IEEE TRANSACTIONS ON COMMUNICATIONS 1 Molecular …

IEEE TRANSACTIONS ON COMMUNICATIONS 1

Molecular Signal Modeling of a Partially CountingAbsorbing Spherical Receiver

Bayram Cevdet Akdeniz, Student Member, IEEE, Nafi Ahmet Turgut, Student Member, IEEE,H. Birkan Yilmaz, Member, IEEE, Chan-Byoung Chae, Senior Member, IEEE,

Tuna Tugcu, Member, IEEE, and Ali Emre Pusane, Member, IEEE

Abstract—To communicate at the nanoscale, researchers haveproposed molecular communication as an energy-efficient solu-tion. The drawback to this solution is that the histogram of themolecules’ hitting times, which constitute the molecular signal atthe receiver, has a heavy tail. Reducing the effects of this heavytail, inter-symbol interference (ISI), has been the focus of mostprior research. In this paper, a novel way of decreasing the ISIby defining a counting region on the spherical receiver’s surfacefacing towards the transmitter node is proposed. The beneficialeffect comes from the fact that the molecules received from theback lobe of the receiver are more likely to be coming throughlonger paths that contribute to ISI. In order to justify this idea,the joint distribution of the arrival molecules with respect toangle and time is derived. Using this distribution, the channelmodel function is approximated for the proposed system, i.e., thepartially counting absorbing spherical receiver. After validatingthe channel model function, the characteristics of the molecularsignal are investigated and improved performance is presented.Moreover, the optimal counting region in terms of bit error rateis found analytically.

Index Terms—Molecular communication, partially countingreceiver.

I. INTRODUCTION

THROUGH billions of years of producing communicationat small scales (i.e., distances of up to a few micro/nano

meters), nature has provided, tested, and improved molecularcommunication (MC). Humans, on the other hand, struggleat this scale to utilize electromagnetic waves due to theconstraints imposed by the ratio of the antenna size to thewavelength of the electromagnetic signal [1], [2]. As an alter-native to electromagnetic signal, molecular signals have beenproposed for nanonetworks in order to overcome the hurdlesimposed by antenna constraint. There are other advantagesto molecular communication– molecular signals are typicallymore bio-compatible and can reach an intended receiver within

B.C. Akdeniz, and A. E. Pusane are with the Department of Electrical andElectronics Engineering, Bogazici University, Istanbul, 34342, Turkey (e-mail:[email protected] and [email protected]).

N.A. Turgut was with the Department of Electrical and Electronics Engi-neering, Bogazici University. He is now with the Electronics Engineering inKoc Univesity (e-mail: [email protected]).

T. Tugcu is with NETLAB, Department of Computer Engineering, BogaziciUniversity, Istanbul, 34342, Turkey (e-mail: [email protected]).

H. B. Yilmaz was with the Yonsei Institute of Convergence Technology,School of Integrated Technology, Yonsei University, Korea. He is now with theDepartment of Telematics Engineering, Universitat Politecnica de Catalunya,Barcelona, Spain (e-mail: [email protected]).

C.-B. Chae is with the Yonsei Institute of Convergence Technol-ogy, School of Integrated Technology, Yonsei University, Korea (e-mail:[email protected]).

challenging environments even on the macro-scale such aspipelines, tunnels, and saline water environments [3]. There-fore, researchers direct their attentions to molecular commu-nication to achieve communication in nanonetworks.

Most of the existing research on MC has focused onchannel modeling, interference mitigation, and modulationissues [4]–[7]. To address the challenges in a methodologicaland inclusive manner, IEEE has established the standardizationgroup IEEE P1906.1 for MC [8].

One of the main challenges in MC is to develop validchannel models capable of representing a time-dependentreceived signal. For the receiver and the reception processin diffusion-based MC models, there are mainly two typesof models– the passive and absorbing receivers. The formerassumes the molecules are unaffected by the receiver whilethe latter assumes the molecules are absorbed whenever theyhit the receiver. In the passive receiver case, the moleculescan pass through the receiver node surface multiple timeswithout interaction [6], [9], [10]. Therefore, the moleculesare allowed to contribute to the received signal multipletimes when the receiver is passive. For the absorbing receivercase, the molecules contribute to the received signal onlyonce and the molecules that hit the receiver are removedfrom the environment [11]–[14]. This process is modeled bythe first-passage process and to model the received signal,we focus on the time-dependent first hitting histogram [15].In [11] and [12], the received molecular signal is modeledin a one-dimensional (1-D) environment with an absorbingreceiver and the system performance is analyzed by utilizingthe received signal model at the physical layer. In [13], areceived signal model is introduced for a point transmitterand a spherical absorbing receiver in a 3-D environment.Since then, researchers have focused on modeling the receivedsignal for an absorbing receiver while relaxing some of theassumptions. Instead of using a fully absorbing receiver, theauthors have incorporated the receptor effect instead of usinga fully absorbing receiver [16]. Similarly, researchers haveutilized machine learning techniques to model the receivedsignal for a spherical reflecting transmitter with single absorb-ing receiver [17] or multiple point transmitters with multipleabsorbing receivers [18], [19]. In [20], the communication be-tween a spherical receiver and a spherical transmitter in whichthe surface is covered with evenly-spaced point transmittershas been modeled and the channel impulse response has beenpresented.

In addition to channel modeling, another major and common

2 IEEE TRANSACTIONS ON COMMUNICATIONS

challenge in MC systems is the inter-symbol interference (ISI),which is caused due to the late reception of some messengermolecules in the channel. Consequently, many recent workshave focused on overcoming this issue by proposing eithermodulation or equalization methods. In [21] and [22], theauthors have used, simultaneously, different types of moleculesas two orthogonal channels. Trying to cope with ISI molecules,the authors in [23] have released an additional type of messen-ger molecule. To reduce the ISI effect, the receiver observesand evaluates the difference in the number of molecules ofboth types. Although these works show promising results forreducing ISI, they require the usage of different types ofmolecules, hence different types of receptors at the receivercausing increased complexity of the system.

There are also other solutions that use one type of mes-senger molecule. For instance in [24], ISI is used as aconstructive component by adjusting the number of releasedmolecules so that the residual molecules lead to a beneficialeffect on decoding of the following symbols. In [25], con-ventional equalization methods like minimum mean squareerror (MMSE), decision feedback equalizers, and maximumlikelihood sequence estimation methods are proposed for MCchannels. While these methods have incremental effects onthe channel, they require a significant amount of additionalcomputational complexity.

In this paper, we model the received molecular signal fora partially counting absorbing receiver. That is, the receiverabsorbs all hitting molecules but those counted are only theones hitting at a specific site. Modeling the time-dependentreceived signal for such a system is an open issue and has thepotential to enhance the communication system performancewithout any significant additional cost. Most of the receivedmolecules are absorbed from the surface area facing towardsthe transmitter side. As the path to the back side of the receiveris longer, the receptors on that side are more likely to receivethe interference molecules. Therefore, limiting the countingarea to the front side with a limited surface area enhances thesignal quality. The main contributions of this paper are listedas follows:• The derivation of the joint distribution of the received

molecules with respect to time and angle• The modeling the received signal• The investigating of the signal properties, and• The finding of the optimal region for counting

for a partially counting and absorbing spherical receiver in adiffusion-based MC system.

II. SYSTEM MODEL

The received signal in a diffusion-based MC system isaffected by three main processes: emission, propagation, andreception. Analytical derivations for the channel model dependon the emitter, the receiver, the environment, and the propa-gation dynamics. Therefore, we give the details of the systembefore deriving the channel model.

A. Topology ModelWe consider a diffusion-based MC system with one point

transmitter and one spherical receiver in a 3-D environment

Tx𝒓𝟎

Absorbing & Counting

Absorbing But Not Counting

𝜽

𝒓𝒓 Rx

Fig. 1. System model of a diffusion-based MC with a point transmitter anda partially counting absorbing spherical receiver.

(Fig. 1). Novel feature of the receiver (Rx) is the abilityto count the molecules absorbed only through a specificregion. This feature complicates the modeling procedure ofthe received signal. In Fig. 1, the circular cap facing towardsthe transmitter node (Tx) counts the absorbed molecules whilethe rest of the surface area absorbs but does not count themolecules.

As shown in Fig. 1, the molecules propagate by the diffusionprocess when they are emitted from the Tx point. The distancebetween the emission point and the center of the receiver isdenoted by r0 and the radius of the absorbing receiver isdenoted by rr. The circular cap that counts is determined bythe θ angle, which we name as the counting region.

It is assumed that Tx and Rx nodes are fully synchronizedin the time domain, and the interactions between diffusingmolecules are ignored. No environmental or counting circuitnoise is considered; only the diffusion noise is consideredto isolate the signaling gain due to partial counting receiver.Furthermore, it is assumed that a mechanism in Rx nodedetermines the direction of Tx and aligns its counting regionfacing towards Tx.

B. Diffusion Model

The emitted molecules propagate subject to Brownian Mo-tion, which is described by the Wiener process [11]. TheWiener process W (t) is characterized as follows:• W (0) = 0,• W (t) is almost surely continuous,• W (t) has independent increments,• W (t2)−W (t1) ∼ N (0, c(t2 − t1)) for 0 ≤ t1 ≤ t2

where N (µ, σ2) is the Gaussian distribution with mean µand variance σ2, which means the variance of random stepsize is a constant (c) times the time difference. Simulatingthe Brownian Motion includes consecutive steps in an n-dimensional space that obeys Wiener process dynamics. For anaccurate simulation, time is divided into sufficiently small timeintervals (∆t), and at each time interval the molecules takerandom steps in all dimensions. In an n-dimensional space, arandom step is given as

∆ζ = (∆ζ1, ...,∆ζn),

∆ζi ∼ N (0, 2D∆t) ∀i ∈ {1, .., n},(1)

AKDENIZ et al.: MOLECULAR SIGNAL MODELING OF A PARTIALLY COUNTING ABSORBING SPHERICAL RECEIVER 3

𝑠[𝑘]

Modulator

Channel

Demodulator

CSK Selector

CSK Thresholding

𝝀𝟎 … 𝝀𝟐𝒎−𝟐

𝑠 [𝑘]

𝑵𝟎𝑻𝒙 … 𝑵𝟐𝒎−𝟏

𝑻𝒙

Fig. 2. Concentration-based modulator and demodulator.

where ∆ζ, ∆ζi, and D correspond to the random displacementvector, the displacement at the ith dimension, and the diffusioncoefficient, respectively.

C. Modulation & Demodulation

In this paper, concentration shift keying (CSK) based mod-ulation technique is used. General form of CSK is introducedin [5], [26]. In CSK based modulation techniques, the informa-tion is modulated on the amount of the transmitted moleculesat the start of each symbol duration (ts).

General structure of CSK based modulations is depictedin Fig. 2. For the kth symbol s[k], the modulator maps thesymbol to the amount of molecules to emit (i.e., ith symbolis mapped to NTx

i ) at the start of the kth symbol duration.Depending on the modulation order (m), the number ofpossible symbols is determined and equals to 2m. In this paper,we use binary CSK where m=1, i.e., it has two symbols s0and s1 which represent bit-0 and bit-1, respectively. After CSKselector maps the symbol to the amount, the molecules areemitted to the channel and they propagate by diffusion. Duringthe symbol duration, the arriving molecules are absorbed andcounted according to the counting logic. At the end of the kth

symbol duration, the final value is thresholded for obtainingthe detected symbol s[k] for the kth symbol.

III. CHANNEL MODEL FOR PARTIAL COUNTINGRECEIVER

The joint cumulative angle and time distribution of absorbedmolecules at the spherical receiver that are released by a pointtransmitter deserves an analytical derivation. This distributionfunction is utilized to determine the channel taps analyticallyfor the proposed receiver.

In the literature, marginal cumulative distribution with re-spect to time is derived for a fully absorbing spherical receiverand introduced to the MC domain from a communicationperspective [13] as

Fhit(t, r0, rr) =rr

r0erfc

(r0 − rr√

4Dt

), (2)

where erfc(·) is the complementary error function. Furthermore,the marginal angular distribution of the received molecules is

𝜽

𝒓𝒓

𝒓𝟎

Tx

𝒓𝟎∗ Rx

𝑨

Fig. 3. An infinitesimally small sphere over the circular region on the surfaceof the sphere. The circular region is determined by the angle θ.

3 - (°)0 30 60 90 120 150 180

p(3

)

0

0.1

0.2

0.3

0.4r0 = 10 7m

r0 = 12 7m

r0 = 14 7m

Max pt. for r0 = 10 7m



Fig. 4. Theta versus p(θ) curves for different r0 values (rr = 5 µm andD = 80 µm2/s). Maximum values are attained at 28.6◦, 34.3◦, and 40.1◦.

given in [15] (6.3.3a) for a 3-D medium when the time goesto infinity as

p(θ) = 2πr2r sin θ ε(θ) , (3)

where

ε(θ) =

(1−

r2rr20

)

4πrrr0

(1−

2rr

r0cos θ +

r2rr20

)3/2. (4)

In particular, p(θ) in (3) gives the distribution of themolecules absorbed by the cap that is defined by the angleθ which is presented in Fig. 3. This function is plotted fordifferent parameters in Fig. 4. As can be seen, the probabilityof absorption has a peak between θ = 0◦ and θ = 90◦.Furthermore, it is zero for θ = 0◦ and θ = 180◦. These are notsurprising since θ = 0◦ and θ = 180◦ represent only a point onthe surface. Therefore, the probability of absorption in theseregions is zero although θ = 0◦ is the closest point to thetransmitter. If we increase θ, we expect to have more receivedmolecules since the circular region gets bigger. However, aftersome point, the rate of increase is not enough compared to thedecrease in the hitting rate that can also be observed in Fig. 4.


Since the communication process occurs in a limited time,we need to obtain the joint distribution of absorbed moleculeswith respect to time and angle to apply partially countingreceiver system in MC. To the best of our knowledge, the jointdistribution with respect to time and angle has not been derivedyet. By utilizing (2) and (3), we find an approximate analyticalclosed-form expression for the joint cumulative distributionwith respect to time and angle. The main concept of ourapproach is to cover the desired region on the surface of thespherical receiver with infinitesimally small spheres and toevaluate the absorption probability of these spheres. In otherwords, each small patch of the spherical receiver is representedby an infinitesimally small sphere. First, we evaluate theprobability of absorption of a molecule by a small sphereplaced with an angle θ, as shown in Fig. 3. Accordingly, weadd up the probability of absorption of the all small spheresthat are placed with the same angle, as shown in Fig. 5, toobtain the overall probability of absorption for a given θ angle.

When we consider an infinitesimally small sphere withradius dr placed at the surface making an angle of θ withthe center of the sphere as shown in Fig. 3, the distance ofthis arbitrarily placed sphere to the point transmitter can becalculated using Cosine rule as

r∗0 =√r20 + r2r − 2r0rr cos θ. (5)

Considering Fig. 3, if only the small sphere at point A isavailable in the environment, the probability of absorption ofmolecules until time t could be obtained using (2) as

FAhit(t) = Fhit(t, r∗0 , dr) =

dr

r∗0erfc

(r∗0 − dr√

4Dt

). (6)

Since r∗0 � dr, we have r∗0 − dr ≈ r∗0 and, therefore, (6) canbe rewritten as

FAhit(t) ≈dr

r∗0erfc

(r∗0√4Dt

). (7)

In particular, (7) is not valid when the small sphere atpoint A is placed on the surface of the spherical receiver (alsocalled the big sphere). Since some part of the small sphere liesbehind the surface of the big sphere, this part does not act as areceiver. Furthermore, the receptors of the big sphere are alsoactive; hence, this event should also be taken into account. LetA∗ be the region of the active receptors of the small sphereplaced at point A. Clearly, our aim is to derive the probabilityof absorption of a molecule until time t with the active regionsof the small sphere at point A, which is FA

∗

hit (t).It is important to note that for different θ angles, the

orientation of the active regions of the small sphere is different,which can be observed from Fig. 6. This implies that, fordifferent θ angles, FA

∗

hit (t) should have a different formula thatdetermines the orientation of the active regions. Therefore,in order to derive FA

∗

hit (t), an angle factor T (θ) and a timefactor φ(t) are multiplied by FAhit(t) as the adjusting factorsto determine this difference. These adjusting factors will be

Fig. 5. Small spheres that make θ angle with the center of the big sphere.Note that, these spheres are lined up on a circle whose radius is rrsin(θ).

derived using marginal cases later on. Therefore, FA∗

hit (t) canbe represented as

FA∗

hit (t) =K FAhit(t)T (θ)φ(t), (8)

where K is the probability that a molecule is not absorbedby the small spheres other than the small sphere at pointA until time t. Since the probability of absorption of amolecule by the whole part of the receiver is Fhit(t, r0, rr),and since the probability of absorbing of a molecule by thesmall sphere at point A is FA

∗

hit (t), K can be written asK = 1−(Fhit(t, r0, rr)−FA

∗

hit (t)). Thus, FA∗

hit (t) can be writtenas

FA∗

hit (t) =[1−Fhit(t, r0, rr) + FA∗

hit (t)]FAhit(t)T (θ)φ(t)

≈ [1−Fhit(t, r0, rr)]dr

r∗0erfc(

r∗0√4Dt

)T (θ)φ(t),

(9)

since FA∗

hit (t)� Fhit(t, r0, rr).After approximating FA

∗

hit (t), for a small sphere placed atpoint A that makes θ angle as shown in Fig. 3, the next stepis to find the total number of small spheres that have same θangle. These small spheres are lined up on a circle with radiusrr sin(θ), as shown in Fig. 5. Since the radius of these spheresare infinitesimal, the total number of spheres on this circle canbe calculated by dividing the circumference of the circle to the

diameter of the small sphere as Nθ =2πrr sin(θ)

2dr. Using this

Nθ, the probability of absorption of a molecule until time tby any small sphere that makes same θ angle as the sphere atpoint A can be obtained by

p(θ, t) = NθFA∗

hit (t)

= πrr sin(θ)[1−Fhit(t, r0, rr)]erfc(

r∗0√4Dt

)r∗0

T (θ)φ(t).(10)

When t goes to infinity, (10) becomes equal to (3), wherelimt→∞

p(θ, t) can be obtained as

limt→∞

p(θ, t) = πrr sin θ(1− rrr0

)1

r∗0T (θ) lim

t→∞φ(t). (11)


Fig. 6. Demonstration for the orientation of the active receptors of the smallsphere (A∗) for different θ values. For the small sphere with θ1 = 0, theactive receptors are towards to the direction of the transmitter. As θ changes,the orientation of the active receptors changes hence this change should betaken into account.

Hence, equalizing (3) and (11) gives us T (θ) as

T (θ) =2rrr

∗0 ε(θ)(

1− rrr0

)limt→∞

φ(t). (12)

Although (12) contains limt→∞

φ(t), in the following steps thisterm is canceled out and p(θ, t) does not involve any limitterm.

The next step is deriving the other compensation function,φ(t). Note that p(θ, t) gives the distribution of molecules withrespect to angle θ until time t. Therefore, taking the integralof p(θ, t) with respect to θ from θ=0 to an arbitrary angle α,gives the cumulative distribution of molecules at the receiverwith respect to time and angle as

F (α, t) =

∫ α

0

p(θ, t) dθ. (13)

In (13), one can easily observe that, when α = π, all ofthe surface of the receiver is absorbing. Therefore, F (π, t)is equal to the marginal cumulative function given in (2) asF (π, t) = Fhit(t, r0, rr). By using this equality, we can obtainφ(t) as

φ(t) =Fhit(t, r0, rr)

πrr[1−Fhit(t, r0, rr)]Sπ, (14)

where

Sπ =

∫ π

0

sin θ

r∗0erfc(

r∗0√4Dt

)T (θ) dθ. (15)

Note that the denominator of φ(t) contains T (θ). Sincelimt→∞

φ(t) term in this integral can be taken outside of the inte-gral, we conclude that φ(t) involves lim

t→∞φ(t) in the numerator

while T (θ) involves this term in the denominator. Therefore,multiplying these two compensation functions together cancelslimt→∞

φ(t) terms in p(θ, t).

Fig. 7. p(θ, t) heat map for rr = 5 µm, r0 = 10 µm, and D = 80 µm2/s.

After finding φ(t), we can write p(θ, t) as

p(θ, t) =sin θ erfc

(r∗0√4Dt

)Fhit(t, r0, rr)

(1−2rrr0

cos θ+r2rr20

)32

∫π

0

sin θ′ erfc(

r∗0√4Dt

)(1− 2rr

r0cos θ′+

r2rr20

)32

dθ′

.

(16)

Fig. 7 is the heat map of p(θ, t) that gives the angular distri-bution of a molecule until time t. Considering this figure, someinteresting inferences can be obtained. Firstly, the moleculeswill accumulate less at the higher angles compared to thelower angles. This is expected since as the angle increases,the distance also increases, which leads to the diminishing ofthe probability of absorption. Second and more interestingly,at very small angles around zero that can also be considered asthe line of sight angles, the probability of absorption is evenlower compared to other angles. This is a consequence of thefact that the number of small spheres is very limited for theseangles (when θ = 0◦, there is only one point and consequentlyone small sphere); hence, the probability of absorption at theseangles is quite low.

Note that p(θ, t) is used for calculating F (α, t) that isgiven in (13) and results in (17) where Ei(.) is an exponentialintegral function.

Once F (α, t) is obtained, the channel tap for the nth

symbol duration, pn, can be obtained (for a given countingregion (α) and symbol duration ts) as

pn(α) = F (α, nts)− F (α, (n−1)ts) . (19)

IV. CHANNEL MODEL VALIDATION AND MOLECULARSIGNAL PROPERTIES

A. Received Signal Validation

Once the analytical distribution of the received moleculesfor partially counting system is obtained, the next step is tocompare it with the simulation results obtained by using (1).As can be seen in Fig. 8, validation is done for various param-eters with different α values, and simulation and theoreticalresults are coherent.


F (α, t) =

erfc(

(r0−rr)√4Dt

)(Dt erfc

(√r20−2r0rr+r2r√

4Dt

)+ 1

2√π

√Dt√r20 − 2r0rr + r2rEi

(− (r20−2r0rr+r

2r)

4Dt

))U(t) Dt

√r20−2r0rr+r2r

r20

−erfc

((r0−rr)√

4Dt

)(Dterfc

(√r20−2r0rr cos(α)+r2r√

4Dt

)+ 1

2√π

√Dt√r20 − 2r0rr cos(α) + r2rEi

(− (r20−2r0rr cos(α)+r2r)

4Dt

))U(t) Dt

√r20−2r0rr cos(α)+r2r

r20

(17)

U(t) =

∫ π

0

erfc(

√r20 + r2r − 2r0rr cos θ

√4Dt

)sin θ(

1−2rr

r0cos θ +

r2rr20

)3/2dθ =

r02(Dt erfc

(r0−rr√

4Dt

)+√Dt (r0 − rr)Ei

(− (r0−rr)2

4Dt

))D rrt (r0 − rr)

−r0

2(Dt erfc

(r0+rr√

4Dt

)+√Dt (r0+rr)Ei

(− (r0+rr)

2

4Dt

))Drrt (r0+rr)

(18)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

time(s)

F(α

,t )

theory, α=π/3

simulation, α=π/3theory, α=π/4

simulation, α=π/4theory, α=π/6

simulation, α=π/6

(a) rr = 5 µm, r0 = 10 µm and D = 80 µm2/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

0.3

time(s)

F(α

,t )

theory, α=π/3

simulation, α=π/3

theory, α=π/4simulation, α=π/4

theory, α=π/6

simulation, α=π/6

(b) rr = 5 µm, r0 = 9 µm and D = 80 µm2/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

0.3

time(s)

F(α

,t )

theory, α=π/3

simulation, α=π/3


theory, α=π/6

simulation, α=π/6

(c) rr = 5 µm, r0 = 10 µm and D = 160 µm2/s

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.05

0.1

0.15

0.2

0.25

time(s)

F(α

,t )

theory, α=π/3

simulation, α=π/3


theory, α=π/6

simulation, α=π/6

(d) rr = 10 µm, r0 = 20 µm and D = 80 µm2/s

Fig. 8. Comparison of the analytical cumulative function obtained in (17) with simulation results for α = π/3, π/4 and π/6 from top to bottom.

B. Peak Time

The communication literature considers the peak time,tpeak, to be a crucial property for characterizing the channel,and defines it as the time that the received signal makes apeak at the receiver. In [13], it is concluded that, for thefully absorbing receiver, tpeak is proportional with the squareof d = r0 − rr, which is the shortest distance from thetransmitter to the receiver’s surface. This is a major drawbackin molecular communication via diffusion (MCvD) channelssince as d increases, the data rate exponentially decreases tocapture the signal until its peak, while in electro-magnetic

communication this decrement is linear. We evaluate tpeak bytaking the derivative of F (α, t) with respect to time, which isthe hitting rate of the molecules for a given α, and examineits maximum value. As can be seen in Fig. 9, tpeak is stilldirectly proportional with d2, which is the same with the fullyabsorbing receiver case.

C. Optimum α for the Given Channel Parameters

The optimum reception angle, α∗, of the receiver in termsof bit error rate (BER) is determined by finding the position ofthe global minimum of BER formula of the CSK modulation


d - (7m)2 3 4 5 6 7 8 9 10

t pea

k

0

0.05

0.1

0.15

0.2, = 20°, = 40°, = 60°, = 80°

Fig. 9. tpeakvsd curves for rr = 5 µm, D = 80 µm2/s.

0 20 40 60 80 10010

−3

10−2

10−1

α (degree)

valu

e of

the

obje

ctiv

e fu

nctio

n

BERSIDmax. SID (anal.), α=31.78max. SID (sim.), α=32min. BER, α=32

Fig. 10. BER vs α curves and corresponding SID curves for rr = 5 µm,r0 = 10 µm, D = 80 µm2/s, ts = 150ms with minimum point of BERfunction obtained via computer simulations and maximum of the SID functionobtained with both simulation and analytical solution of (23).

with respect to α. On the other hand, closed-form of the BERformula in CSK is not a tractable function if the number of thechannel taps is high. Therefore, we use an alternative objectivefunction whose argument of the global maximum is almostthe same as the argument of the global minimum of BERas proposed in [27]. This function is named as the signal tointerference difference (SID), and gives the difference betweenthe first tap and the sum of the other taps:

SID = p1(α)−∞∑n=2

pk(α). (20)

As shown in Fig. 10, the argument of the global maximumof this function is very close to the argument of the globalminimum of BER. Using SID, the corresponding optimizationproblem is written as

α∗ = arg max0≤α≤π

[p1(α)−

∞∑k=2

pk(α)

]. (21)

Since∑∞k=2 pk(α) = F (α,∞) − p1(α), the optimization

problem can be rewritten as

α∗ = arg max0≤α≤π

[2F (α, ts)− F (α,∞)] . (22)

The solution of the optimization problem in (22) can besolved by taking the derivative of the objective functionwith respect to α and equating it to zero with reasonablesimplifications. The corresponding solution can be obtainedas

α∗ = cos−1

r20 + r2r −(√

2a(Y+M)Y

)22r0rr

, (23)

where a = Dts, Y = −2rrFhit(ts,r0,rr)U(ts)

, and M = − r20−r

2r

2r0.

The details of the derivation of α∗ is presented in Appendix.

V. PERFORMANCE ANALYSIS

In this section the performance analysis of the proposedsystem is examined for different parameters. We mainlyevaluate the performance of the system in terms of BER.These evaluations are done using channel taps, both obtainedanalytically by using (19) and Monte Carlo simulations byreleasing 105 molecules from the transmitter and recordingtheir arrival times and angles at the receiver. For all sim-ulations, binary CSK is used. In particular, for transmittingthe kth symbol s [k] ε {0, 1}, the transmitter releases NTx

1

molecules for s [k] = 1, and no molecule is released fors [k] = 0. Then, the number of absorbed molecules amongthose released molecules is determined using the channel tapsand the Gaussian distribution assumption, as stated in [28]. Atthe end of the kth symbol interval, the number of absorbedmolecules is determined. Finally, the number is thresholdedand s[k] is obtained. The threshold is always chosen to obtainthe lowest BER to determine ultimate performance of thesystems. The performed simulations are conducted for 104

times, and for each simulation, 105 consecutive binary symbolsare transmitted in a channel with memory is 100 symbols.

We firstly evaluate the performance of the proposed systemwith respect to d and the diffusion coefficient (D). As canbe observed from Fig. 11, the optimum α in terms of BERincreases as D increases. This is expected since, as themolecules move faster, they can readily reach the furtherpart of the receiver; hence, α should be increased in ordernot to miss the molecules coming during the current symbolslot. Similar results can be observed from Fig. 12 whereoptimum α increases as the distance between the transmitterand the receiver decreases. Especially in the current timeslot, the molecules can move towards the further parts of thespherical receiver as the distance decreases or D increases.Therefore, the relative gain of the first tap compared to othertaps increases by increasing α when the distance is shorteror D is higher. Furthermore, one can deduced from Fig. 11and 12, as ts is increased the optimum α will also increases.This is also expected since optimum α will be 180◦ when tsapproaches to infinity.

In Fig. 13, we present the BER curves of three systems;α = π (conventional receiver), α = π/2 (half sphere), andα = α∗ as well as their corresponding channel taps for bothsimulation and analytical results. Considering this figure, itcan be concluded that the performance of the system willbe significantly improved (BER is 102-104 times lower than


0 10 20 30 40 50 60 70 80 9010−3

10−2

10−1

100

α (degree)

BE

R

D=40 µ m2 /s

D=80 µ m2 /s

D=120 µ m2 /s

(a) ts=100ms

0 10 20 30 40 50 60 70 80 90

10−5

10−4

10−3

10−2

10−1

100

α (degree)

BE

R

D=40 µ m2 /s

D=80 µ m2 /s

D=120 µ m2 /s

(b) ts=150ms

0 10 20 30 40 50 60 70 80 90

10−8

10−6

10−4

10−2

100

α (degree)

BE

R

D=40 µ m2 /s

D=80 µ m2 /s

D=120 µ m2 /s

(c) ts=200ms

Fig. 11. BER vs α curves for rr = 5 µm, r0 = 10 µm and NTx1 = 500 with different diffusion coefficient (D) values

.

0 10 20 30 40 50 60 70 80 90

10−3

10−2

10−1

100

α (degree)

BE

R

d=4 µ md=5 µ md=6 µ m

(a) ts=100ms

0 10 20 30 40 50 60 70 80 90

10−6

10−4

10−2

100

α (degree)

BE

R


(b) ts=150ms

0 10 20 30 40 50 60 70 80 90

10−8

10−6

10−4

10−2

100

α (degree)

BE

R


(c) ts=200ms

Fig. 12. BER vs α curves for rr = 5 µm, D = 80 µm2/s and NTx1 = 500 with different d=r0-rr values.

the conventional receiver) if α is chosen properly. Althoughthe signal tap is also decreased with this method, due to thedecrease in ISI, this reduction is compensated. Furthermore,it can also be seen that analytical solutions using (17) andsimulations are coherent.

VI. CONCLUSION

In this paper, it has been confirmed that a partially countingabsorbing receiver demonstrates a significant improvementover the conventional fully absorbing one. Due to the natureof diffusion, it can be expected that the molecules received inthe back lobe of the receiver will most possibly take longertime to reach that point than the molecules received in thefront lobe. Therefore, the molecules absorbed in the back lobemost likely belong to the previous transmitted symbols. Thus,they contribute to ISI. We, therefore, have proposed a countingregion on the spherical receiver surface that faces towardsthe transmitter node. In order to justify this idea, we havederived the joint cumulative angle and the time distributionof the absorbed molecules at the receiver surface that hadyet to be derived in the literature. Using this function andsimulations, we have observed that the molecules are likely tobe accumulated with a certain range of angles, which satisfiesour claim. We, then, have examined the received signal modelfor various parameters. The optimum counting region to obtainthe lowest BER was also derived. We have presented hereevidence of the improved performance of the proposed system.As future work, our plans are to weight counting regions by an

optimization approach so as to improve the performance of thesystem even better than how it did here. We intend to adoptthis work to nanonetworks that involve one hub and manytransmitters that aim to send their messages to this hub, thecounting region of which should be assigned to the transmittersusing the concepts proposed in this work.

APPENDIXThe objective function SID = [2F (α, ts)− F (α,∞)] can

be written explicitly using (17) and discarding the α indepen-dent terms as

SID =Y(aerfc

( √x√4a

)+ 1√

2π

√a√xEi

(− (x)

4a

))a√x

+M√x(24)

= SID1 +M√x, (25)

where x = r20 − 2r0rr cos(α) + r2r , a = Dts, Y =−2rrFhit(ts,r0,rr)

U(ts), and M = − r

20−r

2r

2r0. Before taking the deriva-

tive of SID with respect to α, it needs to be simplified. SID1

can be expanded to the series around x = 0, and, since xis on the order 10−12, higher order terms can be neglected.Therefore, SID1 can be written as SID1 ≈ Y√

x+ Y log x

2√πa

. Usingthis approximation, SID can be written as

SID ≈ Y√x

+Y log x

2√πa

+M√x. (26)


100 200 300 400 500 600 700 800 900 1000

10−8

10−6

10−4

10−2

100

Number of molecules for bit−1

BE

R

theory, optimum αsimulation, optimum αtheory, α = π/2simulation, α = π/2theory, α = πsimulation, α = π

(a) BER curves for ts=300ms and d = 5 µm

100 200 300 400 500 600 700 800 900 100010−4

10−3

10−2

10−1

100


BE

R


(b) BER curves for ts=300ms and d = 7 µm

100 200 300 400 500 600 700 800 900 1000

10−4

10−3

10−2

10−1


BE

R


(c) BER curves for ts=400ms and d = 7 µm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.05

0.1

0.15

0.2

0.25

time(s)

chan

nel t

aps

theory, optimum αsimulation, optimum αtheory, α = π/2simulation, α = π/2

theory, α = πsimulation, α = π

(d) channel taps for ts=300ms and d = 5 µm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time(s)

chan

nel t

aps



(e) channel taps for ts=300ms and d = 7 µm

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

time(s)

chan

nel t

aps



(f) channel taps for ts=400ms and d = 7 µm

Fig. 13. Top: BER vs number of molecules per bit-1 (NTx1 ) curves for rr = 5 µm, D = 80 µm2/s with different d = r0-rr and ts values with receiver α

= 180◦ (convetional CSK), α = 90◦ and optimum α to obtain lowest BER. Bottom: corresponding channel taps of the communication systems.

Taking the derivative of SID with respect to α and equatingit to 0, we can arrive at

Y√πa(r20 − 2r0rr cos(α) + r2r)

=M + Y

(r20 − 2r0rr cos(α) + r2r)1.5 .

(27)

Hence, α∗ can be obtained by solving (27) as

α∗ = cos−1

r20 + r2r −(√

2a(Y+M)Y

)22r0rr

. (28)

ACKNOWLEDGEMENT

The work of and B.C. Akdeniz, H. B. Yilmaz, C.-B. Chae,T. Tugcu, and A. E. Pusane was supported in part by thejoint project titled MEDUSA between TUBITAK (116E916)of Turkey and NRF (2016K2A9A1A06926542) of Korea. Thework of T. Tugcu was also partially supported by the StatePlanning Organization (DPT) of Turkey under the projectTAM (2007K120610). The work of H. B. Yilmaz was alsopartially supported by the Government of Catalonia’s Secre-tariat for Universities and Research via the Beatriu de Pinospostdoctoral programme. The work of C.-B. Chae was alsopartially supported by the Basic Science Research Programthrough the NRF of Korea (2017R1A1A1A05001439).

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