Performance Limit and Coding Schemes for Resistive Random-Access Memory Channels

The proposed across-array coding strategy is illustrated in Fig. 3. Consider the processing of the data vector $\mbox{\boldmath$b$}=(b_{1},b_{2},...,b_{NR})$, where $N$ is the code length and $R$ is the code rate. Here, $b$ is encoded into codeword $\mbox{\boldmath$x$}=(x_{1},x_{2},...,x_{N})$ which is assigned to $T$ memory arrays, where $N=sT$ for some integer $s$. Thus, we split $x$ into $T$ equal-length vectors, each of which is assigned to an independent memory array. Without loss of generality, we assign $\mbox{\boldmath$x$}^{t}=(x_{1}^{t},x_{2}^{t},....,x_{N/T}^{t})$ with $x_{i}^{t}=x_{(t-1)N/T+i},i=1,...,N/T$, to the $t$-th memory array for $t=1,...,T$. Since each memory array is of size $m\times n$, $mnT/N$ codewords can be stored by these $T$ memory arrays, where $mnT/N$ is assumed to be an integer. As the code rate is $R$, the storage efficiency is $R$ bits/cell.

Each codeword is decoded independently based on its readback signal. The codeword $\mbox{\boldmath$x$}=(\mbox{\boldmath$x$}^{1},...,\mbox{\boldmath$x$}^{T})$ is decoded based on its readback signal $\mbox{\boldmath$y$}=(\mbox{\boldmath$y$}^{1},...,\mbox{\boldmath$y$}^{T})$, where $\mbox{\boldmath$y$}^{t}$ is the readback signal of $\mbox{\boldmath$x$}^{t}$ from the $t$-th memory array, $t=1,...,T$, to obtain the estimated data $\hat{\mbox{\boldmath$b$}}$. Here we employ a suboptimal decoding scheme known as the TIN decoding [6]. The TIN decoder ignores the correlation between the channel input and the sneak paths and treats the sneak-path interference as the i.i.d. noise.

To illustrate the advantage of across-array coding strategy more explicitly, in Fig 4, we evaluate the PMF of the sneak-path rate over one codeword that is stored in $T$ memory arrays. In the figure, we employ the array sizes and the code lengths of $N=m\times n=64\times 64$ and $128\times 128$. Since the coded bits are distributed in $T$ memory arrays, the sneak-path rate as well as its PMF are rewritten as $\sum_{t=1}^{T}\sum_{i=1}^{N/T}e_{i}^{t}/(mn(1-q))$ and $F^{T}_{q}(\epsilon)=\textrm{Pr}(\sum_{t=1}^{T}\sum_{i=1}^{N/T}e_{i}^{t}/(mn(1-q))=\epsilon)$, where $e_{i}^{t}$ is the sneak-path event indicator during the reading of the $i$-th bit that belongs to the $t$-th array. For each case of $m\times n=64\times 64$ and $128\times 128$ and the given input distribution with $q=0.25$ and $0.5$, as $T$ increases, the spread of the PMF of the sneak-path rate gets smaller and concentrates closer to the mean value $\epsilon_{q}$ and the channel becomes more stable. The reason is that since the codeword is assigned to $T$ independent memory arrays, the sneak-path rate is averaged over the $T$ arrays. Based on the law of large numbers, as $T\rightarrow\infty$, the sneak-path rate converges exactly to the mean value with probability 1, and therefore, we can design a code based on this mean value to guarantee error free decoding. Note that the across-array coding strategy does not change the channel correlation. It actually reduces the correlation of the sneak-path interference within one codeword since the readback signals for coded bits from different memory arrays are independent with each other. As all the coded bits are encoded from the same message data, this resembles a “code diversity” strategy for block fading channels [7].

The drawback of a joint coding across $T$ arrays is a potential increase of the read/write latency. Note that such additional latency is unavoidable for many sneak-path mitigating approaches in the literature. For example, the multistage reading technique presented in [8] requires three readings and three writings to get a better estimation of the sneak current; [4] investigates the multiple-read detector and shows that it can achieve near-optimum performance with 10 reads. We remark that for our proposed across-array coding strategy, the additional read/write latency can be minimized if a parallel reading/writing circuit [9] is adopted across different crossbar arrays. Moreover, the additional latency incurred will become negligible if the two-dimensional crossbar arrays are stacked to form a 3D structure, which naturally enables the parallel reading/writing across different arrays [10].

We first define a block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel and show how the ReRAM channel capacity is related to that of the block varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel.

To begin, we first define an $(\epsilon,\sigma)$-channel. As illustrated in Fig. 5, an $(\epsilon,\sigma)$-channel is a concatenation of an i.i.d. asymmetrical discrete channel and an i.i.d. additive Gaussian channel, and therefore, it is also an i.i.d. channel without channel correlation. The asymmetrical discrete channel is with binary-input $X\in\{0,1\}$ and ternary-output from $\{R_{0},R_{0}^{\prime},R_{1}\}$ with $R_{0}^{\prime}=\left(\frac{1}{R_{0}}+\frac{1}{R_{s}}\right)^{-1}$, and the transition property is described by Pr$(R_{0}|0)=1-\epsilon$, Pr$(R_{0}^{\prime}|0)=\epsilon$, and Pr$(R_{1}|1)=1$. The additive Gaussian channel is with noise distribution $\eta\sim\mathcal{N}(0,\sigma^{2})$, whose output $Y\in\mathcal{R}$ serves as the output of the $(\epsilon,\sigma)$-channel.

For given input distribution Pr$(X=0)=1-q$, Pr$(X=1)=q$, the $(\epsilon,\sigma)$-channel capacity can be derived as

where

The $(0,\sigma)$- and $(1,\sigma)$-channels are asymmetrical binary-input AWGN channels, which are two special cases of an $(\epsilon,\sigma)$-channel. Obviously, the $(\epsilon,\sigma)$-channel capacity decreases as $\epsilon$ increases leading to $C_{q}(1,\sigma)<C_{q}(\epsilon,\sigma)<C_{q}(0,\sigma)$.

A $T$-block block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel with parameters $\mbox{\boldmath$\epsilon$}=(\epsilon^{1},\epsilon^{2},...,\epsilon^{T})$ varies from data block to data block, while within the $t$-th data block, the channel is a symbol-wise i.i.d. $(\epsilon^{t},\sigma)$-channel, $t=1,2,...,T$. The block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel is a type of channel with block interference as proposed by [11].

The ReRAM channel over $T$ memory arrays resembles the $T$-block block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel. In particular, the sneak-path rate of the ReRAM channel varies from memory array to memory array resembles the block-varying property (the channel varies from block to block) of the block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel. The channel parameter $\epsilon^{t}=\textrm{Pr}(R_{0}^{\prime}|X=0)$ resembles the instantaneous sneak-path rate of the $t$-th memory array. The main difference is that in the ReRAM channel, the sneak-path interference is dependent on the input data and this data-dependency leads to the channel correlation, while the parameter $\epsilon$ of block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel is data-independent and the channel within each block is i.i.d. However, if we adopt TIN decoding where the decoder ignores this data-dependency and regards the sneak paths as the i.i.d. noise, an ReRAM channel is equivalent to a block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel. The memory block length, denoted by $M$, of the block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel should be identical to the data array size of the ReRAM channel, i.e., $M=mn$. The channel parameters $\epsilon^{t},t=1,2,...,T$ should be i.i.d. generated based on PMF of the sneak-path rate of the ReRAM channel. Therefore, the maximum achievable coding rate over the ReRAM channel under TIN decoding can be approximated by the block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel capacity.

In preparation to give the capacity limit, we define an $(\epsilon,\sigma)$-channel code:

The existence of the $(\epsilon,\sigma)$-channel code ensemble is guaranteed by the conventional channel coding theorem.

Consider a block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel with memory block size $M$. Parameter $\epsilon^{t}$ has the PMF $F(\epsilon^{t}),t=1,2,...,T$, and mean value $\bar{\epsilon}=\sum_{\epsilon}\epsilon F(\epsilon)$, as defined in (4). Since the channel varies from block to block, we consider joint $T$-block encoding and decoding. The code length is therefore $MT$. Let $R$ be the encoding rate, and we then have the following theorem:

Proof: We first show that for a fixed input distribution of Bernoulli $(q)$, $C_{q}(\bar{\epsilon},\sigma)$ is achievable. Let $\mbox{\boldmath$x$}=(\mbox{\boldmath$x$}^{1},...,\mbox{\boldmath$x$}^{T})$ be the joint $T$-block codeword, where $t$-th block $\mbox{\boldmath$x$}^{t}=(x^{t}_{1},...,x^{t}_{M})$ experiences an $(\epsilon^{t},\sigma)$-channel, i.e., each symbol $x^{t}_{j},j=1,...,M$, experiences an i.i.d. $(\epsilon^{t},\sigma)$-channel. We assume that the codeword is encoded in the way that the $i$-th bits, located at different data blocks, i.e., $(x_{i}^{1},x_{i}^{2},...,x_{i}^{T})$, belong to a codeword of a length-$T$ $(\bar{\epsilon},\sigma)$-channel code. In this way, the original length-$MT$ codeword $x$ can be considered as a vector consisting $M$ length-$T$ $(\bar{\epsilon},\sigma)$-channel codewords. This is possible because we can always split the uncoded data vector into $M$ equal-length sub-vectors, and encode each of them independently using an $(\bar{\epsilon},\sigma)$-channel code. As the encoding rate of each $(\bar{\epsilon},\sigma)$-channel code is $C_{q}(\bar{\epsilon},\sigma)$, the overall code rate is $R=C_{q}(\bar{\epsilon},\sigma)$.

During decoding, for each $i=1,...,M$, $(x_{i}^{1},x_{i}^{2},...,x_{i}^{T})$ is decoded based on its channel output $(y_{i}^{1},y_{i}^{2},...,y_{i}^{T})$, where $y_{i}^{t}$ is a channel observation of $x_{i}^{t}$. Since coded bit $x_{i}^{t}$ experiences an $(\epsilon^{t},\sigma)$-channel, where $\epsilon^{t},t=1,2,...,T$, are i.i.d. generated based on the PMF of $F(\epsilon)$, the overall codeword experiences an $(\bar{\epsilon},\sigma)$-channel where $\bar{\epsilon}=\sum_{\epsilon}\epsilon F(\epsilon)$. Since $(x_{i}^{1},x_{i}^{2},...,x_{i}^{T})$ is an $(\bar{\epsilon},\sigma)$-channel codeword, the decoding error probability approaches 0 as $T\rightarrow\infty$ according to Definition 1.

In [11], an upper bound of the block interference channel is proposed. That is, given the channel parameters $\epsilon$, the channel becomes memoryless, leading to $R\leq\frac{1}{MT}I(\mbox{\boldmath$x$};\mbox{\boldmath$y$})\leq\frac{1}{MT}I(\mbox{\boldmath$x$};\mbox{\boldmath$x$}|\mbox{\boldmath$\epsilon$})\leq I(x_{1}^{1};y_{1}^{1}|\epsilon^{1})=\sum_{\epsilon}F(\epsilon)C_{q}(\epsilon,\sigma)=\overline{C}_{q}(\sigma)$. $\Box$

It was also shown in [11] that if the channel state is finite, i.e., $\epsilon$ is from a finite set, the upper bound is tight when the block size $M\rightarrow\infty$.

Based on our channel equivalence, $\max_{q}C_{q}(\epsilon_{q},\sigma)$ is an approximate lower bound of the ReRAM channel capacity with TIN decoding, and $\max_{q}\sum_{\epsilon}F_{q}(\epsilon)C_{q}(\epsilon,\sigma)$ is an upper bound. Since we have the explicit formula (4) for $\epsilon_{q}$, the lower bound is much easier to be calculated than the upper bound, which requires $F_{q}(\epsilon)$. Fortunately, we can show that when the array size $N$ is large, the two bounds are numerically very close with each other, and hence the lower bound $\max_{q}C(\epsilon_{q},\sigma)$ can be a good approximation of the ReRAM channel capacity. Fig. 6 illustrates the capacity upper bound $\sum_{\epsilon}F_{q}(\epsilon)C_{q}(\epsilon,\sigma)$ and lower bound $C(\epsilon_{q},\sigma)$ as functions of $q$, for different memory array sizes $m\times n=32\times 32,64\times 64,128\times 128,256\times 256$ and different noise values of $\sigma=30,50,100$. The resistance parameters are fixed with $R_{1}=100\ \Omega,R_{0}=1000\ \Omega$, and $R_{s}=250\ \Omega$. Observe that when the memory array size $N$ is large, the two bounds are very close with each other.

Fig. 6 also indicates that the ReRAM channel capacity bounds decrease as the data size increases due to the increase of the sneak-path rate, i.e., the larger the data array size the lower the average storage efficiency for each cell, and vice versa. For a very low noise level of $\sigma=30$, the capacity bounds are maximized at about $q=0.5$, and for $\sigma=50,100$, they are typically maximized when $q<0.5$. The optimal value of $q$ that maximizes the channel capacity bounds decreases as noise level increases. This is because noise amplifies the detrimental effect of sneak paths, while reducing $q$ effectively reduces the sneak-path rate.

A system model for the proposed coding scheme is illustrated in Fig. 7. The encoder includes an ECC encoder and a data shaper. The ECC encoder encodes data vector $\mbox{\boldmath$b$}=(b_{1},b_{2},...,b_{NR})$ into codeword $\mbox{\boldmath$c$}=(c_{1},c_{2},...,c_{N})$ whose entries are uniformly distributed on $\{0,1\}$. The data shaper reforms the data distribution into Bernoulli $(q)$. Its output is $\mbox{\boldmath$x$}=(x_{1},x_{2},...,x_{N})$. Here the data shaper in our system has rate-1, and therefore, the overall code rate is still $R$.

The decoder involves a real-time channel estimator, elementary signal estimator (ESE), a de-shaper, and an ECC decoder. Since the decoding is actually a block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channel decoding, the channel estimator first estimates the channel parameters $\mbox{\boldmath$\epsilon$}=(\epsilon^{1},...,\epsilon^{T})$ over the $T$ memory arrays based on readback signal $\mbox{\boldmath$y$}=(\mbox{\boldmath$y$}^{1},...,\mbox{\boldmath$y$}^{T})$. Based on $\hat{\mbox{\boldmath$\epsilon$}}$ and $y$, the ESE calculates a soft estimation $\{L(x_{i}^{t}|y_{i}^{t},\hat{\epsilon}^{t})\}$, i.e., the log-likelihood ratio (LLR), for each coded bit $x_{i}^{t}$ that is used as the decoder input. The decoder consists of a de-shaper and an ECC decoder, both of which use soft-in soft-out (SISO) processings and perform iteratively to improve the decoding reliability. The corresponding decoding is standard message-passing decoding. Specifically, the de-shaper calculates soft LLR $\{L^{e}(c_{i})\}$ for each ECC coded bit, based on which an ECC decoding refines the estimation and feds back an updated LLR $\{L^{a}(c_{i})\}$ to the de-shaper for the next round of decoding iterations. After a fixed maximum number of iterations, a hard decision is performed at the ECC decoder to produce data estimation $\hat{\mbox{\boldmath$b$}}$.

The data shaper consists of a length-$L$ repeater, a length-$NL$ bit interleaver $\pi$, and an $L$-to-1 mapper (Fig. 8). The repeater duplicates each ECC coded bit $L$ times. The bit interleaver permutes the repeater output to relocate the bits. The $L$-to-1 mapper maps every $L$ bits to one bit, i.e., $\mathcal{M}:\{0,1\}^{L}\rightarrow\{0,1\}$. Therefore, the data shaper’s overall rate is 1. For the $L$-to-1 mapper, since there are $2^{L}$ patterns for the mapping inputs, by mapping $i$ of them to 1 and $2^{L}-i$ of them to 0, we obtain the data output with a distribution of Bernoulli $(\frac{i}{2^{L}})$. By choosing $i=1,2,...,2^{L}-1$, we achieve data distributions of Bernoulli $(q)$ with $q=\frac{1}{2^{L}},\frac{2}{2^{L}},...,\frac{2^{L}-1}{2^{L}}$.

The interleaver inside the data shaper is crucial in our scheme. Rather than adopting random interleaving, we propose a structured interleaving scheme, as shown in the data shaper’s factor graph in Fig. 8. The interleaver $\pi$ consists of $L$ sub-interleavers $\pi_{i},i=1,...,L$, each of which can be random. The data-shaping process can be described using a factor graph. Each variable node is associated with an ECC coded bit, where the $i$-th variable node is associated with $c_{i}$. There are $L$ edges from a variable node to the interleavers corresponding to the $L$ repetitions of the ECC coded bit. Each mapping node has $L$ edges from the interleavers corresponding to the $L$ mapping inputs. The $i$-th mapping node is associated with the mapping output $x_{i}$. By using our structured interleaver, the $i$-th repetitions of the ECC coded bits enter a sub-interleaver $\pi_{i}$ whose outputs are used as the $i$-th inputs of the mapping nodes. By doing so, each ECC coded bit has exactly one repetition that occupies the $i$-th input of a mapping node for $i=1,...,L$.

The advantage of employing this structured interleaver can be explained using an example. Consider a data shaper with an $(L=3)$-repeater and a 3-to-1 mapper $\mathcal{M}(\tilde{c}_{1},\tilde{c}_{2},\tilde{c}_{3})=\tilde{x}$ (Fig. 9). There are eight patterns for three binary inputs $\tilde{c}_{1},\tilde{c}_{2},\tilde{c}_{3}$, where only two of them, 110 and 111, are mapped to 1, and the other six are mapped to 0. If $\tilde{c}_{1},\tilde{c}_{2},\tilde{c}_{3}$ are i.i.d. with Pr$(\tilde{c}_{i}=0)$=Pr$(\tilde{c}_{i}=1)=\frac{1}{2},i=1,2,3$, the mapping can realize output data distribution with Pr$(\tilde{x}=0)=\frac{3}{4}$, Pr$(\tilde{x}=1)=\frac{1}{4}$. Next we address the de-mapping. Mapping output $\tilde{x}$ actually contains a different quantity of information about the three input bits $\tilde{c}_{1},\tilde{c}_{2},\tilde{c}_{3}$. By formulating the mapping rule as $\tilde{x}=\mathcal{M}(\tilde{c}_{1},\tilde{c}_{2},\tilde{c}_{3})=\tilde{c}_{1}\cdot\tilde{c}_{2}$, where $\cdot$ is a multiply operation, we evaluate the mutual information between $\tilde{x}$ and each input bit as $I(\tilde{x};\tilde{c}_{1})=I(\tilde{x};\tilde{c}_{2})=\frac{3}{4}\log_{2}\frac{4}{3}$ and $I(\tilde{x};\tilde{c}_{3})=0$. Therefore, if the de-mapper is sufficiently near-optimal, during de-mapping we can obtain information about the first and second bits $\tilde{c}_{1},\tilde{c}_{2}$. Unfortunately, we cannot get any information about the third bit $\tilde{c}_{3}$ since $\tilde{x}$ does not contain any information about $\tilde{c}_{3}$. In other words, one-third of the bits are erased after de-mapping. If random interleaving is employed, with probability $\frac{1}{27}$, all the three repetitions of an ECC coded bit will be assigned as the third input of a mapping node and erased. In other words, with probability $\frac{1}{27}$, an ECC coded bit will be erased after de-shaping, which leads to a poor ECC decoding performance. Our structured interleaving guarantees that all the ECC coded bits can obtain a positive and statistically equal quantity of information from the de-shaper to benefit the ECC decoding.

We propose a maximum likelihood channel estimation to obtain parameters $\mbox{\boldmath$\epsilon$}=(\epsilon^{1},...,\epsilon^{T})$. Since the decoder assumes the channel created by the $t$-th memory array as an i.i.d. $(\epsilon^{t},\sigma)$-channel, with the channel observation of $\mbox{\boldmath$y$}^{t}=(y^{t}_{1},...,y^{t}_{N/T})$, the log-likelihood function of $\epsilon^{t}$ is written as:

Next we consider the approximation for (10). Let $\bar{y}^{t}_{i}=\arg\min_{x\in\{R_{0},R_{0}^{\prime},R_{1}\}}|y^{t}_{i}-x|$ be the hard decision value of $y^{t}_{i}$. Since each term of (10) is in the form of $\log\sum_{x\in\{R_{0},R_{0}^{\prime},R_{1}\}}p_{x}\phi(y^{t}_{i},x)$, when the channel noise level is low, it is dominated by the term of $x=\bar{y}^{t}_{i}$. We thereby apply

and hence have the following approximation:

where $n_{x}\overset{\Delta}{=}\sum_{i=1}^{N/T}1\{\bar{y}^{t}_{i}=x\}$ is the total number of $y^{t}_{i}$ whose hard decision is $x$ and $c$ is a constant term independent of $\epsilon^{t}$. By maximizing $\Lambda(\epsilon^{t};\mbox{\boldmath$y$}^{t})$ we obtain the estimation of $\epsilon^{t}$:

Next, the ESE calculates the LLR for each coded bit of $(\mbox{\boldmath$x$}^{1},...,\mbox{\boldmath$x$}^{T})$ based on the readback signal $(\mbox{\boldmath$y$}^{1},...,\mbox{\boldmath$y$}^{T})$ and the estimated channel parameters:

for $i=1,...,N/T,t=1,...,T$. Note that the implementation of ESE (LABEL:eq:ESE) ignores the correlation between cells in a memory array and regards the sneak-path interference as the i.i.d. noise. A more sophisticated decoding scheme can be developed by utilizing the cell correlation and performing joint data and sneak path detection. It is left as our future work.

To demonstrate the accuracy of the proposed channel estimation, in Fig. 10, we illustrate the mean squared error (MSE) between the estimated and the actual sneak-path rates. We obtain the MSE: $E[(\hat{\epsilon}-\epsilon)^{2}]$ by simulation for $T=1$ and memory array sizes $m\times n=64\times 64,128\times 128$, where $\epsilon$ is the actual sneak-path occurrence rate, and $\hat{\epsilon}$ is the estimated value obtained by (11). In general, the MSE is below $10^{-2}$ and decreases as the channel noise level decreases. For comparison, we also illustrate the MSE: $E[(\epsilon_{q}-\epsilon)^{2}]$ between the average and the actual sneak-path rates, where the average sneak-path rate is employed by the decoder when channel estimation is unavailable. Our proposed channel estimation is much more accurate to predict the channel than the average channel parameter, especially for large array size.

The SISO de-shaper can also be realized by message-passing processing over the factor graph shown in Fig. 8. During the de-shaping, each node performs as a local processor and the edges pass LLR messages. The message passing on the edges is bi-directional. The overall processing is performed iteratively. In each iteration, each node in the factor graph acts once. A mapping node performs de-mapping processing based on the $L$ priori LLRs from its neighboring variable nodes and the LLR from ESE and outputs an extrinsic LLR for each mapping input. The extrinsic LLR is used as an a priori LLR for variable node processings. A variable node combines the $L$ priori LLRs from its neighboring mapping nodes and feeds back an extrinsic LLR to each of its neighboring mapping nodes. After a certain number of iterations the variable nodes output a more reliable LLR for each ECC coded bit as the de-shaper output.

In this section, we present the ECC optimization and BER simulation results for ReRAM systems. With our proposed ECC, we achieve a high storage efficiency with a gap of less than 0.1 bit/cell from the ReRAM channel capacity.

For the ECC, we adopt an IRA code which is a type of irregular low-density parity-check (LDPC) code that is able to approach the i.i.d. channel capacity with low encoding and decoding complexity [12, 13]. We consider the code design for two ReRAM systems with a memory array sizes of $m\times n=64\times 64$ and $128\times 128$, and at a noise level of $\sigma=100$. We adopt the data shapers with a $(L=4)$-repeater and 4-to-1 mappers with Mappings A and B (Fig. 11). Mappings A and B produce data distributions of $q=\frac{5}{16}$ and $q=\frac{3}{16}$, respectively, which approach the maximum storage efficiency for the considered ReRAM systems (Fig. 6). Here the maximum storage efficiencies for $m\times n=64\times 64$ and $128\times 128$ ReRAM systems are $\max_{q}C_{q}(\epsilon_{q},\sigma=100)=0.660$ and $0.494$ bit/cell, respectively. We optimize the IRA code for these two cases over an i.i.d. $(\epsilon_{q},\sigma)$-channel using a density evolution method [14]. The code parameters for these two ReRAM systems are listed in TABLE I. Our codes achieve $R=0.542824$ and $0.414735$ bit/cell, which are close to the capacity bound with gaps of about $0.12$ and $0.08$ bit/cell.

In Figs. 12 and 13, we simulate the BER of the two coded ReRAM systems in TABLE I with the across-array coding strategy over both the ReRAM channels (solid lines) and the equivalent block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channels (labeled by $\circ$). Message-passing decoding were employed for both the de-shaper and the ECC decoder, where the decoding of IRA codes can be found [13]. For the $m\times n=64\times 64$ system, we employ a code length $N=64\times 64=4096$ and for $128\times 128$, we adopt $N=128\times 128=16384$. In both figures, the BERs over the ReRAM channels are almost the same as those over the block-varying $(\mbox{\boldmath$\epsilon$},\sigma)$-channels, thus demonstrating the proposed channel equivalence. The BERs improve as $T$ increases and approach the decoding performance over the i.i.d. $(\epsilon_{q},\sigma)$-channel, which is the performance limit for ReRAM channels as $T\rightarrow\infty$. This performance improvement as $T$ increases can be regarded as a “diversity” gain by assigning the codeword to multiple memory arrays. For comparison, we also illustrate the BER performances of the same IRA codes without data shaping $(q=1/2)$ over i.i.d. $(\epsilon_{q},\sigma)$-channel. The decoding performance deteriorates significantly due to the lack of data shaping.