![Bit error probability for Full Absorption Receiver with parameters Ntx = 5, t b = 0.1s, R = 100µm, D = 1000µm 2 /s, λa = 8 × 10 −6 , de = 20µm, rr = 5µm, and input bit stream of [1101×].](https://www.researchgate.net/profile/Arumugam-Nallanathan-2/publication/330579045/figure/fig2/AS:718531884752903@1548322759337/Bit-error-probability-for-Full-Absorption-Receiver-with-parameters-Ntx-5-t-b-01s-R_Q320.jpg)
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1
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.
2
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
3
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
4
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.
5
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
6
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
7
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.
8
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
9
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.
10
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
11
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)
12
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)
13
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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