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Interference Mitigation in Large-Scale Multiuser

Molecular Communication

Abstract

In recent years, communicating information using molecules via diffusion has attracted signiﬁcant interest in bio-

medical applications. To date, most of research have concentrated on point-to-point molecular communication (MC),

whereas in a realistic environment, multiple MC transmitters are likely to transmit molecular messages simultaneously

sharing the same propagation medium, resulting in signiﬁcant performance variation of the MC system. In this type

of large-scale MC system, the collective signal strength at a desired receiver can be impaired by the interference

caused by other MC transmitters, which may degrade the system reliability and efﬁciency. This paper presents the

ﬁrst tractable analytical framework for the collective signal strength at a partially absorbing receiver due to a desired

transmitter under the impact of a swarm of interfering transmitters in a three-dimensional (3D) large-scale MC

system using stochastic geometry. To combat the multi-user interference (MUI) and the intersymbol interference

(ISI) in the multi-user environment, we propose Reed Solomon error correction coding, due to its high effectiveness

in combating burst and random errors, as well as the two types of information molecule modulating scheme, where

the transmitted bits are encoded using two types of information molecules at consecutive bit intervals. We derive

analytical expressions for the bit error probability (BEP) of the large-scale MC system with the proposed two schemes

to show their effectiveness. The results obtained using Monte Carlo simulations, matched exactly with the analytical

results, justifying the accuracy of the derivations. Results reveal that both schemes improve the BEP by 3 to 4 times

compared to that of a conventional MC system without using any ISI mitigation techniques. Due to the implementation

simplicity, the two-types molecule encoding scheme is better than the RS error correction coding scheme, as the RS

error correction coding scheme involves additional encoding and decoding process at both transmitter and receiver

nodes. Furthermore, the proposed analytical framework can be generalized to the analysis of other types of receiver

designs and performance characterization in multi-user large-scale MC systems. Also, the two types of information

molecule modulating scheme, can be extend to M-type of information molecule modulating scheme without loss of

generality.

Index Terms

Large-scale molecular communication system, partially absorbing receiver, intersymbol interference, multi-user

interference, 3D stochastic geometry, Reed Solomon Codes.

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I. INTRODUCTION

Conveying information over a distance in an environment where set of interconnected micro and nanoscale

devices are communicating, has been a challenge for decades. It requires the classical communication

methods to be revised to meet the requirements of the very small dimensions and speciﬁc transmission

mediums. Molecular communication (MC) is an emerging communication paradigm which has gained

signiﬁcant research attentions in recent years as a solution for the aforementioned research question. It is

inspired by the nature and adopts the prevalent communication mechanism of living cells and organisms to

achieve effective communication [1]. In MC, the information is carried by chemical signals and based on the

propagation channel characteristics, it is generally classiﬁed into walkway-based, ﬂow-based, and diffusion-

based MC types [2]. Recent work in MC research span from theoretical analysis of the performance of

the communication channels, improved transmitter and the receivers designed to more practical design of

suitable modulation and coding techniques [3–8]. The potential applications of MC in micro and nanoscale

level systems, such as in bio-medicine systems for detection, control and treatment of diseases, require

robust and reliable communication capabilities.

In this paper, we limit ourselves to MC via diffusion (MCvD), as it is the simplest, general, and energy

efﬁcient MC paradigm. In a typical point-to-point MCvD system, the information is carried via so-called

messenger molecules. The input information is modulated onto the physical properties of the messenger

molecules, such as their types, quantity, or their release time, using different modulation techniques [3].

Then, the transmitter emits a number of messenger molecules in a time slotted fashion to convey the input bit

sequence. Once emitted by the transmitter, these molecules propagate through the ﬂuid medium via diffusion,

where each molecule undergoes random walk following Brownian motion and the propagation follows a

Wiener process [9]. Finally, at the receiver, it demodulates the received signal based on the properties of the

received molecules, such as the type, number or the received time of the absorbed molecules during each

bit interval. Here, we focus on the quantity of molecules for information modulation and demodulation.

Due to the randomness in the arriving time of molecules, crossover between molecules from different bit

intervals can occur. Thus, delayed molecules from previous symbols arriving at the current bit interval,

cause intersymbol interference (ISI) at the receiver. As shown in [10], ISI has been regarded as one of the

main bottlenecks exist in MC, which degrades the overall system performance and its reliability.

In a realistic 3D large-scale MC environment, there exists a swarm of MC transmitters which shares

the transmission medium and transmits molecular messages simultaneously. This type of large-scale MC

system suffers not only from ISI, but also from multi-user interference (MUI) due to the interfering signals

from the swarm of interfering transmitters on the desired signal. In [11], an analytical framework based on

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Poisson point process was presented to model the collective signal strength due to the joint transmission

of all transmitters in a 3D large-scale MC system. In [12], the exact expressions of ISI and MUI of MC

system with the fully absorbing receiver were presented to reveal the theoretical limits on multi-user MC.

While [11, 12] only modeled the 3D large-scale MC system, active error mitigation in the presence of a

swarm of interfering transmitters is not considered.

Existing works on interference mitigation in MC mostly focused on modeling the ISI in point-to-point

MC system [10, 13–15], and inspired from conventional data communication networks, error correction

codes (ECCs) were employed to overcome this bottleneck. In [13], the bit error performance of Hamming

codes (HC), Euclidean Geometry Low Density Parity Check (EG-LDPC) codes, and Cyclic Reed-Muller

(C-RM) codes were studied in diffusion-based MC systems. In [14], the self-orthogonal convolutional

codes (SOCCs) with majority-logic decoding scheme was proposed as a low energy decoding technique

for MC. Apart from the study of existing channel coding techniques mentioned above, the authors in

[15] designed a low complexity ISI-free code based on inherent characteristics of MC system, speciﬁcally

for diffusion-based MC system. Ref. [16] presents a simple pre-coding technique to reduce the ISI in a

MC system, with an intelligent transmitter consisting of Mbits of Memory. In this proposed technique,

molecular diffusion rate is adjusted for each bit interval by taking advantage of the memory. Another ISI-

free modulation scheme for a given maximum transmission delay is proposed in [17], where symbols of

the same type are released sufﬁciently far apart to reduce the ISI in the system. Inspired from modern

advancements in digital communication applications, the authors in [10] proposed Reed Solomon (RS)

codes as an error correction technique for a diffusion-based MC system, with a point transmitter and a full

absorbing receiver, as they are highly effective against burst and random errors. Their results also shown

that the bit error rate performance improves with the help of RS codes compared to both no-coding scenario

and HC. Moreover, in [3], following the biological behavior of MC system, as an alternative solution for

interference mitigation new type of molecules with kill effect on information carrying molecules, such as

molecules with antibacterial behaviors, have been introduced to the system at regular intervals to destroy

the accumulated information molecules.

From the analysis point of view, most prior works in the literature only consider ISI mitigation in a point-

to-point single transmitter receiver pair, and they fail to address the inherent characteristics and requirements

of large-scale multi-user environments in MC. Therefore, a proper analytical and simulation framework is

required to precisely analyze the effect of ISI and MUI, and quantify the performance improvement of

interference mitigation schemes in a 3D large-scale system with a partially absorbing receiver. In this work,

we study the interfering effect of a swarm of point transmitters on the desired signal of a partially absorbing

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receiver in a large-scale MC system using stochastic geometry. We propose two schemes to mitigate the

adverse effects of both ISI and MUI, which are RS coding scheme and two-types molecule encoding

scheme. To the best of our knowledge, this is the ﬁrst analytical consideration of MUI and ISI with a

partially absorbing receiver in a large-scale multi-user MC system. Moreover, this is the ﬁrst time that

interference mitigation techniques are studied in a large scale MC system under a swarm of transmitters.

Our major contributions can be summarized as follows:

1) We present an analytical framework for a large-scale MC system with a MC point transmitter

delivering information to its associated partially absorbing receiver under the interference from other

MC transmitters. We model the MC interfering transmitters as a homogeneous Poisson point process

(HPPP) and the stochastic geometry is applied to obtain tractable analytical expressions for the average

behavior of the large-scale MC system. To the best of our knowledge, this is the ﬁrst attempt in the

literature that analyzes the behavior of a partially absorbing receiver on large-scale MC systems with

multi-users, and addresses the limitations exist in the previous work [11], [12].

2) We derive a closed-form expression for the collective signal strength at the surface of the partially

absorbing receiver under the adverse effects of swarm of interfering point transmitters in 3D space.

Based on these, we derive a tractable expression for the bit error probability (BEP) of a large-scale

MC system with a partial absorbing receiver.

3) We analyze the impact of both ISI and MUI on the desired signal, and propose two novel schemes to

mitigate the adverse effects of these interference, which are RS coding and two types of information

molecular modulating schemes. In the ﬁrst scheme, we apply RS codes which is a popular ECC

in data communication to the MC system to improve the error recovery capacity of the system

and analyze the performance improvement achieved in a 3D large-scale multiuser environment. The

second scheme is to prevent the ISI and we propose the introduction of two types of information

molecules to the system with the expectation of increasing the time interval between two same type

of molecular pulses. With this presumption, we analyze the feasibility of employing two types of

information molecules to modulate the information, in alternative bit intervals to improve the system

performance in error prone situations.

4) To evaluate the performance improvement, we derive the closed-form expressions for the strength of

the collective signal at the receiver, tractable analytical expressions for the signal-to-interference ratio

(SIR) at the receiver, and the BEP of the MC systems with the two proposed schemes. Our derived

analytical results are veriﬁed via the Monte Carlo simulation.

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II. SY ST EM MO DE L

Fig. 1. Illustration of Our System Model, where the dotted lined boxes are only applicable for the system with RS coding and MC

modulator/demodulator is differently conﬁgured for two types of molecular system

In this paper, we focus on a 3D large-scale diffusion-based MC system, where a typical point transmitter

communicates with a spherical partially absorbing receiver located distance deaway under the interference

from a swarm of active point transmitters with density λaas illustrated in Fig. 1. The interfering point

transmitters are spatially distributed outside the receiver in R3

VΩrr

, following an independent and HPPP, Φa,

with density of λa, and the partially absorbing receiver with a radius rrand volume VΩrrhas a ﬁnite

absorption rate of k1[11]. In the baseline MC system, each transmitter emits molecular signal pulse at the

start of each bit interval, Tband the ﬁrst bit is emitted at t= 0. The emitted molecules diffuse randomly

in the propagation medium until they hit the surface of the receiver. For simpliﬁcation, we assume that

the ﬂow current is absent in the propagation medium and the extension for ﬂow currents can be treated in

future work. The absorbing receiver counts the number of molecules absorbed by its surface during each

Tbinterval. We have considered a partially absorbing receiver, where a fraction of the hitting molecules is

absorbed by the receptors of the receiver and counted for the signal demodulation. Due to the fact that the

receiver has no knowledge whether the received molecules are emitted from the desired transmitter or from

any other interfering transmitters, the molecules from the interfering transmitters cause MUI at the receiver.

It is shown in [11] that the negative effects brought by the joint transmission on collective signal strength

increases with time and is heavily dependent on the density of the swarm of active point transmitters and

the amount of molecules from the past bits arriving at the current bit interval at the receiver. Further, due to

the random nature of molecular propagation, the residual molecules from previous transmissions hitting the

receiver surface during the current bit interval cause ISI in MC. Hence, in large-scale MC systems, heavy

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signal distortion could occur at the receiver due to MUI and ISI, causing erroneous signal demodulation

and it is essential to incorporate interference mitigation techniques in these systems to facilitate error free

transmission.

To address the aforementioned problems, this paper presents two different interference mitigation schemes

for 3D large-scale MC system. Borrowing from the data communication, as our ﬁrst method to overcome

adverse effects of ISI and MUI, we propose channel coding techniques. Speciﬁcally we focus on the RS

code as our previous results [10] shown that RS codes provide a considerable improvement in BER in a

diffusion-based MC system with a point transmitter and a full absorbing receiver compared to HC [13, 14].

This is because the RS codes have the capacity to correct an entire symbol error compared to the one bit

error correction achieved by HC. Although, from the computational complexity point of view, RS scheme

costs more than simple channel coding scheme, it provides ﬂexibility in error correction capacity, and it is

possible to tune nand kof the RS(n, k)coder to match the system requirement.

Second method for interference mitigation is to design two-types of information molecule modulation

scheme for the ISI and MUI reduction inspired from [18]. Note that these two types of molecules can

be recognized separately at the receiver in practical scenario, for instance the GABA-A receptors can

distinguish different types of ligands [19]. In our proposed scheme, type A and B molecules are alternated

as the messenger molecules between each consecutive bit interval, so that effectively there exist 2Tbgap

between the same type of molecules. In return, this technique will reduce the ISI and MUI in the system and

will act as a bit error mitigation technique. As shown in Table I, we ﬁrst divide the input sequence into odd

and even numbered sequences by considering the original position of the bit, then the entire even numbered

sequence is modulated using type B molecules, whereas the odd numbered sequence is modulated using

type A molecules. Since this modulation scheme introduces minimum 2Tbtime gap between emission of

each type of molecules to the medium, the ISI from the the nearest previous bit, which contributes largely

to the overall ISI, is not present in the system inherently. Additionally, for a odd sequence the ISI from the

bits transmitted by the even sequence, and the vice versa is also eliminated. Hence, the proposed system

modulated with two types of information molecules, inherently exhibits low level of ISI and MUI. The

other advantage of the proposed scheme is the low complexity associated with the implementation at both

transmitter and the receiver, as the proposed technique doesn’t require prior knowledge of the molecular

type assignment pattern to modulate and demodulate the current bit.

Even though this work considers only two types of information molecules, it provides the fundamental

insights to be extended to a M-type molecular modulation scheme. Yet, having a larger value for Mwill

complicate the design of the receiver in practice. This is because one of the key requirement of the propose

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TABLE I

PROP OSE D EN COD ING T EC HNI QUE ,ILLUSTRATING THE ASSIGNMENT OF TY PE AMOLECULES,NA

tx,AND TY PE BMOLECULES,NB

tx ,WH EN

jth BIT IS ODD AND bi∈1,0.

Bit Sequence b1b2b3b4... bj−1bj

single type of information molecules Ntxb1Ntxb2Ntxb3Ntxb4... Ntxbj−1Ntxbj

Two types of information Even NB

txb2NB

txb4... N B

txbj−1

molecules Odd NA

txb1NA

txb3... N A

txbj

system is the ability of the receiver to differentiate between the M-types of molecules. In such scenario,

creating a receptor with different types of sensitivities to many types of molecules is not practically possible,

compared to designing a receptor with two or three different types of sensitivities. Hence, there exist an

optimum value for Mdepending on the application scenario.

Furthermore, we follow the global synchronization assumption as [11, 12, 20], where all transmitters are

assumed with synchronous transmission. This facilitates simple analysis and leads to tractable results. Yet,

our system can be extended to asynchronous transmission by following [21]. In a asynchronous system,

molecules will be emitted randomly within the bit duration, Tb. ISI can be signiﬁcant in asynchronous

transmission, due to the spreading of signiﬁcant number of molecules from molecular pulses emitted nearer

to the end of the bit interval to the next bit interval at the receiver. Moreover, the ISI due to asynchronous

transmission is higher in the system with no coding and with one type of molecules, compared to the system

with two types of molecules. Yet, in the system with RS(n, k)codes with one type of molecules, the nand

kcan be ﬁne tuned to recover from the bit errors caused by ISI. Hence, the effect of the synchronization

errors on RS coded system can be minimized. In the system with two types of molecules, as we alternate

the type of molecules used in every other bit interval, the spreading of the molecules emitted towards the

end of the bit interval may not affect the next bit when the synchronization error is less than Tb, as the

carrier molecular type for the next bit will be different from the previous bit. Yet, if the synchronization

error is more than Tb, the ISI becomes signiﬁcant and the system would not have the capacity to correct

these errors.

Our proposed MC system consists of three key modules as illustrated in Fig. 1; the transmitter with a

modulator and RS encoder, the propagation channel, and the receiver with a demodulator and RS decoder,

which are presented in details in the following subsections.

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A. Transmitters

1) Single Type of Information Molecule: In terms of the modulation scheme, we adopt the Binary

Concentration Shift Keying (BCSK) as in [2], where information is conveyed through the number of type

S molecules transmitted at the start of each bit interval, Tb. In order to transmit bit-1signal, NS

tx number

of type S molecules are transmitted, and bit-0is delivered with the absence of type S molecules.

2) Two Types of Information Molecules: In this scheme, we introduce two distinguishable types of

molecules as information carriers, namely, type A and type B molecules. As illustrated in Table. I, the

input bit sequence is divided into two sequences, where the bits in the odd numbered positions constitute

the ﬁrst sequence (odd sequence), and the bits in the even numbered positions constitute the second sequence

(even sequence). Both sequences are modulated using the BCSK as in [2]. In more detail, NB

tx number

of type Bmolecules are emitted at t= (2m−2)Tbtime instance to represent the bit-1 in the even bit

sequence, whereas NA

tx number of type Amolecules are emitted at t= (2m−1)Tbto represent the bit-1

in the odd bit sequence, here mindicates the position of the bit in the even or odd sequence. Similar to

the system with single type of information molecule, bit-0 in either sequences are signaled using absence

of molecules.

B. Propagation

The Brownian motion governs the movement of information molecules in ﬂuid environment. In the

system modeled, the collision between molecules are ignored, which is reasonable assumption when the

ﬂuid medium is signiﬁcantly bigger compared to the size of the molecules, i.e when the molecular density

in the transmission medium is low [10]. The molecular propagation in these type of environment follows

Fick’s Second law [22], which is described using

∂(rC(r, t|r0))

∂t =D∂2(rC(r, t|r0))

∂r2,(1)

where C(r, t|r0)is the molecule concentration at time tat distance r,r0is the distance between transmitter

and the center of the receiver, and Dis the diffusion coefﬁcient. The value of D depends on the temperature,

viscosity of the ﬂuid, and the Stokes’ radius of the molecule [3].

C. Receiver

From the perspective of receiver design in MC, majority have considered two types of receivers, namely

passive and active receivers. Passive receivers only observe and count the number of molecules inside

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the receiver without interfering with the molecule propagation, while active receivers absorb fraction of

molecules that hit the receiver surface. In nature, most receivers commonly remove information molecules

from the propagation medium once they bind to a receptor. Further, some molecules which hit the receiver

surface can bounce back without binding to the receptors [23]. Also, some molecules may get bound to

inactive receptors. Hence, we consider a molecule as received by a receiver for the demodulation process,

only if it binds to one of the active receptors on the surface of the receiver [23, 24]. This motivates us to

employ partial absorbing receiver, which absorbs only a fraction of the bound molecules in our system.

The reception process of an absorbing receiver can be described as [9]

D∂C (r, t|r0))

∂r |r=r+

r=k1C(rr, t|r0),(2)

where k1is the absorption rate. Eq. (2) describes the boundary condition for the partially absorbing receiver

when k1is a non zero ﬁnite constant. Further, when k1→ ∞, it deﬁnes the boundary condition for the

fully absorbing receiver. Moreover, in the case of multiple absorbing receivers the number of molecules

absorbed by the receiver depends on the total number of absorbing receivers present. In the work presented,

we consider the single partially absorbing receiver, which can reduce to fully absorbing receiver when the

absorption rate, k1→ ∞.

1) Single Type of Information Molecules: The demodulation of the BCSK signal is performed at the

receiver to identify the bit pattern being transmitted. The total number of molecules intersecting the surface

of the absorbing receiver during each [(j−1)Tb, jTb]interval is counted at the demodulator for information

demodulation. For instance, to demodulate the jth bit, the number of molecules received during [(j−

1)Tb, jTb]interval NS

net is compared against a predeﬁned demodulation threshold value Nth. If Nth < N S

net

the received bit is interpreted as bit-1, otherwise as bit-0.

2) Two Type of Information Molecules: The receiver is assumed to have the capability to differentiate

between the type A molecule and the type B molecule. The partially absorbing receiver counts the net

number of molecules absorbed at every Tbinterval and compares it against a pre-set threshold level for

each molecular type for information demodulation. In detail, the number of type B molecules absorbed

during the time interval [(2m−2)Tb,(2m−1)Tb], (NB

net), is compared against the threshold level NB

th to

demodulate the bits in the even bit sequence, whereas the number of type A molecules absorbed during

[(2m−1)Tb,(2m)Tb], (NA

net), is compared against the threshold level NA

th to demodulate the bits in the odd

bit sequence. If NA

th ≤NA

net or NB

th ≤NB

net, the received bit is demodulated as bit-1, otherwise as bit-0.

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D. Reed Solomon Coder

According to [11], it is evident that 3D large-scale MC system experiences a signiﬁcant amount of bit

errors and it requires a powerful error correcting mechanism, which has the capacity to correct more than

one bit in a single codeword. As a conventional practice in data communication, ECCs are used to achieve

an acceptable level of reliability for the received message, when information is transmitted through error

prone transmission channels. Hence, to combat the negative impact brought by the high MUI and ISI,

following the observations of [10], we propose RS codes [25], which is a non-binary Bose, Chaudhuri, and

Hocquenghem (BCH) code, as an ECC for 3D MC systems with a swarm of interfering transmitters.

In RS codes, the redundant data is embedded with the input message to facilitate the error recovery at

the receiver. RS(n, k)is deﬁned over a Galois Field (GF (pm)), where pis a prime, and mis a positive

integer. The RS(n, k)encoder, expands the input message (M) of klength into codeword (C) of nlength

by adding redundant parity bits (P) of n−klength. These parity bits are generated, so that every codeword,

C, is a multiple of the generator polynomial of the RS code, G. In special circumstances, the parity bits

generated to satisfy the above condition, could be a null sequence.

The RS decoder, ﬁrst computes the syndrome (S) by multiplying the received codeword, C0by a

predeﬁned check polynomial [26]. If the syndrome computation results in a non zero value, it treats the

received codeword as in error and attempts to correct the identiﬁed erroneous codeword, by ﬁrst determining

the symbol error locator polynomial using Berlekamp-Massey algorithm, then ﬁnding the roots of the error

locator polynomial using the Chien search algorithm, and then estimating the error magnitudes using Forney

algorithm. Once the error location and the error values are estimated, a correction is applied to the received

codeword to recover from the error. An extensive review of RS codes is given in [25, 26].

In general, a RS code with n−klength of parity bits has the capacity to correct up to n−k

2number

of errors per codeword [26]. Hence, the higher the redundancy is, the more errors can be corrected. Due

to this effectiveness of RS codes in combating both random multiple bit errors and burst errors, they

are more popular in data communication applications compared to HC, which only provides single bit

error correction. On the other hand, as the size of the codeword nincreases, the implementation grows in

complexity. Hence, the code length, nand the message length, kof the RS(n, k)code should be carefully

selected according to the channel error rate and the required level of BEP improvement.

III. CHA NN EL IM PU LS E RES PONSE

In this section, we derive the analytical expression for the channel impulse response at the partially

absorbing receiver due to a jlength bit stream emission at each point transmitter in a large scale MC

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system, where information is modulated using single type of information molecules.

A. Reaction Rate of the Receiver

As the stationary spherical receiver with radius rris located at the origin of an unbounded 3D space,

the reaction can occur at anywhere on its surface. If the transmitter is located at r0and emits an impulse

signal at t=t0, the reaction rate at distance r=rrin general is given as

K(t|kr0k, t0)=4πr2

rD∂C (r, t|kr0k, t0)

∂r |r=rr,(3)

where kr0kdeﬁnes the norm of the vector r0, and is a function that assigns a strictly positive length or

size to the vector.

According to [9, Eq. (3.99)], the concentration of molecules at distance rand time tin a MC system

with a partially absorbing receiver is derived as

C(r, t|kr0k, t0) = 1

4πrkr0k

1

p4πD(t−t0)

exp n−(r− kr0k)2

4D(t−t0)o−exp n−(r+kr0k − 2rr)2

4D(t−t0)o

−1

4πrkr0kα

exp nα2D(t−t0) + α(r+r0−2rr)oerfc nαpD(t−t0) + r+r0−2rr

p4D(t−t0)o

,

(4)

where α=k1rr+D

Drr

. Substituting (4) into (3), we can derive the reaction rate with impulse signal emitted

at t0as

KPA(t|kr0k, t0) = k1rr

r0

1 + erf n(rr− kr0k)

4D(t−t0)o−exp nα2D(t−t0) + α(r0−rr)o

×erfc nαpD(t−t0) + r0−rr

p4D(t−t0)o

,

(5)

where k1=D(rrα−1)

rr

.

In a fully absorbing receiver, a fully absorption of molecule that collide with the surface of the receiver

occurs. This is a special case of partial absorption receiver, and the boundary condition for the full absorption

receiver is obtained from Eq. (2), when k1→ ∞. Hence, its rate of absorption can be obtained from (5)

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by employing the limit k1, α → ∞ as

KFA (t|kr0k,0) = rr

kr0kkr0k − rr

√4πDt3exp(−(kr0k − rr)2

4Dt ),(6)

when t0= 0.

B. Channel Impulse Response

The fraction of absorbed molecules at the receiver during [(j−1)Tb, jTb]sampling interval due to the

desired transmitter located kdekdistance from the origin with a single pulse emitted at every Tbinterval is

given as

FD(Ωrr,(j−1)Tb, jTb| kdek) = FD(Ωrr,0, j Tb|kdek)−FD(Ωrr,0,(j−1)Tb|kdek),(7)

where FD(Ωrr,0, jTb| kdek)is the total fraction of molecules contributed by all the bits emitted by the

transmitter till t=jTb. In absence of molecular degradation, the FD(Ωrr,0, jTb|kdek)can be expressed as

FD(Ωrr,0, jTb| kdek) =

j

X

i=1

biZjTb

0

K((t−(i−1)Tb)|kdek) dt, (8)

where biis the ith bit of the transmitted sequence and K((t−(i−1)Tb)|kdek)is the reaction rate due to

a transmitter located at de, and it can be evaluated using (5) for partially absorbing receiver, and (6) for

fully absorbing receiver, respectively.

Note that the propagation of each molecules is independent, thus the fraction of molecules absorbed at

the receiver due to all the interfering transmitters located at kxkdistance from the origin in the 3D space

R3during [(j−1)Tb, jTb]interval can be written using Slivnyak Meekes’ Theorem [27] as

Fall

I(Ωrr,(j−1)Tb, jTb| kxk) = X

x∈Φa

FI(Ωrr,(j−1)Tb, jTb| kxk),(9)

where Fall

I(Ωrr,(j−1)Tb, jTb| kxk)is the total fraction of molecules absorbed at the receiver due to a

swarm of interfering transmitters, during the time interval [(j−1)Tb, jTb], and FI(Ωrr,(j−1)Tb, jTb|kxk)

is the fraction of molecules absorbed at the receiver due to single interfering transmitter at location x,

emitting jsequence of bits. FI(Ωrr,(j−1)Tb, jTb| kxk)can be formulated similar to (8) with bibeing the

value of the ith bit of the interfering transmitter at xposition as

FI(Ωrr,(j−1)Tb, jTb| kxk) =

j

X

i=1

biZjTb

(j−1)Tb

K((t−(i−1)Tb)|kxk) dt. (10)

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Depending on the type of receiver used, we can use either (5) or (6) to evaluate for K((t−(i−1)Tb)|kxk).

Furthermore, the total fraction of molecules absorbed at the receiver due to both the desired and the

interfering transmitters can be written as

Fall(Ωrr,(j−1)Tb, jTb) = FD(Ωrr,(j−1)Tb, jTb| kdek) + X

x∈Φa

FI(Ωrr,(j−1)Tb, jTb| kxk),(11)

where, FD(Ωrr,(j−1)Tb, jTb| kdek)can be evaluated using (7) and FI(Ωrr,(j−1)Tb, jTb| kxk)can be

evaluated using (9).

IV. RECEIVER OBSERVATION

In this section, we focus on the receiver observation due to jlength bit stream emitted by each active

transmitter at Tbbit intervals, and derive the exact expression for expected number of molecules at the

absorbing receiver in large-scale MC system.

A. Single Type of Information Molecule

When modulated using the single type of information molecule, due to independent propagation of each

molecule in the medium, the expected net number of molecules absorbed during [(j−1)Tb, jTb]interval

due to desired transmitter can be formulated as

ES

DNS

D(Ωrr,(j−1)Tb, jTb)=NS

txEFS

D(Ωrr,(j−1)Tb, jTb| kdek)

=NS

tx

j

X

i=1

biZjTb

(j−1)Tb

K((t−(i−1)Tb)|kdek) dt,

(12)

where K((t−(i−1)Tb)|kdek)for partially absorbing receiver is given in (5) and that of full absorbing

receiver is given in (6).

In MC system, all the transmitters are located outside the surface of the spherical receiver, which is

unlike in the wireless communication system where transmitter can be located randomly. Hence, in MC

system there exists a minimum distance of rrbetween a point transmitter and the receiver center. Therefore,

we express the expected number of molecules observed at the receiver due to interfering transmitters for

the time interval [(j−1)Tb, jTb]as

ES

INS

I(Ωrr,(j−1)Tb, jTb)=NS

txE(X

x∈Φa

FS

I(Ωrr,(j−1)Tb, jTb| kxk))

=λaNS

tx ZR

FS

I(Ωrr,(j−1)Tb, jTb| kxk) dx,

(13)

14

which can be further simpliﬁed applying Campbell’s theorem in 3D space as [27, Eq.(1.18)]

ES

INS

I(Ωrr,(j−1)Tb, jTb)= 4πλaNS

tx Z∞

rr

j

X

i=1

biZjTb

(j−1)Tb

K((t−(i−1)Tb)|kxk) dt r2dr. (14)

B. Two Types of Information Molecules

In this section, we analyze the receiver observation of the MC system modulating using two types of

information molecules.

1) For odd numbered bits: The bits emitted at odd Tbbit intervals are transmitted using type A molecules,

with NA

tx number of molecules to signal bit-1 and absence of molecules to signal bit-0. Taking this into

account, the expression for the expected number of molecules at the partially absorbing receiver derived in

(12) and (14) can be modiﬁed as

EA

DNA

D(Ωrr,(j−1)Tb, jTb)=NA

tx

dj/2e

X

i=1

b(2i−1) ZjTb

(j−1)Tb

K((t−(i−1)2Tb)|kdek) dt, (15)

and

EA

INA

I(Ωrr,(j−1)Tb, jTb)= 4πλaNA

tx Z∞

rr

dj/2e

X

i=1

b(2i−1) ZjTb

(j−1)Tb

K((t−(i−1)2Tb)|kxk) dt r2dr, (16)

where K((t−(i−1)2Tb)|kdek)and K((t−(i−1)2Tb)|kxk)can be evaluated using (5) for partially

absorbing receiver and (6) for fully absorbing receiver.

2) For even numbered bits: The expression for the expected number of molecules at the receiver due

to even numbered sequence, with the knowledge that bit-1 is represented using NB

tx number of type B

molecules and bit-0 by the absence of molecules, can be formulated as

EB

DNB

D(Ωrr,(j−1)Tb, jTb)=NB

tx

bj/2c

X

i=1

b2iZjTb

(j−1)Tb

K((t−(2i−1)Tb)|kdek) dt, (17)

and

EB

INB

I(Ωrr,(j−1)Tb, jTb)= 4πλaNB

tx Z∞

rr

bj/2c

X

i=1

b2iZjTb

(j−1)Tb

K((t−(2i−1)Tb)|kxk) dt r2dr. (18)

where K((t−(2i−1)Tb)|kdek)and K((t−(2i−1)Tb)|kxk)can be evaluated using (5) for partially

absorbing receiver and (6) for fully absorbing receiver.

15

C. Cumulative Expectation

1) Single Type of Information Molecule: The total expected net number of absorbed molecules at the

receiver in the MC system with single type of information molecule, during the time interval [(j−1)Tb, jTb]

due to jlength bit sequence can be written by combining (12) and (14) as

ES

all NS

all(Ωrr,(j−1)Tb, jTb)=ES

D{ND(Ωrr,(j−1)Tb, jTb)}+ES

I{NI(Ωrr,(j−1)Tb, jTb)}.

(19)

2) Two Types of Information Molecules: The cumulative expected number of molecules at the receiver

in the MC system with two types of information molecules, can be formulated as

EA+B

all {N(Ωrr,(j−1)Tb, jTb)}

=NA

txhEFA

D(Ωrr,(j−1)Tb, jTb| kdek)+EFA

I(Ωrr,(j−1)Tb, jTb| kxk)i

+NB

txhEFB

D(Ωrr,(j−1)Tb, jTb| kdek)+EFB

I(Ωrr,(j−1)Tb, jTb| kxk)i,

(20)

where EFA

D(Ωrr,(j−1)Tb, jTb| kdek),EFA

I(Ωrr,(j−1)Tb, jTb| kxk),E{FB

D(Ωrr,(j−1)Tb,

jTb| kdek)}, and EFB

I(Ωrr,(j−1)Tb, jTb| kxk)can be evaluated using (15), (16), (17) and (18) re-

spectively.

D. Signal to Interference Ratio

The Signal to Interference Ratio (SIR) for the MC system presented in this paper is deﬁned as the

ratio of molecules that arrive in the current symbol duration from the desired transmitter to the ratio of

interfering molecules from the swarm of transmitters. Hence, the impact of the proposed systems can be

further realized through analysis of SIR for the bit interval [(j−1)Tb, jTb].

1) Single Type of Information Molecule: The SIR of the MC system modulating using single type of

information molecule is deﬁned as

SI RS[j] = NS

txEFS

D(Ωrr,(j−1)Tb, jTb| kdek)

NS

txEPx∈ΦaFS

I(Ωrr,(j−1)Tb, jTb| kxk),(21)

where EFS

D(Ωrr,(j−1)Tb, jTb| kdek)and EPx∈ΦaFS

I(Ωrr,(j−1)Tb, jTb| kxk)can be evaluated us-

ing (12) and (14), respectively.

2) Two Types of Information Molecules: The SIR of the MC system modulating using two types of

information molecules changes with the position of the jth bit in the sequence. Hence, the SIR can be

16

written as

SI RA+B

A[j] = NA

txEFA

D(Ωrr,(j−1)Tb, jTb| kdek)

NA

txEPx∈ΦaFA

I(Ωrr,(j−1)Tb, jTb| kxk),(22)

when jth bit is in an odd numbered position, and as

SI RA+B

B[j] = NB

txEFB

D(Ωrr,(j−1)Tb, jTb| kdek)

NB

txEPx∈ΦaFB

I(Ωrr,(j−1)Tb, jTb| kxk),(23)

when jth bit is in an even numbered position. The EFA

D(Ωrr,(j−1)Tb, jTb| kdek),E{FA

I(Ωrr,(j−1)Tb,

jTb| kxk)},EFB

D(Ωrr,(j−1)Tb, jTb| kdek)and EFB

I(Ωrr,(j−1)Tb, jTb| kxk)can be evaluated using

(15), (16), (17) and (18), respectively. It should be noted that, the interference is caused by molecules of same

type only, and molecules from two different types don’t create interference at the receiver. For instance, if

the desired signal is transmitted using type A molecules, the interference is only from the type A molecules

transmitted by the swarm of interfering transmitters.

V. ERROR PROBAB IL IT Y

In this section, we derive the bit error probability of a 3D large-scale MC system deﬁned in Section

II with the two schemes to reduce the effect of MUI and ISI interference. The net number of absorbed

molecules at the surface of the partially absorbing receiver and the fully absorbing receiver is sampled at

the detector and one sample per bit is used for the information demodulation purpose.

A. Bit Error Probability of the MC System

The net number of absorbed molecules at the receiver in the jth bit interval due to all the transmitters

in the space R3with multiple transmitted bit sequences can be written as

Nnet[j] = ND

net[j] + NI

net[j],(24)

where

ND

net[j]∼BNtx , FD(Ωrr,(j−1)Tb, jTb|kdek),(25)

NI

net[j]∼X

x∈Φa

BNtx, FI(Ωrr,(j−1)Tb, jTb|kxk),(26)

For the simplicity of the analysis, (25) and (26) can be approximated using a Poisson model as in [2].

17

Therefore Eq. (24) can be expressed using Poisson approximation as

Nnet[j]∼PNtxFD(Ωrr,(j−1)Tb, jTb|kdek)+PNtx X

x∈Φa

FI(Ωrr,(j−1)Tb, jTb| kxk)

∼PNtxFD(Ωrr,(j−1)Tb, jTb|kdek) + Ntx X

x∈Φa

FI(Ωrr,(j−1)Tb, jTb| kxk).

(27)

The BEP of the large-scale MC system, for the jth bit can be written as

Pe[j] = Pr[e|bj= 0, b1:j−1]Pr[bj= 0] + Pr[e|bj= 1b1:j−1]Pr[bj= 1],(28)

where Pr[e|bj= 0, b1:j−1]and Pr[e|bj= 1b1:j−1]are the conditional probabilities.

1) Single Type of Information Molecule: For the MC system modulating using single type of information

molecule, the net number of absorbed molecules at the receiver in the jth bit interval due to all the

transmitters in the space R3with multiple transmitted bit sequences can be approximated as

NS

net[j]∼PNS

txRS

Tot(Ωrr, j),(29)

where

RS

Tot(Ωrr, j) = FD(Ωrr,(j−1)Tb, jTb|kdek) + X

x∈Φa

FI(Ωrr,(j−1)Tb, jTb| kxk),

=RS

D(Ωrr, j| kdek) + X

x∈Φa

RS

I(Ωrr, j| kxk).

(30)

Closed-from expressions for conditional probabilities in (28) for single type of information molecules is

derived in the following lemma.

Lemma 1. The conditional probabilities in (28) in the jthbit are written as

Pr[e|bj= 1, b1:j−1] = PrhNS

net[j]< N S

thi,(31)

and

Pr[e|bj= 0, b1:j−1] = PrhNnetS[j]≥NS

thi

= 1 −PrhNnetS[j]< N S

thi,

(32)

where

PrhNS

net[j]< N S

thi=Enexp{−NS

txRS

Tot}

NS

th−1

X

n=0

[NS

txRS

Tot]n

n!o.(33)

18

Proof. See Appendix A for the derivation of the closed-form expression for PrhNS

net[j]< N S

thi.

By substituting (31) and (32) into (28), we can obtain the expression for the bit error probability for the

uncoded system with single type of information molecules.

2) Two Types of Information Molecules: The net number of absorbed molecules at the receiver in the

jth bit interval due to all the transmitters with multiple transmitted bit sequences modulated using type A

and B information molecules, can be expressed using Poisson approximation as

NA+B

net [j]∼PNAorB

tx RT otA+B(Ωrr, j),(34)

where

RA+B

T ot (Ωrr, j)

=FA

D(Ωrr,(j−1)Tb, jTb| kdek) + X

x∈Φa

FA

I(Ωrr,(j−1)Tb, jTb| kxk)

+FB

D(Ωrr,(j−1)(Tb), j(Tb)| kdek) + X

x∈Φa

FB

I(Ωrr,(j−1)(Tb), j(Tb)| kxk)

=RA

T ot(Ωrr, j) + RB

T ot(Ωrr, j).

(35)

With the assumption of no interaction and reaction between type A and type B information molecules, the

conditional probabilities given in (28) can be rewritten as

Pr[e|bj= 1, b1:j−1] = PrhNA+B

net [j]< Nthi,(36)

and

Pr[e|bj= 0, b1:j−1] = PrhNA+B

net [j]> Nthi,(37)

where Nth can be either NA

th or NB

th depending on the position of the jth bit in the input sequence.

Let’s ﬁrst evaluate the conditional probability for the even numbered sequence, when jis an even number

and Nth =NthB.

Pr[e|bj= 1, b1:j−1] = PrhNA+B

net [j]< NB

thi,

=PrhNB

net[j]< N B

thi,

=Enexp{−NB

txRB

T oteven }

NB

th−1

X

n=0

[NB

txRB

T oteven ]n

n!o.

(38)

19

Proof. Eq.(38) can be further simpliﬁed as in Lemma 1.

We can follow a similar method to derive Pr[e|bj= 0, b1:j−1]for the even numbered sequence, which

can be written as

Pr[e|bj= 0, b1:j−1] = PrhNA+B

net [j]≥NB

thi

=PrhNB

net[j]≥NB

thi

= 1 −PrhNB

net[j]< N B

thi.

(39)

By substituting (38) and (39) into (28), we obtain the expression for the bit error probability of the uncoded

MC system with the even numbered sequence. As the evaluation of conditional probabilities of the odd

numbered sequence takes a similar approach as that of even numbered sequence, we can derive the BEP

of the odd numbered sequence by following the same procedure as that presented above by accounting for

the bit interval of the odd sequence as (2n−1)Tb.

Note that our derivations can be easily generalized for any type of receiver, by evaluating the term

K((t−(i−1)Tb)|kxk)in Eq. (7) and (9), which deﬁnes the reaction rate at the receiver due to a transmitter

located at x, considering the type of receiver used. For instance, we can expand this derivation for reversible

adsorption receiver, simply by using, [2, Eq. (8)] for K((t−(i−1)Tb)| kxk)to determine the fraction of

information molecules received.

B. Bit Error Probability of the Reed Solomon Coded System

We propose RS codes as channel codes to mitigate the interference caused by the interfering transmitters

and previously transmitted bits. Following [26], the BEP for the jth bit of a RS(n, k)coded system can

be written as

Pb[j] =

n

X

i=te+1 n

iPe[j]i(1 −Pe[j])n−i,(40)

where te=n−k

2is the the error correction capacity, and Pe[j]is the bit error probability for the uncoded

system given by (28).

VI. RE SU LTS AND ANA LYSI S

This section examines the expected number of molecules observed and the bit error probability at the

partially absorbing receiver with the proposed two distortion mitigation techniques. It has been shown

in [11], that results obtained from both Monte Carlo simulations and Particle-Based simulations are well

matched in 3D large-scale molecular communication system with absorbing receiver. Due to extensive

20

computational capacity required to simulate large scale MC environment in particle-based simulations, in

this paper, we use Monte Carlo simulation to validate the analytical derivations presented in earlier sections.

In all the ﬁgures presented in this paper, analytical curves are abbreviated as ”Ana” and that of Monte Carlo

simulations as ”Sim”. Furthermore, the results of the MC system modulated with single type of information

molecules is abbreviated as ”One” and that of type A and B information molecules as ”Two”.

In Fig. 2-11 the simulation results are obtained using the Monte Carlo simulation method, which was

performed by averaging the expected number of observed molecules due to all active interfering transmitters

with randomly generated locations and a desirable transmitter located at a predeﬁned position, de, as

calculated from (5) and (6) over 106realizations. In Fig. 2 and Fig. 3, the analytical curves of the expected

net number of absorbed molecules at the receiver in the MC system with single type of information molecule,

during the time interval [(j−1)Tb, jTb]are plotted using (12) for desirable transmitter, (14) for interfering

transmitter, and (19) for the entire system and are abbreviated as “Ana. Desired”, “Ana. Interfering”, and

“Ana. Net”, respectively. Similarly in Fig. 4 and Fig 5, analytical curves “Ana. Desired”, “Ana. Interfering”,

and “Ana. Net” are generated using (15) with (16), (17) with (18), and (20) respectively. Eq. (21), and

(22) with (23), are used in Fig. 6 to generate analytical curves for the SIR for partially absorption receiver

and are abbreviated as “Ana. SIR One” and “Ana. SIR Two”. The analytical curves of the BEP due to

single type of information molecules without coding, two types of information molecules without coding

and RS coding with single type of information molecules, are plotted in Fig. 7-11 using, (28) with (31) and

(32), (28) with (38) and (39), and (40) with (28),(31) and (32), and are abbreviated as “Ana. One without

coding”, “Ana. Two without coding”, and “Ana. One with RS(n, k)”, respectively.

For the results presented in this paper, we set the receiver radius as rr= 10µm for Fig. 8 , 10 and 11,

and rr= 5µm for the rest of the ﬁgures. The desired transmitter is ﬁxed at de= 20µm for Fig. 3,7, 9, 10,

and 11, and at de= 15µm for the rest of the ﬁgures. The density of interfering transmitters is set in the

range of 5×10−7< λa<5×10−5, with these transmitters positioned randomly outside the receiver surface

in a 3D space of radius, R= 100µm. All the parameter values used in both theoretical and simulation

results are set similar as in [2, 10–12, 28].

A. Expected Number of Molecules

1) Single Type of Information Molecule: Fig. 2 and Fig. 3 plot the cumulative number of molecules

observed at both fully absorbing and partially absorbing receivers, when the input sequence takes the form

of [1 0 0 0 ... ]. It can be seen from Fig. 2 that as the density of the swarm of transmitters, λa, increases,

the strength of interference increases relative to the desired signal. This interference can create a signiﬁcant

21

0 0.05 0.1 0.15 0.2 0.25 0.3

0

10

20

30

40

50

Cumilative Number of molecules

0 0.05 0.1 0.15 0.2 0.25 0.3

t(s)

0

100

200

300

400 Sim. Net

Sim. Interfereing

Sim. Desired

Ana. Net

Ana. Interfereing

Ana. Desired

λa = 8 x10-07

λa = 8 x10-05

Fig. 2. Expected number of molecules observed at the Full absorption Receiver with parameters Ntx = 200, tb= 0.1s, R = 100µm, de=

15µm, rr= 5µm, and D= 800µm2/s.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

20

40

60

Cumulative number of molecules

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

20

40

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t (s)

0

10

20 Sim. Net

Sim. Interfereing

Sim. Desired

Ana. Net

Ana. Interfereing

Ana. Desired

D=250, k1=120

D=800, k1=250

D=150, k1=50

Fig. 3. Expected number of molecules observed at the Partial absorption Receiver with parameters Ntx = 1000, tb= 0.5s, R = 100µm, λa=

8×10−6, de= 20µm, and rr= 5µm.

error at demodulation especially when decoding bit-0.

According to Fig. 3, we can see that for partially absorbing receiver, the interference from the swarm of

transmitters in the 3D space remains high at all the absorption rates, k1, used. Furthermore, we noticed that

the cumulative number of molecules received at the receiver increases gradually with time, with a sharp

increase at the beginning of the bit interval. As illustrated in the Fig. 3 top subplot, although the molecules

from the desired transmitter tend to stabilize with time, that of interfering transmitters increase gradually.

Hence, from these observations it can be concluded that in 3D large-scale MC system the interference from

rest of the transmitters is signiﬁcantly high, and it is necessary to introduce error recovery techniques to

22

facilitate error free information reception at the receiver. To solve this bottleneck, we have applied channel

coding as well as a novel molecular modulating technique based on two distinguishable types of molecules

as information carriers.

2) Two Types of Information Molecules: Fig. 4 and 5 plot the expected number of molecules observed

at the receiver as a function of time averaged over the number of molecules transmitted to signal bit-1, NA

tx

or NB

tx, for both full absorption and partial absorption receivers. According to Fig. 2 and Fig. 3, it is evident

that the interference cause maximum distortion to the desired signal at the receiver when the transmitter

emits bit-0 precedes by series of bit-1s. Therefore, we have used [1 1 0 0 0 ...] as the input sequence, and

have observed the receiver response from the 3rd bit interval onwards. It is observed that the percentage

net number of molecules at both full absorbing and partial absorbing receivers is nearly halved for the MC

system based on two types of information molecules compared to that of the system with single type of

information molecules, there by reducing the ISI signiﬁcantly and improving the system reliability by a

notable level.

0 0.5 1 1.5

t (s)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Percentage of Net Number of Molecules

Sim. One Net

Sim. Two Net

Sim. One Desired

Sim. Two Desired

Sim. One Interfering

Sim. Two Interfering

Ana. One Net

Ana. Two Net

Ana. One Desired

Ana. Two Desired

Ana. One Interfering

Ana. Two Interfering

Fig. 4. Expected number of molecules observed at the Full Absorption Receiver with parameters Ntx = 1, tb= 1s, R = 100µm, λa=

5×10−07, de= 15µm, rr= 5µm and D= 800µm2/s.

3) Signal to Interference Ratio: Fig. 6 presents the SIR for partially absorption receiver. It compares

the SIR of both ISI mitigation systems discussed in this manuscript. From the results presented it is clear

that the MC system with single type of molecules experiences heavy SIR level than the MC system with

two types of information molecules. Hence, the MC system with two types of information molecules will

results in a signiﬁcant reduction of ISI at the receiver, in turn achieving a higher BEP performance.

23

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t(s)

0

0.005

0.01

0.015

0.02

0.025

0.03

Percentage of number of molecules

Sim. One Net

Sim. Two Net

Sim. One Desired

Sim. Two Desired

Sim. One Interfering

Sim. Two Interfering

Ana. One Net

Ana. Two Net

Ana. One Desired

Ana. Two Desired

Ana. One Interfering

Ana. Two Interfering

Fig. 5. Expected number of molecules observed at the Partial Absorption Receiver with parameters Ntx = 1, tb= 1s, R = 100µm, k1=

60µm/s, λa= 5 ×10−07 , de= 15µm, rr= 5µm and D= 120µm2/s,.

0 0.2 0.4 0.6 0.8 1 1.2

t(s)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

SIR

Sim. SIR One

Sim. SIR Two

Ana. SIR One

Ana. SIR Two

Fig. 6. SIR for Partial Absorption Receiver with parameter tb= 1.2s, R = 100µm, k1= 120µm/s, λa= 5 ×10−07 , de= 15µm, rr=

5µm, and D= 250µm2/s.

B. Bit Error Probability

In this subsection we exploit the BEP performance variation of the two proposed systems with the

demodulation threshold value Nth for both fully absorbing and partially absorbing receivers.

1) Single Type of Information Molecule: Fig. 7 and Fig. 8 plot the bit error probabilities of the fully

and partially absorbing receivers respectively, in the proposed 3D large-scale MC system based on single

type of information molecules, with RS(n, k)codes.

In both ﬁgures we see a good match between the theoretical and the simulation results, which validates

our derivations. Furthermore, we observe that there exist an optimal Nth threshold level for a given set

24

12345678910

Nth

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Bit Error Probability

Sim. without coding

Sim. with RS(16,2)

Sim. with RS(10,2)

Sim. with RS(8,4)

Ana. without coding

Ana. with RS(16,2)

Ana. with RS(10,2)

Ana. with RS(8,4)

Fig. 7. Bit error probability for Full Absorption Receiver with parameters Ntx = 15, tb= 0.1s, R = 100µm, λa= 8 ×10−6, de=

15µm, rr= 5µm, and D= 800µm2/s.

1 2 3 4 5 6 7 8 9

Nth

-7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Bit Error Probability

Sim. without coding

Sim. with RS(16,2)

Sim. with RS(10,2)

Sim. with RS(8,4)

Ana. without coding

Ana. with RS(16,2)

Ana. with RS(10,2)

Ana. with RS(8,4)

Fig. 8. Bit error probability for Partial Absorption Receiver Ntx = 100, tb= 0.25s, R = 100µm, k1= 120µm/s, λa= 8 ×10−7, de=

20µm, rr= 10µm, and D= 250µm2/s.

of system parameters of the 3D large-scale MC system. Fig. 7 and 8 shown that, systems with RS(16,2)

and RS(10,2) clearly outperforms the system with no coding in terms of improvement achieved in BEP,

while RS(8,4) exhibits a severe loss of performance in terms of BEP. The severe high BEP of RS(8,4)

when decision threshold level is set to a smaller value like Nth = 1 is because, in this case almost all

the bits in the received signal are demodulated as bit 1 due to considerably low Nth. Further, the error

correction capacity, denoted by te=n−k

2, of RS(8,4) is not sufﬁcient to handle the requirement of this

particular system. In such scenarios, incorporation of RS codes only increase the system impairment due to

the extra redundant data introduced to the system by the channel coding. In overall, it is evident from results

25

presented in Fig. 7 and 8, that the RS codes are capable of providing an acceptable level of improvement

to the BEP. Hence, the code length, nand the message length, kof the RS(n, k)code should be carefully

selected according to the system settings and the required BEP.

2) Two Types of Information Molecules: Fig. 9, 10 and 11 plot the variation of the BEP with Nth for both

fully and partially absorbing receivers, for the three systems, two types of information molecules without

coding, single type of information molecules without coding, and RS(16,2) coding with single type of

information molecules. According to Fig. 10 and Fig. 11 the proposed MC system based on two types

of information molecules exhibits a signiﬁcant improvement in BEP performance compared to the system

which employs only one type of information molecule. Furthermore, the proposed system outperforms the

MC system with single type of information molecules and RS(16,2) coding, at certain Nth levels in the

case of partial absorption receiver and till the threshold level reach Nth = 3 in the case of full absorption

receiver. It should be noted that in terms of complexity, it is easier to implement the new MC system

based on two types of information carrier molecules with simple modiﬁcation at transmitter and receiver,

compared to the MC system with RS coding which requires signiﬁcant processing power at the receiver

to perform the error recovery process. Additionally, RS coding introduces n−kamount of redundancy for

each codeword of nlength, whereas the MC system based on two types of information molecules does not

introduce any redundancy to the transmitted data stream. In a word, compared to MC system with single

type of information molecules and RS codes as ECCs, the MC system based on two types of information

molecules achieves a sufﬁcient level of BEP improvement with a considerable level of low complexity, low

redundancy, and low memory management.

1 2 3 4 5 6 7 8 9

Nth

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Bit Error Probability

Sim. One without coding

Sim. Two without coding

Sim. One with RS(16,2)

Ana. One without coding

Ana. Two without coding

Ana. One with RS(16,2)

Fig. 9. Bit error probability for Full Absorption Receiver with parameters Ntx = 5, tb= 0.1s, R = 100µm, D = 1000µm2/s, λa=

8×10−6, de= 20µm, rr= 5µm, and input bit stream of [1101×].

26

12345678910

Nth

0

0.1

0.2

0.3

0.4

0.5

0.6

Bit Error Probability

Sim. One without coding

Sim. Two without coding

Sim. One with RS(16,2)

Ana. One without coding

Ana. Two without coding

Ana. One with RS(16,2)

Fig. 10. Bit error probability for Partial Absorption Receiver with parameters k1= 350µm/s,Ntx = 25, tb= 0.1s, R = 100µm, D =

600µm2/s, λa= 5 ×10−5, de= 20µm, rr= 10µm, and input bit stream of [1001×].

1 2 3 4 5 6 7 8 9 10

Nth

0

0.1

0.2

0.3

0.4

0.5

0.6

Bit Error Probability

Sim. One without coding

Sim. Two without coding

Sim. One with RS(16,2)

Ana. One without coding

Ana. Two without coding

Ana. One with RS(16,2)

Fig. 11. Bit error probability for Partial Absorption Receiver with parameters k1= 500µm2/s,Ntx = 50, tb= 0.1s, R = 100µm, D =

1000µm2/s, λa= 5x10−6, de= 20µm, rr= 10µm, and input bit stream of [1001×].

VII. CONCLUSIONS AND FUTURE WOR K

In this paper, we proposed RS coding based error correction, and a novel molecular modulating scheme

which employs two distinguishable types of information molecules as information carriers, as interference

mitigation schemes for ISI and MUI mitigation in a large-scale molecular communication system with a

swarm of interfering transmitters. We ﬁrst provided an analytical framework to quantify the performance

improvement with the help of the proposed two schemes using stochastic geometry. We then derived closed-

form expressions for the collective signal strength and the BEP at a partially absorbing receiver due to desired

transmitter in the presence of swarm of interfering transmitters. We have extended our analytical derivations

27

to analyze the performance gain achieved in terms of BEP, with the proposed two schemes, and validated

via Monte Carlo simulation. We observed that there exists an optimal demodulation threshold level for a

given set of system parameters of our proposed 3D large-scale MC system. Furthermore, in comparison to

MC system with single type of information molecules and RS codes as ECCs, the MC system modulating

with two types of information molecules achieves a sufﬁcient level of BEP improvement with a considerable

level of low complexity, low redundancy, and low memory management. Yet, RS coding provides ﬂexibility

in error correction capacity, cause it is possible to tune the parameters of the RS coder to match the system

requirement.

APPENDIX A

PROOF OF LEMMA1

In order to derive a closed-form expression for the BEP of the uncoded system deﬁned by (28), the

expressions for conditional probabilities in (31) and (32) should be evaluated using (33).

Based on the fact that

∂nexp −NS

txφxτ

∂xnx=φ−1

= exp −NS

txτ−NS

txφτ n,(A.1)

we further simplify (31) as

PrhNS

net[j]< N S

thi

=Z∞

0

exp −NS

txτfRS

Tot(τ)dτ+

NS

th−1

X

n=1

1

(−φ)nn!Z∞

0

∂nexp −NS

txφxτ

∂xnx=φ−1

fRS

Tot (τ) dτ

=LRS

Tot(NS

tx) +

NS

th−1

X

n=1

1

(−φ)nn!

∂nhLRS

Tot(NS

txφx)i

∂xnx=φ−1

,

(A.2)

where fRS

Tot (τ)is the PDF of RS

Tot, and LRS

Tot (·)is the Laplace transform of RS

Tot.

Using the deﬁnition of RS

Tot presented in (30), the Laplace transform of RS

Tot can be evaluated as

LRS

Tot NS

tx=Eexp −NS

txnRS

D(Ωrr, j| kdek) + X

x∈Φa

RS

I(Ωrr, j| kxk)o,

= exp n−NS

txRS

D(Ωrr, j| kdek)o×exp −λaZ∞

rr1−exp −NS

txRS

I(Ωrr, j|r)4πr2dr,

=LRS

DNS

tx.LRS

INS

tx.

(A.3)

28

Based on (A.3), we can express

∂nL

Rj

Tot

(Ntxφx)

∂xnx=φ−1

as

∂nhLRj

Tot (Ntxφx)i

∂xnx=φ−1

=

∂n

exp n−NtxφxRj

D(Ωrr, j| kdek)o×

exp n−λaR∞

rr1−exp −NtxφxRj

I(Ωrr, j|r)4πr2dro

∂xn|x=φ−1

,

(A.4)

which can be further simpliﬁed using General Leibniz rule [29] as

∂nhLRS

Tot NS

txφxi

∂xnx=φ−1

=

n

X

l=0 n

l∂lhLRS

INS

txφxi

∂xlx=φ−1

×exp{−NS

txRS

D(Ωrr, j| kdek)}(−NS

txφRS

D(Ωrr, j| kdek))(n−l),

(A.5)

where

∂lLRS

I(NS

txφx)

∂xl

can be further simpliﬁed using Fa`

adi Bruno’s formula [30] as

∂lhLRS

INS

txφxi

∂xlx=φ−1

= exp −λaZ∞

rr1−exp −NS

txRS

I(Ωrr, j|r)4πr2dr

×

l

Xl!

l

Q

k=1

lk!k!lk

l

Y

k=1 −λaZ∞

rrh−−NS

txφRS

I(Ωrr, j|r)k×exp −NS

txRS

I(Ωrr, j|r)i4πr2drlk

,

(A.6)

where the summation

l

Pis over all l-tuples of non-negative integers (l1, ..., ll) satisfying the constraint

1·l1+2·l2+·· ·+k·lk+· · ·+l·ll=l. Noting that

l

Q

k=1

(−φ)klk=(−φ)l, we ﬁnally derive PrhNS

net[j]< N S

thi

by ﬁrst substituting (A.6) into (A.5) and then using the resulting expression and (A.3) in (A.2).

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