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Capacity analysis for diffusive molecular communication with ISI channel
Accepted Manuscript Capacity analysis for diffusive molecular communication with ISI channel Zhen Cheng, Yihua Zhu, Kaikai Chi, Yanjun Li, Ming Xia
PII: DOI: Reference:
S1878-7789(17)30091-1 http://dx.doi.org/10.1016/j.nancom.2017.07.002 NANCOM 187
To appear in:
Nano Communication Networks
Received date : 23 May 2016 Revised date : 5 January 2017 Accepted date : 24 July 2017 Please cite this article as: Z. Cheng, Y. Zhu, K. Chi, Y. Li, M. Xia, Capacity analysis for diffusive molecular communication with ISI channel, Nano Communication Networks (2017), http://dx.doi.org/10.1016/j.nancom.2017.07.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Capacity analysis for diffusive molecular communication with ISI channel Zhen Cheng*, Yihua Zhu, Kaikai Chi, Yanjun Li, Ming Xia School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] *Corresponding author
Abstract: The Inter-Symbol Interference (ISI) in Diffusive Molecular Communication (DMC) is unavoidable due to the stochastic behavior of molecular Brownian motion in a biological environment. In this paper, we analyze the channel capacity of DMC model by considering the ISI from all the previous time slots and the channel transmission probability in each time slot. We first use the Poisson distribution to approximate the Binomial distribution in order to obtain the total number of molecules received in the current time slot. Then we derive the mathematical expressions of optimal decision threshold by using the Skellam distribution. On this basis, we formulate the mutual information and bit error probability. Our numerical results are presented to show the channel capacity can be maximized with optimal lower bit error probability by controlling the channel transmission probability in each time slot and using the Skellam distribution. More importantly, under the same physical parameters, a higher capacity is achieved with using fewer molecules in our work. Keywords: Diffusive molecular communication, channel capacity, mutual information, optimal decision threshold, channel transmission probability, bit error probability 1. Introduction The rapid development in the field of nanotechnology enables manufacturing nanomachines with size ranging from one to few hundreds of nanometers [1]. Nanomachines can be interconnected to form a nanonetwork to share information and perform complex and collaborative tasks [2]-[3]. More important applications of nanonetworks are being considered in biomedical, industrial and environmental areas [4]-[6]. In particular, molecular communication is a promising communication technique to realize nanonetworks [7]. In Diffusive Molecular Communication (DMC) model [8], the information is transmitted, propagated and received in a biological environment based on the exchange of molecules [9]. The molecules released by the Transmitter Nanomachine (TN) obey the laws of molecules diffusion,
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and some of them may reach the Receiver Nanomachine (RN) [10]. TN and RN refer to the sender and receiver as in the traditional communication, respectively. In recent years, there have been growing interests and research efforts dedicated to DMC. Pierobon and Akyildiz proposed a physical end-to-end molecular communication model [11], and then they made noise analysis for DMC systems by assuming continuous molecular emissions at TN [12]. Ahmadzadeh et al. proposed three different relaying schemes for multi-hop networks to improve the communication range of DMC [13]. Singhal et al. investigated modulation techniques for end-to-end communication and made performance analysis of amplitude modulation schemes for DMC [14]. Noel demonstrated the accuracy of transformation model between active and passive receiver for DMC [15]. Deng presented an analytical model for DMC system with a reversible adsorption receiver based on concentration shift keying modulation [16]. However, DMC poses unique challenges based on the mechanism of molecules propagation with the Inter-Symbol Interference (ISI) from the residual molecules diffusion in previous time slots. One of the important challenges of DMC is how to improve the capacity with ISI channel. Some researchers have investigated the analysis of capacity. The binary modulation mechanism was presented to analyze the capacity performance of DMC model [17]. A scheme of average capacity analysis of DMC model by using finite life expectancy of molecules was proposed in [18]. The authors designed a binary digital DMC system for maximizing the mutual information in [19], and also discussed a detection technique for DMC model in the presence of ISI [20]. Atakan [21] proposed optimal transmission probability to maximize the molecular communication rate by using binary DMC model. Each of these works leaves out one or another critical effect. For example, the paper [17] only considered the ISI resulting from the symbol from previous one time slot, and the works in [18]-[20] made analysis only considering the same transmission probability of bit 1 in each time slot. The molecular communication model [21] was relatively simple, in which, only one molecule was emitted in each time slot which would result in high unreliability of receiving molecules for RN. In addition, the ISI from all the previous time slots unavoidably exists due to the reason that the propagation of the molecules is governed by the laws of Brownian motion. At the same time, different channel transmission probability can be used to determine the average number of bits 1 in the code words to ensure high capacity. Based on the above considerations, our work provides
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analysis of the channel capacity of the DMC model by considering the ISI from all the previous time slots and different channel transmission probability in each time slot. The main contributions of our paper are summarized as follows: (1) We use the Poisson distribution to approximate the Binomial distribution to obtain the total number of molecules received in the current time slot by RN considering the ISI. Thus we derive the mathematical expressions of optimal decision threshold by using the Skellam distribution. (2) On the basis of optimal decision threshold, we give the formulas of mutual information. Our numerical results are presented to show how the parameters including the distance between TN and RN, diffusion coefficient, the number of slots, each time slot duration and channel transmission probability have impacts on the capacity. (3) More importantly, the channel capacity can be maximized by controlling the channel transmission probability in each time slot by using the Skellam distribution. Under the same physical parameters, a higher capacity is obtained in comparison with the previous works in [19] and [20]. At the same time, fewer molecules are needed in our paper. (4) We formulate the bit error probability based on the derivation of optimal decision threshold. Our results show that the bit error probability is optimal lower with the number of the molecules emitted by TN in each time slot. The remainder of our paper is organized as follows: Section 2 will describe the DMC model. Section 3 makes analysis of the capacity and bit error probability based on the Skellam distribution. Numerical results are given in Section 4. Finally, this paper is concluded in Section 5. 2. DMC model In this section, we examine the DMC model between TN and RN in a biological environment. We consider a time-slotted system model with stochastic molecules diffusion and assume all events occur at some discrete time T=nTs. Here n is an integer which is the number of time slots, and Ts is one time slot duration. We also assume that TN and RN are perfectly synchronized in terms of time. The five main processes of DMC model include information encoding process, transmission process, propagation process, reception process and information decoding process which can be implemented in the biological environment. In the transmission process, the binary channel is used to transmit binary information from TN to RN in time T. Both input and output are bit 1 or 0 in each transmission. When TN wants to
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transmit a single message of one bit 1, TN instantaneously emits M molecules which are prepared in the information encoding process at the beginning of each time slot, while releases no molecule to signify bit 0. This modulation method is known as ON/OFF keying. The information of the address carried by each molecule in its DNA sequence indicates the logical address of the destination which is to arrive at. In the propagation process, the movement of molecules is assumed to be governed by a Brownian motion. Then, the time t experienced by any molecule to reach RN obeys the following probability density function [4] which characterizes the delay for one dimensional motion of molecules:
f (t )
d 4 Dt 3
e
d2 4 Dt
, t 0
(1)
where D is the environment diffusion coefficient which describes the tendency of the propagating molecules to diffuse through the medium, and d is the distance between TN and RN. The probability P(d,t) that a molecule is received by RN within time t can be calculated as follows:
P(d , t ) erfc(
d ), t 0 2 Dt
(2)
After the molecules propagate in the biological environment, the reception event by RN may be occurred. If RN receives at least one molecule, it can recognize and interpret the message by decoding the received molecules. Once the address from the decoded information is the same as the address of RN, the molecule is considered to be successfully received by RN. Then we propose an approach which can decide the output (the bit received) is 0 or 1. From the independence of molecules emitted at the beginning of each time slot, RN counts the number of the received molecules, which is simply the sum of the number of molecules in each emission. The number of received molecules in the current time slot n which is denoted by Nc is compared with some predefined decision threshold θ. If Nc≥θ, the output is 1. If Nc≤θ, the output is 0. Thus the value of optimal decision threshold plays an important role in the estimation of channel capacity. For a practical bio-inspired system, the molecules would need to bind to receptors on the surface of RN. Note that all molecules which arrive at RN are absorbed immediately and removed from the biological environment upon their first hits at the receiver.
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3. Capacity analysis 3.1 The ISI analysis At the beginning of time slot k (1≤k≤n), TN releases M molecules to represent transmission of bit 1 with the channel transmission probability βk, while releases no molecule to signify bit 0 with the probability (1-βk). The probability that the emitted molecule is successfully received in time slot k is βkP(d,Ts) and it can’t be successfully received in time slot k is βk(1-P(d,Ts)). Now we consider the molecules propagated from the previous (n-1) time slots have impacts on the number of molecules received by RN in time slot n. The probability Pkn (1≤k≤n) that one molecule of the M molecules emitted in time slot k is received in time slot n can be computed as the following formulas:
Pkn k [ P(d ,(n k 1)Ts )) P(d ,(n k )Ts ))]
(3)
As the system model introduced in Section 2, all the molecules are independent of each other and each molecule reaches RN with probability βkP(d,Ts) in time slot k (1≤k≤n). In particular, let βc be the probability of transmission of bit 1 in the current time slot n, and Mc is denoted as the number of the molecules which are emitted by TN and received by RN both in the current time slot n. Thus Mc follows the Binomial distribution [22]:
M c ~ Binomial ( M , c P(d , Ts ))
(4)
Fig.1 shows the probability Pkn (1≤k≤n) in (3) is varying with the number of time slots. Here, we set d=30μm, D=100μm2/s, Ts=5s, n=20. The value of the probability Pkn (1≤k≤n) is around 0.1. It is so small that the random variables which represent the number of molecules emitted in time slot k and received in time slot n follow the Binomial distribution, which can be approximated by the Poisson distribution according to the probabilistic theory.
0.18 0.16 0.14
Probability(Pkn)
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0.12 0.10 0.08 0.06 0.04 0.02 0
0
2
4
6
8
10 n
12
14
16
18
20
Fig.1 Probability Pkn versus the number of time slots n Now we use the Poisson distribution to approximate the Binomial distribution, then the expression in (4) can be written as M c ~ Poisson(M c P(d , Ts ))
(5)
Based on the DMC model, RN can receive the residual molecules from all the previous time slots, which results in the interferences to the subsequent bit reception. Such interference is defined as ISI in molecular communication. Thus, in the current time slot n, the number of molecules from total interferences caused by all the previous (n-1) time slots, which is denoted by MISI, follows the Poisson distribution as follows: n 1
M ISI ~ k ( Poisson( MP(d , ( n k 1)Ts )) Poisson( MP( d , ( n k )Ts )))
(6)
k 1
3.2 Deriving the optimal decision threshold 1-PF
H0 : X=0
Y=0
PF 1-P H1 : X=1
D
PD
Y=1
Fig.2 DMC model of bit transmission as a binary hypothesis testing channel In Fig.2, X and Y denote the input and output in the current time slot, respectively. H0 and H1 represent the cases that TN transmits bit 0 and 1 in the current time slot, respectively. PF is the false alarm probability and PD is the detection probability which can be defined as follows:
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PF Pr(Y 1| X 0) PD Pr(Y 1| X 1) 1 PF Pr(Y 0 | X 0)
(7)
1 PD Pr(Y 0 | X 1) Under the hypothesis H0 and H1, the numbers of molecules received in the current time slot n labeled M H 0 and M H1 follows the Poisson distributions, respectively. n 1
M H 0 ~ k ( Poisson( MP(d , ( n k 1)Ts )) Poisson( MP( d , ( n k )Ts )))
(8)
k 1
M H1 ~ Poisson( MP(d , Ts )) n 1
k ( Poisson( MP(d , (n k 1)Ts )) Poisson(MP (d , (n k )Ts )))
(9)
k 1
Let
1 >0 and 2 >0. We say that the random variable V follows the Skellam distribution,
denoted by Skellam( 1 , 2 ) [23], if and only if V=U1-U2
(10)
where U1 and U2 are two independent random variables such that Ui∼Poisson( i ) for any i {1, 2} . The probability mass function (pmf) of Skellam distribution which is Skellam( 1 , 2 ) is
a function of l, and it is expressed [23]-[24] as follows:
f (l ; 1 , 2 ) e
( 1 2 )
1 2
l /2
I l (2 12 )
(11)
2 k l
x 2 I l ( x) k 0 k ! ( k l 1)
(12)
where l represents the value of the difference between two Poisson random variables U1 and U2,
1 and 2 are the means, or expected values of the two Poisson distributions Ui∼Poisson( i ) ( i {1, 2} ). Il(x) in (12) is the modified Bessel function of the first kind and x is a real number. Here, ( p 1) p( p) for any positive integer p. According to (8) and (9), we use Skellam distribution to describe the difference of two Poisson variables which represents the number of molecules received by RN. It is labeled the random variable Z which follows Skellam distribution based on the distribution of M H 0 and M H1 in (8)
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and (9) respectively as follows:
H 0 : Z ~Skellam( 1 , 2 )
(13)
H1 : Z ~Skellam(1 , 2 ) where the parameters of Skellam distribution which can be obtained by (8) and (9) are n 1
1 k MP (d , (n k 1)Ts )
(14)
k 1
n 1
2 k MP(d , (n k )Ts )
(15)
k 1
n 1
1 =MP(d , Ts ) k MP(d , (n k 1)Ts )
(16)
k 1
n 1
2 k MP(d , (n k )Ts )
(17)
k 1
With the hypothesis test model introduced above, we can derive the optimal decision using the maximum-a-posterior (MAP) detection method [25]. The MAP test is equivalent to the likelihood-ratio test as follows: P ( z | H1 ) P( z | H 0 )
(18)
where the value of η can be obtained from [25] as follows:
1 c
(19)
c
According to (11), the formulas (18) can be written as z /2
e ( 1 2 ) 1 I z (2 12 ) 2 z /2 ( 1 2 ) 1 e I z (2 12 ) 2 According to the reference [26], if x>> z 2 1/ 4 , I z ( x)
(20)
1 e x is a well-known 2 x
asymptotic approximation. Thus we can get the following formulas
I z (2 12 )
I z (2 12 )
1 2 ( 12 )
1/4
1 2 (12 )
1/4
e2
e2
12
12
(21)
(22)
There are two reasons that we use the asymptotic approximation of I l ( x) instead of exact one.
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On one hand, in order to get the value of optimal decision threshold θ, we need to obtain the solution of equation (20) which is also the value of random variable Z in (13). Thus we should focus on the computation of I z (2 12 ) and I z (2 12 ) . On the other hand, because the computation ability of RN is limited, it is difficult to compute the exact values of I z (2 12 ) and I z (2 12 ) by the complex formulas of I l ( x) which is given in (12). From equations (20) to (22), we obtain the following equation z /2
1 1 2 e e 12 1/4 2 ( ) 2 1 2 z /2 1 2 e ( 1 2 ) 1 e 12 1/4 2 ( ) 2 1 2 ( 1 2 )
(23)
The solution of equation (23) is expressed as follows:
2 ln (1 2 ) ( 1 2 ) 2 12 2 12 1/ 4 ln 1 2 12 z round 2 1 ln 1 2
(24)
where round represents the rounding operation since the decision threshold should be a positive integer. Thus the value of optimal decision threshold θ can be calculated by (24). In order to test a hypothesis, we must have a cumulative distribution function (cdf) of a Skellam distribution. Because the pmf of the Skellam distribution is defined at integer values, we have expressions for the cdf
F ( ; 1 , 2 )
e
( 1 2 )
l
1 2
l /2
I l (2 12 )
(25)
Especially, we can see that l which represents the difference of two Poisson random variables, is a non-negative integer. Thus we calculate the cdf as follows:
F ( ; 1 , 2 ) e l 0
l /2
( 1 2 )
1 I l (2 12 ) 2
(26)
On this basis, the false alarm probability PF and the detection probability PD are computed as follows:
PF Pr(M c | X 0) 1 F ( ; 1 , 2 )
(27)
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PD Pr(M c | X 1) 1 F ( ; 1 , 2 )
(28)
Thus we calculate the channel capacity of DMC model using the formulas below C max I ( X ; Y )
(29)
where 1
1
I ( X ; Y ) P(Y | X )P( X ) log X 0 Y 0
P(Y | X ) P(Y )
(30)
Furthermore, the error probability of transmitting the random bit in the current time slot n is expressed by
Pe c (1 PD ) (1 c ) PF
(31)
4. Numerical results In this section, we first use MATLAB to obtain numerical results of the channel capacity. The aim is to investigate how the different parameters including the probability of transmission of bit 1 in each time slot, distance d between two nanomachines, diffusion coefficient D, the number of time slots n and each time slot duration Ts have impacts on mutual information. Then we also show that the bit error probability is lower varying with the parameter M which is the number of the molecules emitted by TN in each time slot and demonstrate that the selection of value of M can ensure the reliability for RN. 4.1 The probability βk (1≤k≤n-1) We first observe the effect of βk (1≤k≤n-1) on the mutual information. Suppose the current time slot to be the time slot n. When we fix the probability βc=0.6, the mutual information is varying with the value of βk (1≤k≤n-1). In Fig.3, the curve shows a monotonically decreasing trend of mutual information as a function of the value of βk (1≤k≤n-1). Here, we suppose the probability of transmission of bit 1 in each of the previous (n-1) time slot to be equivalent. We can see that the smaller the value of βk (1≤k≤n-1), the larger the capacity. It is based on the fact that more residual molecules from the previous (n-1) time slots will result in more interferences to the current time slot n. Here, we set the parameters as D=100μm2/s, d=20μm, Ts=100s, n=10, M=50.
0.8 0.7 0.6 0.5
I(X;Y)
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0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
βk (1≤k≤n-1)
0.7
0.8
0.9
1
Fig.3 Attainable mutual information versus βk (1≤k≤n-1) The parameters used to make analysis of the channel capacity from Section 4.2 to Section 4.3 are given in Table 1 as follows: Table 1. Simulation parameters Symbol
Description
Value
d
Distance between two nanomachines
20 μm
D
Environment diffusion coefficient
100 μm2/s
M
Number of the molecules emitted by TN in each time slot
10
n
Number of time slots
20
Ts
Each time slot duration
100 s
Note that although the parameters d, D, M, n and Ts are not completely consistent in all figures in the numerical results, they are mainly used to better reflect their effects on our different research targets including analysis of capacity, bit error probability and reliability, and especially show the corresponding change trends in DMC model. 4.2 Distance d between two nanomachines and diffusion coefficient D We plot the achievable mutual information in relation to the channel transmission probability in the current time slot βc with different values of distance d and diffusion coefficient D in Fig.4 and Fig.5, respectively. For the same value of βc, the shorter the distance d and the larger the diffusion coefficient D, the larger the capacity. It is concluded that on one hand, the molecules with shorter distance between two nanomachines can be easily received. On the other hand, the environment
with a larger diffusion coefficient results in faster and further molecules propagation. In Fig.4 and Fig.5, we set Ts=100s, n=20, M=10, βk=0.0001 (1≤k≤n-1). In addition, we set D=100μm2/s in Fig.4 and d=20μm in Fig.5. However, the capacity reaches its peak at some particular value of βc, and then the capacity is decreasing after some value of βc. The numerical result is coincided with the fact that the more the molecules in the biological environment, the more effects of ISI and the slower diffusion, thus the capacity is observed to decrease after some value of βc.
0.8 d=10μm d=20μm d=30μm
0.7 0.6
I(X;Y)
0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
βc
0.6
0.7
0.8
0.9
1
Fig.4 Attainable mutual information versus βc with different value of d 0.7
D=30m2/s D=50m2/s D=100m2/s
0.6 0.5
I(X;Y)
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0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
βc Fig.5 Attainable mutual information versus βc with different value of D
4.3 Number of time slots n and each time slot duration Ts The number of time slots and each time slot duration are also critical parameters for the channel capacity. Fig.6 and Fig.7 show the mutual information is varying with βc for different values of number of time slots n and the time slot duration Ts, respectively. In Fig.6 and Fig.7, we set d=20μm, D=100μm2/s, M=10, and βk=0.0001 (1≤k≤n-1). Furthermore, we set Ts=100s in Fig.6. For the same value of βc, the less number of time slots, the larger the capacity. It is based on the fact that the more the number of time slots, the more the ISI to the current time slot. The parameters are set as n=20 in Fig.7. It is obvious that the molecules can be easily received with longer time slot duration. For the same value of βc, the longer the time slot duration Ts, the larger the capacity. In Fig.6 and Fig.7, we can see that the channel capacity is increasing with βc. However, the capacity reaches its peak at some particular value of βc. This incidence can be explained as follows: when the diffusion environment is filled with more molecules after some time, the propagation becomes more slowly, thus the capacity is found to start decreasing after some value of βc.
0.7 n =10 0.6
n =20 n =30
0.5
I(X;Y)
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0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
βc
0.6
0.7
0.8
0.9
1
Fig.6 Attainable mutual information versus βc with different value of n
0.7 TS =10s 0.6
TS =50s TS =100s
0.5
I(X;Y)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
βc
0.6
0.7
0.8
0.9
1
Fig.7 Attainable mutual information versus βc with different value of Ts The main reasons of the asymmetry of the curves from Fig.4 to Fig.7 are as follows: (1) In our paper, we derive the mathematical expressions of optimal decision threshold and the mutual information by using the Skellam distribution. The trends of the curves from Fig.4 to Fig.7 are similar to the curve of cdf of the Skellam distribution. (2) When the value of βc increases and is smaller than the particular value at which the capacity has its peak, the number of molecules is also increasing in the biological environment, then the probability that the molecules are received in the current time slot n by RN also increases. In such a case, the channel capacity is increasing with βc. With the increasing number of molecules, the speed of capacity growth is slowly. (3) However, the capacity reaches its peak at some particular value of βc which has the range of [0.6, 0.7]. There are more and more molecules in the biological environment. Then the capacity is decreasing after some value of βc. Therefore, more effects of ISI are also generated which will result in the fact that the diffusion is slower and the speed of capacity decreasing is fast. 4.4 Bit error probability Now we give the numerical results of error probability of one bit transmission in the current time slot on the basis of calculation of optimal decision threshold in (24) by using the Skellam distribution. Fig.8, Fig.9 and Fig.10 show the error probability is varying with the number of the molecules emitted by TN in each time slot by the formulas (27), (28) and (31) when the current transmit bit is bit-0, bit-1 and random transmit bit, respectively. In these three figures, we set the
parameters d=20μm, D=100μm2/s, Ts=2s, n=10, βk=0.0001 (1≤k≤n-1), βc=0.5.
Error Probability
10
10
10
-1
-2
-3
0
2
4
6
8
10 M
12
14
16
18
20
Fig.8 The error probability of transmitting bit-0 in the current time slot In Fig.8, we can see that the error probability of transmitting bit-0 in current time slot increases monotonically with the increasing number of the molecules emitted by TN in each time slot. When the bit 0 is transmitted in the current time slot n, TN releases no molecules according to the mechanism of modulation. In such a case, the molecules from the previous time slots are easily received by RN in the current time slot with the increasing value of M. 10
Error Probability
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
10
10
0
-1
-2
0
2
4
6
8
10 M
12
14
16
18
20
Fig.9 The error probability of transmitting bit-1 in the current time slot Fig.9 plots the error probability of transmitting bit-1 in the current time slot. We see that the error probability of transmitting bit-1 decreases monotonically with the increasing value of M. It is based on the following fact: when the bit 1 is transmitted in the current time slot with emission of
M molecules by TN, the molecules from the current time slot and all the previous time slots can be easily received by RN in the current time slot. Therefore the probability that no molecules are received by RN is very low when the number of the molecules emitted by TN in each time slot is increasing. 10
0
-1
Error Probability
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10
-2
10
-3
10
0
2
4
6
8
10
12
14
16
18
20
M
Fig.10 The error probability of transmitting random bit in the current time slot Fig.10 shows the error probability of transmitting random bit in the current time slot is decreasing with the value of M. According to the formulas (31), the value of Pe depends on the optimal decision threshold , M and the physical parameters including D, d, n and Ts. For the same value of and the physical parameters, Pe is decreasing according to the numerical results in Fig.8 and Fig.9 when M is increasing. Interestingly, when the value of M is larger than 100 and continues to increase, the error probability of transmitting random bit in the current time slot will be increasing. This can be explained by the fact that larger number of molecules released in each time slot by RN will result in more residual molecules existed in the biological environment, which will lead to more ISI for RN. 4.5 Comparison to related work For the related work, the authors in [17] only consider the ISI from previous one time slot, and the mean mutual information for n time slots not for the n-th time slot is maximized in [18]. The work in [21] is primarily aiming at ultra shorter-range communication in which the distance between two nanomachines is 0.1μm. Thus we mainly compare our results with the work in [19] and [20] respectively.
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For the parameters d=20μm, D=100μm2/s, n=50, a peak in capacity is 0.61 bit/s and 0.23 bit/s with the time duration Ts=100s and Ts=8s in [20], respectively. However, we can get the higher capacity which is 0.64 bit/s and 0.41 bit/s with Ts=100s and Ts=8s, respectively. The reason for this comparison result is due to the interferences from the back time slots to the current time slots which are also considered. For the parameters d=20μm, D=100μm2/s, n=20, the capacity reaches a peak value of 0.28 and 0.15 with the time duration of Ts=100s and Ts=10s in [19], respectively. For the same parameters, from the numerical result in Fig.7, we can see that the capacity reaches a peak value of 0.68 bit/s and 0.49 bit/s with the time duration of Ts=100s and Ts=10s in our paper, respectively. In addition, in [19], the mean number of molecules emitted at each time slot is 1mol, and it is much larger than our work in which the number of molecules emitted at each time slot is 10. Therefore, the number of the molecules needed in our work is significantly smaller than [19] to achieve higher capacity. Using fewer molecules has two benefits. On one hand, fewer molecules result in fewer interferences. On the other hand, the cost of molecular device and the time for preparing smaller number of molecules is less than the case for preparing larger number of molecules which can improve the transmission efficiency. However, the selection of number of molecules emitted in each time slot should ensure that RN can receive at least one molecule of the M molecules. This case can be measured by the reliability. Here, the reliability labeled r is defined as the probability that at least one molecule of the M molecules is received by RN in the current time slot n. On the basis of the formulas Pkn in (3), we can obtain the formulas of computing r by the following equation n 1
r 1 (1 c F (Ts )) M (1 Pkn ) M k 1
(32)
1 0.9 0.8 0.7
Reliability(r)
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0.6 0.5 0.4 0.3 0.2 0.1 0
0
10
20
30 M
40
50
60
Fig.11 Reliability versus M In Fig.11, we set D=100μm2/s, d=20μm, Ts=10s, n=20. We can see that the reliability is increasing with the number of molecules emitted in each time slot. When the value of M is larger than 10, the reliability is close to 1. Thus, the value of parameter M which is set in our paper can ensure the reliability for RN. 5. Conclusions The objective of our paper is to make analysis of capacity of DMC model with ISI channel. The expression of optimal decision threshold is first derived by considering the effect of ISI from all the previous time slots and using the Skellam distribution. Then a closed form of channel capacity is proposed as a function of physical parameters as well as the channel transmission probability. We also formulate the bit error probability under the optimal decision threshold. Our numerical results show that how the parameters including the distance between TN and RN, the diffusion coefficient, the number of slots, each time slot duration and channel transmission probability have impacts on the capacity. In particular, the proposed channel transmission probability in each time slot plays an important role in maximizing the capacity. Therefore, we can maximize the channel capacity better by using Skellam distribution and controlling the channel transmission probability with lower bit error probability. At the same time, fewer molecules are needed in our paper to achieve higher capacity under the same physical parameters, and we also demonstrate that the selection of value of M can ensure the reliability for RN. As the future work, we plan to extend our work by using other modulation techniques proposed
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for the DMC system in the literatures. Also we aim to provide analysis of the channel capacity by considering the ISI under more complex communication topologies. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 61472367, 61401397) and Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY15F020029, LY14F020020) References [1] I.F. Akyildiz, J.M. Jornet, M. Pierobon, Nanonetworks: a new frontier in communications, Commun. ACM 54 (11) (2011) 84-89. [2] O.C. Farokhzad, R. Langer, Impact of nanotechnology on drug delivery, ACS Nano 3 (1) (2009) 16-20. [3] J. Wood, Integrated circuit built on single nanotube: nanotechnology, Mater. Today 9 (5) (2006) 13-13. [4] J.W. Yoo, D.J. Irvine, D.E. Discher, S. Mitragotri, Bio-inspired, bioengineered and biomimetic drug delivery carriers, Nat. Rev. Drug. Discov. 10 (2011) 521-535. [5] P. Bogdan, G. Wei, R. Marculescu, Modeling populations of microrobots for biological applications, IEEE International Conference on Communications, Ottawa, Canada, 2012. [6] T. Nakano, M. Moore, F. Wei, A. Vasilakos, and J. Shuai, Molecular communication and networking: opportunities and challenges, IEEE Trans. Nanobiosci. 11(2) (2012) 135-148. [7] I.F. Akyildiz, F. Brunetti, C. Blzquez, Nanonetworks: a new communication paradigm, Comput. Networks 52 (12) 2260 (2008) [8] M. Pierobon, I.F. Akyildiz, Capacity of a diffusion-based molecular communication system with channel memory and molecular noise, IEEE Trans. Inform. Theory 59(2) (2013) 942-954. [9] T. Nakano, A. Eckford, T. Haraguchi, Molecular communication, Cambridge University Press, 2013. [10] L.P. Gine, I.F. Akyildiz, Molecular communication options for long range nanonetworks, Comput. Networks 53 (2009) 2753-2766. [11] M. Pieroborn, I.F. Akyildiz, A physical end-to-end model for molecular communication in nanonetworks, IEEE J. Sel. Areas Commun. 28 (4) (2010) 602-611. [12] M. Pierobon, I.F. Akyildiz, Diffusion-based noise analysis for molecular communication in
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nanonetworks, IEEE Trans. Signal Process 59 (6) (2011) 2532-2547. [13] A. Ahmadzadeh, A. Noel, R. Schober, Analysis and design of multi-hop diffusion-based molecular communication networks, IEEE Transactions on Molecular, Biological and Multi-Scale Communications 1(2) (2015) 144-157. [14] A. Singhal, R. K. Mallik, B. Lall, Performance analysis of amplitude modulation schemes for diffusion-based molecular communication, IEEE Trans. Wirel. Commun. 14(10) (2015) 5681-5691. [15] A. Noel, Y. Deng, D. Makrakis, A. Hafid, Active versus passive: receiver model transforms for diffusive molecular communication, IEEE Glebecom, Washington DC, USA, 2016. [16] Y. Deng, A. Noel, M. Elkashlan, A. Nallanathanet, K.C. Cheung, Modeling and simulation of molecular communication systems with a reversible adsorption receiver, IEEE Transactions on Molecular, Biological and Multi-Scale Communications 1(4) (2016) 1-15. [17] M.H. Kabir, K.S. Kwak, Molecular nanonetwork channel model, Electron. Lett. 49(20) (2013) 1285-1287. [18] T. Nakano, O. Yutaka, L. Jian-Qin, Channel model and capacity analysis of molecular communication with Brownian motion, IEEE Commun. Lett. 16 (6) (2012) 797-800. [19] M. Ling-San, Y. Ping-Cheng, C. Kwang-Cheng, I.F. Akyildiz, A diffusion-based binary digital communication system, Proc. Int. Conf. Communications, Ottawa, Canada, 2012. [20] M. Ling-San, Y. Ping-Cheng, C. Kwang-Cheng, I.F. Akyildiz, Optimal detection for diffusion-based communications in the presence of ISI, IEEE Globecom, California, USA, 2012. [21] B. Atakan, Optimal transmission probability in binary molecular communication, IEEE Commun. Lett. 17(6) (2013) 1152-1155. [22] M.Ş. Kuran, H.B. Yilmaz, T. Tugcu, I.F. Akyildiz, Interference effects on modulation techniques in diffusion based nanonetworks, Nano Commun. Netw. 2(2012) 65-73. [23] J.G. Skellam, The frequency distribution of the difference between two Poisson variates belonging to different populations, Journal of the Royal Statistical Society: Series A 109(3) (1946) 296-296. [24] Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind. Expo. Math. 26 (2008) 279-293.
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[25] H.V. Poor, An introduction to signal detection and estimation, 2nd ed. Springer, 1994. [26] P. Kasperkovitz, Asymptotic approximations for modified Bessel functions. J. Math. Phys. 21(1) (1980) 6-13.
*Brief Biography Click here to download Brief Biography: Brief Biography.doc
Author Biography Zhen Cheng received B.S. degree from Huanggang Normal University, Hubei, China, in 2004, and received M.S. and PH.D degrees from Huazhong University of Science and Technology, Hubei, China, in 2007 and 2010 respectively. She is currently an associate professor in the School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, China. Her current research interests include wireless networks, nanonetworks, molecular communication. She has published more than 30 technical papers in international proceedings and journals.
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Received date : 23 May 2016 Revised date : 5 January 2017 Accepted date : 24 July 2017 Please cite this article as: Z. Cheng, Y. Zhu, K. Chi, Y. Li, M. Xia, Capacity analysis for diffusive molecular communication with ISI channel, Nano Communication Networks (2017), http://dx.doi.org/10.1016/j.nancom.2017.07.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Capacity analysis for diffusive molecular communication with ISI channel Zhen Cheng*, Yihua Zhu, Kaikai Chi, Yanjun Li, Ming Xia School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] *Corresponding author
Abstract: The Inter-Symbol Interference (ISI) in Diffusive Molecular Communication (DMC) is unavoidable due to the stochastic behavior of molecular Brownian motion in a biological environment. In this paper, we analyze the channel capacity of DMC model by considering the ISI from all the previous time slots and the channel transmission probability in each time slot. We first use the Poisson distribution to approximate the Binomial distribution in order to obtain the total number of molecules received in the current time slot. Then we derive the mathematical expressions of optimal decision threshold by using the Skellam distribution. On this basis, we formulate the mutual information and bit error probability. Our numerical results are presented to show the channel capacity can be maximized with optimal lower bit error probability by controlling the channel transmission probability in each time slot and using the Skellam distribution. More importantly, under the same physical parameters, a higher capacity is achieved with using fewer molecules in our work. Keywords: Diffusive molecular communication, channel capacity, mutual information, optimal decision threshold, channel transmission probability, bit error probability 1. Introduction The rapid development in the field of nanotechnology enables manufacturing nanomachines with size ranging from one to few hundreds of nanometers [1]. Nanomachines can be interconnected to form a nanonetwork to share information and perform complex and collaborative tasks [2]-[3]. More important applications of nanonetworks are being considered in biomedical, industrial and environmental areas [4]-[6]. In particular, molecular communication is a promising communication technique to realize nanonetworks [7]. In Diffusive Molecular Communication (DMC) model [8], the information is transmitted, propagated and received in a biological environment based on the exchange of molecules [9]. The molecules released by the Transmitter Nanomachine (TN) obey the laws of molecules diffusion,
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and some of them may reach the Receiver Nanomachine (RN) [10]. TN and RN refer to the sender and receiver as in the traditional communication, respectively. In recent years, there have been growing interests and research efforts dedicated to DMC. Pierobon and Akyildiz proposed a physical end-to-end molecular communication model [11], and then they made noise analysis for DMC systems by assuming continuous molecular emissions at TN [12]. Ahmadzadeh et al. proposed three different relaying schemes for multi-hop networks to improve the communication range of DMC [13]. Singhal et al. investigated modulation techniques for end-to-end communication and made performance analysis of amplitude modulation schemes for DMC [14]. Noel demonstrated the accuracy of transformation model between active and passive receiver for DMC [15]. Deng presented an analytical model for DMC system with a reversible adsorption receiver based on concentration shift keying modulation [16]. However, DMC poses unique challenges based on the mechanism of molecules propagation with the Inter-Symbol Interference (ISI) from the residual molecules diffusion in previous time slots. One of the important challenges of DMC is how to improve the capacity with ISI channel. Some researchers have investigated the analysis of capacity. The binary modulation mechanism was presented to analyze the capacity performance of DMC model [17]. A scheme of average capacity analysis of DMC model by using finite life expectancy of molecules was proposed in [18]. The authors designed a binary digital DMC system for maximizing the mutual information in [19], and also discussed a detection technique for DMC model in the presence of ISI [20]. Atakan [21] proposed optimal transmission probability to maximize the molecular communication rate by using binary DMC model. Each of these works leaves out one or another critical effect. For example, the paper [17] only considered the ISI resulting from the symbol from previous one time slot, and the works in [18]-[20] made analysis only considering the same transmission probability of bit 1 in each time slot. The molecular communication model [21] was relatively simple, in which, only one molecule was emitted in each time slot which would result in high unreliability of receiving molecules for RN. In addition, the ISI from all the previous time slots unavoidably exists due to the reason that the propagation of the molecules is governed by the laws of Brownian motion. At the same time, different channel transmission probability can be used to determine the average number of bits 1 in the code words to ensure high capacity. Based on the above considerations, our work provides
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analysis of the channel capacity of the DMC model by considering the ISI from all the previous time slots and different channel transmission probability in each time slot. The main contributions of our paper are summarized as follows: (1) We use the Poisson distribution to approximate the Binomial distribution to obtain the total number of molecules received in the current time slot by RN considering the ISI. Thus we derive the mathematical expressions of optimal decision threshold by using the Skellam distribution. (2) On the basis of optimal decision threshold, we give the formulas of mutual information. Our numerical results are presented to show how the parameters including the distance between TN and RN, diffusion coefficient, the number of slots, each time slot duration and channel transmission probability have impacts on the capacity. (3) More importantly, the channel capacity can be maximized by controlling the channel transmission probability in each time slot by using the Skellam distribution. Under the same physical parameters, a higher capacity is obtained in comparison with the previous works in [19] and [20]. At the same time, fewer molecules are needed in our paper. (4) We formulate the bit error probability based on the derivation of optimal decision threshold. Our results show that the bit error probability is optimal lower with the number of the molecules emitted by TN in each time slot. The remainder of our paper is organized as follows: Section 2 will describe the DMC model. Section 3 makes analysis of the capacity and bit error probability based on the Skellam distribution. Numerical results are given in Section 4. Finally, this paper is concluded in Section 5. 2. DMC model In this section, we examine the DMC model between TN and RN in a biological environment. We consider a time-slotted system model with stochastic molecules diffusion and assume all events occur at some discrete time T=nTs. Here n is an integer which is the number of time slots, and Ts is one time slot duration. We also assume that TN and RN are perfectly synchronized in terms of time. The five main processes of DMC model include information encoding process, transmission process, propagation process, reception process and information decoding process which can be implemented in the biological environment. In the transmission process, the binary channel is used to transmit binary information from TN to RN in time T. Both input and output are bit 1 or 0 in each transmission. When TN wants to
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transmit a single message of one bit 1, TN instantaneously emits M molecules which are prepared in the information encoding process at the beginning of each time slot, while releases no molecule to signify bit 0. This modulation method is known as ON/OFF keying. The information of the address carried by each molecule in its DNA sequence indicates the logical address of the destination which is to arrive at. In the propagation process, the movement of molecules is assumed to be governed by a Brownian motion. Then, the time t experienced by any molecule to reach RN obeys the following probability density function [4] which characterizes the delay for one dimensional motion of molecules:
f (t )
d 4 Dt 3
e
d2 4 Dt
, t 0
(1)
where D is the environment diffusion coefficient which describes the tendency of the propagating molecules to diffuse through the medium, and d is the distance between TN and RN. The probability P(d,t) that a molecule is received by RN within time t can be calculated as follows:
P(d , t ) erfc(
d ), t 0 2 Dt
(2)
After the molecules propagate in the biological environment, the reception event by RN may be occurred. If RN receives at least one molecule, it can recognize and interpret the message by decoding the received molecules. Once the address from the decoded information is the same as the address of RN, the molecule is considered to be successfully received by RN. Then we propose an approach which can decide the output (the bit received) is 0 or 1. From the independence of molecules emitted at the beginning of each time slot, RN counts the number of the received molecules, which is simply the sum of the number of molecules in each emission. The number of received molecules in the current time slot n which is denoted by Nc is compared with some predefined decision threshold θ. If Nc≥θ, the output is 1. If Nc≤θ, the output is 0. Thus the value of optimal decision threshold plays an important role in the estimation of channel capacity. For a practical bio-inspired system, the molecules would need to bind to receptors on the surface of RN. Note that all molecules which arrive at RN are absorbed immediately and removed from the biological environment upon their first hits at the receiver.
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3. Capacity analysis 3.1 The ISI analysis At the beginning of time slot k (1≤k≤n), TN releases M molecules to represent transmission of bit 1 with the channel transmission probability βk, while releases no molecule to signify bit 0 with the probability (1-βk). The probability that the emitted molecule is successfully received in time slot k is βkP(d,Ts) and it can’t be successfully received in time slot k is βk(1-P(d,Ts)). Now we consider the molecules propagated from the previous (n-1) time slots have impacts on the number of molecules received by RN in time slot n. The probability Pkn (1≤k≤n) that one molecule of the M molecules emitted in time slot k is received in time slot n can be computed as the following formulas:
Pkn k [ P(d ,(n k 1)Ts )) P(d ,(n k )Ts ))]
(3)
As the system model introduced in Section 2, all the molecules are independent of each other and each molecule reaches RN with probability βkP(d,Ts) in time slot k (1≤k≤n). In particular, let βc be the probability of transmission of bit 1 in the current time slot n, and Mc is denoted as the number of the molecules which are emitted by TN and received by RN both in the current time slot n. Thus Mc follows the Binomial distribution [22]:
M c ~ Binomial ( M , c P(d , Ts ))
(4)
Fig.1 shows the probability Pkn (1≤k≤n) in (3) is varying with the number of time slots. Here, we set d=30μm, D=100μm2/s, Ts=5s, n=20. The value of the probability Pkn (1≤k≤n) is around 0.1. It is so small that the random variables which represent the number of molecules emitted in time slot k and received in time slot n follow the Binomial distribution, which can be approximated by the Poisson distribution according to the probabilistic theory.
0.18 0.16 0.14
Probability(Pkn)
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0.12 0.10 0.08 0.06 0.04 0.02 0
0
2
4
6
8
10 n
12
14
16
18
20
Fig.1 Probability Pkn versus the number of time slots n Now we use the Poisson distribution to approximate the Binomial distribution, then the expression in (4) can be written as M c ~ Poisson(M c P(d , Ts ))
(5)
Based on the DMC model, RN can receive the residual molecules from all the previous time slots, which results in the interferences to the subsequent bit reception. Such interference is defined as ISI in molecular communication. Thus, in the current time slot n, the number of molecules from total interferences caused by all the previous (n-1) time slots, which is denoted by MISI, follows the Poisson distribution as follows: n 1
M ISI ~ k ( Poisson( MP(d , ( n k 1)Ts )) Poisson( MP( d , ( n k )Ts )))
(6)
k 1
3.2 Deriving the optimal decision threshold 1-PF
H0 : X=0
Y=0
PF 1-P H1 : X=1
D
PD
Y=1
Fig.2 DMC model of bit transmission as a binary hypothesis testing channel In Fig.2, X and Y denote the input and output in the current time slot, respectively. H0 and H1 represent the cases that TN transmits bit 0 and 1 in the current time slot, respectively. PF is the false alarm probability and PD is the detection probability which can be defined as follows:
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PF Pr(Y 1| X 0) PD Pr(Y 1| X 1) 1 PF Pr(Y 0 | X 0)
(7)
1 PD Pr(Y 0 | X 1) Under the hypothesis H0 and H1, the numbers of molecules received in the current time slot n labeled M H 0 and M H1 follows the Poisson distributions, respectively. n 1
M H 0 ~ k ( Poisson( MP(d , ( n k 1)Ts )) Poisson( MP( d , ( n k )Ts )))
(8)
k 1
M H1 ~ Poisson( MP(d , Ts )) n 1
k ( Poisson( MP(d , (n k 1)Ts )) Poisson(MP (d , (n k )Ts )))
(9)
k 1
Let
1 >0 and 2 >0. We say that the random variable V follows the Skellam distribution,
denoted by Skellam( 1 , 2 ) [23], if and only if V=U1-U2
(10)
where U1 and U2 are two independent random variables such that Ui∼Poisson( i ) for any i {1, 2} . The probability mass function (pmf) of Skellam distribution which is Skellam( 1 , 2 ) is
a function of l, and it is expressed [23]-[24] as follows:
f (l ; 1 , 2 ) e
( 1 2 )
1 2
l /2
I l (2 12 )
(11)
2 k l
x 2 I l ( x) k 0 k ! ( k l 1)
(12)
where l represents the value of the difference between two Poisson random variables U1 and U2,
1 and 2 are the means, or expected values of the two Poisson distributions Ui∼Poisson( i ) ( i {1, 2} ). Il(x) in (12) is the modified Bessel function of the first kind and x is a real number. Here, ( p 1) p( p) for any positive integer p. According to (8) and (9), we use Skellam distribution to describe the difference of two Poisson variables which represents the number of molecules received by RN. It is labeled the random variable Z which follows Skellam distribution based on the distribution of M H 0 and M H1 in (8)
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and (9) respectively as follows:
H 0 : Z ~Skellam( 1 , 2 )
(13)
H1 : Z ~Skellam(1 , 2 ) where the parameters of Skellam distribution which can be obtained by (8) and (9) are n 1
1 k MP (d , (n k 1)Ts )
(14)
k 1
n 1
2 k MP(d , (n k )Ts )
(15)
k 1
n 1
1 =MP(d , Ts ) k MP(d , (n k 1)Ts )
(16)
k 1
n 1
2 k MP(d , (n k )Ts )
(17)
k 1
With the hypothesis test model introduced above, we can derive the optimal decision using the maximum-a-posterior (MAP) detection method [25]. The MAP test is equivalent to the likelihood-ratio test as follows: P ( z | H1 ) P( z | H 0 )
(18)
where the value of η can be obtained from [25] as follows:
1 c
(19)
c
According to (11), the formulas (18) can be written as z /2
e ( 1 2 ) 1 I z (2 12 ) 2 z /2 ( 1 2 ) 1 e I z (2 12 ) 2 According to the reference [26], if x>> z 2 1/ 4 , I z ( x)
(20)
1 e x is a well-known 2 x
asymptotic approximation. Thus we can get the following formulas
I z (2 12 )
I z (2 12 )
1 2 ( 12 )
1/4
1 2 (12 )
1/4
e2
e2
12
12
(21)
(22)
There are two reasons that we use the asymptotic approximation of I l ( x) instead of exact one.
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On one hand, in order to get the value of optimal decision threshold θ, we need to obtain the solution of equation (20) which is also the value of random variable Z in (13). Thus we should focus on the computation of I z (2 12 ) and I z (2 12 ) . On the other hand, because the computation ability of RN is limited, it is difficult to compute the exact values of I z (2 12 ) and I z (2 12 ) by the complex formulas of I l ( x) which is given in (12). From equations (20) to (22), we obtain the following equation z /2
1 1 2 e e 12 1/4 2 ( ) 2 1 2 z /2 1 2 e ( 1 2 ) 1 e 12 1/4 2 ( ) 2 1 2 ( 1 2 )
(23)
The solution of equation (23) is expressed as follows:
2 ln (1 2 ) ( 1 2 ) 2 12 2 12 1/ 4 ln 1 2 12 z round 2 1 ln 1 2
(24)
where round represents the rounding operation since the decision threshold should be a positive integer. Thus the value of optimal decision threshold θ can be calculated by (24). In order to test a hypothesis, we must have a cumulative distribution function (cdf) of a Skellam distribution. Because the pmf of the Skellam distribution is defined at integer values, we have expressions for the cdf
F ( ; 1 , 2 )
e
( 1 2 )
l
1 2
l /2
I l (2 12 )
(25)
Especially, we can see that l which represents the difference of two Poisson random variables, is a non-negative integer. Thus we calculate the cdf as follows:
F ( ; 1 , 2 ) e l 0
l /2
( 1 2 )
1 I l (2 12 ) 2
(26)
On this basis, the false alarm probability PF and the detection probability PD are computed as follows:
PF Pr(M c | X 0) 1 F ( ; 1 , 2 )
(27)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
PD Pr(M c | X 1) 1 F ( ; 1 , 2 )
(28)
Thus we calculate the channel capacity of DMC model using the formulas below C max I ( X ; Y )
(29)
where 1
1
I ( X ; Y ) P(Y | X )P( X ) log X 0 Y 0
P(Y | X ) P(Y )
(30)
Furthermore, the error probability of transmitting the random bit in the current time slot n is expressed by
Pe c (1 PD ) (1 c ) PF
(31)
4. Numerical results In this section, we first use MATLAB to obtain numerical results of the channel capacity. The aim is to investigate how the different parameters including the probability of transmission of bit 1 in each time slot, distance d between two nanomachines, diffusion coefficient D, the number of time slots n and each time slot duration Ts have impacts on mutual information. Then we also show that the bit error probability is lower varying with the parameter M which is the number of the molecules emitted by TN in each time slot and demonstrate that the selection of value of M can ensure the reliability for RN. 4.1 The probability βk (1≤k≤n-1) We first observe the effect of βk (1≤k≤n-1) on the mutual information. Suppose the current time slot to be the time slot n. When we fix the probability βc=0.6, the mutual information is varying with the value of βk (1≤k≤n-1). In Fig.3, the curve shows a monotonically decreasing trend of mutual information as a function of the value of βk (1≤k≤n-1). Here, we suppose the probability of transmission of bit 1 in each of the previous (n-1) time slot to be equivalent. We can see that the smaller the value of βk (1≤k≤n-1), the larger the capacity. It is based on the fact that more residual molecules from the previous (n-1) time slots will result in more interferences to the current time slot n. Here, we set the parameters as D=100μm2/s, d=20μm, Ts=100s, n=10, M=50.
0.8 0.7 0.6 0.5
I(X;Y)
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0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
βk (1≤k≤n-1)
0.7
0.8
0.9
1
Fig.3 Attainable mutual information versus βk (1≤k≤n-1) The parameters used to make analysis of the channel capacity from Section 4.2 to Section 4.3 are given in Table 1 as follows: Table 1. Simulation parameters Symbol
Description
Value
d
Distance between two nanomachines
20 μm
D
Environment diffusion coefficient
100 μm2/s
M
Number of the molecules emitted by TN in each time slot
10
n
Number of time slots
20
Ts
Each time slot duration
100 s
Note that although the parameters d, D, M, n and Ts are not completely consistent in all figures in the numerical results, they are mainly used to better reflect their effects on our different research targets including analysis of capacity, bit error probability and reliability, and especially show the corresponding change trends in DMC model. 4.2 Distance d between two nanomachines and diffusion coefficient D We plot the achievable mutual information in relation to the channel transmission probability in the current time slot βc with different values of distance d and diffusion coefficient D in Fig.4 and Fig.5, respectively. For the same value of βc, the shorter the distance d and the larger the diffusion coefficient D, the larger the capacity. It is concluded that on one hand, the molecules with shorter distance between two nanomachines can be easily received. On the other hand, the environment
with a larger diffusion coefficient results in faster and further molecules propagation. In Fig.4 and Fig.5, we set Ts=100s, n=20, M=10, βk=0.0001 (1≤k≤n-1). In addition, we set D=100μm2/s in Fig.4 and d=20μm in Fig.5. However, the capacity reaches its peak at some particular value of βc, and then the capacity is decreasing after some value of βc. The numerical result is coincided with the fact that the more the molecules in the biological environment, the more effects of ISI and the slower diffusion, thus the capacity is observed to decrease after some value of βc.
0.8 d=10μm d=20μm d=30μm
0.7 0.6
I(X;Y)
0.5 0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
βc
0.6
0.7
0.8
0.9
1
Fig.4 Attainable mutual information versus βc with different value of d 0.7
D=30m2/s D=50m2/s D=100m2/s
0.6 0.5
I(X;Y)
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0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
βc Fig.5 Attainable mutual information versus βc with different value of D
4.3 Number of time slots n and each time slot duration Ts The number of time slots and each time slot duration are also critical parameters for the channel capacity. Fig.6 and Fig.7 show the mutual information is varying with βc for different values of number of time slots n and the time slot duration Ts, respectively. In Fig.6 and Fig.7, we set d=20μm, D=100μm2/s, M=10, and βk=0.0001 (1≤k≤n-1). Furthermore, we set Ts=100s in Fig.6. For the same value of βc, the less number of time slots, the larger the capacity. It is based on the fact that the more the number of time slots, the more the ISI to the current time slot. The parameters are set as n=20 in Fig.7. It is obvious that the molecules can be easily received with longer time slot duration. For the same value of βc, the longer the time slot duration Ts, the larger the capacity. In Fig.6 and Fig.7, we can see that the channel capacity is increasing with βc. However, the capacity reaches its peak at some particular value of βc. This incidence can be explained as follows: when the diffusion environment is filled with more molecules after some time, the propagation becomes more slowly, thus the capacity is found to start decreasing after some value of βc.
0.7 n =10 0.6
n =20 n =30
0.5
I(X;Y)
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0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
βc
0.6
0.7
0.8
0.9
1
Fig.6 Attainable mutual information versus βc with different value of n
0.7 TS =10s 0.6
TS =50s TS =100s
0.5
I(X;Y)
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0.4 0.3 0.2 0.1 0
0
0.1
0.2
0.3
0.4
0.5
βc
0.6
0.7
0.8
0.9
1
Fig.7 Attainable mutual information versus βc with different value of Ts The main reasons of the asymmetry of the curves from Fig.4 to Fig.7 are as follows: (1) In our paper, we derive the mathematical expressions of optimal decision threshold and the mutual information by using the Skellam distribution. The trends of the curves from Fig.4 to Fig.7 are similar to the curve of cdf of the Skellam distribution. (2) When the value of βc increases and is smaller than the particular value at which the capacity has its peak, the number of molecules is also increasing in the biological environment, then the probability that the molecules are received in the current time slot n by RN also increases. In such a case, the channel capacity is increasing with βc. With the increasing number of molecules, the speed of capacity growth is slowly. (3) However, the capacity reaches its peak at some particular value of βc which has the range of [0.6, 0.7]. There are more and more molecules in the biological environment. Then the capacity is decreasing after some value of βc. Therefore, more effects of ISI are also generated which will result in the fact that the diffusion is slower and the speed of capacity decreasing is fast. 4.4 Bit error probability Now we give the numerical results of error probability of one bit transmission in the current time slot on the basis of calculation of optimal decision threshold in (24) by using the Skellam distribution. Fig.8, Fig.9 and Fig.10 show the error probability is varying with the number of the molecules emitted by TN in each time slot by the formulas (27), (28) and (31) when the current transmit bit is bit-0, bit-1 and random transmit bit, respectively. In these three figures, we set the
parameters d=20μm, D=100μm2/s, Ts=2s, n=10, βk=0.0001 (1≤k≤n-1), βc=0.5.
Error Probability
10
10
10
-1
-2
-3
0
2
4
6
8
10 M
12
14
16
18
20
Fig.8 The error probability of transmitting bit-0 in the current time slot In Fig.8, we can see that the error probability of transmitting bit-0 in current time slot increases monotonically with the increasing number of the molecules emitted by TN in each time slot. When the bit 0 is transmitted in the current time slot n, TN releases no molecules according to the mechanism of modulation. In such a case, the molecules from the previous time slots are easily received by RN in the current time slot with the increasing value of M. 10
Error Probability
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10
10
0
-1
-2
0
2
4
6
8
10 M
12
14
16
18
20
Fig.9 The error probability of transmitting bit-1 in the current time slot Fig.9 plots the error probability of transmitting bit-1 in the current time slot. We see that the error probability of transmitting bit-1 decreases monotonically with the increasing value of M. It is based on the following fact: when the bit 1 is transmitted in the current time slot with emission of
M molecules by TN, the molecules from the current time slot and all the previous time slots can be easily received by RN in the current time slot. Therefore the probability that no molecules are received by RN is very low when the number of the molecules emitted by TN in each time slot is increasing. 10
0
-1
Error Probability
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10
-2
10
-3
10
0
2
4
6
8
10
12
14
16
18
20
M
Fig.10 The error probability of transmitting random bit in the current time slot Fig.10 shows the error probability of transmitting random bit in the current time slot is decreasing with the value of M. According to the formulas (31), the value of Pe depends on the optimal decision threshold , M and the physical parameters including D, d, n and Ts. For the same value of and the physical parameters, Pe is decreasing according to the numerical results in Fig.8 and Fig.9 when M is increasing. Interestingly, when the value of M is larger than 100 and continues to increase, the error probability of transmitting random bit in the current time slot will be increasing. This can be explained by the fact that larger number of molecules released in each time slot by RN will result in more residual molecules existed in the biological environment, which will lead to more ISI for RN. 4.5 Comparison to related work For the related work, the authors in [17] only consider the ISI from previous one time slot, and the mean mutual information for n time slots not for the n-th time slot is maximized in [18]. The work in [21] is primarily aiming at ultra shorter-range communication in which the distance between two nanomachines is 0.1μm. Thus we mainly compare our results with the work in [19] and [20] respectively.
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For the parameters d=20μm, D=100μm2/s, n=50, a peak in capacity is 0.61 bit/s and 0.23 bit/s with the time duration Ts=100s and Ts=8s in [20], respectively. However, we can get the higher capacity which is 0.64 bit/s and 0.41 bit/s with Ts=100s and Ts=8s, respectively. The reason for this comparison result is due to the interferences from the back time slots to the current time slots which are also considered. For the parameters d=20μm, D=100μm2/s, n=20, the capacity reaches a peak value of 0.28 and 0.15 with the time duration of Ts=100s and Ts=10s in [19], respectively. For the same parameters, from the numerical result in Fig.7, we can see that the capacity reaches a peak value of 0.68 bit/s and 0.49 bit/s with the time duration of Ts=100s and Ts=10s in our paper, respectively. In addition, in [19], the mean number of molecules emitted at each time slot is 1mol, and it is much larger than our work in which the number of molecules emitted at each time slot is 10. Therefore, the number of the molecules needed in our work is significantly smaller than [19] to achieve higher capacity. Using fewer molecules has two benefits. On one hand, fewer molecules result in fewer interferences. On the other hand, the cost of molecular device and the time for preparing smaller number of molecules is less than the case for preparing larger number of molecules which can improve the transmission efficiency. However, the selection of number of molecules emitted in each time slot should ensure that RN can receive at least one molecule of the M molecules. This case can be measured by the reliability. Here, the reliability labeled r is defined as the probability that at least one molecule of the M molecules is received by RN in the current time slot n. On the basis of the formulas Pkn in (3), we can obtain the formulas of computing r by the following equation n 1
r 1 (1 c F (Ts )) M (1 Pkn ) M k 1
(32)
1 0.9 0.8 0.7
Reliability(r)
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0.6 0.5 0.4 0.3 0.2 0.1 0
0
10
20
30 M
40
50
60
Fig.11 Reliability versus M In Fig.11, we set D=100μm2/s, d=20μm, Ts=10s, n=20. We can see that the reliability is increasing with the number of molecules emitted in each time slot. When the value of M is larger than 10, the reliability is close to 1. Thus, the value of parameter M which is set in our paper can ensure the reliability for RN. 5. Conclusions The objective of our paper is to make analysis of capacity of DMC model with ISI channel. The expression of optimal decision threshold is first derived by considering the effect of ISI from all the previous time slots and using the Skellam distribution. Then a closed form of channel capacity is proposed as a function of physical parameters as well as the channel transmission probability. We also formulate the bit error probability under the optimal decision threshold. Our numerical results show that how the parameters including the distance between TN and RN, the diffusion coefficient, the number of slots, each time slot duration and channel transmission probability have impacts on the capacity. In particular, the proposed channel transmission probability in each time slot plays an important role in maximizing the capacity. Therefore, we can maximize the channel capacity better by using Skellam distribution and controlling the channel transmission probability with lower bit error probability. At the same time, fewer molecules are needed in our paper to achieve higher capacity under the same physical parameters, and we also demonstrate that the selection of value of M can ensure the reliability for RN. As the future work, we plan to extend our work by using other modulation techniques proposed
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for the DMC system in the literatures. Also we aim to provide analysis of the channel capacity by considering the ISI under more complex communication topologies. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant Nos. 61472367, 61401397) and Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY15F020029, LY14F020020) References [1] I.F. Akyildiz, J.M. Jornet, M. Pierobon, Nanonetworks: a new frontier in communications, Commun. ACM 54 (11) (2011) 84-89. [2] O.C. Farokhzad, R. Langer, Impact of nanotechnology on drug delivery, ACS Nano 3 (1) (2009) 16-20. [3] J. Wood, Integrated circuit built on single nanotube: nanotechnology, Mater. Today 9 (5) (2006) 13-13. [4] J.W. Yoo, D.J. Irvine, D.E. Discher, S. Mitragotri, Bio-inspired, bioengineered and biomimetic drug delivery carriers, Nat. Rev. Drug. Discov. 10 (2011) 521-535. [5] P. Bogdan, G. Wei, R. Marculescu, Modeling populations of microrobots for biological applications, IEEE International Conference on Communications, Ottawa, Canada, 2012. [6] T. Nakano, M. Moore, F. Wei, A. Vasilakos, and J. Shuai, Molecular communication and networking: opportunities and challenges, IEEE Trans. Nanobiosci. 11(2) (2012) 135-148. [7] I.F. Akyildiz, F. Brunetti, C. Blzquez, Nanonetworks: a new communication paradigm, Comput. Networks 52 (12) 2260 (2008) [8] M. Pierobon, I.F. Akyildiz, Capacity of a diffusion-based molecular communication system with channel memory and molecular noise, IEEE Trans. Inform. Theory 59(2) (2013) 942-954. [9] T. Nakano, A. Eckford, T. Haraguchi, Molecular communication, Cambridge University Press, 2013. [10] L.P. Gine, I.F. Akyildiz, Molecular communication options for long range nanonetworks, Comput. Networks 53 (2009) 2753-2766. [11] M. Pieroborn, I.F. Akyildiz, A physical end-to-end model for molecular communication in nanonetworks, IEEE J. Sel. Areas Commun. 28 (4) (2010) 602-611. [12] M. Pierobon, I.F. Akyildiz, Diffusion-based noise analysis for molecular communication in
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*Brief Biography Click here to download Brief Biography: Brief Biography.doc
Author Biography Zhen Cheng received B.S. degree from Huanggang Normal University, Hubei, China, in 2004, and received M.S. and PH.D degrees from Huazhong University of Science and Technology, Hubei, China, in 2007 and 2010 respectively. She is currently an associate professor in the School of Computer Science and Technology, Zhejiang University of Technology, Hangzhou, China. Her current research interests include wireless networks, nanonetworks, molecular communication. She has published more than 30 technical papers in international proceedings and journals.
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