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Molecular communication (MC) has recently emerged as a novel paradigm for nano-scale communication utilizing molecules as information carriers. In diffusion-based molecular communication, the system performance is constrained by the inter-symbol-interference (ISI) caused by crossover of information carrying molecules in consecutive bits. To cope with this, we propose the Reed-Solomon (RS) codes as an error recovery tool, to improve the transmission reliability in diffusion-based MC systems. To quantify the performance improvement due to RS codes, we derive the analytical expression for the approximate bit error probability (BEP) of the diffusion-based MC system with the full absorption receiver. We further develop the particle-based simulation framework to simulate the proposed system with RS code to verify the accuracy of our derived analytical results. Our results show that, as the number of molecules per bit increases, the BEP of the system with RS codes exhibits a substantial improvement than that of non-coded systems. Furthermore, the BEP of the proposed system with RS codes can be greatly improved by increasing the minimum distance of the codeword.
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1
Reed Solomon Codes for Molecular
Communication with a Full Absorption Receiver
Maheshi B. Dissanayake, Yansha Deng, Arumugam Nallanathan, E. M. N. Ekanayake, and Maged Elkashlan
Abstract—Molecular communication (MC) has recently e-
merged as a novel paradigm for nano-scale communication
utilizing molecules as information carriers. In diffusion-based
molecular communication, the system performance is constrained
by the inter-symbol-interference (ISI) caused by crossover of
information carrying molecules in consecutive bits. To cope with
this, we propose the Reed-Solomon (RS) codes as an error
recovery tool, to improve the transmission reliability in diffusion-
based MC systems. To quantify the performance improvement
due to RS codes, we derive the analytical expression for the
approximate bit error probability (BEP) of the diffusion-based
MC system with the full absorption receiver. We further develop
the particle-based simulation framework to simulate the proposed
system with RS code to verify the accuracy of our derived
analytical results. Our results show that, as the number of
molecules per bit increases, the BEP of the system with RS
codes exhibits a substantial improvement than that of non-coded
systems. Furthermore, the BEP of the proposed system with
RS codes can be greatly improved by increasing the minimum
distance of the codeword.
Index Terms—Molecular communication, error correction
codes, Reed Solomon codes, particle-based simulation,
I. INTRODUCTION
WITH the advancements in nano-technology, new multi-
disciplinary research on communication between bio-
nanomachines via chemical signals is emerging [1]. This new
paradigm, namely Molecular Communication (MC), adopts
biologically-inspired techniques for information transmission
among nano-machines. The nature inspired molecular com-
munication can be observed in calcium signalling among
cells, and Deoxyribonucleic Acid (DNA) signalling among
DNA segments [2] In this paper, we concentrate on the
molecular communication via diffusion (MCvD) [3] to provide
the baseline for the communication design, which will serve as
an important foundation for designing more complicated MC
systems with drift. In MCvD, the transmitter modulates the
information onto the physical properties of the information
carrying molecules. Once emitted by the transmitter, the
molecules diffuse in the medium and certain portion of the
Manuscript received Sep. 6, 2016; revised Jan. 9, 2017 and Feb. 9, 2017;
accepted Feb. 16, 2017. The editor coordinating the review of this manuscript
and approving it for publication was Prof. Nghi Tran.
Maheshi B. Dissanayake is with the Department of Electrical and Electronic
Engineering, Faculty of Engineering, University of Peradeniya, Sri Lanka and
part of this work was carried out while at King0sCollege London, London,
WC2R 2LS, UK. (e-mail: maheshid@ee.pdn.ac.lk).
E.M.N. Ekanayake is with the Faculty of Engineering, University of
Peradeniya, Sri Lanka. (e-mail: neka@ee.pdn.ac.lk).
Y. Deng and A. Nallanathan are with King0sCollege London, London,
WC2R 2LS, UK (email:yansha.deng,arumugam.nallanathan@kcl.ac.uk).
M. Elkashlan is with Queen Mary University of London, London E1 4NS,
UK (email: maged.elkashlan@qmul.ac.uk).
transmitted molecules arrive at the receiver during the current
bit interval. The receiver then demodulates the transmitted
information using the number of molecules captured during
that bit interval. Due to the random walk of each molecule in
the fluid medium, there exists high probability of crossover
between the molecules emitted in neighboring bits. These
molecules which arrive during different bit intervals, result
in the so-called inter-symbol-interference (ISI) [4]. Note that
ISI is regarded as one of the fundamental bottlenecks of MC
systems that degrades the system reliability.
In the literature, there exist many different approaches
for ISI mitigation in MC. As summarized in [1], borrowing
from traditional communication system designs, equalization
techniques like decision feedback filter technique, transmitter-
based novel modulation techniques such as molecular transi-
tion shift keying and Molecular ARray-based COmmunication
(MARCO), and energy efficient transmitter and receiver-side
signal power adjustment methods have been proposed as
ISI mitigation techniques for MC environment. Although,
some of these techniques increase the complexity of the MC
system, they provide a significant improvement in terms of ISI
mitigation. Furthermore, inspired by the biological behaviour
of MC system, different types of molecules with resistance or
kill effect on information carrying molecules have also been
proposed as another method to overcome the ISI effect [1].
Yet, this technique requires extra resources and intelligence
to handle the resistance molecular type and tends to increase
the amount of molecules in the medium most likely creating
collision.
Error correction codes (ECC) have been proposed as an
alternative way to mitigate the adverse effects of ISI in MC
systems by providing error correction capability at the receiver.
Hence, ECC can results in easy manipulation of the system
compared to using resistance molecules to overcome ISI. More
specifically, the bit error performance of established codes,
such as Hamming codes (HC) [5], Euclidean Geometry Low
Density Parity Check, and Cyclic Reed-Muller codes [5], have
been applied and investigated in MC systems. A new type
of ECC, called ISI free code, was proposed and designed
to prevent occurrence of ISI via the predefined redundant bit
patterns in MC systems [6]. Furthermore, in [7], the authors
concentrate on improving the bit error performance via self-
orthogonal convolutional codes, which takes into account the
energy requirements at the receiver for decoding the error
correction information.
Different from existing literature, we introduce the Reed-
Solomon (RS) codes for error correction in the point-to-point
MC systems with a point transmitter and a full absorption
2
Fig. 1. Block Diagram of the Proposed MC system
receiver. The RS codes are non-binary block codes and are
highly effective against burst and random errors in many real
channels [8] compared to HC that only capable of correcting
single bit error. Also RS code is one of the few forward
error correction code, which can attain the theoretical limit
known as the Singleton bound. Since, RS codes utilize the
symbol based arithmetic, it can decode codewords with longer
block lengths with less decoding time, compared to HC [8].
Another advantage of RS codes is the existence of efficient
decoding algorithm due to vast applications of RS codes in
data communication. More importantly, simplified RS codes
are relatively easy to implement in terms of hardware [9].
In this paper, we present an analytical model for the
point-to-point MC systems with RS codes and derive an
expression for the approximated bit error probability (BEP)
of the proposed system. Furthermore, the derived analytical
result is verified by the proposed simulation framework, which
captures the random Brownian motion of each molecule in
the diffusion-based MC system. The results obtained from
simulation are in close agreement with the analytical values.
Furthermore, it is observed that the BEP performance is greatly
improved via RS codes, and the performance gain increases
with the increase of the number of molecules per bit and
the minimum distance of the codeword. The remainder of the
paper is organized as follows. Section II presents the system
model, and Section III briefly introduce RS codes. Section
IV presents an expression for the BEP of the proposed MC
system. Section V evaluates the performance improvement
achieved by the proposed RS codes using numerical and
simulation results. Conclusions are given in SectionVI.
II. SYSTEM MODEL
We consider a point-to-point diffusion-based MC system
with a full absorption receiver with radius rand a point
transmitter located distance daway from the surface of the
receiver. The block diagram of the proposed MC system is
outlined in Fig. 1. In the diffusion-based MC system, the
information is modulated on the number of the molecules
emitted by the transmitter at start of each bit interval, Tb.
Given that the molecules are released into a medium of large
extent compared to their size, collisions between messenger
molecules are neglected. Each molecule randomly diffuses
in the medium following Brownian motion, with constant
diffusion coefficient D. Once a molecule reaches the surface of
the receiver, it will be absorbed by the receptors at the surface
of the receiver. The process of messenger molecules hitting
the body of the receiver is named as the hitting process.
For the modulation scheme, we adopt the Binary Concen-
tration Shift Keying (BCSK) as in [2]. At the transmitter,
the information is conveyed through the number of molecules
being emitted at the start of each bit interval. At the receiver, if
the total number of molecules absorbed during a bit interval
is above a threshold, the received bit is demodulated as 1,
and otherwise as 0. To facilitate error correction, RS encoded
redundant data is embedded into the message bit pattern at
the transmitter. These extra data is utilized at the receiver to
recover from bit errors, in turn improving the overall BEP.
III. REED SOLOMON CODES
In communication theory, ECC is used for error detection
and correction to achieve an acceptable level of accuracy for
the received information, when data is transmitted through
error prone transmission channels. As one type of ECC, the
Reed Solomon codes was first proposed by I. S. Reed and G.
Solomon, which is a non-binary Bose, Chaudhuri, and Hoc-
quenghem (BCH) code, with a simple algorithm for the error
detection and correction [9]. Considering that the RS codes
are well suited for correcting burst errors, they are widely
used in digital data transmission and storage applications. We
denote RS code with nlength codeword and klength input
message block, as RS(n,k) and it is defined over a Galois Field
(GF (pm)), where pis a prime, and mis a positive integer. The
length of the parity bits is nkbits, which has the capacity
to correct up to nk
2number of errors per codeword [9].
The RS codes can be implemented very easily at the encoder
using Linear Feedback Shift Registers and at the decoder using
Berlekamp algorithm. The systematic RS encoder, appends
parity information to the original message in such a way that
the constructed codeword is completely divisible by the gen-
erator polynomial using Galois Field algebra. This generator
polynomial is shared by both the RS encoder and decoder.
The RS decoding operates in two main stages. In the
first stage, the decoder inspects the received codeword for
errors. The received codeword is identified as in error, if
it is not completely divisible by the generator polynomial.
This technique is called syndrome computation. If syndrome
calculation results in zero, the decoder terminates, otherwise
in the second stage, the system attempts to correct the error
by detecting the error position and error value. To correct the
identified errors, the decoder first determines the error locator
polynomial using either Berlekamp-Massey algorithm [10] or
Euclidean algorithm [10]. The former leads to a more efficient
implementation, while the latter is easier to implement. In the
next stage, the Chien search algorithm [10] is used to solve
the roots of the error locator polynomial gernerated, which
indicate the error locations in the receieved codeword. Then,
Forney algorithm [10] is used to estimate the error magnitudes.
Once the error location and the error values are estimated, then
a correction is applied to the received codeword to recover
from the error. In depth description of each algorithm used in
the implementation of RS codes can be found in [10] while RS
3
decoder implementation presented in this paper can be found
in [9].
One of the key bottlenecks in the implementation of ECC
in MC environment is the complexity and the processing
power requirements. Inspired by the extensive application of
RS codes in battery powered devices, many research are
conducted on optimizing the energy performance of the RS
coders. For instance, it is reported in [11] that the energy
consumption of RS coding can be reduced by 40% in low
activity environments and can be synthesized by using 45 nm
technology. The reduced complexity algorithms with accept-
able level of bit error rate are another method adopted by
many researchers to achieve low complexity and low power
requirements. For instance, [12] proposes to apply the soft
information based decoding to achieve better results in error
correction rates compared to hard decision decoding, while
keeping the complexity at lower level. In summary, it is
safer to assume that RS codes can be implemented with
reduced complexity in MC with the advanced research on low
complexity implementation of RS coders.
IV. BIT ER ROR PROBABI LI TY
It is known that in MC, the number of absorbed molecules
during one bit interval, Tb, can be modeled as binomial distri-
bution. For computational tractability, the binomial distribution
can be approximated using the normal distribution, when Nm
the number of molecules sent at the start of each bit interval
is large, and Phit (d,t) the fraction of molecules absorbed by
the receiver is not near 0 or 1. Thus, we express the total
number of absorbed molecules, in the ith bit Nhi t [i], during
[(i1)Tb,iTb]in the absence of ISI as
Nhit [i]NNmPhit (d,Tb),NmPhit (d,Tb)[1 Phit (d,Tb)].
(1)
Yet, due to ISI, the actual number of molecules absorbed
during Tb, account for the number of molecules recieved from
the current bit (ai), Nai, and from the previous bits, Na1:i1.
However, prior work [13] shows that the most prominent effect
on ISI is contributed by the nearest past bit, ai1, with an
appropriate bit interval, Tb. As such, in the presence of ISI,
the Nhit [i]can be approximated as
Nhit [i]=aiNai+ai1Nai1(2)
where
NaiNNmPh1,NmPh1[1 Ph1](3)
and
Nai1NNmPh2,NmPh2[1 Ph2]
NNmPh1,NmPh1[1 Ph1].
(4)
In (3) and (4), Ph j denotes Phit (d,jTb) and it can be derived
as in [1], using Eq. (6), Eq. (7) and Eq. (8).
For an uncoded binary channel, the bit error probability, Pe,
over all possible combinations of aican be expressed as
Pe=Pr[e|ai=0]Pr[ai=0] +Pr[e|ai=1]Pr[ai=1],(5)
This can be further expanded by considering the possible
outcomes for both aiand ai1as
Pe=Pr[e|ai=0,ai1=0].Pr[ai=0].Pr[ai1=0]
+Pr[e|ai=0,ai1=1].Pr[ai=0].Pr[ai1=1]
+Pr[e|ai=1,ai1=0].Pr[ai=1].Pr[ai1=0]
+Pr[e|ai=1,ai1=1].Pr[ai=1].Pr[ai1=1]
(6)
where Pr[e|ai,ai1]is the conditional probability of error for
ith bit with known transmitted bits aiand ai1. The conditional
probabilities in equation (6) can be approximated as
Pr[e|ai=0,ai1=1] =Pr[Nhit τ]=Pr[Nai1τ]
Qτ[(NmPh2)(NmPh1)]
(NmPh1[1 Ph1])+(NmPh2[1 Ph2])(7)
Pr[e|ai=1,ai1=0] =Pr[Nhit < τ]=Pr[Nai< τ]
1Qτ(NmPh1)
NmPh1[1 Ph1](8)
Pr[e|ai=1,ai1=1] =Pr[Nhit < τ]=Pr[Nai+Nai1< τ]
1Qτ(NmPh2)
(2NmPh1[1 Ph1])+(NmPh2[1 Ph2])(9)
and
Pr[e|ai=0,ai1=0] =Pr[Nhit τ]=0(10)
where τis the detection threshold and Q(.) denotes the tail
probability of the standard normal distribution.
The evaluation of the post-decoding BEP of block codes,
Pb, is complex, even classical coding books like [9] and [10]
fail to present a closed-form expression. As such, bounds for
post-decoding BEP of block codes can be obtained as
1
kPwPbPw(11)
following [10], where Pwrepresents the block error proba-
bility. When the complete weight distribution of the specific
RS (n,k) code is not available, the upper bound is used as
the post-decoding BEP. As such, we approximate the post-
decoding BEP of RS codes as [10]
Pb=
n
X
i=te+1 n
i!Pi
e(1Pe)ni(12)
where te=nk
2and Peis the bit error probability for the un-
coded channel. Substituting (7), (8), (9), and (10) into (6), we
obtain the Pein (12).
V. PERFORMANCE EVAL UATION
A. Simulation Framework
To validate the expression of BEP derived in Section IV, we
present particle-based simulation framework extended from [2]
and [14]. This simulation framework takes into account the bit
pattern generation, the RS encoding, the signal modulation, the
propagation, the molecule reception, the signal demodulation,
and the RS decoding as outlined in Fig. 1. We simulate
the 2D environment due to the computational complexity in
4
10 20 30 40 50 60 70 80 90 100
10−7
10−6
10−5
10−4
10−3
10−2
10
1
Bit Error Probability
Analytical with RS coding
Analytical without coding
Simulation with RS coding
Simulation without coding
Nm
Fig. 2. Bit error probability of RS(4,2) coded and uncoded bit stream.
particle-based simulation for 3D MC system with RS codes.
The information bits are randomly generated with simulation
parameters set as: d=1µm,D=79.4µm2/s,Tb=0.032 s,
τ=Nm
2, and the length of the bit sequence is 2400000 with
100 repetitions per each iteration. The analytical results are
plotted using (12), with the assumption that the occurrence of
bit 1 and 0 of the transmitted bit is equally likely.
B. Numerical Results
In this subsection, we present the simulation and analytical
results of the bit error probability of our proposed MC
system. The accuracy of the theoretical BEP approximation
is evaluated. Fig. 2 plots the impact of RS coding on the
BEP of proposed MC system. It is shown that the theoretical
results match with that of particle-based simulations, which
validate the accuracy of our derived expression. Compared
with uncoded case, the BEP is greatly improved with the help
of RS(4,2) applied in our proposed MC system. For instance,
with Nm=100, the BEP achieved by the RS(4,2) is 5×106,
which is a negligible level.
Fig. 3 plots the BEP of the proposed system with various
minimum distance of the RS code and HC. According to Fig.
3, RS codes is shown to provide a significant level of BEP
improvement compared to HC. We also observe that the BEP
of the RS codes can be greatly improved by increasing the
minimum distance and the coding gain, k
n. This is because
increasing nkincreases the error correction capacity of
the RS code given by nk
2, in comparison to HC which is
only capable of correcting single bit error. In addition, since
particle-based simulations for very large input bit streams are
time consuming, only analytical results are presented.
VI. CONCLUSION
In this paper, we introduced RS code for error correction
of the diffusion-based MC system. We derived the bit error
probability of the proposed MC system with full absorption
receiver using theoretical approximations. The derived analyt-
ical results for the bit error probability are verified by particle-
based simulations. It is shown that the RS code substantially
without coding
with RS(4,2)
with RS(8,4)
with RS(10,2)
with RS(10,4)
with RS(16,8)
with HC(4,2)
with HC(8,4)
with HC(10,2)
with HC(10,4)
with HC(16,8)
10−8
10−6
10−4
10−2
100
Bit Error Probability
20 40 60 80 100 120 140
Nm
Fig. 3. Bit error probability of RS codes and Hamming Codes (HC).
improves the bit error probability of MC systems, compared to
uncoded MC systems. The bit error probability can be further
enhanced by increasing, the number of molecules emitted per
bit and the minimum distance of the RS codeword. These
results justify the effectiveness of RS codes in diffusion-
based MC systems. The extension of this work under adaptive
weighted threshold detection scheme and finding the optimum
combinations of (n,k) can be considered in future work.
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