, respectively; ${\Delta }_x$ and ${\Delta }_q$ are the down- and cross-track bit-location fluctuations, respectively; c is the constant, which represents the relationship between the standard deviation of the Gaussian function and PW₅₀ (c = 1/2.3548) [23]; and PW_x and PW_q are the PW₅₀ components of the down- and cross-track pulses, respectively. Finally, we defined TMR for the BPMR system as

$$TMR\left(\%\right)=\frac{{\Delta }_{off}}{T_q}.$$

(35)

3.2. Detection Scheme

s_v[j,k] and s_h[j,k] are detected based on the 1D VA algorithm. After detection, the output of the 1D Viterbi detection is $\overline{a}$[j,k], which is similar to the original signal, a[j,k]. Thus, the signal, $\overline{a}$[j,k], can pass through the interference target to regenerate $\overline{ISI}$[j,k] or $\overline{ITI}$[j,k], which are the ISI or ITI, respectively, with a higher accuracy compared to ISI[j,k] or ITI[j,k].

$$\begin{array}{lll}\overline{ISI}[j,k]& =& t\overline{a}\left[j-1,k-1\right]+t\overline{a}\left[j-1,k+1\right]\\ & +& r\overline{a}\left[j,k-1\right]+r\overline{a}\left[j,k+1\right]\\ & +& t\overline{a}\left[j+1,k-1\right]+t\overline{a}\left[j+1,k+1\right].\end{array}$$

(36)

$$\begin{array}{lll}\overline{ITI}[j,k]& =& t\overline{a}\left[j-1,k-1\right]+p\overline{a}\left[j-1,k\right]+t\overline{a}\left[j-1,k+1\right]\\ & +& t\overline{a}\left[j+1,k-1\right]+p\overline{a}\left[j+1,k\right]+t\overline{a}\left[j+1,k+1\right].\end{array}$$

(37)

Then, either $\overline{ISI}$[j,k] or $\overline{ITI}$[j,k] is again subtracted from z[j,k] to create a signal with almost no ISI or ITI, which has only 1D interference ($\overline{s_v}$[j,k] or $\overline{s_h}$[j,k]). Finally, either $\overline{s_v}$[j,k] or $\overline{s_h}$[j,k] is detected by 1D VA to restore the original signal, $\widehat{a}$[j,k].

4. Simulation and Results

For simulation, we generated random data for 10 pages. Each page includes a 1 $\times$ 1,440,000-bits block u[k]. First, u[k] is converted into a[j,k] with a size of 1200 $\times$ 1200 bits. We used the first page to estimate parameters of the GPR target, G, and the equalizer, F, using the model shown in Figure 1. The remaining pages were applied to the proposed detection model, as shown in Figure 2, to evaluate the bit error rate (BER) performance. The BPMR channel was set up with an AD of 3 Tb/in² (0.465 Tb/cm²) (T_x = T_q = 14.5 nm) [30]. In this article, the signal-to-noise ratio (SNR) is definite as SNR = 10log₁₀(1/${\sigma }^2$). First, we experimented with the proposed model on the BPMR channel without the TMR effect (0% TMR). The coefficient of the channel, H, is given in [23]. As shown in Figure 2, the ISI estimator (ISI[j,k] is the output of the estimator) or the ITI estimator (ITI[j,k] is the output of the estimator) can be used. For the second 1D Viterbi detection, either horizontal or vertical detection (depending on the interference target). If the ITI target is applied, horizontal detection is used. If the ISI target is applied, vertical detection is used. Therefore, t there are four cases, including the combinations of ITI estimator and vertical direction detector (ITI-VD), ITI estimator and horizontal direction detection (ITI-HD), ISI estimator and vertical direction detector (ISI-VD), and ISI estimator and horizontal direction detector (ISI-HD).

In the first experiment, we determined the best estimator–detector combination for the proposed model. From Figure 3 shows that the ITI-VD structure has the most favorable BER. ITI-VD includes the ITI estimator and ISI target. Then second 1D Viterbi detection was applied to the vertical direction.

Since the ITI component is larger than the ISI in channel, H, of the simulation for the BPMR, the BER performance of this combination is outstanding compared with the that of other cases. In addition, based on this experiment, we can see how to choose the estimator between ITI[j,k] and ISI[j,k]. When the ITI component is larger than the ISI component in the channel, ITI[j,k] is chosen, and when the ISI component is larger than the ITI component in the channel, ISI[j,k] is chosen.

We then compared the proposed model with the previous studies in the channel without TMR effect (${\Delta }_{off}=0$) and media noise (${\Delta }_x={\Delta }_q=0$). The results are presented in Figure 4.

Figure 4 shows that the proposed model with the optimal structure can improve the gain from 0.5 to 2.5 dB at a BER of 10⁻⁴. For the three-way GPR target with feedback [18], the model estimates either the upper or lower interference. Our proposed model estimates both the upper and lower interference. Thus, our proposed model can achieve higher performance compared to the model in [18]. Comparing the serial soft output [24], our proposed model removes the interference from the equalized signal. Thus, it preserves the noise information better than the serial soft output, which estimates noise information by using the feedback from the hard output. Finally, with the ITI subtraction technique [25] and estimation with a neural network [28], a general filter to estimate the ITI. For ITI subtraction technique, the authors used one-layered filter to predict the ITI. For neural networks [28], a multi-layered filter can be used to predict the ITI. In our proposed model, we used the 2D VA, which gives a more specific rule to predict the ITI (or ISI). Thus, the proposed model achieves better optimization compared to the models in [25,28].

Next, we considered the BER performance of the proposed model in the BPMR channel with the TMR effect (${\Delta }_{off}\ne 0$). We assumed that our model suffers from a 10% TMR. We compared the proposed model with serial detection [24] and the model used in [18]. The results are shown in Figure 5. When the TMR effect occurs, the ITI is becomes asymmetric. However, our proposed model can estimate and compensate for the change in ITI and still achieve the best performance.

To investigate resistance to the TMR effect, we changed the level of TMR from 10% to 30%, with an SNR of 15 dB to compare the BER performance. The results are presented in Figure 6.

Based on the results in Figure 5 and Figure 6, we can see that the proposed model can achieve the best performance when the level of the TMR effect is less than 20%. When the TMR effect is more than 20%, the performance of our proposed model is the same as that of serial soft output [24]. Therefore, the proposed model can resist a TMR effect of less than 20%.

In the next experiment, we simulated our proposed model on the BPMR channel with 6% position fluctuation. Figure 7 shows that our proposed model improved the BER performance compared to the serial soft output in [24] and three-way GPR target with feedback in [18] with 6% position fluctuation. When position fluctuation occurs, both ITI and ISI change. However, in this case, the ITI is still larger than the ISI. Therefore, estimating and compensating for the change in ITI is a major factor for improving the BER performance. In addition, we simulated according to the degree of position fluctuation, and the results are shown in Figure 8. We can see that the proposed model can resist position fluctuation of less than 18%. When the position fluctuation is more than 18%, the performances of all models converge.

5. Conclusions

We proposed an interference estimator that uses 2D VA to improve detection performance. In our proposed model, we grouped the sums of products between the signal and the interference into the symbols to apply the 2D VA. In the 2D VA, because each symbol has six levels, we used a trellis with 36 states and 6 input branches. The detected symbols were chosen to estimate the ITI or ISI (depending on the directions and the parameters of the channel). The simulation results show that our proposed model can improve the performance of previous studies. Especially with the optimal structure, the proposed model can achieve gains of 0.5 and 2.5 dB at a BER of 10⁻⁴ compared to the models in [18,24], respectively. In addition, the simulation shows that the proposed model can achieve a better BER performance of serial detection with a TMR effect lower than 20% and a position fluctuation less than 18%.

In the current research, we exploited the signal from a given direction (horizontal or vertical directions) to estimate interference (ITI or ISI). Thus, in the near future, we will develop a model to exploit and combine the information from both directions (horizontal and vertical).