PERFORMANCE OF MIMO MOLECULAR COMMUNICATIONS IN
DIFFUSION-BASED CHANNELS
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Musaab Saeed
August 2017 PERFORMANCE OF MIMO MOLECULAR COMMUNICATIONS IN
DIFFUSION-BASED CHANNELS
Musaab Saeed
Thesis
Approved: Accepted:
Advisor Interim Department Chair Dr. Hamid Bahrami Dr. Joan Carletta
Committee Member Dean of the College Dr. Nghi Tran Dr. Donald P. Visco, Jr.
Committee Member Dean of the Graduate School Dr. Jin Kocsis Dr. Chand K. Midha
Date
ii ABSTRACT
Nanotechnology provides a promising solution for nano-scale communications consid- ering the nano-machine as the basic unit at this scale. The communication among the nano-machines is achieved by means of molecular communication. The most widespread model for a molecular communication channel is the diffusion-based chan- nel, where the information-carrying molecules propagate randomly in the medium based on Brownian motion. Other models include a positive drift from the transmit- ter nano-machine to the receiver nano-machine taking into account the life expectancy of the molecules, that is, the degradation over time. Due to this random motion, some molecules may arrive at the receiver after their intended time-slot, which can lead to interference with the molecules of the subsequent time-slots causing inter-symbol interference (ISI). Another source of interference is the inter-link interference (ILI), which emerges when multiple transmitters and receivers exist. In this work, we study the bit error rate (BER) performance of a molecular communications system having two transmitters and two receivers considering both ISI and ILI. Numerical results bring out the dependency of the BER performance of the system on the number of released molecules and distances between transmitters and receivers as well as the degradation parameter, drift velocity and the diffusion constant of the environment.
iii TABLE OF CONTENTS Page
LIST OF TABLES ...... v
LIST OF FIGURES ...... vi
CHAPTER
I. INTRODUCTION ...... 1
II. PERFORMANCE OF MIMO MOLECULAR COMMUNICA- TIONS IN DIFFUSION-BASED CHANNELS ...... 4 2.1 Introduction ...... 4
2.2 System Model ...... 5
2.3 Bit Error Rate Analysis ...... 12
2.4 Numerical Results ...... 19
III. MIMO MOLECULAR COMMUNICATIONS VIA DIFFUSION WITH DRIFT AND DEGRADATION ...... 22 3.1 Introduction ...... 22
3.2 Modeling The Molecular Channel ...... 23
3.3 Bit Error Rate Analysis ...... 33
3.4 Numerical Results ...... 36
IV. CONCLUSIONS AND FUTURE WORK ...... 40
BIBLIOGRAPHY ...... 42
iv LIST OF TABLES
Table Page
3.1 Fitted model parameters for the system topology (d = 8 µm, R = 3 2 µm, h = 2 µm, D = 40 µm/sec , ts = 1 sec, α = 0.3) ...... 29
v LIST OF FIGURES
Figure Page
2.1 Binomial pdf for n = 100 and p = 1/2...... 6
2.2 A simple MIMO molecular communication system ...... 9
2.3 BER performance versus the number of released molecules ...... 19
2.4 BER performance versus distance ...... 20
3.1 Proposed MIMO molecular communication system ...... 23
3.2 BER performance versus the number of released molecules for three values of α ...... 37 3.3 BER performance versus distance for different values of α, v and D . . . 37
vi CHAPTER I
INTRODUCTION
Nanotechnology is a newly emerging field dealing with the development of nanoscale machines, or simply nano-machines. Nanonetworks, the interconnection of several nano-machines, is envisaged to expand the capabilities and applications of such ma- chines in many ways, such as allowing them to cooperate and perform more complex tasks, allowing dense deployments of interconnected nano-machines, and, in some application scenarios, enabling the interaction with remote nano-machines by means of broadcasting mechanisms [1]. The classical communication techniques (e.g., radio, optical or acoustic), cannot be applied to nanonetworks by merely reducing conven- tional networks dimensions. Therefore, the concept of molecular communication has received increasing attention in the recent years as an interdisciplinary research area that spans the applications of nanotechnology and provide a prominent approach for establishing communications at nanoscales [22]. The Molecular communication is not competitive, but complementary to the classical communication since it has unique communication features including low energy consumption, high compatibility with biological systems and diffusion in aqueous environment even though it has low speed, short range, and is stochastic in nature [6].
Inspired by the communication mechanisms that naturally occur amongst
1 living cells, molecular communication is defined as the transmission and reception of information by means of nano-machines using molecules as carriers [1, 22]. The trans- mitter encodes the intended messages onto the molecules using different properties, such as concentration [18, 14], type [3], releasing time [27, 5] and/or ratio [11] of the molecules. In the literature, there are two different propagation schemes in molecular communication: passive transport and active transport [8]. In passive transport, the information-carrying particles propagate through the environment via the diffusion mechanism, where the molecules follow Brownian motion to diffuse through a fluid medium connecting the transmitter and the receiver without using external energy
[17, 23, 25, 4]. In active transport, information-carrying particles are transported by an external means including drift [10, 26, 27], molecular motors [7] and bacte- ria [3]. A considerable portion of the research efforts has been dedicated to passive molecular communication, i.e., diffusion-based molecular communication systems as in the aforementioned works, since it forms the backbone of the study of molecular communication field. However, some diffusion-based molecular communication chan- nels are characterized by Brownian motion with positive drift from the transmitter to the receiver [10, 26, 27]. Finally, the receiver attempts to recover the message by observing the pattern of the received molecules which are generally considered to be removed from the environment upon reception.
Throughout this thesis, it is assumed that the information is encoded based on the concentration of the released molecules and we consider diffusion-based molecular communication channels characterized by Brownian motion with positive drift assum-
2 ing perfectly-absorbing spherical receivers and taking into account the life expectancy of the molecules as it plays a significant role in such communication scenarios. One issue that should be addressed when dealing with a diffusive channel is the inter- symbol interference (ISI) since some of the messenger molecules may fail to arrive at the receiver within their intended time-slots, due to the diffusion nature, and interfere with the messenger molecules of subsequent transmissions, causing ISI. Also, in the presence of multiple transmitters and multiple receivers, inter-link interference (ILI) emerges as another source of interference. In this work, we study the performance of a multi-input multi-output (MIMO) molecular communication system.
The main focus of the thesis is on the study of the bit error rate (BER) performance of such molecular communication systems having two transmitters and two receivers and by considering both ISI and ILI. Numerical results show the depen- dency of the BER performance of the system on the number of released molecules and distances between transmitters and receivers as well as the degradation parameter, drift velocity and the diffusion constant of the environment.
3 CHAPTER II
PERFORMANCE OF MIMO MOLECULAR COMMUNICATIONS IN
DIFFUSION-BASED CHANNELS
2.1 Introduction
In this chapter, we consider the diffusion-based propagation as one of the widely adopted propagation models for molecular communication. It refers to the random motion of the molecules to propagate from the transmitter to the receiver according to the properties of the diffusion channel [6]. We study the effect of inter-symbol inter- ference (ISI) and inter-link interference (ILI) [19] on the performance of a molecular communication systems using CSK. [9] studies the performance of a multiple-input single-output (MISO) system considering both ISI and ILI. However, the intended receiver for both the transmitters is the same. The system considered in our study can be thought of as a multi-input multi-output (MIMO) molecular communication system, in which two transmitters and a receiver with two receptors communicate simultaneously.
Based on a diffusion-based channel model, we analyze the bit error rate (BER) performance of a system composed of two separate transmitters and a receiver with two collocated receptors, where the detection is based on the joint observation of
4 molecular concentration from the two receptors and using the maximum-a-posteriori
(MAP) detection. Through numerical simulations, we verify the derived theoretical results and compare the performance of the MIMO system with that of the SISO and
MISO molecular communications systems.
2.2 System Model
In a diffusion-based channel, due to the random motion of the released molecules in the fluid medium, the time that the molecules arrive at the receiver is probabilistic.
Assuming the transmitter is located at the origin and a molecule is released at time t = 0, the position of the released molecule at any time t is denoted by X(t), and the probability density function of X(t) can be written as [24]
1 x2 PX (x, t) = exp − (2.1) q 3 4Dt (4πDt) where x is the distance from the transmitting node and D is the diffusion constant of information molecules in the medium in µm2/sec unit. The probability that a molecule is absorbed by the receiver within time-slot duration ts is given by [16]
R d p(d, ts) = erfc √ (2.2) R + d 4Dts where erfc(x) is the complementary error function, d is the distance from the trans- mitter to the surface of the receiver and R is the radius of the receiver.
5 0.08 Binomial Distribution 0.07 Normal Distribution
0.06
0.05
0.04
0.03
0.02
0.01
0 0 20 40 60 80 100
Figure 2.1: Binomial pdf for n = 100 and p = 1/2
We assume that the transmitter releases n molecules into the medium for transmitting a bit 1 and no molecules to transmit a 0. Let N denote the number of molecules that are absorbed by the receiver within the time ts. N is a random variable with binomial distribution (with n trials and p(d, ts) as a success probability)
[20], [2]; i.e.
N ∼ B n, p (d, ts) (2.3)
As shown in Fig. 2.1, a Binomial distribution B n, p can be approximated with a normal distribution N ∼ N np, np(1 − p) when p is not close to one or zero
6 and np is large enough [15]. Under such conditions (2.3) can be approximated as
N ∼ N np(d, ts), np(d, ts) 1 − p(d, ts) (2.4)
2.2.1 Inter-Symbol Interference
Due to the nature of the diffusion-based channel, some of the released molecules may arrive at the receiver after their intended time-slots, which can lead to interference with the information molecules of the subsequent transmission intervals, causing ISI.
A potential solution to reduce the amount of ISI at the receiver is to keep the symbol duration long and; thus, give the information molecules a longer time to arrive at the receiver. This can effectively reduce the number of residual molecules left in the channel. On the other hand, increasing the symbol duration would also decrease the data rate, which is already low compared to the classical electromagnetic channels
[28]. As an alternative, signal processing and coding techniques can also be used in
[29] to further mitigate the ISI.
Suppose that Ni denotes the number of molecules that were emitted i time- slots before, i.e., at i × ts seconds before, and leak into the current time-slot. Then, according to [9], Ni is a random variable following the subtraction of two normal
7 distributions as follows
1 N ∼ N np d, (i + 1)t , np d, (i + 1)t i 2 s s 1 1 − p d, (i + 1)t − N np d, it , np d, it s 2 s s 1 − p d, its (2.5)
1 where the factor 2 is due to equal probability of transmission of bits 0 and 1. The first term indicates the total number of molecules that are emitted at that time-slot and absorbed by the receiver within all subsequent i + 1 time-slots and the second term indicates those molecules that were absorbed within the subsequent i time-slots. The total ISI can be written as the sum of interference due to all previous transmissions; i.e.
∞ X NISI = Ni (2.6) i=1
A reasonable approximation for the ISI is obtained by only considering the interfer- ence from the previous time-slot [15] as
1 N ∼ N np(d, 2t ), np(d, 2t ) 1 − p(d, 2t ) 1 2 s s s 1 − N np(d, t ), np(d, t ) 1 − p(d, t ) (2.7) 2 s s s
8 Figure 2.2: A simple MIMO molecular communication system
2.2.2 Inter-Link Interference
As shown in Fig. 2.2, the considered MIMO molecular communication system is composed of two transmitters (Tx1 and Tx2) and a receiver with two receptors (Rx1 and Rx2). Each receptor is assumed to be spherical with radius R. It is assumed that d1 and d2 are significantly larger than the distance between the receptors. The first transmitter Tx1 is assumed to be placed at distance d1 while the second transmitter
Tx2 is placed at distances d2 from receiver. Also, each transmitter is assumed to perfectly control the emission process of the molecules into the environment.
Once the molecules are released from the transmitter, they propagate ran- domly in the fluid medium based on diffusion. Both transmitters are considered to transmit independent information and similar to the SISO system, the arrival times of the molecules at receiving side follow the Brownian motion model. Hence, it is possible for each receiver that the molecules emitted by its corresponding transmit- ter in the previous transmission intervals to arrive late causing ISI for both links of the molecular MIMO system. Also, since that the transmitters release the molecules
9 simultaneously and they are identical and indistinguishable, each receiving node also suffers from the ILI. The ILI is defined as the interference due to the current and the previously transmitted symbols of the other link.
Let us assume that the two transmitters Tx1 and Tx2 release n1 and n2 molecules for transmitting the bit 1, respectively. Assuming that only the interference from an adjacent time-slot is significant, there are four possible scenarios for the ILI at receiver depending on the current and subsequent transmitted bits from Tx2. Let us define Nji,2 as the distribution of the ILI from Tx2 to the first receptor, if the current and previously transmitted bits from Tx2 are i and j (i, j ∈ {0, 1}), respectively.
Then, N00,2 = 0. If the current transmitted bit from Tx2 is 1 and the previous bit is zero, then the distribution of the ILI becomes:
N01,2 ∼ N n2p(d2, ts), n2p(d2, ts) 1 − p(d2, ts) (2.8)
where p(d2, ts), as defined in (2.2), is the probability of success for the molecules released by Tx2 at the current time-slot and to be absorbed by Rx1 also at the current time-slot. Likewise, if the current transmitted symbol from Tx2 is zero and the previous one is one, then the distribution of the molecules arrived at Rx1 can be written as the subtraction of two normal distributions as
10 N10,2 ∼N n2p(d2, 2ts), n2p(d2, 2ts) 1 − p(d2, 2ts)
− N n2p(d2, ts), n2p(d2, ts) 1 − p(d2, ts) (2.9)
where p(d2, 2ts) represents the absorbing probability of molecules release by Tx2 at the previous time-slot and absorbed by Rx1 within both the previous and current time- slots. It is worth mentioning that the ILI in this case is similar to ISI in (2.7). In the case that Tx2 transmits ones in the current and previous time-slot, the distribution of the ILI at Rx1 follows a normal distribution as follows
N11,2 ∼N n2p(d2, ts), n2p(d2, ts) 1 − p(d2, ts)
+ N n2p(d2, 2ts), n2p(d2, 2ts) 1 − p(d2, 2ts)
− N n2p(d2, ts), n2p(d2, ts) 1 − p(d2, ts) (2.10)
Assuming equal bit probabilities, the ILI from Tx2 to Rx1, denoted by NILI,2 is given by the following distribution
1 N = N + N + N + N ILI,2 4 00,2 01,2 10,2 11,2 1 = N 2n p(d , 2t ), 4n p(d , t ) 1 − p(d , t ) 4 2 2 s 2 2 s 2 s + 2n2p(d2, 2ts) 1 − p(d2, 2ts) (2.11)
11 Similarly, the distribution of the ILI from Tx1 to the signal of Tx2 can be easily obtained from (2.11) by replacing n2 and d2 with n1 and d1, respectively.
2.3 Bit Error Rate Analysis
When there is only one transmitter sending bit 0, i.e., no molecule is released, then the number of the absorbed molecules by the receiver within the current time-slot which includes only the ISI from the previous symbol, denoted by N0, is normally distributed as
2 N0 ∼ N (µ0, σ0) (2.12)
where
1 µ = n p(d, 2t ) − p(d, t ) (2.13) 0 2 s s and
1 1 σ2 = np(d, t ) 1 − p(d, t ) + np(d, 2t ) 1 − p(d, 2t ) (2.14) 0 4 s s 4 s s
12 Similarly, when the transmitter is transmitting bit 1, it releases n molecules, and the distribution of the number of the absorbed molecules, N1, is given by
N1 =N0 + N np(d, ts), np(d, ts) 1 − p(d, ts)
2 ∼ N (µ1, σ1) (2.15)
where the second term takes care of the number of absorbed molecules among n released within the current time-slot, and
1 µ = n p(d, 2t ) + p(d, t ) (2.16) 1 2 s s
5 1 σ2 = np(d, t ) 1 − p(d, t ) + np(d, 2t ) 1 − p(d, 2t ) (2.17) 1 4 s s 4 s s
The detection at the receiver is based on the comparison of the number of absorbed molecules in one time-slot with some predefined threshold. Thus, the threshold must be derived properly to minimize the bit error rate of the transmitter.
Let Z denotes the number of absorbed molecules at the receiver; then, we have two hypothesis H0 and H1 for transmission of 0 and 1 as follows
2 H0 : Z = N0 ∼ N (µ0, σ0)
2 H1 : Z = N1 ∼ N (µ1, σ1) (2.18)
13 To minimize the BER, the threshold is derived using the MAP detection, which is the same as the likelihood-ratio test (LRT) in this case; i.e.
( 2 2 ) p(Z | H1) σ0 (Z − µ1) (Z − µ0) = exp − 2 + 2 p(Z | H0) σ1 2σ1 2σ0 ( σ0 1 1 1 2 = exp − 2 − 2 Z + σ1 2 σ1 σ0 2 2 ) µ1 µ0 1 µ1 µ0 2 − 2 Z − 2 − 2 (2.19) σ1 σ0 2 σ1 σ0
By taking logarithm and setting to zero, the optimal decision threshold is derived as
√ −B + B2 − 4AC τ = (2.20) 2A where
1 1 1 µ1 µ0 A = − 2 − 2 B = 2 − 2 2 σ1 σ0 σ1 σ0 2 2 σ0 1 µ1 µ0 C = ln − 2 − 2 (2.21) σ1 2 σ1 σ0
The BER can then be derived as
1 P = p(N > τ) + p(N < τ) e 2 0 1 1 τ − µ τ − µ = Q 0 + 1 − Q 1 (2.22) 2 σ0 σ1
14 where Q(x) is the standard Q-function defined as
1 Z ∞ y2 Q(x) = √ exp − dy (2.23) 2π x 2
For the case of the MIMO system, when Tx1 is transmitting bit 0, the number of the absorbed molecules by the first receptor within the current time-slot, denoted by N0,1, includes both the ISI from the previous symbol transmitted by Tx1 and ILI from Tx2. Therefore, N0,1 has the following normal distribution
2 N0,1 = NISI,1 + NILI,2 ∼ N (µ0, σ0) (2.24)
where
1 µ = n p(d , 2t ) − n p(d , t ) + n p(d , 2t ) (2.25) 0 2 1 1 s 1 1 s 2 2 s and
1 σ2 = n p(d , t ) 1 − p(d , t )+ 0 4 1 1 s 1 s 1 n p(d , 2t ) 1 − p(d , 2t )+ 4 1 1 s 1 s 1 n p(d , t ) 1 − p(d , t )+ 4 2 2 s 2 s 1 n p(d , 2t ) 1 − p(d , 2t ) (2.26) 8 2 2 s 2 s
When Tx1 is transmitting bit 1, it releases n1 molecules. The number of absorbed
15 molecules by Rx1, in this case, N1,1, has the following normal distribution
N1,1 =N0,1 + N n1p(d1, ts), n1p(d1, ts) 1 − p(d1, ts)
2 ∼ N (µ1, σ1) (2.27)
where
1 µ = n p(d , 2t ) + n p(d , t ) + n p(d , 2t ) (2.28) 1 2 1 1 s 1 1 s 2 2 s and
5 σ2 = n p(d , t ) 1 − p(d , t )+ 1 4 1 1 s 1 s 1 n p(d , 2t ) 1 − p(d , 2t )+ 4 1 1 s 1 s 1 n p(d , t ) 1 − p(d , t )+ 4 2 2 s 2 s 1 n p(d , 2t ) 1 − p(d , 2t ) (2.29) 8 2 2 s 2 s
On the other hand, when Tx1 is transmitting bit 0, the number of absorbed molecules at Rx2, i.e. N0,2, has the same distribution as N0,1, and when Tx1 is transmitting bit 1, the distribution of the number of molecules at Rx2, i.e. N1,2 is the same as
N1,1. It worth mentioning that in practice, the existence of the second receptor affects the distribution of the molecules at the first receptor, i.e., some of the molecules are absorbed by the second receptor. For simplicity, we don’t take this effect into account
16 in our analysis.
In the MIMO system, the detection is based on the observations made at the two receivers. Let Z1 and Z2 denote the number of molecules observed at Rx1 and
Rx2 respectively. Then, the two detection hypotheses are
2 H0 : Z1,Z2 ∼ N (µ0, σ0)
2 H1 : Z1,Z2 ∼ N (µ1, σ1) (2.30)
Applying LRT results in the following equation
2 ( 2 2 p(Z | H1) σ0 (Z1 − µ1) (Z2 − µ1) = 2 exp − 2 − 2 p(Z | H0) σ1 2σ1 2σ1
2 2 ) (Z1 − µ0) (Z2 − µ0) + 2 + 2 (2.31) 2σ0 2σ0
By taking logarithm and setting to zero, the optimal decision threshold becomes
√ −B + B2 − 4AC τ = (2.32) 1 2A where
2 2 2 2 σ1 − σ0 µ1σ0 − µ0σ1 A = 2 2 B = 2 2 2σ0σ1 σ0σ1 2 2 2 2 σ1 − σ0 2 µ1σ0 − µ0σ1 C = 2 2 Z1 + 2 2 Z1+ 2σ0σ1 σ0σ1 2 2 2 2 2 µ0σ1 − µ1σ0 σ0 2 2 + ln 2 (2.33) σ0σ1 σ1
17 The BER for the information transmitted from Tx1 can be written as
1 P = (P + P ) (2.34) e1 2 F1 M1
where PF1 and PM1 are the probability of false alarm and probability of misdetection, respectively, and are derived as
Z ∞ Z ∞
PF1 = p(Z1,Z2|H0)dZ2dZ1 0 τ1 Z ∞ Z ∞ ( 2 1 1 (Z1 − µ0) = 2 exp − 2 0 τ1 2πσ0 2 σ0 2 ) (Z2 − µ0) + 2 dZ2dZ1 σ0 Z ∞ ( 2 ) 1 (Z1 − µ0) τ1 − µ0 = √ exp − 2 Q dZ1 (2.35) 0 2πσ0 2σ0 σ0
Z ∞ Z τ1
PM1 = p(Z1,Z2|H1)dZ2dZ1 0 0 ( Z ∞ Z τ1 2 1 1 (Z1 − µ1) = 2 exp − 2 0 0 2πσ1 2 σ1 2 ) (Z2 − µ1) + 2 dZ2dZ1 σ1 Z ∞ ( 2 ) 1 (Z1 − µ1) = √ exp − 2 0 2πσ1 2σ1 " # τ1 − µ1 1 − Q dZ1 (2.36) σ1
Similarly, we can write the BER for Tx2 in terms of PF2 and PM2 by replacing τ1 with
18 100
10-1
10-2
10-3 BER 10-4 SIMO MIMO ( Analytical result ) 10-5 MIMO ( Simulation result ) MISO -6 SISO (slot duration t = 10 sec ) 10 s SISO (slot duration t = 5 sec ) s 10-7 0 20 40 60 80 100 120 140 160 180 200 Number of Released Molecules
Figure 2.3: BER performance versus the number of released molecules
its corresponding τ2 in (2.35) and (2.36), respectively. Note that in the special case when one of the transmitters is located far away from the two receptors, the system represents a single-input multiple-output (SIMO) scenario. Therefore, the studied
MIMO system encompasses this special case.
2.4 Numerical Results
In this section, we present the numerical results for the BER of different communica- tion schemes over a diffusion-based channel. Throughout the simulations, we set the radius of the receptors at R = 10 µm and the diffusion constant of the environment at D = 79.4 µm2/sec.
19 100
10-1
10-2
10-3 BER SIMO ( 300 molecules ) SIMO ( 250 molecules ) 10-4 MIMO ( 300 molecules ) MIMO ( 250 molecules ) 10-5 SISO ( 300 molecules ) SISO ( 250 molecules )
10-6 40 45 50 55 60 65 70 Distance in µm Figure 2.4: BER performance versus distance
In Fig. 2.3, we plot the BER performance of different schemes versus the number of released molecules which represents the transmission power. For the SISO system, the distance between the transmit and receive nano-machines is set at d = 30
µm, while for the MIMO system, d1 = 30 µm and d2 = 40 µm. For the MIMO system we plot the BER of the stream transmitted from Tx1, while keeping the number of molecules from Tx2 to transmit bit one 100 (n2 = 100). As expected, the BER of all schemes is inversely proportional to the molecule emission power. For the SISO case, to show the effect of the time-slot duration, we plot the performance for two cases of ts = 5 sec and ts = 10 sec. As shown, the system performs ten times better in the case of longer symbol duration at reasonably large number of released molecules.
This is due to the fact that in this case the probability that the released molecules
20 are absorbed at the receptor within the current time-slot is higher. For the rest of the cases we set the time-slot duration at ts = 10 sec. For the MIMO case, we have plotted both the numerical results (results obtained from Montecarlo simulations, i.e., the number of correctly-detected bits over the total number of transmitted bits) and theoretical results (results obtained using derivations in (2.34) - (2.36)) which show the exactness of the theoretical analysis. As a special case, we have also plotted the
BER performances of the SIMO system by letting d2 to be very large. As expected, the performance of the SIMO system is better than the MIMO system as there is no ILI. For the sake of comparison, we have also plotted the performance of the
MISO system derived in[9]. As seen, both the MIMO and SIMO systems achieve an increased diversity order compared to the SISO and MISO cases.
In Fig. 2.4, we plot the BER performance of different schemes versus the distance d1 for two different numbers of released molecules (250 and 300 molecules), while keeping d2 = 40 µm. We set the time-slot duration at ts = 10 sec, and n2, i.e. the number of released molecules from Tx2 for transmitting bit one 100 (i.e. n2 = 100) as before. As expected, the BER is an increasing function of the distance, i.e., as the distance between the transmitter and the receptor increases, the BER performance also increases. The performance of the SIMO system is generally superior to that of the MIMO and SISO systems; however, the performance gaps reduce by increasing the distance.
21 CHAPTER III
MIMO MOLECULAR COMMUNICATIONS VIA DIFFUSION WITH DRIFT
AND DEGRADATION
3.1 Introduction
In this chapter, it is assumed that the molecules travel according to Brownian mo- tion model with a positive drift from the transmitter nano-machine to the receiver nano-machine, that is, the molecular channel is governed by two parameters of the
fluid medium: the diffusion coefficient D and the drift velocity v. Furthermore, we assume that the molecules can degrade over time and, therefore, we account for the life expectancy of the molecules as a very important factor in such communication scenarios. Finally, the receiver attempts to recover the message by observing the pattern of the received molecules which are generally considered to be removed from the environment upon reception.
The main focus of this chapter is on the study of the performance of a multi- input multi-output (MIMO) molecular communication system considering both ISI and ILI. In [12, 13], a MIMO system is proposed considering only the diffusion mech- anism. However, the drift velocity and the degradation of the molecules haven’t been considered. The channel impulse response is obtained by modifying the single-
22 Figure 3.1: Proposed MIMO molecular communication system input-single-output (SISO) channel model and consequently, the system performance is analyzed based on the developed MIMO simulator.
3.2 Modeling The Molecular Channel
The communication system considered in this chapter consists of two point sources that are assumed to be perfectly synchronized with two spherical receivers in a 3-D environment. As shown in Fig. 3.1, the two transmitters Tx1 and Tx2 are placed d1 distance apart from their corresponding receivers Rx1 and Rx2 and the distance from Tx1 to the surface of Rx2 is d2. Each receiver is a spherical surface of radius
R, placed h distance from the other receiver and is considered to act as a perfect absorber, that is, every molecule gets absorbed by the receiver once it arrives at its surface. The molecular channel is divided into time-slots of duration ts where the ith
23 slot is defined within the time period [(i−1)ts, its]. To encode information, we use the binary concentration shift keying (BCSK) modulation technique, i.e., at the beginning of each time-slot, each transmitter releases a certain number of information-carrying molecules n to represent bit-1 and no molecules to send bit-0, to reduce the energy consumption. Then, the emitted molecules travel to the other side under the effect of
Brownian motion along with positive drift from the transmitter to the receiver and, therefore, the propagation of the molecules is governed by two parameters of the fluid medium: the diffusion coefficient D and the drift velocity v.
For the case of no-drift, the first hitting probability to a spherical receiver is given by [30]
R d d2 y(t) = √ exp − (3.1) R + d 4πDt3 4Dt where d is assumed as the transmitter-receiver distance. Introducing drift v, the hitting probability in (3.1) turns out to be
R d (vt − d)2 h(t) = √ exp − (3.2) R + d 4πDt3 4Dt
Furthermore, we assume that the molecules have a life expectancy and they can degrade and, consequently, become useless in terms of communication. To ac- count for the degradation of molecules over time, an exponential distribution is con-
24 sidered for the lifetime τ of the molecules which pdf is given by [21]
g(τ) = αe−ατ , τ > 0 (3.3)
where α is the degradation parameter. A molecules contributes to the signal if it hits the receiver surface prior to degradation, i.e, its lifetime is more than t. Therefore, the expected fraction of molecules, emitted at time t = 0, hitting the receiver until time t before getting degraded can be written as follows
Z t Z ∞ p(d, t) = h(t) g(τ)dτdt 0 t Z t = h(t)e−αtdt (3.4) 0 which gives the expected number of molecules to be received until time t before getting degraded, when multiplied by the number of molecules emitted at t = 0.
As mentioned, each molecule emitted from the point transmitter has two com- ponents in its movement dynamics; one is due to the Brownian motion and the other arises from the drift in addition to the degradation of molecules over time. Moreover, we cannot merely use (3.4) to obtain the total number of absorbed molecules by a single receiver before time t since we have two receivers in our system. Therefore, we simulate the molecular channel for the given MIMO setup using the following
25 propagation model
(xt, yt, zt) =(xt−∆t, yt−∆t, zt−∆t) + (∆x, ∆y, ∆z)
∆x ∼ N (0, 2D∆t) + vx∆t
∆y ∼ N (0, 2D∆t) + vy∆t
∆z ∼ N (0, 2D∆t) + vz∆t (3.5)
2 where xt, yt, zt, vx, vy, vz, N (µ, σ ) are the molecules positions at each dimension at time t, the drift velocities at each dimension and the normal distribution with mean
µ and variance σ2 respectively. In this work, we assume drift only in x-dimension, i.e., vx = v, vy = vz = 0. Based on our system topology, we have four different p(d, t) depending on the molecules emission transmitter and hitting receiver. Thereafter, we use the simulation data to obtain p(d, t) for the MIMO system and utilize it for the analysis.
From (3.4), we can obtain the fraction of absorbed molecules within the time t, p(d, t), and then the number of molecules that are absorbed by the receiver within the time t, denoted by N, is a random variable that follows a binomial distribution with n trials and p(d, t) as a success probability [20, 2]
N ∼ B n, p (d, t) (3.6)
A Binomial distribution B(n, p) can be approximated with a normal distribution
26 M ∼ N np, np(1 − p) when p is not close to one or zero and np is large enough [15].
Then, (3.6) can be approximated as
N ∼ N np(d, t), np(d, t) 1 − p(d, t) (3.7)
3.2.1 Fitting Channel Parameters
The pseudo code of our simulator is given in Algorithm 1. The required parameters are d, R, h, D, ts, v, α, ∆t, iter where ∆t corresponds to simulation step time and iter is the number of iterations used in the simulator. We develop and run our simulator through MatLab for a given topology parameters to estimate the fraction of the molecules that are absorbed within the time-slot duration ts before degradation.
In our simulator, we use a model function that is related to (3.4) with some controllable parameters to fit the simulation data. However, we first need to charac- terize the exponentials using the power series as follows
2 Let x1 = −(vt − d) /(4Dt), x2 = −αt and so we can write x = x1 + x2. Then, (3.4) can be rewritten as
Z ts R d p(d, ts) = √ 1 + x dt (3.8) 3 0 R + d 4πDt
Since we cannot directly evaluate (3.8) in the interval (0, ts), we evaluate it in the interval (0.005, ts) as almost no molecules arrive at the destination in the time period
(0, 0.005). As stated, we have four different estimated probabilities resulting from our
27 Algorithm 1 Simulator pseudo code
Require: d, R, h, D, ts, v, α, ∆t, iter Output: prob of getting absorbed within the time-slot duration before getting de- graded.
1: Determine Time Step (ts/1000). 2: Determine coordinations of the transmitters and the receivers in the environment. 3: for all iter do 4: for all Time Step do 5: for all emitted molecule do 6: for all drift value do 7: Evaluate molecule displacement in each direction: 8:∆ x ∼ N (0, 2D∆t) + vx∆t 9:∆ y ∼ N (0, 2D∆t) + vy∆t 10:∆ z ∼ N (0, 2D∆t) + vz∆t 11: for all receiver do 12: Evaluate distance of the molecule to the receiver 13: if distance is less radius of the receiver then 14: Increase number of molecules hitting the receiver by e−αt. 15: Remove the molecule from the environment. 16: end if 17: end for 18: end for 19: end for 20: end for 21: end for 22: prob = number of absorbed molecules / (number of emitted molecules × iter).
28 parameter A1 parameter A2 p11 0.0475 −18.3838 p12 0.0115 15.6201 Table 3.1: Fitted model parameters for the system topology (d = 8 µm, R = 3 2 µm, h = 2 µm, D = 40 µm/sec , ts = 1 sec, α = 0.3)
simulator pij, i, j ∈ {1, 2} where i represents the transmitter index while j indicates the receiver. Since the topology is symmetric, then p11 is very close to p22 and similarly p12 is almost p21. Then, we set the slot duration at one second and hence, from (3.8), we can obtain p(d, ts), along with the controllable parameters A1 and A2, as follows
R d n 1/2 p(d, ts) = √ 20 ∗ 2 − A1 4dv R + d 4πD
− 2001/2 v2/100 + 4dv − (400d2)/3 − (2d2)/3
2 1/2 o + 2v /(4D) + A2α(2 /10 − 2) − 2 (3.9)
Note that we replace d = d1 and d = d2 for p11 and p12 respectively. We use nonlin- ear model in MatLab for fitting the simulator data and estimating the controllable parameters. Table 3.1 shows the fitted model parameters for the selected system topology.
3.2.2 Interference Analysis
Due to the nature of the molecular channel, some of the released molecules may fail to arrive at the receiver in their intended time-slots and remain in the medium leading
29 to inter-symbol interference (ISI), i.e., interference from the previous symbols of the corresponding transmitter. One way to reduce the amount of the ISI is to increase the slot duration giving the molecules more chance to reach to the receiver before the end of their intended time-slot. On the other hand, increasing the symbol duration leads to decreasing in data rate. Moreover, [29] proposed coding techniques and signal processing to further mitigate the ISI.
Let Ni denotes the number of molecules that were emitted i time-slots before and leak into the current time-slot. Then Ni is a random variable which follows the subtraction of two normal distributions as follows
1 N ∼ N np d, (i + 1)t , np d, (i + 1)t i 2 s s 1 1 − p d, (i + 1)t − N np d, it , np d, it s 2 s s 1 − p d, its (3.10)
1 where the factor 2 is to indicate the equal probability of transmission of bits 0 and 1.
The first term indicates the total number of released molecules at the beginning of that time-slot and absorbed within all subsequent i + 1 time-slots while the second term represents those molecules that were absorbed within the subsequent i time- slots. The total ISI caused by all previous transmissions can be written as follows
∞ X NISI = Ni (3.11) i=1
30 Moreover, since only the interference from the previous time-slot is significant [15], then we can write
1 N ∼ N np d, 2t , np d, 2t 1 − p d, 2t 1 2 s s s 1 − N np d, t , np d, t 1 − p d, t (3.12) 2 s s s
In addition, there exists inter-link interference (ILI) as another source of in- terference which emerges when multiple transmitters and receivers exist since that the transmitters release the molecules simultaneously and they are identical and in- distinguishable. We define the ILI as the interference from both the current and the previous symbols of the other communication link.
Let us assume that both the two transmitters Tx1 and Tx2 release n molecules for transmitting the bit 1 and moreover, the significant interference arises only from the previous time-slot. Then, for the data transmitted from Tx1, there are four possible scenarios for the ILI at receiver depending on the current and previously transmitted bits from the Tx2. Let us define Nji,2 as the distribution of the ILI from
Tx2 to Tx1 where i and j (i, j ∈ {0, 1}) are the current and previously transmitted bits from Tx2 respectively. Then, N00,2 = 0. If the current transmitted bit from Tx2 is 1 and the previous bit is zero, then the distribution of the ILI becomes:
N01,2 ∼ N np(d2, ts), np(d2, ts) 1 − p(d2, ts) (3.13)
31 where p(d2, ts), as defined in (3.8), is the probability that the molecules released by Tx2 at the current time-slot are absorbed by Rx1 within the current time-slot.
Likewise, if the current transmitted symbol from Tx2 is zero and the previous one is one, the distribution of the molecules arrived at Rx1 can be written as the subtraction of two normal distributions as
N10,2 ∼N np(d2, 2ts), np(d2, 2ts) 1 − p(d2, 2ts)
− N np(d2, ts), np(d2, ts) 1 − p(d2, ts) (3.14)
where p(d2, 2ts) represents the probability of molecules release by Tx2 at the previous time-slot and absorbed by Rx1 within both the previous and current time-slots. Note that the ILI in this case is similar to ISI in (3.12). In the case that Tx2 transmits ones in the current and previous time-slot, the distribution of the ILI at Rx1 follows a normal distribution as follows
N11,2 ∼N np(d2, ts), np(d2, ts) 1 − p(d2, ts)
+ N np(d2, 2ts), np(d2, 2ts) 1 − p(d2, 2ts)
− N np(d2, ts), np(d2, ts) 1 − p(d2, ts) (3.15)
Assuming equal probabilities for all scenarios, the ILI from Tx2 to Rx1, denoted by
32 NILI,2 is given by the following distribution
1 N = N + N + N + N ILI,2 4 00,2 01,2 10,2 11,2 1 = N 2np(d , 2t ), 4p(d , t ) 1 − p(d , t ) 4 2 s 2 s 2 s + 2p(d2, 2ts) 1 − p(d2, 2ts) (3.16)
Due to the symmetry of the topology, this distribution can be considered as the distribution of the ILI from Tx1 to the signal of Tx2 as well.
3.3 Bit Error Rate Analysis
When the transmitter Tx1 is transmitting bit 0, the number of the absorbed molecules by its corresponding receiver, R1, within the current time-slot, denoted by N0, in- cludes both the ISI from the previous symbol transmitted by Tx1 and ILI from Tx2.
Therefore, N0 has the following normal distribution
2 N0 = NISI,1 + NILI,2 ∼ N (µ0, σ0) (3.17)
where
1 µ = n p(d , 2t ) − p(d , t ) + p(d , 2t ) (3.18) 0 2 1 s 1 s 2 s
33 and
1 σ2 = np(d , t ) 1 − p(d , t )+ 0 4 1 s 1 s 1 np(d , 2t ) 1 − p(d , 2t )+ 4 1 s 1 s 1 np(d , t ) 1 − p(d , t )+ 4 2 s 2 s 1 np(d , 2t ) 1 − p(d , 2t ) (3.19) 8 2 s 2 s
Similarly, when Tx1 is transmitting bit 1, the number of the absorbed molecules by
R1, denoted by N1, has the following normal distribution
N1 =N0 + N np(d1, ts), np(d1, ts) 1 − p(d1, ts)
2 ∼ N (µ1, σ1) (3.20)
where
1 µ = n p(d , 2t ) + p(d , t ) + p(d , 2t ) (3.21) 1 2 1 s 1 s 2 s
34 and
5 σ2 = np(d , t ) 1 − p(d , t )+ 1 4 1 s 1 s 1 np(d , 2t ) 1 − p(d , 2t )+ 4 1 s 1 s 1 np(d , t ) 1 − p(d , t )+ 4 2 s 2 s 1 np(d , 2t ) 1 − p(d , 2t ) (3.22) 8 2 s 2 s
For the detection process, let us assume the number of molecules observed at
Rx1 is Z1. Then, the two detection hypotheses are
2 H0 : Z1 ∼ N (µ0, σ0)
2 H1 : Z1 ∼ N (µ1, σ1) (3.23)
Applying LRT results in the following equation
2 2 p(Z1 | H1) σ0 (Z1 − µ1) (Z1 − µ0) = exp − 2 + 2 (3.24) p(Z1 | H0) σ1 2σ1 2σ0
By taking logarithm and setting to zero, the optimal decision threshold becomes
√ −B + B2 − 4AC τ = (3.25) 2A
35 where
1 1 1 µ1 µ0 A = − 2 − 2 B = 2 − 2 2 σ1 σ0 σ1 σ0 2 2 σ0 1 µ1 µ0 C = ln − 2 − 2 (3.26) σ1 2 σ1 σ0
Then, the BER can then be derived as
1h i P = p(N > τ) + p(N < τ) e 2 0 1 1 τ − µ τ − µ = Q 0 + 1 − Q 1 (3.27) 2 σ0 σ1 where Q(x) is the standard Q-function defined as
1 Z ∞ y2 Q(x) = √ exp − dy (3.28) 2π x 2
Due to the symmetry of the topology, this BER is also equivalent to the BER for
Tx2.
3.4 Numerical Results
In this section, we present the numerical results for the performance of the MIMO scheme over a diffusion-based channel with a drift from the transmitter to the receiver under the effect of degradation. Throughout the simulations, we set the radius of each
36 10-1 Degradation parameter α = 0 Degradation parameter α = 0.1 -2 10 Degradation parameter α = 0.3
10-3
10-4 BER
10-5
10-6
10-7 100 200 300 400 500 600 Number of Released Molecules Figure 3.2: BER performance versus the number of released molecules for three values of α
100
10-5
BER α = 0 , v = 0 , D = 30 µm2/s α = 0 , v = 0 , D = 60 µm2/s 10-10 α = 0 , v = 8 µm/s , D = 30 µm2/s α = 0 , v = 8 µm/s , D = 60 µm2/s α = 0.3 , v = 0 , D = 30 µm2/s α = 0.3 , v = 8 µm/s , D = 30 µm2/s 10-15 4 6 8 10 12 14 16 18 20 Distance in µm Figure 3.3: BER performance versus distance for different values of α, v and D
37 receiver at R = 3 µm, the distance between the two receivers at h = 2 µm, the time- slot duration at ts = 1 sec and the number of iterations in the simulator at 100.
To capture the effect of increasing the number of released molecules and the degradation parameter α, we plot the BER versus the number of released molecules which represents the transmission power for three values of α as in Fig. 3.2. The distance between the first transmission pair is set at d1 = 8 µm and consequently d2 = 11.31 µm. In addition, we set the diffusion constant of the environment at
D = 40 µm2/sec and the drift at v = 6 µm/sec. As expected, the BER is inversely proportional to the molecule emission power. Moreover, It also depends on the value of the degradation parameter; the performance deteriorates as the parameter value increases. It is observed that the effect of degradation is larger with increase in the number of released molecules as the number of degrading molecules, before arriving at the receiver, also increases.
As illustrated in Fig. 3.3, we plot the BER performance versus the distance d1 (note that the distance d2 changes correspondingly) for different values of the degradation parameterα, the drift v and the diffusion constant D, fixing the number of released molecules at 400 molecules. As expected, the BER is an increasing function of the distance and moreover, the BER decreases with increase in drift velocity.
Additionally, the degradation process worsens the performance. As an outcome of the interactions between the drift and the diffusion components, we can observe a nice behavior of the BER versus the distance. At smaller distances; if the drift velocity is reasonably high, then the motion of the molecule is dominated by the drift and varying
38 the diffusion constant wouldn’t affect much the performance, while for reasonably lower drift velocities, the dominance is switched to the diffusion process. On the other hand, at higher distances and for higher values of the drift, the dominating component is the diffusion component and consequently, varying the diffusion constant would result in a noticeable effect on the performance. However, the performance gaps generally reduce by increasing the distance since the molecules require more time to arrive at the receiver before getting degraded.
39 CHAPTER IV
CONCLUSIONS AND FUTURE WORK
In the second chapter, we have studied the performance of a MIMO molecular commu- nications system in a diffusion-based environment governed by the Brownian motion of molecules, and compared the results with that of the SISO, SIMO and MISO systems. In particular, we analyzed the BER of the MIMO system in the presence of ISI and ILI. The optimal detection threshold was derived using MAP detection method such that the BER is minimized. It was shown that the analytical results closely follow the simulation results. The dependency of the BER on the number of released molecules and distances between transmitters and receptors was studied in the simulation results. It was shown that the MIMO molecular communication sys- tem significantly outperforms the SISO and MISO systems in all the scenarios, while its performance is inferior to that of the SIMO molecular communication system.
In the third chapter, we have studied the performance of a MIMO molecular communications system. The environment is governed by the diffusion of molecules and a net drift from the transmitter to the receiver taking into account the effect of life expectancy of molecules and they can degrade over time. The objective of this work is to model the channels response via fitting the data generated from our simulator which is developed through MatLab to obtain the fraction of the molecules that
40 are absorbed at the receiver within the time-slot duration before getting degraded.
Thereafter, we utilized the estimated function with some controllable parameters to determine the ISI and ILI and hence analyze the BER of the MIMO scheme. The optimal detection threshold was derived using MAP detection method such that the
BER is minimized. The dependency of the BER performance on the number of released molecules and distances between transmitters and receivers along with the degradation parameter, drift velocity and the diffusion constant was studied in the simulation results. It was shown that the degradation process has significant effect with increase in the number of released molecules. Apparently, the performance gaps generally reduce by increasing the distance.
As a future work, we can consider the process of encoding the intended mes- sages onto the molecules using different properties such as type and/or ratio of the molecules. In addition, we can also study other practical propagation models for
MIMO molecular communication systems including molecular motors and bacteria to name a few. In addition, we can elaborate more on the potential applications of molecular communication in biomedical, environmental and manufacturing areas.
However, bio-inspired approaches can open up new opportunities and research di- rections in terms of security and privacy for molecular communication systems since they are bio-inspired and there exist in nature several defense mechanisms against enormous types of attacks that are really effective.
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