Symbol Synchronization for Diffusion-Based Molecular Communications

One of the crucial requirements for establishing a reliable communication link is symbol synchronization where the start of a symbol interval is determined at the receiver. Most works available in the literature on MC assume perfect symbol synchronization for data detection, see e.g. [4, 5, 6, 7]. First studies on establishing a synchronization mechanism for MC systems have been conducted in [8, 9, 10, 11, 12, 13, 14, 15]. In particular, in [8, 9, 10], the authors proposed a scheme for synchronizing multiple molecular machines that have to carry out a common task, e.g., coordinate their behavior as in quorum sensing among bacteria. Symbol synchronization was investigated in [11, 12, 13, 14, 15]. In [11, 12, 13, 14], the authors proposed a two-way message exchange protocol between the transmitter and the receiver facilitating the estimation and correction of constant frequency and delay offsets between the clocks of the transmitter and the receiver. However, to achieve high performance, the synchronization protocols in [11, 12, 13, 14] require several rounds of two-way message exchange between the transmitter and the receiver which leads to a large overhead considering the slow propagation of molecules in MC channels. Furthermore, for cases when flow is present in the environment, e.g. in the direction from the transmitter to the receiver, it may not be possible to establish a feedback link from the receiver to the transmitter. To reduce the synchronization overhead, the authors in [15] proposed a blind synchronization scheme based on a sequence of data molecules observed at the receiver. However, in [15], the clocks of the transmitter and the receiver are assumed to have identical frequencies and only a constant clock offset may exist.

In this paper, we develop two new symbol synchronization frameworks that take into account some practical challenges of MC systems which have not been considered in [11, 12, 13, 14, 15]. In particular, in [11, 12, 13, 14, 15], similar to wireless communications [16], it is assumed that the nodes are equipped with internal clocks and accurate oscillators. Thereby, the problem of synchronization was reduced to the elimination of possible frequency and delay offsets between the clocks. Furthermore, in [11, 12, 13, 14, 15], it is assumed that the transmitter emits molecules with a fixed release frequency, i.e., the symbol durations are constant and identical. However, in a real MC system, the transmitter will be a biological or electronic nanomachine, e.g. a modified cell, which controls the release of the information molecules into the channel using e.g. electrical, chemical, or optical signals [3, 17]. Because of the inherent randomness in the availability of food and energy for molecule generation, the process for molecule production, and the release process, see [3, Chapters 12 and 13], in practical MC systems, the lengths of the symbol intervals may vary.

To cope with the aforementioned practical challenges, in this paper, we develop symbol synchronization schemes for MC systems where the transmitter is not required to be equipped with an internal clock nor restricted to release the molecules with a constant frequency. To facilitate simultaneous symbol synchronization and data detection, we employ two types of molecule, and propose two different synchronization-detection frameworks: i) Framework 1 (independent symbol synchronization and data detection): In this framework, one type of molecule is used for symbol synchronization and the other one is used for data detection. ii) Framework 2 (joint symbol synchronization and data detection): Here, both types of molecule are used for joint symbol synchronization and data detection. Both frameworks entail different degrees of synchronization-detection accuracy, complexity, and applicability as will be discussed in detail throughout the paper. For both proposed frameworks, we first derive the optimal maximum likelihood (ML) symbol synchronization scheme as performance upper bound. Since ML synchronization entails high complexity, for each framework, we also propose three suboptimal schemes, namely a linear filter-based (LF) scheme, a peak observation-based (PO) scheme, and a threshold-trigger (TT) scheme, which are suitable for MC systems with limited computational capabilities. We further discuss the advantages and disadvantages of the proposed synchronization schemes in terms of their required a priori knowledge, relative complexity, and applicability. In addition, we study insertion and deletion errors, which arise due to imperfect synchronization. We further apply an error-correction code from [18] to mitigate these errors. Our simulation results unveil the effectiveness of the proposed synchronization schemes and suggest that the end-to-end performance of MC systems significantly depends on the accuracy of the symbol synchronization.

We note that this paper expands its conference version [1] in several directions. First, in [1], we studied only independent symbol synchronization and data detection, i.e., Framework 1. Second, the ML symbol synchronization problem formulated in this paper is different from the one in [1]. Third, the proposed LF symbol synchronization scheme was not considered in [1]. Finally, many of the extensive discussions and simulation results are not included in [1].

The remainder of this paper is organized as follows. In Section II, the considered system and signal models are presented. The proposed synchronization and detection schemes are introduced in Section III, and their properties are discussed in more detail in Section IV. Numerical results are reported in Section V, and conclusions are drawn in Section VI.

We consider an MC system consisting of a transmitter, a channel, and a receiver, see Fig. 1. The transmitter is able to release two types of molecules, namely type-$A$ and type-$B$ molecules which facilitates simultaneous symbol synchronization and data detection. Exploiting these two types of molecule, we consider the following two synchronization and detection frameworks:

• i)

  Framework 1 (Independent Synchronization and Detection): Type-$B$ molecules are employed for synchronization and type-$A$ molecules are used for information transmission.

• ii)

  Framework 2 (Joint Synchronization and Detection): Type-$A$ and type-$B$ molecules are used for joint information transmission and synchronization.

The advantageous and disadvantageous of the above two frameworks are discussed in detail in Section IV-A. For instance, joint ML synchronization-detection under Framework 2 outperforms the independent ML synchronization and ML detection under Framework 1 in terms of the end-to-end bit error rate (BER). However, Framework 1 offers more flexibility in the sense that any arbitrary modulation and detection schemes can be employed for data transmission independent of the adopted symbol synchronization scheme.

The details of how the molecules are released by the transmitter for the above two frameworks will be explained in the next subsection. The released molecules diffuse through the fluid medium between the transmitter and the receiver. The movements of individual molecules are assumed to be independent from each other. Furthermore, we assume that the molecules of types $A$ and $B$ have identical diffusion coefficients denoted by $D$ [2]. We consider a spherical receiver whose surface is partially covered by two different types of receptors for detecting type-$A$ and type-$B$ molecules, respectively [19]. Molecules that reach the receiver can participate in a reversible bimolecular reaction with receiver receptor proteins. Thereby, the receiver treats the time-varying numbers of type-$A$ and type-$B$ molecules bound to the receptors as the received signals for data detection and synchronization.

The MC channel is characterized by the following two quantities. i) The expected number of type-$x$ molecules bound to the corresponding receptors at the receiver at time $t$ due to the release of molecules by the transmitter in one symbol interval starting at $t=0$, which is denoted by $P_{x}(t),\,\,x\in\{A,B\}$. ii) The expected number of external noise molecules bound to the receptors, denoted by $z_{x},\,\,x\in\{A,B\}$. In general, $P_{x}(t),\,\,x\in\{A,B\}$, depends on the release mechanism at the transmitter, the MC environment, and the properties of the receiver such as its size, the number of receptors, etc. For instance, assuming instantaneous molecule release and a point source transmitter, expressions for $P_{x}(t)$ can be found in [19] for a general reactive receiver and in [5] for an absorbing receiver. On the other hand, the external noise molecules originate from other MC links or natural sources which also employ type-$A$ or type-$B$ molecules. We emphasize that the synchronization and detection schemes proposed in this paper are general and are applicable for any given expression for $P_{x}(t)$ and any value of $z_{x}$. For future reference, we refer to ${\mathrm{SNR}}_{x}=\frac{{\mathrm{max}}_{t\geq 0}\,\,P_{x}(t)}{z_{x}},\,\,x\in\{A,B\}$, as the signal-to-noise ratio (SNR) for type-$x$ molecules.

Let $w[k]\in\{0,1\}$ denote the binary data symbol in the $k$-th symbol interval. We assume $\mathrm{Pr}\{w[k]=1\}=\mathrm{Pr}\{w[k]=0\}=0.5$ where $\mathrm{Pr}\{X\}$ denotes the probability of event $X$. The transmitter wishes to continuously send data symbols; however, the release time of the molecules at the transmitter may vary from one symbol interval to the next due to variations in the availability of food and energy for molecule generation, the rate for molecule production, and the release process over time, see [3, Chapters 12 and 13]. To model the aforementioned effects, let $t_{s}[k]\in\mathcal{T}[k]$ denote a random variable (RV) whose realization specifies the start of the $k$-th symbol interval where $\mathcal{T}[k]$ is given by

The duration of the $k$-th symbol interval is the time elapsed between $t_{s}[k]$ and $t_{s}[k+1]$; hence, in (1), $T^{\min}$ and $T^{\max}$ are in fact the minimum and maximum possible lengths of a symbol interval, respectively. In other words, the length of each symbol interval is an RV in $[T^{\min},T^{\max}]$. Note that the symbol rate of the considered MC system, denoted by $R$, is bounded by $\frac{1}{T^{\max}}\leq R\leq\frac{1}{T^{\min}}$.

Let $a[k]$ be a binary variable which is equal to one if type-$A$ molecules are released at the beginning of the $k$-th symbol interval, and equal to zero otherwise. Similarly, let $b[k]$ denote a binary variable which is equal to one if type-$B$ molecules are released at the beginning of the $k$-th symbol interval, and equal to zero otherwise. In the following, we specify $a[k]$ and $b[k]$ for the two considered transmission frameworks.

• i)

  Framework 1: To establish symbol synchronization, at the beginning of each symbol interval, the transmitter releases $N_{B}$ type-$B$ molecules, i.e., $b[k]=1,\,\,\forall k$. Moreover, depending on whether $w[k]=1$ or $w[k]=0$ holds, the transmitter releases either $N_{A}$ or zero type-$A$ molecules, respectively, i.e., $a[k]=w[k],\,\,\forall k$ ¹¹1In practical MC systems, the number of molecules released by the transmitter may not be constant and may also vary from one symbol interval to the next. For simplicity, in this paper, we assume that the transmitter waits until a sufficient number of molecules is available and then releases exactly $N_{B}$ synchronization and/or $N_{A}$ information molecules, respectively. The extension of the proposed synchronization schemes to account for varying numbers of released molecules is an interesting topic for future research.. In other words, ON-OFF keying modulation is performed [2].

• ii)

  Framework 2: Here, we employ type-$A$ and type-$B$ molecules for joint synchronization and data transmission. In particular, depending on whether $w[k]=1$ or $w[k]=0$ holds, the transmitter releases either $N_{A}$ type-$A$ molecules or $N_{B}$ type-$B$ molecules, respectively, i.e., $a[k]=w[k]$ and $b[k]=1-w[k],\,\,\forall k$. We note that using different types of moelcules for modulation has already been proposed in [6] and is referred to as molecule shift keying (MoSK) modulation. However, for MoSK, perfect synchronization is assumed in [6], whereas under the proposed Framework 2, we aim to develop a joint synchronization and detection scheme.

To model the received signal, we assume that the receiver periodically counts the numbers of type-$A$ and type-$B$ molecules bound to the respective receptors on its surface with a frequency of $\Delta t$ seconds. Therefore, time can be discretized into a sequence of observation time samples $t_{n}=(n-1)\Delta t,\,\,n=1,2,\dots$, at the receiver. Moreover, let us define $r_{x}(t_{n})$ as the number of type-$x$ molecules bound to the respective receptors at sample time $t_{n}$. Since at any given time after the release of the molecules by the transmitter, the molecules are either bound to a receptor or not, a binary state model applies and the number of bound molecules follows a binomial distribution. We note that the binomial distribution converges to the Poisson distribution when the number of trials is high and the success probability is small [20]. These assumptions are justified for MC since the number of released molecules is typically very large and the probability that any given molecule released by the transmitter reaches the receiver is typically very small [21]. Therefore, $r_{x}(t_{n})$ can be modeled as follows [22, 23, 24]

where $\mathcal{P}(\lambda)$ denotes a Poisson RV with mean $\lambda$ and $\bar{r}_{x}(t_{n})$ is the mean number of type-$x$ molecules bound to the receiver’s receptors at time $t_{n}$, i.e., $\bar{r}_{x}(t_{n})=\mathcal{E}\{r_{x}(t_{n})\}$ where $\mathcal{E}\{\cdot\}$ denotes expectation. Hence, we obtain

In the following, we first develop optimal and suboptimal synchronization schemes. Subsequently, we present the adopted detection scheme.

Similar to Framework 1, for Framework 2, we first derive the joint ML synchronization and detection scheme. Then, we present several suboptimal low-complexity schemes for practical MC systems with limited computational capabilities. However, unlike Framework 1, which uses observations $r_{A}(t_{n}),\,\forall t_{n}$, for data detection and observations $r_{B}(t_{n}),\,\forall t_{n}$, for symbol synchronization, Framework 2 employs both observations $r_{A}(t_{n})$ and $r_{B}(t_{n}),\,\forall t_{n}$, for simultaneous symbol synchronization and data detection, cf. Section II-B.

In the following, we compare several different aspects of the proposed synchronization schemes.

A common challenge of imperfect symbol synchronization are deletion and insertion errors [18, 26]. A deletion error occurs if the adopted synchronization protocol fails to identify the start of a symbol interval, and an insertion error occurs if a false alarm introduces an additional symbol interval. We note that deletion and insertion errors may have a severe impact on the BER performance. For instance, a single insertion or deletion error shifts the positions of all subsequent symbols and significantly deteriorates the BER calculated based on a symbol-by-symbol comparison of the transmitted and detected data. Insertion and deletion errors are schematically illustrated in Fig. 4 and mathematically defined as

where $\hat{t}_{s}[k]$ is an estimate for $t_{s}[k]$ for one of the proposed synchronization schemes and $k^{\mathrm{offset}}$ is the number of deletion events minus the number of insertion events that have occurred for $\forall k^{\prime}<k$. To cope with this challenge in conventional communication systems, special codes were designed which are capable of correcting codewords corrupted by insertions and deletions [18, 26]. For instance, in [18], F. Sellers proposed to periodically insert a synchronizing marker sequence at the beginning/end of each data block. By searching for the markers in their expected positions, the decoder can detect and subsequently correct single insertions or deletions between successive markers. In a similar manner, multiple synchronization errors can be corrected by using longer markers, at the expense of additional redundancy [26].

In this paper, to illustrate the impact and mitigation of insertion and deletion errors in MC systems, we adopt the codes proposed in [18]. In particular, let $\mathbf{W}$ be the original data sequence and $\mathbf{M}$ be a marker of length $l$ which is inserted periodically at the end of each $L$ data bits to create the encoded data symbols $\bar{\mathbf{W}}$. For instance, if the data sequence is $\mathbf{W}=101010\,\,001011$, $L=6$, and $\mathbf{M}=100$, the encoded data symbols will be $\bar{\mathbf{W}}=101010\textbf{100}\,\,001011\textbf{100}$. The decoder for this example code is given in Table II. It was shown in [18] that this code can correct one deletion or one insertion error. In Section V, we study the effectiveness of the aforementioned simple deletion/insertion error correction code in MC systems employing the proposed synchronization schemes. Designing deletion/insertion codes specifically for MC systems is an interesting topic for future research.

For simplicity, we assume instantaneous molecule release by a point source transmitter, consider an unbounded three-dimensional environment, and employ the reactive receiver model recently developed in [19] for the calculation of $P_{x}(t),\,\,x\in\{A,B\}$. Moreover, we assume that $t_{s}[k]$ is uniformly distributed in $\mathcal{T}[k]$, i.e., the length of each symbol interval is an RV uniformly distributed in the interval $[T^{\min},T^{\max}]$. Let $\bar{T}^{\mathrm{symb}}$ be the average symbol duration, i.e., $\bar{T}^{\mathrm{symb}}=\frac{T^{\max}+T^{\min}}{2}$. To be able to easily control the variability of the symbol intervals with a single parameter, we assume that $T^{\min}=(1-\alpha)\bar{T}^{\mathrm{symb}}$ and $T^{\max}=(1+\alpha)\bar{T}^{\mathrm{symb}}$ hold, where $\alpha\in[0,1]$ in general²²2In order to enforce the constraint $T^{\max}\leq 2T^{\min}$ for the proposed ML and LF schemes, $\alpha\leq\frac{1}{3}$ has to hold.. Furthermore, we consider blocks of $K=20$ symbol intervals and average our results over $10^{6}$ blocks. Unless stated otherwise, we adopt the default values of the system parameters given in Table III. Moreover, we assume $z_{x}=5,\,\,x\in\{A,B\}$, and change the number of molecules released by the transmitter to obtain different SNRs. For instance, for the default system parameters in Table III, we obtain $\text{SNR}_{A}=\text{SNR}_{B}=3$ dB.

In order to compare the performances of the considered synchronization schemes, we define the normalized synchronization error as

Moreover, the end-to-end BER in one block is computed as

where $|\cdot|$ denotes the absolute value of a number. Finally, we note that for the proposed TT schemes, the value of threshold $\xi$ is optimized for optimal performance, i.e., minimum BER in Figs. 8, 9, and 11 and minimum $\mathcal{E}\{|\bar{e}_{t}[k]|\}$ for the remaining figures.

In Fig. 5, we show the histogram of $\bar{e}_{t}[k]$ for $\alpha=0.2$, $\bar{T}^{\mathrm{symb}}=2$ ms, and $\text{SNR}_{A}=\text{SNR}_{B}=3$ dB. In the following, we highlight some interesting observations from this figure.

• •

  First, for both frameworks, we observe that the peaks of the probability density function (PDF) for the ML, LF, and PO synchronization schemes are centered at $\bar{e}_{t}[k]=0$ whereas for the TT synchronization scheme, the peak of the PDF occurs at a positive value of $\bar{e}_{t}[k]$. This is expected since the TT synchronization scheme does not aim to estimate the start of the symbol intervals and only determines when the number of molecules bound to the receptors is above threshold $\xi$.

• •

  Fig. 5 also reveals the presence of insertion and deletion errors for the proposed synchronization schemes for both frameworks, cf. Subsection IV-A. In particular, small values of $|\bar{e}_{t}[k]|$ mean that there are no deletion and no insertion errors, whereas large and small values of $\bar{e}_{t}[k]$ (i.e., $\bar{e}_{t}[k]>0.5$ and $\bar{e}_{t}[k]<-0.5$) correspond to deletion and insertion errors, respectively. Fig. 5 shows that the probabilities of insertion and deletion errors are not equal for the proposed synchronization schemes. Moreover, we see from Fig. 5 that for both frameworks, deletion errors are more likely to occur for the TT synchronization scheme than for the PO synchronization scheme since the probability that large values of $\bar{e}_{t}[k]$ occur is higher for the PO scheme than for the TT scheme. On the other hand, we observe that error events $\bar{e}_{t}[k]>0.5$ are unlikely for the ML and LF synchronization schemes which suggests that deletion errors do not occur for these schemes. In contrast, by considering the range $\bar{e}_{t}[k]<-0.5$ in Fig. 5, we note that the insertion error probabilities increase from ML to TT to LF to PO synchronization for both frameworks.

• •

  We note that a direct comparison of the two frameworks based on Fig. 5 is not straightforward. Nevertheless, by visually comparing the curves in Figs. 5 a) and b), we can observe that the synchronization error PDFs for the respective synchronization schemes for Framework 1 are similar to those for Framework 2. As will be verified in Figs. 6 and 7, in terms of synchronization error, Framework 1 indeed achieves a similar performance as Framework 2.

In order to compare the proposed synchronization schemes more quantitatively, in Fig. 6 a), we plot the mean absolute error (MAE), $\mathcal{E}\{|\bar{e}_{t}[k]|\}$, versus symbol index $k$ for $\alpha=0.2$, $\bar{T}^{\mathrm{symb}}=2$ ms, and $\text{SNR}_{A}=\text{SNR}_{B}=3$ dB. We observe from Fig. 6 a) that the MAE increases with increasing symbol index for the proposed suboptimal schemes which is due to the accumulation of errors. In contrast, for the ML scheme, the MAE remains almost constant in successive symbol intervals which indicates that even if an error occurs in one symbol interval, the ML scheme is able to reasonably recover synchronization for the next symbols without a noticeable performance degradation. Moreover, except for the TT scheme which achieves a better performance under Framework 1 than under Framework 2, the MAE performance of the proposed synchronization schemes is almost identical for both frameworks.

In order to study the bias of the proposed estimators, in Fig. 6 b), we show the absolute mean error, $|\mathcal{E}\{\bar{e}_{t}[k]\}|$, versus symbol index $k$ for $\alpha=0.2$, $\bar{T}^{\mathrm{symb}}=2$ ms, and $\text{SNR}_{A}=\text{SNR}_{B}=3$ dB. From this figure, we conclude that all proposed estimators are biased. We note that for the ML and LF schemes, $\mathcal{E}\{\bar{e}_{t}[k]\}$ is positive for $k=1$ and negative for $k\geq 2$; for the PO schemes, $\mathcal{E}\{\bar{e}_{t}[k]\}$ is positive for $k\leq 4$ and negative for $k\geq 5$; and for the TT scheme, $\mathcal{E}\{\bar{e}_{t}[k]\}$ is positive for all $k$. Overall, for large $k$, the mean of the synchronization error, $\mathcal{E}\{\bar{e}_{t}[k]\}$, is negative for all schemes except the TT scheme and $|\mathcal{E}\{\bar{e}_{t}[k]\}|$ increases with symbol index $k$ particularly for the suboptimal LF and PO schemes.

Next, we investigate the effect of the variability of the symbol duration on the synchronization error performance. In particular, in Fig. 7 a), we plot the MAE, $\mathcal{E}\{|\bar{e}_{t}[k]|\}$, versus parameter $\alpha$ for $\bar{T}^{\mathrm{symb}}=2$ ms and $\text{SNR}_{A}=\text{SNR}_{B}=3$ dB. We observe that for the ML schemes, the MAE remains almost constant for $\alpha\leq\frac{1}{3}$ and considerably deteriorates for $\alpha>\frac{1}{3}$. The reason for this is that for $\alpha>\frac{1}{3}$, the constraint $T^{\max}\leq 2T^{\min}$, which is required for the proposed ML schemes, does not hold. Another interesting observation from Fig. 7 a) is that the LF scheme outperforms the less complex PO and TT schemes only if $\alpha<0.25$ holds.

In order to study the effect of ISI on the error performance, in Fig. 7 b), we show MAE, $\mathcal{E}\{|\bar{e}_{t}[k]|\}$, versus symbol duration $\bar{T}^{\mathrm{symb}}$ for $\alpha=0.2$ and $\text{SNR}_{A}=\text{SNR}_{B}=3$ dB. As expected, the error performance of all proposed schemes improves with increasing symbol duration since the ISI decreases as $\bar{T}^{\mathrm{symb}}$ increases. Moreover, the performance of the LF schemes is very close to that of the ML schemes when the ISI is negligible. On the other hand, if the ISI is severe, the LF schemes may be outperformed even by the simple PO and TT schemes. The higher sensitivity of the LF schemes to small $\bar{T}^{\mathrm{symb}}$ compared to the PO and TT schemes can be attributed to the fact that the filtering operation increases the length of the overall impulse response, and hence introduces additional ISI. Therefore, a decrease of $\bar{T}^{\mathrm{symb}}$ deteriorates the performance of the LF scheme more severely than that of the PO and TT schemes.

Next, we study the performance of the proposed synchronization schemes in terms of the end-to-end BER. As performance upper bound, we consider the case of perfect symbol synchronization where the beginning of the symbol intervals is perfectly known at the receiver and the ML detection in (17) and (20) is adopted for Frameworks 1 and 2, respectively. In Fig. 8, the BER is shown versus $\text{SNR}=\text{SNR}_{A}=\text{SNR}_{B}$ for $\alpha=0.2$ and a) $\bar{T}^{\mathrm{symb}}=1$ ms (strong ISI), b) $\bar{T}^{\mathrm{symb}}=2$ ms (weak ISI). We highlight the following observations from Fig. 8.

• •

  From Fig. 8 a), we observe that there is an SNR gap of approximately $2$ dB between the BERs of the ML synchronization schemes and the upper bound which is solely due to imperfect synchronization. This gap becomes smaller in Fig. 8 b), particularly for Framework 1, as there is less ISI and hence the quality of synchronization improves. The BER of the LF scheme in Fig. 8 b) is significantly improved compared to that in Fig. 8 a) due to the weaker ISI. This trend was also expected from Fig. 7 b) based on the synchronization errors for $\bar{T}^{\mathrm{symb}}=1$ and $\bar{T}^{\mathrm{symb}}=2$ ms.

• •

  As expected, joint ML synchronization and detection in (20) under Framework 2 outperforms independent ML synchronization in (7) and ML detection in (17) under Framework 1. However, the suboptimal schemes for Framework 2 are all outperformed by the respective suboptimal schemes for Framework 1. This is due to the fact that for the suboptimal synchronization schemes for Framework 1, we employ optimal ML detection in (17) whereas for the suboptimal schemes for Framework 2, synchronization and detection are performed jointly and are strictly suboptimal.

• •

  Figs. 8 a) and b) reveal that the TT scheme for Framework 1 offers a good performance for both ISI scenarios. Therefore, the TT scheme for Framework 1 might be a good option for practical MC systems with limited computational capabilities, since it provides a favorable tradeoff between complexity and performance.

Recall that in the system model, we assume that the maximum number of type-$A$ and type-$B$ molecules that the transmitter can release in a given symbol interval is $N_{A}$ and $N_{B}$, respectively. Moreover, following the discussion in Section IV-B-2, one can also constrain the average number of molecules that the transmitter releases by $\bar{N}$. Here, we consider both constraints and aim to optimize the fractions of type-$A$ molecules, i.e., $\beta$, and type-$B$ molecules, i.e., $1-\beta$, see Section IV-B-2 for details. In Fig. 9, we show the BER versus parameter $\beta$ for $\alpha=0.2$, $N^{\max}=10^{3}$ per symbol, and a) $\bar{T}^{\mathrm{symb}}=1$ ms (strong ISI) and b) $\bar{T}^{\mathrm{symb}}=2$ ms (weak ISI). As expected, the optimal $\beta$ for all schemes in Framework 2 is $0.5$ which is due to the symmetry of the system with respect to type-$A$ and type-$B$ molecules. Interestingly, for all schemes in Framework 1, the optimal value of $\beta$ is strictly lower than $0.5$. This suggests that for the overall BER performance, it is beneficial to allocate more molecules to symbol synchronization than to data detection. Moreover, the BER performance is considerably better for the case of weak ISI (Fig. 9 b)) compared to that of strong ISI (Fig. 9 a)).