Performance Analysis of Diffusive Molecular Communication in Three Dimensional Channel

Prabhat Kumar Sharma and Lokendra Chouhan Department of Electronics and Communication Engineering Visvesvaraya National Institute of Technology Nagpur - 440010, India (e-mail:[email protected], [email protected])

Abstract—In this paper, we analyze the performance of the effect of inter-symbol-interference (ISI). In [8], the authors 3-dimensional diffusive molecular communication in terms of examined the different modulation techniques in terms of receiver operating characteristics, the average probability of the bit-error-rate over time-varying channels considering error and capacity. The information is encoded through the number of molecules emitted from transmitter nanomachine, 3-dimensional (3-D). which follows the Brownian motion degrade over time, and Till date, most of the researches focused on molecular reach at the spherical receiver nanomachine in random time. communication in the 1-D environment where first hitting In this work, the expressions for the probability of detection, distribution for 1-D is used for analysis. However, the false alarm, the probability of error and capacity are derived. medium inside the blood capillary is 3-D space and Our simulations show that the radius of the receiver has a strong effect over the receiver operating characteristic and molecules can travel in any direction. Therefore, it is more the average probability of error, for considered system model. practical to consider the 3-D environment where RN is The expectancy of molecules have a great impact on the considered to be spherical with some finite radius. The arrival probability in a 3-D environment and hence on the expression for first hitting distribution for 3-D is derived capacity of the considered system. in [9]. The authors in [10] also investigated the different Index Terms—Molecular communication, first hitting dis- types of receivers (active, passive, reversible adsorption tribution, capacity, receiver operating characteristic (ROC). and desorption receivers) in 1-D and 3-D environment. In our analysis, we use the first hitting distribution derived in [9]. I. INTRODUCTION To the best of our knowledge, none of the earlier works Molecular communication is a nano-technology which in the existing literature analyzed the probability of error attracts a biologist, a physicist, and a communication and upper bound on the rate at which information can engineer for research [1]. Similar to the traditional com- be reliably transmitted over a 3-D diffusive molecular munication system, molecular communication consist of communication channel. Hence, in this paper, we analyze transmitter nanomachines (TN), propagation medium and the performance of molecular communication system in receiver nanomachines (RN). Recently molecular commu- the 3-D environment in terms of ROC, capacity and the nication has gained more attention by bio-medical field average probability of error. The closed form expressions for different applications such as nanomachines are used of optimum threshold, probability of error and capacity to transmit and gather information about malignant tumor are derived. We also consider the effect of practical inside the human body. The biological nanomachines have disturbances such as multiple-source interference (MSI) in-built sensing and actuating mechanism i.e., TN has and ISI at RN. the ability to actuate the transmission of information through molecules, by sensing the surrounding conditions. II. SYSTEM MODEL [2]. However, the information exchange is an important Consider a molecular communication system where a phenomenon for the identification of malignant tumor and point TN and a spherical RN of radius r are placed in send the drug to the target tumor without affecting healthy the 3-D environment as shown in Fig.1. Let d denotes tissues. Hence, molecular communication has attracted the distance between TN and surface of the spherical RN. significant research attention in recent years [3]. TN has the capability of the emitting fixed number of Authors in [4] modeled the channel and analyzed the molecules which propagates in the aqueous environment capacity for the 1-D environment (1-D) for extended and with diffusion coefficient Dm. The RN assumed to be naive modulations technique. The probability of error perfectly absorbing in nature and has an inbuilt mechanism and capacity are also analyzed for M-ary and extended of counting the number of molecules 1. For transmission, modulation schemes 1-D in [5]. The performance of amplitude shift keying (ASK) method is used i.e., M direct diffusive molecular communication with mobile number of molecules are transmitted for bit 1 and, no nanomachines is recently examined in terms of the ROC, molecules are transmitted for bit 0. The channel is divided the error probability and capacity [6]. In [7], relay as- into K number of time slots where information bits are sisted diffusion with drift based molecular communication system analyzed the probability of error considering the 1The TN and RN assume to be perfectly synchronized. 2

molecules and small arrival probability, the Binomial distribution can be approximated well by Poisson distribution [14]. By applying Poisson approximation, the term Nc[j] is approximated with average rate λc[j]=Mx[j]p0, i.e., Nc[j] ∼P(Mx[j]p0). Similarly, each term NI [i] in ISI can also be approximated as a Poisson distributed random variable i.e., NI [i] ∼P(Mx[j−i]pi). For MSI, we assume that the number of received unintended molecules for Fig. 1: Molecular communication system in 3-D scenario. different time-slots are independent and no two unintended molecules are captured at same time [15]. Therefore, No[j] transmitted in each time slot for τ duration. If K denotes also follows Poisson distribution with parameter λo[j], i.e., the total number of time slots, then jth time slot is defined No[j] ∼P(λo[j]). as the time periods [(j − 1)τ,jτ], where j ∈{1, 2, ...K}. The molecules emitted from TN follows the 3-D Brownian motion and propagate through the diffusion process and III. DETECTION PERFORMANCE ANALYSIS hit the RN surface. At RN after legend-receptor binding, The symbol detection problem at the RN can be formu- activated receptor are counted and the corresponding sym- lated as the binary hypothesis testing problem bol is detected [12]. At the beginning of each time-slot, j−1 the TN emits M molecules in the propagation medium H0 : N[j]= NI [i]+No[j], (7) with prior probability β for transmission of information i=1 symbol 1 or remains silent for transmission of information j−1 symbol 0. The molecules emitted from TN flow inside H1 : N[j]=Nc[j]+ NI [i]+No[j], (8) the aqueous environment get attenuated and have the i=1 life expectancy which follow the exponential distribution where H0 and H1 denote the null and alternative hy- with degradation parameter γ. The density function for potheses corresponding to the transmission of symbol 0 exponential distribution is given by and 1, respectively. Using the property of sum of inde- g(u)=γe−γu, (1) pendent Poisson distributed random variables, the number of molecules N[j] under H0 and H1 are distributed as, where, u referred as the lifetime of molecules. The first H0 : P(λ0[j]), H1 : P(λ1[j]), (9) hitting distribution for 3-D environment is already derived in [9] and is given by j−1 where, the parameters, λ0[j]= i=1 x[j − i]Mpi + λo[j] j−1 2 − r d −d and λ1[j]=Mp0 + i=1 x[j i]Mpi + λo[j]. Further f(t, i)= √ exp . (2) to find optimum threshold at RN log- likelihood-ratio-test r + d 4πDmt 4Dmt (LLRT) can be formulated as Due to the stochastic nature of the diffusive channel, the H Pr[N[j]]|H1 1 1 − β times of arrival at the RN of the molecules emitted by log ≷ log , (10) Pr[N[j]]|H0 H β the TN, are random in nature and can span multiple time- 0 slots. Let pj−i denote the probability that molecules are where, using the Poisson distributions under H1 and H0 transmitted in ith time-slot from TN and are arrived in −λ [j] N[j] e 1 (λ1[j]) jth time-slot at RN. The probability pj−i can be derived p(N[j]|H1)= , (11) using the first hitting time distribution [13] for mobile N[j]! −λ [j] N[j] nanomachines as e 0 (λ0[j]) p(N[j]|H0)= . (12) (j−i+1)τ ∞ N[j]! pm = pj−i = f(t; i) g(u) du dt, (5) (j−i)τ t Now the optimal decision rule at the RN can be obtained H1 as, N[j] ≷ η[j], where the optimal threshold η[j] is given Let N[j] is the total number of molecules received at RN H0 in jth time-slot and is expressed as by ln(1/β − 1) + λ1[j] − λ0[j] N[j]=Nc[j]+NI [j]+No[j], (6) η[j]= , (13) ln(λ1[j]) − ln(λ0[j]) where Nc[j], is the number of molecules observed from Let X[j] and Y [j] be two discrete random variables that current time-slot, which follows Binomial distribution represent the transmitted and received symbols in the jth B(Mx[j],p0), where p0 denotes the probability of arrival time-slot respectively. j−1 in the same time-slot. The quantity NI [j]= i=1 NI [i] Then detection probability (PD[j]) and false alarm denotes the ISI, i.e., number of molecules received from probability (PFA[j])atRNare − ∼B − η[j] previous 1 to (j 1) time-slots, where NI [i] (Mx[j −λ1[j] l e (λ1[j]) i],pi), with arrival probability of previous ith time slot pi. PD[j]=Pr(y[j]=1|x[j]=1)=1− . (14) l! The last term No[j] denotes the MSI. For large number of l=1 3

y j |x j I X j ,Y j y j |x j x j × Pr( [ ]=0 [ ]=0) y j |x j × x j ( [ ] [ ]) = Pr( [ ]=0 [ ]=0)Pr( [ ]=0) log2 y j + Pr( [ ]=1 [ ]=0) Pr( [ ]=0) Pr( [ ]=0) y j |x j y j |x j × Pr( [ ]=1 [ ]=0) y j |x j × x j × Pr( [ ]=0 [ ]=1) log2 y j + Pr( [ ]=0 [ ]=1) Pr( [ ]=1) log2 y j Pr( [ ]=1) Pr( [ ]=0) y j |x j y j |x j × x j × Pr( [ ]=1 [ ]=1) , + Pr( [ ]=1 [ ]=1) Pr( [ ]=1) log2 y j (3) Pr( [ ]=1) − P j P j − P j − β (1 FA[ ]) P j − β FA[ ] = (1 FA[ ])(1 )log2 y j + FA[ ](1 )log2 y j Pr( [ ]=0) Pr( [ ]=1) (1 − PD[j]) PD[j] + (1 − PD[j])β log + PD[j]β log , (4) 2 Pr(y[j]=0) 2 Pr(y[j]=1)

η[j] −λ0[j] l e (λ0[j]) for the given values of PFA[j] and PD[j] in jth time slot, PFA[j]=Pr(y[j]=1|x[j]=0)=1− . (15) l! the capacity is obtained by maximizing the I(X[j],Y[j]) l=1 as a function of prior probability β. Differentiating Eq. (4) where α gives the largest integer less than or equal to with respect to the prior probability β and equating zero

α. q0[j] H(PFA[j])−H(PD[j])+(PD[j]−PFA[j]) log2( )=0, q1[j] IV. AVERAGE PROBABILITY OF ERROR AND CAPACITY (23) The expression for probability of error for jth time slot where, H(a) is binary entropy function and deﬁned as, is derived as follows H(a)=−a log2(a) − (1 − a)log2(1 − a). Also, q0[j]= Pr(y[j]=0)and q1[j]=Pr(y[j]=1). Using (23) the Pe[j]=Pr(y[j]=0|x[j]=1)Pr(x[j]=1) expression of prior probability βm at maximum mutual + (y[j]=1|x[j]=0) (x[j]=0), information is derived as Pr Pr (16) H(PD [j])−H(PFA[j]) P [j]−P [j] using (14)-(15), Pe[j] can be written as PFA[j] 2 D FA +1 −1 β , m = H(P [j])−H(P [j]) (24) Pe[j]=(1− PD[j]) β + PFA[j](1− β). (17) D FA ) P [j]−P []j (PFA[j]−PD[j]) 2 D FA +1 The average probability of error can be written as K The closed form expression for the channel capacity is 1 derived using (4) and (24) as follows P¯e = Pe[j]. (18) K j=1 C[j]=H(q0[j])−(1−βm)H(PFA[j])−βmH(PD[j]). (25) Let X[j] and Y [j] be two discrete random variables that represent the transmitted and received symbols in the jth time-slot respectively. The channel capacity is found by V. N UMERICAL RESULTS maximizing the end to end mutual information in each time slots. The mutual information between X[j] and Y [j] In this section, we numerically analyze the system using 5 can be obtained as Monte-Carlo simulations for 10 samples. The system I(X[j],Y[j]) = Pr(y[j]|x[j]) x[j]∈{0,1} y[j]∈{0,1} Pr(y[j]|x[j]) × Pr(x[j]) log , (19) 2 (y[j]) Pr Analytical r = 1 nm, d = 20 nm (Simulation) The eq. (19) is simpliﬁed in eq. (3)-(4). where, 10-1 r = 2.5 nm, d = 20 nm (Simulation) r = 5 nm. d = 20 nm (Simulation) Pr(y[j] = 0) = Pr(y[j]=0|x[j]=0)Pr(x[j]=0) + Pr(y[j]=0|x[j]=1)Pr(x[j]=1) =(1−PFA[j])(1−β)+(1−PD[j])β, (20) Average probabilty of error Similarly, 10-2 Pr(y[j] = 1) = PFA[j](1 − β)+PD[j]β. (21) 0 1020304050607080 Threshold (# of molecules) Using the above results with number of time slots Fig. 2: P¯e vs η. →∞ K [13], the capacity of a diffusive molecular com- parameters are chosen such that the number of molecules munication channel in 3-D environment can be obtained M and the arrival probability pm satisfy the Binomial to by maximizing the mutual information I(X[j],Y[j]) as Poisson approximation conditions, i.e., Mpm<10. C[j]=max(I(X[j],Y[j]) bits/slot, (22) Fig. 2 shows the average probability of error per- β formance as a function detection threshold for different 4

values of radius of spherical RN. For the simulations, the 1 parameters are chosen as: K =5, M = 100, d =20nm, 0.9 −10 λo[j]=10, ∀j, γ =10, β =0.5 and Dm =5× 10 0.8 2 m /s. The parameters remain same until unless stated. 0.7 Two observations can be made, ﬁrst, is that the average 0.6 probability of error achieves its lower bound at optimum 0.5 threshold value η[j]. Second, for the large value of r the 0.4

¯e Capacity in bits/slot 0.3 lower-bound of P is less, compare to the case when the Analytical r r =2.5 0.2 r = 0.5 m (Simulation) value of is small. For example at nm the lower- r = 1 m (Simulation) P¯e r =1 0.1 r = 1.5 m (Simulation) bound of is .085, while, at nm the lower-bound r = 2.5 m (Simulation) P¯e 0 of increases to 0.25. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time slot duration (s)

1 Fig. 4: Capacity of 3-D molecular communication system

0.9 vs time slot duration

0.8 0.7 is also derived. Simulation results are perfectly matched 0.6 with all the numerical results we have developed. Future

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