Keywords

1 Introduction

Nanotechnology can be defined as the science of engineering functional systems at an atomic and molecular scale, the prefix ‘nano -’ denotes a factor of 10−9 and means a billionth [1]. The rapid evolution in nanotechnology has provided appropriate development in miniaturization and fabrication of nanomachines with simple sensing, computation, data storing, and communication and action capability [2]. Further capabilities and applications can be enabled if multiple nanomachines communicate to perform collaborative and synchronous functions in a distributed manner to form a nanonetwork [3]. Molecular communication is considered a bio-inspired paradigm, in which molecules are transmitted, propagated and received between nanomachines [4].

1.1 Related Work

In [5] an estimation of the achievable information rates is presented for a diffusion based molecular communication, and information is encoded as a set of distinct molecules. Through extending the framework and results in [6], the outcomes in [5] show large gains in the information rate, compared to the case where the emitted molecules are of the same type. In the literature, various studies have aimed to model the physical channel of the diffusion based molecular communication [7], governed by Fick’s laws, in particular some research have explored the channel transfer function [8], while other research focused on channel capacity from information theoretical aspects [9,10,11,12,13]. The noise effects on channel capacity have been investigated in [14,15,16], concluding that diffusing a larger number of molecules increases the signal to noise ratio and could reduce the noise impact. The authors in [17] presented the design challenges and principles in diffusion based molecular communication, considering the propagation delay and channel distortion to be the main challenges. Synchronization between the transmitter nanomachine and receiver nanomachine is considered one of the challenges in molecular communication systems. Synchronization is important, as it can affect the error rate performance of the receiver nanomachine [18]. Mostly in literature related to molecular communication, authors assume that the system is synchronized; however, studying biological mechanisms brings opportunity to find different tools that can be used to overcome challenges. In biology, there is a mechanism known as quorum sensing [19, 20], in which bacteria can utilize to synchronize their behaviour, through the emission and sensing of a certain type of molecules called autoinducer. The authors in [19, 21] proposed quorum sensing as a tool to achieve synchronization among nanomachines in diffusion based molecular communication system.

In [22] performance analysis of the flow assisted diffusion channel molecular communication is investigated. The effect of flow velocity, interference of time slotted transmission and diffusion coefficient on average symbol error rate and achievable mutual information rate of the channel is derived.

The authors in [23] proposed an improved non-coherent signal detection scheme, aiming at addressing ISI and noise contamination. Through utilizing three non-coherent metrics to characterize transient features of received signals, and furthermore, designed an optimized combination scheme for more reliable signal detections.

In [24] the performance of molecular communication model has been studied, by presenting an algorithm to find the maximum distance that diffused molecules can reach, and an energy harvesting model for the nanonodes has been proposed. Then the pattern of diffusion issue has been explored, to find out how could nanonodes can distinguish the pattern of diffusion of one nanonode in a distance d0 from it in the same network. Where the information molecules represented as two bytes: 00, 01, 10, 11. The diffusing nanonode would continue to diffuse unites of molecules through 8 time slots, each slot is of length t0.

1.2 The Work in This Paper

The main objective is to study a diffusion based molecular communication model, through the analysis of diffusion propagation medium. Taking in consideration the parameters that impact communication in diffusion based systems. The pattern of diffusion has been explored, by inspecting how a receiver nanomachine could distinguish a message from one transmitter nanomachine at a distance d or two transmitter nanomachines at different distances. The information molecules were represented as 2-bits by using 22 different values with 22 − 1 thresholds. taking into consideration the effects of the following parameters: distance, time, sensed molecular concentration, interference.

In the literature, the performance of the molecular propagation channel has been explored in [24,25,26,27,28,29,30,31]. Through the evaluation of their results, the factors that can affect the sensed molecular concentration at the receiver nanomachine are concluded. These factors have facilitate the model assumptions in this paper. Including the assumption that the estimation of the interference from previous diffused information molecules on the current diffused information molecules, depends on distance, time and sensed molecular concentration. The experiment results of diffusion pattern recognition show that:

  • If the duration of symbol was relatively short, then the diffused symbols were not recognized correctly due to the short symbol duration. (Symbol duration is the time duration between two consequent transmissions). In this experiment the symbol duration was 0.02 ms.

  • The receiver nanomachine was able to recognize the symbols correctly after increasing the symbol duration. The symbol duration increased to 3.2 ms.

Experiments with two different transmitter nanomachines at different distances from the receiver nanomachines diffusing information molecules. The experiment results to check how the receiver nanomachine can recognize the pattern of diffusion show that:

  • The receiver nanomachine can recognize the diffused symbols correctly. However, it cannot distinguish whether symbol came from one nanomachine or another. Due to the overlapping in the values of the sensed molecular concentration which come from each nanomachine in this experiment. Where the reason of the overlapping in these values came from the close distance between the two transmitter nanomachines.

  • Changing the distance of the two transmitter nanomachines from the receiver nanomachine can make the overlapping quite low.

In the case when one transmitter nanomachine is in a close distance from the receiver nanomachine, and the other nanomachine is in a far distance from the receiver nanomachine, the results show:

  • The receiver nanomachine cannot recognize the diffused symbols correctly, because of the higher data rate of the sensed molecular concentration from the close transmitter nanomachine. Thus, the sensed molecular concentration from the far nanomachine is affected by the interference of molecules from the previous symbol duration,

  • By increasing the symbol duration, the receiver nanomachine can recognize the diffused symbols correctly, distinguishing also if the symbol came from one nanomachine or another.

There are research which study the effects of Inter Symbol Interference on the channel capacity and the channel performance. However, we study the effects of the interference on recognizing the pattern of diffusion. Besides that, through experiments we study the effects of symbol duration, data rate, and distance on reducing the effects of the interference.

The paper is organized as follows. In Sect. 2 the model is described. The proposed pattern of diffusion algorithm, the factors that affect distinguishing the sensed molecular concentration, and the experiment results are demonstrated in Sect. 3. Finally, Sect. 5 presents the conclusions from the experiments results.

2 Model

Network Environment:

A system of n nanomachines is considered, these nanomachines communicate according to diffusion based molecular communication. The environment of the communication might contain residual molecules from previous diffusion, and also contain molecules from other nanomachines (that are not among n) and these molecules can be considered as noise. In [32] the diffusion based communication system has showed that only the molecules last previous diffusion can affect the current diffusion. Nanomachines are assumed to have simple computational capability, and storage space for the needed computations.

Time Slots:

The model is time slotted with length t; thus, a transmitter nanomachine can keep diffusing molecules during this t time slot, where \( t = \frac{{d^{2} }}{D} \).

Information Molecule Encoding:

The concentration of received molecules is considered to be the information molecules (though it is also called the transmitted symbol). Thus, a receiver nanomachine decode the received symbol as ‘1’ in case the number of molecules sensde by the receiver nanomachine during t time slot is higher than a given threshold τ; if not, then the symbol is decoded as ‘0’. Generally nanomachines are assumed to sense at least (ε + µ) molecular concentration, where ε represents the residual molecular concentration and µ can represent the environmental noise. If the nanomachine sense molecular concentration that is higher than (ε + µ); i.e., τ, it can recognize that at least one nanomachine is diffusing molecules. In [33] modulation techniques have been explored, one of which is Concentration Shift Keying, where symbols (information molecules) can be represented as b bits through 2b different values, and the levels of threshold can be 2b − 1.

Network Communication:

The nanomachines n are assumed to communicate through diffusion; thus, once the molecules are released by a transmitter nanomachine into the propagation medium, these molecules shall be diffused freely according to the Brownian motion dynamics. The function of the molecular concentration at the receiver nodes in response to an impulse of information molecules (symbol) emission from the transmitter with Q molecules is of the form [34]:

$$ Qh(t)\, = \,\int_{0}^{T} {(Q\, \times \,\frac{1}{(4\pi Dt)}\, \times \,{ \exp }(\frac{{ - d^{2} }}{4Dt}))} $$
(1)

where, t represents time and it is the integral variable and range from 0 to T, d denotes the distance between the receiver and the source, D is the diffusion coefficient, Q number of diffused molecules. Thus, the receiver nanomachine sense the accumulated molecular concentration diffused through the time slot t.

Pattern of Diffusion:

Assuming that the information molecules (symbol) are represented as 2 bits with 22 different values and 22−1 thresholds, the aim is to check how a receiver nanomachine can distinguish a message transmitted from a transmitter nanomachine at distance d. A receiver nanomachine can sense the transmitted information molecules through the following expression, where [34] stated that the molecular concentration peak at a receiver nanomachine is obtained through:

$$ Q(p) = Q\left( {\frac{3}{2\pi }} \right)^{3/2} \frac{1}{{d^{3} }} $$
(2)

where Q(p) means the function to compute the peak of the diffused Q molecules.

From Eq. (2) the peak of molecular concentration at a receiver nanomachine is inversely proportional to the cube of the distance d, and it is not affected by the diffusion coefficient D of the medium. However, the time that it takes diffused molecules to reach their peak can be affected by D; thus a receiver nanomachine at distance d from a transmitter nanomachine can sense that then molecular concentration at d peaks at \( t^{{\prime }} = \frac{{d^{2} }}{6D} \), where t0 here is computed from the derivative of Eq. (1) [25].

3 Pattern of Diffusion Recognition in Molecular Communication Model

This section examines how a nanomachine at distance d can recognize the information molecule (symbol) that a transmitter nanomachine has been diffusing. Where the information molecules (symbol) is represented as 2 bits with 22 different values and 22 − 1 thresholds, i.e., the symbol is encoded according to the Quadruple Concentration Shift Keying(QCSK) [29, 33], as Fig. 1 shows.

Fig. 1.
figure 1

QCSK technique for 2 bits per symbol [33]

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Thus, the diffused symbol is represented as two bits in different forms: 00, 01, 10, and 11, and there are three different thresholds for the receiver nanomachine to distinguish the symbol. In order to give values to these thresholds, the transmitter nanomachine in this section is assumed to diffuse units of molecules throughout 8 time slots, each slot of length t; and the accumulated value of the diffused units throughout 8 t would encode a symbol either s0 = (00), s1 = (01), s2 = (10), and s3 = (11). To encode a symbol, a transmitter nanomachine diffuses either Q molecules to represent 1 or 0 molecules to represent 0, during each t of the 8 t. Here, it is assumed that symbols are represented as the following:

In Fig. 2 the second column represents the ‘Number of Molecules’, while the third column represents’ Molecules diffused through each t of 8 time slots’. Although s0 means that no information molecules would be diffused, but to distinguish between no diffusing and diffusing 0, it is assumed that n0 is diffused to represent s0. The number of molecules, for example n1 equals to (1 × Q) + (1 × Q) + (1 × Q) + (0 × Q) + (0 × Q) + (0 × Q) + (0 × Q) + (0 × Q). The expression in Eq. (2) represents the peak of diffused information molecules, the values of thresholds can be computed through [34]:

$$ \tau = X \times Q\left( {\frac{3}{2\pi }} \right)^{3/2} \frac{1}{{d^{3} }} + I $$
(3)

where X represents the diffused units in each time slot t0, i.e., 0 or 1. The Inter Symbol Interference (ISI), which means the residue molecules from the previous symbol that can affect the current symbol is denoted by I. The ISI in [34] is assumed to come from a sufficient number of interfering sources (nanomachines which are diffusing), in a way that I follows a normal distribution. However, the experiments which have been carried out in this paper, study the pattern of diffusion of one transmitter nanomachine and a receiver nanomachine, then assume that there are two transmitter nanomachines and one receiver. Thus, the diffused information molecules (symbol) will be affected by ISI from at least one nanomachine.

Fig. 2.
figure 2

Symbols representation through 8 time slot

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ISI can affect the successful detection of the (signal) diffused molecular concentration [25]. The effects of ISI can vary with the temporal spreading properties of molecules in the diffusion channel [26]. Thus, the increased distance between the transmitter and receiver nanomachines, and/or the higher data rate (sensed molecular concentration) can cause increased effects of ISI [25]. The authors in [27] proposed a scheme to reduce the effects of the ISI, through reducing pulse-width of the transmitted information molecules, as the molecules encoding technique discussed in the paper is on-off keying (OOK). The authors in [28] proposed an enzyme-based scheme to reduce the effects of ISI, through diffusing enzymes which chemically interact with the ISI information molecules (from previous symbol) and form intermediate products; thus, in this way the information molecules from the previous symbol would not cause ISI in the current symbol. Symbol duration can be defined as the time duration between two consequent transmissions. Symbol duration ts represents the required time to transmit a symbol (information molecule), can also affect ISI, and a longer symbol duration can help to reduce the ISI caused by the previous symbol [33].

In the experiments to check the pattern of diffusion in this section, the value of ISI was assumed to vary depending on the distance, data rate and symbol duration, in a way that ISI \( { \le }\,{ \log }(\frac{d}{{t_{s} }} \times {\text{date}}\,{\text{rate}}) \).

The rest of this section includes an algorithm to distinguish the pattern of diffusion, followed by the results of different experiments based on the algorithm. In order for a receiver nanomachine to distinguish the pattern of information molecules diffused by a transmitter at distance d, it should compute τ1, τ2, τ3 and τ4, through Eq. (3) and follow the steps in Algorithm 1 [24].

In Algorithm 1 [24], it is assumed that each bit of the transmitter message is diffused during t time slot. For simplicity, the time needed for molecular concentration to reach its peak near the receiver nanomachine t0 and symbol duration ts are assumed to equal (8 × t). Recall that X represents the diffused units in each time slot. As there are 8 time slots, X(i) represents the diffused unit at a specific t from the 8 time slots.

Different experiments have been carried out to distinguish the pattern diffusion at a receiver nanomachine following the steps in Algorithm 1 [24]:

figure a

4 Experiment Results

4.1 Recognizing the Diffusion Pattern of One Nanomachine

The first experiment is to check how a receiver nanomachine can recognize the received information molecules. Taking in consideration the effects the time slot length and the inter symbol interference on the recognizing the received molecules correctly.

Figure 3 represents an experiment where four symbols were diffused by a transmitter nanomachine at distance 5 from a receiver nanomachine. In the experiments, Q is assumed to equal 1000, and (ε + µ) is assumed to equal 0.25. The assumed time slot t to diffuse one bit of the 8bits symbol is 0.02 ms, and t0, ts = 0.16 ms. The ticked points at the y-axis in Fig. 3 represents the values of thresholds, where, τ0 = 1.525, τ1 = 4.575, τ2 = 7.625 and τ3 = 12.2. The x-axis represents the time required before the diffused symbol reaches to its peak at the receiver nanomachine. The y-axis represents the sensed molecular concentration (peak of the diffused symbol) at certain t0. Even though the receiver nanomachine can sense at least (ε + µ) in each time slot t, the figures which represent the experiments results have an initial value of the sensed molecules by the receiver that equals 0.

Fig. 3.
figure 3

Sensed symbols by receiver from transmitter at d = 5 and ts = 0.16

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The four diffused symbols in Fig. 3 are: 10, 01, 00, 11. The effects of ISI during the first symbol duration is assumed to be quite low. Thus, it can be seen that the first diffused symbol sensed and distinguished correctly. However, the effects of ISI start from the second symbol; therefore, in Fig. 3, the sensed molecular concentration during the second, third, and fourth ts is higher than the actual diffused molecular concentration, and there is a chance of error in the process of distinguishing a symbol. Thus, symbols 01, 00, 11 have not distinguished correctly in Fig. 3. This can be due to the short symbol duration ts.

In Fig. 4 the assumed time slot t to diffuse one bit of the 8bit symbol is 0.4 ms, and t0, ts= 3.2 ms. The diffused symbols in this figure are: 01, 11, 10, 00, 10, and the results show that the receiver nanomachine sensed and distinguished the correct symbols. Thus, Fig. 4 shows that the diffused symbols were not recognized correctly due to the short symbol duration.

Fig. 4.
figure 4

Sensed symbols by receiver from transmitter at d = 5 and ts= 3.2

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Thus, the diffused symbols were not recognized correctly due to the short symbol duration in Fig. 3. However, The receiver nanomachine was able to recognize the symbols correctly after increasing the symbol duration in Fig. 4.

4.2 Recognizing the Diffusion Pattern of Two Nanomachines

In case there are two different transmitter nanomachines at different d from a receiver nanomachine. These two transmitter nanomachines are assumed to be synchronized. In a way that, one nanomachine starts diffusing at a certain symbol duration, and the second one waits, then at the next symbol duration the second nanomachine diffuses and so on. The receiver nanomachine is assumed to have 8 thresholds to recognize the diffused symbol and to distinguish from which transmitter nanomachine it has come.

Figure 5 shows the results of the sensed molecular concentration by a receiver nanomachine that was diffused from two transmitter nanomachines in d = 3, and 5.

Fig. 5.
figure 5

Sensed symbols by receiver during ts = 3.2 from transmitters at d = 3 and d = 5

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The number of molecules diffused by each transmitter nanomachine is assumed to equal Q = 1000. The assumed time slot t to diffuse one bit of the 8bit symbol is 0.4 ms, and t0, ts = 3.2 ms. The threshold values of the transmitter nanomachine at d = 3 are: (τ0 = 7.060, τ1 = 21.180, τ2 = 35.301, τ3 = 56.481). The remaining thresholds are as described in Fig. 3. Thus, the ticks on the y-axis represent all 8 thresholds. The symbols diffused by the nanomachine at distance d = 3 are: 01, 00, 10, 11, and the symbols from the nanomachine at d = 5 are: 01, 11, 10, 00. The results in Fig. 5, seem to show that the receiver nanomachine can distinguish the symbols correctly, but, as some threshold values overlap, it is difficult for the receiver nanomachine to distinguish whether a symbol comes from one nanomachine or another. However, the figure differentiates between the symbols of each nanomachine (as the sensed molecular concentration of each nanomachine saved in a different array), but the receiver mainly just compares the sensed molecular concentration with the thresholds.

The experiment is repeated, but with different values of d, as Fig. 6 shows:

Fig. 6.
figure 6

Sensed symbols by receiver during ts = 3.2 from transmitters at d = 3 and d = 7

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The overlapping of the threshold values in Fig. 6 might look quite low, but most values of one nanomachine are quite close to one threshold of the other nanomachine. The two transmitters are assumed to be at distances d = 3, and d = 7. The threshold values of d = 7 are: (τ0 = 0.962, τ1 = 2.887, τ2 = 4.813, τ3 = 7.700). The diffused symbols of the transmitter nanomachine at d = 3 are: 01, 00, 10, 11, and symbols of transmitter nanomachine at d = 7 are: 00, 10, 11, 01.

Thus, in Fig. 5 the receiver nanomachine can recognize the diffused symbols correctly. However, it cannot distinguish whether symbol came from one nanomachine or another. It is due to the overlapping in the values of the sensed molecular concentration which come from each nanomachine in this experiment. Where the reason of the overlapping in these values came from the close distance between the two transmitter nanomachines. However, in Fig. 6 changing the distance of the two transmitter nanomachines from the receiver nanomachine can make the overlapping quite low.

In case one transmitter nanomachine is in a close distance from the receiver nanomachine. And the other nanomachine is in a far distance from the receiver nanomachine. This case is presented to avoid overlapping between thresholds values. Thus, different distances were selected in the next experiment.

Figure 7 shows the sensed molecular concentration by a receiver nanomachine, when two transmitter nanomachines at d = 0.3 and d = 3 diffuse information molecules.

Fig. 7.
figure 7

Sensed symbols by receiver during ts = 3.2 from transmitters at d = 0.3 and d = 3

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In Fig. 7, the thresholds values of the sensed molecular concentration which come from the transmitter nanomachine at d = 0.3 are: (τ0 = 12228, τ1 = 36685, τ2 = 61143, τ3 = 97829), which shows a higher data rate at this distance. The symbols diffused by the nanomachine at d = 0.3 are: 00, 01, 10, 11. The symbols diffused by the nanomachine at d = 3 are: 01, 10, 00, 11. The diffused symbols from the nanomachine at d = 3 are affected by ISI. However, the diffused symbols from the nanomachine at d = 0.3 are not affected that much by ISI, as the range between its thresholds is high (for example, the difference between τ0 and τ1 is almost 24457). Besides this, the data rate of the diffused symbols from the nanomachine at d = 3 is quite low.

In Fig. 8, the assumed time slot t to diffuse one bit of the 8bit symbol is 1 ms, and t0, ts= 8 ms. The diffused symbols from both nanomachine at d = 0.3 and 3, are the same diffused symbols in Fig. 7, as symbol duration increased the effects of ISI on the received symbols from the nanomachine at d = 3 is decreased in Fig. 8.

Fig. 8.
figure 8

Sensed symbols by receiver during ts = 8 from transmitters at d = 0.3 and d = 3

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Thus, in Fig. 7 the receiver nanomachine cannot recognize the diffused symbols correctly. Due to the higher data rate of the sensed molecular concentration from the close transmitter nanomachine. Thus, the sensed molecular concentration from the far nanomachine is affected by the interference of molecules from the previous symbol duration. However, in Fig. 8 by increasing the symbol duration, the receiver nanomachine can recognize the diffused symbols correctly. Beside distinguishing if the symbol came from one nanomachine or another.

5 Conclusions

The propagation of molecules in the communication medium is a significant topic to be explored. In order to study the effects of noise, the residual molecules from previous communications and properties of the medium itself, on the sensed molecules by the receiver nanomachine(s). The pattern of diffusion has been explored, by defining the factors which can affect the sensed molecular concentration by the receiver, such as distance, interference, symbol duration and data rate. Experiment results showed the effects of increasing the symbol duration of diffused symbols on the sensed molecular concentration at the receiver nanomachine.

Two cases were considered in the experiment. The first case is how a receiver nanomachine can distinguish the pattern of diffusion of one transmitter nanomachine at distance d. The experiment results of diffusion pattern recognition show that the diffused symbols were not recognized correctly due to the short symbol duration. In this experiment the symbol duration was 0.02 ms. Then, the receiver nanomachine was able to recognize the symbols correctly after increasing the symbol duration. The symbol duration increased to be 3.2 ms.

The second case is related to recognizing the pattern of diffusion by the receiver nanomachine, when there are two transmitter nanomachines at different distances, diffuse molecular concentration. The experiment results to check how the receiver nanomachine can recognize the pattern of diffusion show that, the receiver nanomachine can recognize the diffused symbols correctly. However, it cannot distinguish whether symbol came from one nanomachine or another. It is due to the overlapping in the values of the sensed molecular concentration which come from each nanomachine in this experiment. Where the reason of the overlapping in these values came from the close distance between the two transmitter nanomachines. However, changing the distance of the two transmitter nanomachines from the receiver nanomachine can make the overlapping quite low. In case one transmitter nanomachine is in a close distance from the receiver nanomachine. And the other nanomachine is in a far distance from the receiver nanomachine. The results show that the receiver nanomachine cannot recognize the diffused symbols correctly. Due to the higher data rate of the sensed molecular concentration from the close transmitter nanomachine. Thus, the sensed molecular concentration from the far nanomachine is affected by the interference of molecules from the previous symbol duration. By increasing the symbol duration, the receiver nanomachine can recognize the diffused symbols correctly, beside distinguishing if the symbol came from one nanomachine or another.

As a future work, it is possible to think about exploring models of nanonetworks taking into consideration the drift of molecules in the medium. Research studies that address flow-based molecular communication are currently quite limited. The propagation medium in some of the nanonetworks applications can be in motion, as an example, nanomachines in bio-medical applications are placed in human blood. Thus, it is not feasible to assume that the propagation medium is always stable. In some cases, a drift velocity may be applied on purpose to increase a molecular communication systems low throughput. Thus, it would be good to study model of molecular communication with drift as well as channel capacity, noise effects and other related issues.