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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017
Equivalent Discrete-Time Channel Modeling for
Molecular Communication With Emphasize
on an Absorbing Receiver
Martin Damrath,∗ Student Member, IEEE , Sebastian Korte, and Peter Adam Hoeher, Fellow, IEEE
Abstract — This paper introduces the equivalent discretetime channel model (EDTCM) to the area of diffusionbased molecular communication (DBMC). Emphasis is on
an absorbing receiver, which is based on the so-called
first passage time concept. In the wireless communications
community the EDTCM is well known. Therefore, it is anticipated that the EDTCM improves the accessibility of DBMC
and supports the adaptation of classical wireless communication algorithms to the area of DBMC. Furthermore, the
EDTCM has the capability to provide a remarkable reduction
of computational complexity compared to random walk
based DBMC simulators. Besides the exact EDTCM, three
approximations thereof based on binomial, Gaussian, and
Poisson approximation are proposed and analyzed in order
to further reduce computational complexity. In addition, the
Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm is adapted to
all four channel models. Numerical results show the performance of the exact EDTCM, illustrate the performance of
the adapted BCJR algorithm, and demonstrate the accuracy
of the approximations.
Index Terms — Equivalent discrete-time channel model,
molecular communication, nanomachines.
I. I NTRODUCTION
ECENTLY, a remarkable progress in the field of nanotechnology has been achieved, providing new opportunities in many different research areas [1]. One important
area is dealing with the realization of so called nanomachines (NMs), which are autonomous machines in the size
of several nanometers up to a few micrometers [2]. NMs are
likely to enable new application scenarios in the future.
Besides industrial and consumer usage like food and water
quality control or intelligent textile fabrics, and the environmental usage like biodegradation or air pollution control, the
main application is anticipated to be in the medical sector,
where NMs can be used for applications like targeted drug
delivery, tissue engineering, or health monitoring [1], [3].
However, NMs are very limited with respect to complexity
and energy due to their size [3], [4]. In fact, it is assumed that
a single NM can perform only simple tasks [5]. Thus, a swarm
of NMs has to be realized to provide even more complex functions [1], [6]. The key technology for enabling an autonomous
R
Manuscript received September 20, 2016; revised December 2, 2016;
accepted January 2, 2017. Date of publication January 10, 2017; date of
current version March 2, 2017. Asterisk indicates corresponding author.
∗ M. Damrath is with the Faculty of Engineering, University of Kiel, Kiel,
Germany (e-mail: md@tf.uni-kiel.de).
S. Korte and P. A. Hoeher are with the Faculty of Engineering, University
of Kiel, Kiel, Germany.
Digital Object Identifier 10.1109/TNB.2017.2648042
swarm is communication between several NMs. Inspired by
nature, molecular communication (MC) [7]–[10] seems to be
the most promising communication technique. Like in the
case of calcium signaling in epithelial cells [8], [11], quorum
sensing between bacteria [12], [13], or neurotransmitters in the
synaptic cleft [3], molecules can be used as information carrier
which passively diffuse through the media. MC comes with the
benefit that it is inherently biocompatible, energy efficient, and
has been evolved over the last billions of years in nature [1].
The diffusion-based MC (DBMC) channel, e.g. studied
in [14]–[17], is fundamentally different to a classical wireless
communication channel. Radio waves propagate deterministically in a given environment, whereas molecules perform a
random walk. Due to the diffusive random propagation of the
molecules, the channel impulse response is slowly decreasing,
which causes intersymbol interference (ISI) and unreliable
transmission [18]. Thus, it is of great importance to invent new
or to adapt existing communication algorithms to the DBMC
channel. However, existing practical implementations and
numerical simulations are often time-consuming and abstract,
which complicates an exhaustive analysis of the impact of
DBMC model on transmitter and/or receiver algorithms.
In [19] a first initiative towards a practical implementation
is given, by introducing a macroscopic MC testbed based on
an electrical sprayer and an alcohol sensor. While it provides
a proof of principle, it has to deal with hardware challenges,
like the non-linearity of the sensor [20]. Furthermore,
it is difficult to control environmental parameters. In [10]
a layered architecture of MC is presented. Furthermore,
the relation between a transmitters’ input and a receivers’
output is generally formulated and specified for a targeted
drug delivery scenario. In [21] a system model including
multiple transmitters and multiple receivers is described,
assuming detection by Michaelis-Menten enzymatic kinetics.
In [22] the communication channel model is described by a
binomial distribution, which is approximated in [15], [23] as
a Gaussian and a Poisson distribution, respectively. In [24] a
simulation environment for three-dimensional MC based on
Brownian motion is proposed with focus on parallelization to
enhance execution time. Therefore, a high-level architecture
is designed to distribute the molecular simulation scenario
load. Another method to decrease the simulation time of
a 3D Brownian motion system is given in [25], where the
authors propose a dual time-step approach which uses larger
simulation time steps for particles that are far away from
the target. In [26] a framework for molecular nanonetworks
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DAMRATH et al.: EQUIVALENT DISCRETE-TIME CHANNEL MODELING FOR MOLECULAR COMMUNICATION
called NanoNS is build on top of the well-known network
simulator NS-2. NanoNS is able to simulate the diffusive
molecular communication channel, where the diffusion is
based on the multiparticle lattice gas automata algorithm. The
medium is divided into lattice sides, which reduces simulation
complexity. In [27] the so-called N3Sim simulation framework
for nanonetworks in two- and three-dimensional bounded and
unbounded environments is introduced. While it simulates
the diffusion-based MC by Brownian motion, it offers
different simulation settings like multiple transmitters with
different transmit pulses, absorbing or transparent receivers or
molecule harvesters. Furthermore, the inertia and interaction of
molecules can be simulated. However, it computes the position
and velocity of each molecule for each time step, which makes
simulations with a very high number of molecules impossible.
In [28] BiNS, a simulation platform designed for information
exchange at nanoscale, is presented. Thereby, propagation of
molecules is based on a Brownian molecular diffusion process
in a three-dimensional environment. Elastic collision between
molecules is taken into account. Furthermore, nanomachines
can be simulated with a specific mass, size, lifetime and
mobility model. The signal transduction is performed by
receptors placed on the outer surface of a nanomachine.
In [29] BiNS is extended to BiNS2 which is used to simulate
diffusion-based molecular communication with drift in blood
vessels. In addition, a grid approach is given to enable
distributed computations, which improves simulation time.
In [30] a three-dimensional end-to-end simulator for diffusionbased molecular communication called MUCIN is introduced.
Besides the ability of sending consecutive symbols,
frequency-based and concentration-based modulation,
imperfect reception, and ISI mitigation function, it also
offers a fast simulation based on the characteristic function
and binomial distributed random variables. In [31] the
authors analyze the effect of approximating the binomial
distribution inside the channel model by Gaussian and Poisson
distributions.
In this contribution the characteristic function is used to
derive an equivalent discrete-time channel model (EDTCM)
for DBMC with emphasize on an absorbing receiver. The
EDTCM is familiar in the wireless communications community to model noisy channels with memory in the discrete-time
domain. Therefore, it supports the understanding of the DBMC
channel and makes it easier to adapt classical wireless communication algorithms to the area of MC. Furthermore, it offers
a simple implementation for numerical simulations, which
provides a faster tool for analyzing system performances when
applying long bit sequences or large number of molecules.
In addition, three approximations are presented, which further
improve simulation time. Moreover, the Bahl-Cocke-JelinekRaviv (BCJR) algorithm is presented for each channel model.
In contrast to [23], the channel model in this contribution
is extended to arbitrary memory lengths and visualized as
an equivalent discrete-time channel model. Furthermore, it is
directly applied by means of numerical simulations to give
an alternative to random walk based simulations. In addition,
the accuracy and computational complexity of each channel
approximation is analyzed.
61
Fig. 1. Model of the diffusion-based molecular communication system
under investigation.
The remainder of this article is organized as follows: In
Section II the system model that is assumed throughout this
contribution is summarized. Based on the system model the
exact EDTCM and three approximations thereof are derived in
Section III. Additionally, the accuracy of the approximations
is analyzed and the computational complexity of all system
models under investigation is discussed. In Section IV the
detection algorithms that are applied in the numerical analysis
in Section V are briefly presented. Thereby, the focus is on
adapting the transition probability of the BCJR algorithm to
the channel models under investigation. Section VI finally
summarizes the work and gives an outlook for future work.
II. S YSTEM M ODEL
The system model under investigation is similar to the
system model introduced in [32]. Thus, in the following, only
a brief description is given. For more detailed information the
reader is referred to [32].
As shown in Fig. 1, the system model consists of a
single point source transmitter (Tx) and a single spherical
receiver (Rx) with radius r in an infinite three-dimensional
homogeneous fluid medium without drift, where d > r
is the distance between Tx and Rx. The fluid medium is
described by the diffusion coefficient D, which is assumed
to be constant. The modulation scheme under investigation is
on-off keying (OOK). The Tx emits either no molecules or
N molecules at the beginning of a bit period of length Tb to
represent bit u[k] = 0 or u[k] = 1, respectively:
if u[k] = 1,
N δ(t − kTb )
x(t) =
(1)
0
otherwise,
where δ(.) is the Dirac delta function. Molecules emitted by
Tx are the only molecules in the medium and propagate by
Brownian motion. Rx is assumed to be synchronized with
Tx in time domain [33], [34] and behaves perfectly absorbing, while performing strength/energy detection [35]–[37].
Consequently, a molecule is detected and removed from the
environment whenever it hits the surface of the receiver.
By means of strength/energy detection, the number of hitting
molecules is accumulated for each bit period separately. As a
result Rx follows the first passage time concept. The probability per unit of time that a molecule hits Rx at time instant t
after being released at t = 0 can be described as an inverse
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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017
Fig. 2. Block diagrams for exact and approximated EDTCM.
Gaussian distribution [38]:
r (d − r )
f hit (t) = √
e
d 4π D t 3
2
− (d−r)
4D t
.
(2)
Consequently, the probability that a molecule hits Rx until t
is given by
t
d −r
r
.
(3)
f hit (τ )dτ = erfc √
Fhit (t) =
d
4D t
0
III. E QUIVALENT D ISCRETE -T IME C HANNEL M ODELS
Numerical simulation of the random propagation of information molecules is of significant computational complexity,
especially for a large number of molecules and for long channel memories. Those simulations lead to a trade-off between
simulation step size and simulation precision. For those cases
an EDTCM based on stochastic distributions can improve
simulation time and precision. Furthermore, an EDTCM provides an easier access to the main properties of the channel
and makes it easier to adapt classical wireless communication
algorithms to MC channels. In the following the exact EDTCM
is derived and three approximations thereof are proposed to
further reduce complexity. Subsequently, the accuracy of the
approximations are analyzed and the computational complexity of all channel models is discussed. Furthermore, the signal
to interference plus noise ratio is defined and investigated.
A. Exact EDTCM
The basis for the EDTCM is given by (3) and the assumption of a perfectly absorbing and summing receiver. As a
result, the probability for a molecule to reach Rx during the
lth bit duration [lTb , (l + 1)Tb ] after release can be calculated
by
h l = Fhit ((l + 1)Tb ) − Fhit (lTb ),
(4)
where h l can be interpreted as the lth channel coefficient
of the EDTCM. The scenario that a single molecule arrives
the Rx during a certain bit duration can be represented by a
Bernoulli experiment with two possible events of either hitting
Rx or not. For N molecules released at the same time and
under the assumption that molecules propagate independently
and not change the hitting probability at Rx, the N Bernoulli
experiments can be described by the binomial distribution
z l ∝ B (N, h l ) ,
(5)
where z l is the number of received molecules during the lth
bit duration after their release. As depicted in Fig. 2(a), the
number y[k] of received molecules at time instant k when
transmitting a bit sequence is a summation over all z l [22]:
y[k] =
L
B (x[k − l], h l ) ,
(6)
l=0
where L is the effective channel memory length and x[k] is
the discrete representation of the OOK modulated symbol:
N
if u[k] = 1,
x[k] =
(7)
0
otherwise.
The summation over binomial distributions is exactly
described by a Poisson binomial distribution.
Notice that an infinite lifetime of molecules leads to
L → ∞. However, in simulations L cannot exceed the length
of the bit sequence when the receiving process is truncated
with the last transmitted bit. Furthermore, the main energy is
usually located in the first channel coefficients, thus L can be
further reduced while still achieving meaningful results [23].
B. Binomial Approximation
For a large memory length L it is time-consuming to create
a single binomial random variable for each branch in Fig. 2(a).
DAMRATH et al.: EQUIVALENT DISCRETE-TIME CHANNEL MODELING FOR MOLECULAR COMMUNICATION
63
However, there is no simple solution to describe the sum
of binomial distributions with different probabilities exactly.
Therefore, we apply different approximations to further reduce
the complexity. The effect of these approximations is analyzed
in Section III-E, while the effect on the error performance is
analyzed in Section V.
The sum of different binomial distributions can be
approximated by a new binomial distribution. If h l is sufficient
small y[k] can be approximated as
L
(8)
y[k] ≈ B N,
u[k − l]h l .
l=0
The corresponding EDTCM is shown in Fig. 2(b). This kind
of approximation is rather unusual, but as it will be shown
in Section III-E, it is superior to other approximations under
certain conditions.
C. Gaussian Approximation
Especially if the number of draws is high and the success
probability is not close to zero or one, the binomial distribution
can be well approximated by the Gaussian distribution [23].
The relation between the two distributions can be approximated as
B (N, h l ) ≈ N μl = Nh l , σl2 = Nh l (1 − h l ) ,
(9)
with mean μl and variance σl2 . Since the sum of independent
Gaussian distributions can be described by a single Gaussian
distribution, y[k] can be approximated as
L
2
y[k] ≈ N z[k], σ [k] =
x[k − l]h l (1 − h l )
= z[k] + n[k],
where z[k] =
L
l=0
(10)
x[k − l]h l describes the expected number
l=0
of molecules at Rx, and n[k] ∝ N (0, σ 2 [k]) represents
an additive amplitude-dependent noise term, which models
the uncertainty of the number of hitting molecules at Rx,
which is known as Brownian noise [39]. This channel model
visualized in Fig. 2(c) is similar to the additive white Gaussian
noise (AWGN) model with memory, which is well-known
in the wireless communications community [40]. The only
difference lies in the amplitude-dependent white Gaussian
noise. Please note that the EDTCM might be extended by
a second noise process that models the background noise
caused by molecules present in the propagation medium before
transmission, which is related to the AWGN process frequently
assumed in wireless communications. The big advantage of
EDTCM with Gaussian approximation is the simplicity in
implementation, because z[k] and σ 2 [k] can be efficiently
calculated by two convolutions (see Fig. 3) and there is only
one access to a Gaussian random generator per time index k
(see Fig 2(c)).
While the number of received molecules is an unsigned
integer value, the Gaussian distribution is defined for real
numbers. Therefore, y[k] needs to be postprocessed:
y [k] = max {0, y[k]} .
(11)
Fig. 3. Variance calculation for the EDTCM with Gaussian approximation.
This postprocessing step causes an increasing approximation
error if the mean of y[k] is close to zero.
D. Poisson Approximation
Especially for unlikely events the binomial distribution can
be well approximated by the Poisson distribution. The relation
between both distributions can be approximated as
B (N, h l ) ≈ P (λl = Nh l ) ,
(12)
where P(λ) describes a Poisson distribution with parameter λ
which is equal to the mean and variance of the distribution [15]. Similar to the Gaussian distribution, the sum of
several independent Poisson distributions can be described by
a single Poisson distribution, thus y[k] can be approximated
as
L
x[k − l]h l .
(13)
y[k] ≈ P λ[k] = z[k] =
l=0
The resulting EDTCM is shown in Fig. 2(d).
E. Accuracy of Approximations
The channel models introduced in Sections III-B, III-C,
and III-D approximate the sum of binomial random variables defined for the exact channel model introduced in
Section III-A. The accuracy of approximation can be analyzed by investigating the mean Kullback-Leibler divergence (MKLD) between the exact Poisson binomial probability mass function PR and the proposed approximations thereof
over all possible input sequences s:
1
D̄(R||B/N /P) = L+1
2
s∈{0,1} L+1
⎡
⎤
wH
(s)N
(n)
P
R
⎦,
×⎣
PR (n) log2
PB/N /P (n)
n=0
(14)
where wH (·) is the Hamming weight. In Fig. 4(a)–4(c)
the MKLD for the system parameters given in Table I
is shown. For the binomial approximation, the MKLD is
decreasing with an increasing distance, which is similar to a
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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017
Fig. 4. Mean Kullback-Leibler divergence between Poisson binomial distribution and approximations (Tb = 0.5 s, L = 5).
TABLE I
S IMULATION PARAMETERS U SED FOR A NALYSIS ;
THE D EFAULT PARAMETERS A RE IN B OLDFACE
of the binomial Poisson distribution except for the Poisson distribution for small distances and the binomial distribution for a
small number of molecules. A practical comparison by means
of bit error simulations will confirm this result in Section V.
F. Complexity Analysis
decreasing hitting probability. Furthermore, it is decreasing
with an increasing number of molecules, similar to the
case of Gaussian approximation. An increasing number of
molecules reduces the uncertainty of the channel output which
is represented by a decreasing variance. For the Poisson
approximation, the MKLD is decreasing when the distance
is increased. This confirms the statement that the Poisson
approximation fits well for unlikely events. In Fig. 4(e)–4(f)
the differences between the MKLD of the approximations
is shown. Therefore, it is possible to define specific regions
where an approximation is superior with respect to another
approximation. Often the difference between the MKLD is
small in the scenario under investigation. In case of a small
number of molecules it can be stated that the binomial approximation lead to a higher approximation error. Furthermore, the
Poisson approximation should be avoided for small distances.
Overall, all three approximations lead to a fair approximation
The main parameters (number of molecules, channel coefficients, mean, variance) of the channel models introduced
in Sections III-A, III-B, III-C, and III-D can be calculated
efficiently by convolutions or can be stored in look-up tables.
Therefore, the main computational complexity is due to the
random number generators (RNGs). There exist efficient algorithms for Gaussian RNG, Poisson RNG [41] and binomial
RNG [42] with constant complexity O(1) independent of
N and h l . Thus, the complexity of all approximation models
is O(1), while the complexity of the exact channel model
is O(L). Therefore, the suggested approximations reduce the
complexity by a factor of L. As a result, the complexity
reduction of the proposed approximations improves with the
channel memory length L.
G. Signal to Interference Plus Noise Ratio
Due to the fact that there is amplitude-dependent noise
present in the MC system, it is useful to define an averaged
signal to interference plus noise ratio (SINR). Because the
mean and variance of the sum of binomial distributions are
equal to the mean and variance of the Gaussian approximation,
DAMRATH et al.: EQUIVALENT DISCRETE-TIME CHANNEL MODELING FOR MOLECULAR COMMUNICATION
the averaged SINR is defined as
μ2
SINR = 2
μISI + σ 2N
=
N h0
l=1
l=0
h 20
2
L
hl
+
N (h l − h l2 )
1
p(u=1)N
l=1
L
l=0
,
(15)
(h l − h l2 )
where μ is the averaged number of molecules that reach Rx
in their first bit interval after release, μISI is the averaged
number of molecules that reach Rx after their first bit interval
after release and causes ISI, σ 2N is the averaged diffusion
noise variance, and N = p(u = 1)N is the averaged number
of emitted molecules. The definition of SINR visualizes the
fact that the diffusion noise can be reduced by increasing the
number of molecules N, while the ISI is only dependent on
the channel coefficients.
For the bit error analysis throughout this paper four different
detection algorithms are applied and compared. First of all,
the common fixed threshold detector (FTD) is used, where the
fixed threshold is chosen to be optimal in terms of minimizing
the BER by means of an exhaustive search. Second, the
low-complexity adaptive threshold detector (ATD) introduced
in [32] is applied, where the threshold is determined by the
previous number of received molecules. Third, the maximumlikelihood sequence estimator (MLSE) based on the squared
Euclidean distance metric is applied [43]. As a fourth detector
we adapt the a posteriori probability (APP) detector, which can
be implemented by means of the BCJR algorithm [44]. While
the latter two algorithms are too complex to be implemented
inside a NM (at least in the near future), they serve as a
benchmark in the simulations. The MLSE and the BCJR
algorithm work on a trellis diagram with M L states [43], [44],
where M is the symbol cardinality (in case of OOK M = 2).
For adapting the BCJR algorithm to the EDTCM it is necessary to define the transition probability γ (si , s j , k) from
state s i = [si,1 , · · · , si,L ] to state s j at time index k. For
the EDTCM proposed in Section III-A, the probability mass
function (PMF) of the sum over binomial distributions can be
described as a Poisson binomial distribution, which can be
calculated with help of the discrete Fourier transformation as
ξs
,s
N
i j
1
C −my[k]
γ si , s j , k =
1 + ξsi ,s j N
N
1 + C m − 1 sl h l .
(18)
According to the channel models introduced in Sections III-B,
III-C, and III-D, the PMF can be approximated by binomial,
Gaussian, and Poisson distributions, respectively. Therefore,
we can approximate γ (si , s j , k) as follows:
N−y[k]
N
y[k]
γB s i , s j , k =
psi ,s j 1 − psi ,s j
,
(19)
y[k]
2
y[k] − μsi ,s j
1
exp −
,
γN s i , s j , k =
2σs2i ,s j
2πσ 2
s i ,s j
(20)
γP s i , s j , k =
y[k]
λsi ,s j
y[k]!
exp −λsi ,s j ,
(21)
where
psi ,s j = ũh 0 +
L
sl h l ,
(22)
l=1
IV. D ETECTORS
L
l=1
2
L
+
N hl
=
N
(m) = 1 + C m − 1 ũh 0
·
2
L
s i ,s j
65
s i ,s j
(m) , (16)
m=0
where ξsi ,s j = wH (si ) + wH (ũ) is the Hamming weight of the
state si plus the Hamming weight of the bit hypothesis ũ from
state s i to state s j , and
2π
,
(17)
C = exp j
1 + ξsi ,s j N
μsi ,s j = λsi ,s j = N psi ,s j ,
σs2i ,s j = μsi ,s j − N ũh 20 −
L
sl h l2 .
(23)
(24)
l=1
Note that it is straightforward to adapt the MLSE metric to
the transition probabilities γ (si , s j , k). However, in this work
the adaptation is applied only to the BCJR algorithm, which is
optimal in the sense that it minimizes the bit error probability.
V. S IMULATION R ESULTS
For simulative analysis throughout this paper the scenario
under investigation is inspired by [32] and summarized in
Table I, where K is the bit sequence length for one channel
realization and R is the total number of channel realizations.
In contrast to [32] the denaturation of the molecules is fixed
to the channel memory length L = 5. In order to verify the
introduced EDTCM and approximations thereof, the results
of [32] are reproduced by applying the detectors described
in Section IV to the proposed channel models. During the
simulations the impact of transmission distance d, number of
molecules N, and bit duration Tb on the BER is investigated.
This is done by varying one system parameter, while fixing
the other ones to N = 1000, d = 30 µm, and Tb = 0.5 s. The
simulation results based on the exact EDTCM can be seen
in Fig. 5 and show similar findings as in [32]. All detection
algorithms perform better if N is increased, d is decreased,
and/or Tb is increased, which is equivalent to increase the
signal to interference plus noise ratio in classical wireless
communications. While an increasing number of molecules
additionally attenuate the stochastic behavior of the diffusion
process, a decreasing transmission distance and a longer bit
duration weaken the effect of ISI. The only detection algorithm
that is not improving the performance with an increase of Tb in
Fig. 5(c) is ATD. In fact, the ATD algorithm benefits from ISI
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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017
Fig. 5. Bit error rate performance as a function of the number of molecules (a), transmitting distance (b), and bit duration (c). If the corresponding
parameter is not varying, it is fixed to N = 1000, d = 30 µm, and Tb = 0.5 s.
Fig. 6. Normalized mean squared error between the bit error rate of the exact channel model (see Fig. 5) and approximations thereof (fixed threshold
detection and adaptive threshold detection).
and struggles if ISI is decreased. Furthermore, the results show
that MLSE and BCJR outperforms simple detection algorithms
like FTD and ATD, while ATD is superior to FTD for a
certain range of Tb and d (for a more detailed analysis the
reader is referred to [32]). The BCJR algorithm introduced
in Section IV achieves a slightly better performance than the
MLSE algorithm by adapting the transition probability to the
channel statistics.
However, the obtained BER in Fig. 5 are smaller and especially the MLSE detector performance is remarkably improved
compared to [32]. This is due to two main reasons. Firstly, the
simulation results using the exact channel model are equal to
a random walk simulation with a sufficient small simulation
time step, which provides more accurate results. In fact, the
simulation time step has to be as small as possible, but much
larger than τp = m/(6πηrs ) to model the propagation as
Brownian motion [45]. Thereby m and rs are the mass and
the Stokes’ radius of the propagating molecule, and η is the
viscosity of the fluid medium. Secondly, the channel memory
length in this work is set to L = 5, while there is an
infinite channel memory length assumed in [32]. Nevertheless,
Fig. 5 confirms the qualitative statements obtained in [32].
In order to analyze the impact of the approximations
compared to the exact channel model, Figs. 6 and 7 show
the normalized mean squared error (MSE) between the BER
obtained by the exact channel model and the BER obtained by
the approximated channel models, where the squared error is
normalized to the BER of the exact channel model. The MSE
is shown for the cases of FTD and ATD (Fig. 6) and for the
case of BCJR detection (Fig. 7). Overall, for the system under
consideration in Fig. 6, the approximations do not deviate
remarkably from the exact channel model (for almost every
point MSEBER < 10−3 ). In Fig. 6(a) the effect of the number
of molecules on the MSE is shown. For N ≥ 5000, the
MSE of the binomial and Poisson approximation is nearly
constant, while the MSE of the Gaussian approximation is still
DAMRATH et al.: EQUIVALENT DISCRETE-TIME CHANNEL MODELING FOR MOLECULAR COMMUNICATION
67
Fig. 7. Normalized mean squared error between the bit error rate for BCJR with Poisson binomial distributed transition probabilities and BCJR
algorithm with approximated transition probabilities, respectively. Each BCJR algorithm is applied to the exact channel model and to the related
approximated channel model.
decreasing for the FTD. This result supports the fact that the
binomial distribution converges to the Gaussian distribution
for n → ∞. From the trend in Fig. 6(b) and Fig. 6(c) it
can be concluded that the approximation accuracy gets better
if the transmission distance is increased and/or the bit duration is decreased. An increasing transmission distance and/or
decreasing bit duration leads to smaller channel coefficients.
This is especially beneficial for the Poisson distribution, which
is a good approximation for unlikely events. As a result, the
performance of the Poisson approximated channel gets worse
if the channel coefficients are increased. This can be observed
in Fig. 6(b) and 6(c) at d = 20 µm and Tb = 1.5 s.
In Fig. 7 the normalized MSE of the BCJR algorithm
applied to the channel approximations and to the exact channel
model is shown. Thereby, the transition probabilities for the
BCJR algorithm are adapted to the various approximated
channel models described in Section IV. Each of those approximated BCJR algorithms is applied to the exact channel
model and to the approximated version of the channel model
related to the applied transition probabilities. Furthermore,
the abscissae are limited to the area of Fig. 5, where the
BCJR algorithm produces reasonable BER for the simulation
parameters under investigation. In general, the MSE follows
the same qualitative behavior of ATD in Fig. 6, except for the
performance of ATD applied to the Gaussian approximated
channel model. If N or Tb is increased, the MSE increases
as well, while an increasing d leads to a decreasing error.
From Fig. 7 it can be observed that applying the binomial
BCJR and the Poisson BCJR to the exact channel model give
a better MSE performance than applying them to the related
approximated channel models. In contrast, the Gaussian BCJR
behaves differently. For N ≥ 1000, d ≤ 40 µm and over the
whole range of simulated bit durations, applying the Gaussian
BCJR to the Gaussian approximated channel model gives a
lower MSE than applying the Gaussian BCJR to the exact
channel model.
As a result, Figs. 6 and 7 give an insight, which approximated channel model and which approximated BCJR detector
can be applied. Thereby, the choice is affected by the acceptable MSE and the given simulation parameters. As an example,
(i) assuming MSEBER ≤ 10−3 , (ii) assuming simulation
parameters according to Table I with N = 1000, d = 30 µm
and Tb = 0.5 s, and (iii) assuming that the BCJR algorithm
should not run with the Poisson binomial distributed transition
probability (in order to save computational complexity at
the detection step). In that case the Gaussian channel model
together with the Gaussian BCJR should be applied.
VI. C ONCLUSION
In this paper an equivalent discrete-time channel model for
three-dimensional diffusion-based molecular communication
using the first passage time concept is introduced. Due to
the fact that the EDTCM is well-known in the wireless communications community, the proposed model facilitates the
understanding of diffusion-based molecular communication
and supports the adaptation of classical wireless communication algorithms to the area of MC. Furthermore, it implies
a great potential for reducing simulation complexity compared to random walk based MC simulators. While the exact
EDTCM is based on the summation of binomial distributions,
three channel models are proposed which approximate the
summation by binomial, Gaussian, and Poisson distributions,
respectively, in order to reduce computational complexity. The
approximation accuracy is analyzed by means of the KullbackLeibler divergence. Favored regions for each approximation
are highlighted. In addition, the transition probability for the
BCJR algorithm is adapted to each proposed channel model. In
numerical BER simulations the exact EDTCM in conjunction
with FTD, ATD, MLSE, and BCJR is studied. In addition, the
improvement of the adapted BCJR algorithm in comparison to
MLSE is shown. Furthermore, the difference between the exact
EDTCM and the approximations thereof in conjunction with
68
IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017
the proposed detection methods are analyzed. The results show
an acceptable small error over a wide range of simulation parameters for the scenario under consideration. While OOK and
energy detection is assumed throughout this work, the channel
models can be extended easily to other modulation schemes
like molecule shift keying or concentration shift keying, and
different detection strategies like amplitude detection.
Future work includes expanding the EDTCM to additional
assumptions, e. g. including the process of ligand receptor
binding, or to consider transmitters with non-zero spatial size.
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