CHAPTER 1 INTRODUCTION
1.4 Thesis Organization
− d2 4Dt
, (1.1)
where D is the diffusion coefficient. However, this form simply describes the expec-tation of the molecule behavior. The fluctuation or randomness of concentration level should be taken into account. This thesis is the first study that provides stochastic model for diffusion channel when continuous waveform is applied at TN. Based on the model, we also propose methods for solving a set of basis func-tions in order to achieve orthogonal expansion for detection.
1.4 Thesis Organization
The rest of the thesis is organized as follows. In Chapter 2, we provide a one-dimensional modulation and detection scheme, under which we design an ISI cancellation algorithm that dramatically improves the system performance. In Chapter 3, we consider a more complicated modulation which embeds information on both quantity and type of molecules. This modulation combats both ISI effect and background noise simultaneously. We also provide principles to choose system parameters to achieve good performance. In Chapter 4 of this thesis, we construct a stochastic model for the diffusion channel starting from Brownian motion of each
1.4. THESIS ORGANIZATION 5
molecule. Different from the results in [10,12], this model applies to any continuous wave transmitted by TN. We also solve a set of basis functions that can expand the received signals into observation vectors, which enables us to design detection criterion for different modulations. Finally, conclusions and future works are given in Chapter 5.
CHAPTER 2
QUANTITY MODULATION
2.1 Introduction
In this chapter, we study the communication between two nano-machines with information embedded in different molecular quantity [13]. In the rest of this chapter, we call this kind of modulation as quantity modulation (QM). It is known that in diffusion-based molecular communications, molecules are emitted by the transmitter and move towards the receiver following the laws of molecule diffu-sion. Recent studies on diffusion-based molecular communications often model the statistical behavior of molecule diffusion as a Brownian motion [8]. Due to the random nature of Brownian motions, molecules that released earlier by TN may arrive late. Therefore, messages carried in current molecules may be interfered by those delayed molecules that were transmitted earlier. This phenomenon is known as the ISI effect in diffusion-based molecular communications. Studies about this effect can be found in [14] and [15].
There are lots of ways to design filters to eliminate the effect of ISI in con-ventional communication such as linear equalizer, adaptive equalizer and decision-feedback equalizer [16]. However, both linear equalizer and adaptive equalizer do not work well in molecular communication due to time-varying channel response.
In this chapter, we utilize the concept of decision feedback and introduce a method to mitigate the effect of ISI.
The rest of this chapter is organized as follows: In Sec. 2.2, we introduce the settings of a binary QM molecular communication system in details. We also describe the characteristics and the mathematical model of a Brownian motion channel. Sec. 2.3 focuses on deriving the decision rule for one-shot transmission of the binary QM system and extending it to the M-ary transmission case. In
6
2.2. SYSTEM MODEL 7
Sec. 2.4, we consider serial transmission and take ISI effect into account. ISI cancellation method is also described in this section. Numerical results are shown in Sec. 2.5.
2.2 System Model
In this section, we first give a general model for transmitter, receiver, and chan-nel in molecular communication. We then describe a QM system of M quantity levels (M -ary modulation) bearing log2M information bits, which will be used later to apply our ISI cancellation method.
2.2.1 Transmission Nano-machine
Fig. 3 illustrates a transmitter nano-machine TN transmitting molecules to a receiver nano-machine RN. When TN obtains information (e.g. bit pattern) to be transmitted, it starts storing certain number of molecules in a vesicle (container that stores molecules) and releases these molecules simultaneously into the envi-ronment. The number of released molecules differs according to the transmitting information. In practical situations, molecules leaves the vesicle with random tim-ing which is discussed in [17]. In this thesis, we simply assume that molecules exit the vesicle simultaneously.
Figure 3: Transmission from TN to RN through a fluid medium.
2.2.2 Reception Nano-machine
RN is located at a position d > 0 apart from TN. There are several recep-tors capturing molecules on RN. RN counts the number of molecules it captures and perform detection according to the number. We assume the molecules can
2.2. SYSTEM MODEL 8
be perfectly captured by the receptors and RN does not have counting errors.
Furthermore, once a molecule arrives at RN, it will be removed from the commu-nication medium.
2.2.3 Channel
Consider a fluid medium between TN and RN with positive drift velocity v.
The molecules are all constrained to move in a one-dimensional space. We assume that the trajectory of emitted molecules can be modeled with independent Brow-nian motions [8]. Let X denote the random variable representing the first hitting time of a molecule. If v > 0, it can be shown that the probability density function (PDF) of X is given by the inverse Gaussian (IG) distribution [18]
fX(x) =
where D denotes the diffusion coefficient which is given by
D = kBT 6πτ r,
where kB is the Boltzmann constant, Ta is the absolute temperature, τ is the viscosity of the fluid medium, and r is the radius of molecules. For simplicity, we assume that the radii for all molecules are the same so that the diffusion coefficients are the same.
2.2.4 QM System
Consider a time-slotted M -ary communication with signaling interval TB, TN can release M different quantities of molecules into the channel. Denote those M quantities by Lm, where m ∈ {0, 1, 2, · · · , (M −1)}. Assume the a priori probabil-ity for releasing Lm molecules to be qm. At the starting time of each transmission
2.3. DETECTION IN ONE-SHOT TRANSMISSION 9
time slot, Lm molecules are emitted simultaneously from the transmitter to indi-cate the transmission of a symbol. We assume perfect synchronization between the transmitter and the receiver. During each time slot, RN counts the total num-ber of arriving molecules. An appropriate decision rule, proposed in Sec. 2.3, is then applied to determine the transmitted data bit at the end of each time slot.
The molecules which fail to arrive within the corresponding time slot become a source of interference, which will cause performance degradation to the detections of later coming symbols. Fig. 4 is an example of QM system with M = 4 and uniform quantity levels.
Perfect synchronization between TN and RN is assumed in this chapter, a possible realization which is based on sending training molecular impulses are introduced in [BoKai].
Figure 4: Illustration of quantity-based modulation scheme with L0 = 2, L1 = 4, L2 = 6, L3 = 8.
2.3 Detection in One-shot Transmission
In this section, we discuss the detection rule of the system for one-shot trans-mission. The main contribution of our work is that we separate the ISI cancella-tion problem from the deteccancella-tion to achieve a more flexible and modularized design, which is different from previous works [19].
2.3.1 Binary Detection
We define two hypotheses H0 and H1. H0 is the hypothesis that L0 molecules are transmitted (indicating bit 0), and H1 is the hypothesis that L1 molecules are transmitted (indicating bit 1). Denote the conditional PDF of the number of
2.3. DETECTION IN ONE-SHOT TRANSMISSION 10
received molecules in a particular time slot, given that hypothesis Hm is true, by Pr{N = n|Hm}, m ∈ {0, 1}. Using the inverse Gaussian PDF given in (2.1), we define the probabilities pj as:
pj = to be the indicator random variable showing whether the k-th molecule emitted in a one-shot transmission arrives within Ts given that Hm is true. That is,
Yk =
1, if the k-th molecule arrives within Ts,
0, otherwise.
(2.3)
Let N be the random variable denoting the total number of molecules arriving at the receiver within a particular time slot. We have the following relation:
Pr{N = n|Hm} = Pr{Y1+ Y2+ ... + YLm = n|Hm}. (2.4)
Given the number of the transmitted molecules, N thus follows Binomial(Lm, p0).
For large Lm, we approximate the binomial distribution by a Gaussian distribution with the knowledge of the mean and variance of N . Namely, we have
Pr{N = n|Hm} ≈ As a special case when M = 2, the distributions of N under two hypotheses can thus be obtained as
2.3. DETECTION IN ONE-SHOT TRANSMISSION 11
According to the conventional hypothesis testing theory [20, 21], the decision rule can be expressed using the likelihood ratio test Λ(N ) as
Λ(N ) = P (N |H1)
If we assume equal a priori probabilities q0 = q1 = 1/2, due to the characteristic of Gaussian distribution shown in Fig. 5, the decision rule can be further reduced to
N
H1
H≷0
η (2.9)
for some threshold η, where η is the solution of the following equation:
Pr{N = η|H0} = Pr{N = η|H1}. (2.10)
Taking logarithms to both sides, the equation becomes
η = s
L1L0ln(L1/L0) L1− L0
p0(1 − p0) + p20L0L1. (2.12) In other words, if the received number of molecules is greater than the threshold η, the receiver will determine H1 as the hypothesis testing result; otherwise H0 will be decided.
2.3.2 M-ary Detection
The detection rule can be extended to M -ary detection with only a few adjust-ments. Suppose we have multiple hypotheses Hm where m ∈ {0, 1, 2, · · · , (M −1)}
which represent the transmission of Lm molecules. The goal is to decide which ˆm we should choose. The maximum a priori (MAP) detection rule is:
ˆ
m(N ) = argmax
m
P (N |Hm). (2.13)
Due to the properties of Gaussian distribution, the above MAP detection rule can be simplified to pairwise comparisons between the “neighboring” conditional probability distributions.
2.3. DETECTION IN ONE-SHOT TRANSMISSION 12
Figure 5: Demonstration of the process of finding η in a binary QM system, where f is the conditional PDF of N given Hm is true.
To write down the expressions explicitly, we define a set of thresholds E = {ηj ∈ [−∞, ∞] : j = 0, 1, 2, · · · , M }, and let η0 = −∞ and ηM = ∞. For j = 1, 2, · · · , (M − 1), ηj can be obtained by solving the equations
Pr{N = ηj|Hj−1} = Pr{N = ηj|Hj}. (2.14)
With the thresholds determined, the detection rule for M -ary transmission can be expressed as
ˆ
m(N ) =
M −1
X
k=0
k · u [−(N − ηk)(N − ηk+1)] . (2.15) where u(·) denotes the unit step function.
2.3.3 Error Rate Analysis
After the construction of the transmission and decision rules, we then analyze how it performs in terms of symbol or bit error rate. Consider a specific case for M = 2 and q0 = q1 = 1/2. Denote the false alarm probability as PF and the missing probability as PM. The error rate can be written as
Pe = q0PF+ q1PM= 1
2(PF+ PM). (2.16)
From Fig. 5 and the decision rule derived in Sec. 2.3.1, it can be shown that
PF = Pr{N > η|H0} = Q η − L0p0 pL0p0(1 − p0)
!
, (2.17)
2.4. SERIAL TRANSMISION AND ISI CANCELLATION 13
PM = Pr{N < η|H1} = Q L1p0 − η pL1p0(1 − p0)
!
, (2.18)
where Q(·) denotes the Q-function. By substituting (2.12) into equation (2.17) and (2.18), we can evaluate the error rate Pein (2.16). Fig. 6 shows the comparison between our analysis and numerical results.
10 20 30 40 50 60 70
10
−810
−610
−410
−210
0L
1(Maximum number of molecules transmitted per symbol)
SER
Numerical curve Theoretical curve
Figure 6: Theoretical result versus numerical result for one-shot binary quantity-based modulation.
2.4 Serial Transmision and ISI Cancellation
The above described QM molecular communication system seems to work al-ready. However, in practical situations, we need to perform serial transmissions rather than one-shot transmission. Thus the ISI effect must be taken into account.
Our results show that if we do not modify our one-shot detection rule, the system performance will fall dramatically under serial transmission environments due to the severe ISI effect. To solve this problem, we propose a method to mitigate the ISI effect.
In order to mitigate the ISI effect, we first need an estimation of the number of
2.4. SERIAL TRANSMISION AND ISI CANCELLATION 14
the delayed molecules that come from former time slots. If we know the conditional probability distribution of the number of ISI molecules conditioned on the current received number, then we can estimate the ISI effect as the conditional mean.
However, the conditional distribution does not have a closed-form solution for inverse Gaussian random variables. Here, we proposed another intuitive way to do this estimation.
First, we define “memory-Γ cancellation” to mean that the ISI effect during the past Γ time slots are taken into account when making decision. We first use memory-1 cancellation as a demonstrative example. If the number of molecules received during the (i−1)-th time slot is ni−1and the decided transmission quantity level is ˆli−1, where ˆli−1 ∈ {L0, L1, · · · , LM −1}, we then subtract ˆli−1· [FX(2Ts) − FX(Ts)] (the a priori expected received number in the i-th time slot from the (i − 1)-th time slot) from ni before making the i-th decision, where FX(·) is the cumulative distribution function (CDF) of the first-hitting time of molecules. In other words, the actual number ˜ni used in making decision is
˜
ni = ni− ˆli−1· [FX(2Ts) − FX(Ts)]. (2.19) Likewise, we can perform memory-Γ cancellation if we have enough buffer at the receiver end to memorize temporarily the recently received numbers of molecules.
More explicitly, denote the probabilities that a single transmitted molecule ar-rives during the time interval [jTs, (j + 1)Ts] by pj for j ∈ N ∪ {0} as before.
If the decided transmission quantity level of the current time slot is ˆl, where ˆl ∈ {L0, L1, · · · , LM −1}, then the received number of molecules j time slots later should be subtracted by ˆl · pj+1 before making decision. In other words, if the number of molecules received in i-th time slot is ni, the actual number ˜ni used in making decision is
For binary QM systems, the decision rule can be written as
˜
2.5. NUMERICAL RESULTS 15
The extension to M-ary QM systems is straightforward.
2.5 Numerical Results
In this section, we first discuss the binary and M -ary QM modulation systems with and without performing ISI cancellation. After that, we make comparisons of the system performance under different time slot durations.
The number of molecules is one of the main resources utilized in molecular communications. Analogous to the “power” concept in conventional communi-cations, we need to take this number into account when comparing the system performances. In the following subsections, we present the results under differ-ent maximum number of molecules allowed per symbol, and the quantity levels are uniformly spaced. The simulation parameters are d = 0.2 cm, drift velocity v = 0.01 cm/sec, diffusion coefficient D = 0.05 cm2/sec , and time slot duration Ts = 5 sec.
2.5.1 SER Comparison with and without ISI Cancellation
In Fig. 7, a binary transmission system with memory-1 and memory-2 cancel-lation is considered. The SER drops from 0.04 to 0.01 when L1 = 30, and drops from 10−2 to 10−4 when L1 = 90. The improvement grows as L1 increases, which means that by choosing L1 properly, a reliable end-to-end transmission can be achieved. We also observe that even without ISI cancellation, the error rate will drop as L1 increases. The reason is that the spacing between symbols is increased.
However, as shown in Fig. 8, it is not the case for the quaternary transmission system. It can be seen that even though L3 becomes large, the error rate is still high without ISI cancellation, which means we cannot rely solely on increasing the maximum number of molecules without ISI cancellation.
It is worth mentioning that the ISI cancellation method can be performed not only in such quantity-based modulation systems, but it can also be used in other systems like on-off keying1 with slight modifications.
1Transmitting zero or a single molecule.
2.5. NUMERICAL RESULTS 16
10 20 30 40 50 60 70 80 90
10
−510
−410
−310
−210
−1L
1(Maximum number of molecules transmitted per symbol)
SER
No ISI cancellation
Memory−1 ISI cancellation Memory−2 ISI cancellation
Figure 7: Binary quantity-based modulation with ISI cancellation.
100 150 200 250 300 350 400 450 500
10
−510
−410
−310
−210
−110
0L
3(Maximum number of molecules transmitted per symbol)
SER
No ISI cancellation Memory−1 cancellation Memory−2 cancellation
Figure 8: Quaternary quantity-based modulation with ISI cancellation.
2.5. NUMERICAL RESULTS 17
2.5.2 Performance Under Different Duration of Time Slot
In this subsection, we consider a binary transmission system with and without ISI cancellation for different time slot durations. From Fig. 9, we can observe that the SER decreases as the duration Ts increases. In other words, to improve performance, one can increase the duration of the time slot as shown in Fig. 9.
Although the error rate is already quite acceptable, it can be further improved by the ISI cancellation approach. The improvements is about 10 times better when Ts = 10 sec and L1 = 70. Note that when Ts is small, say Ts = 1 sec, compared to the expected first-hitting time d/v, the error rate increases even if we increase L1 when no cancellation is performed. This is because when Ts is small, molecules tend not to arrive in one symbol time but stay in the background, and that a larger L1 will cause a larger amount of molecules to be in the background and hence larger interference.
10 20 30 40 50 60 70
10
−410
−310
−210
−110
0L
1(Maximum number of molecules transmitted per symbol)
SER
Without ISI cancellation With memory−1 cancellation
T
s= 10 sec
T
s= 5 sec T
s= 1 sec
Figure 9: Binary quantity-based modulation with ISI cancellation under different Ts.
CHAPTER 3
QUANTITY-TYPE MODULATION
3.1 Introduction
To communicate between two nano-machines, several modulation techniques have been proposed to bear information [10, 12, 22–25]. Among various kinds of modulation, quantity [22] and type [23, 24] modulation is of our interests. Based on the design paradigms of both works, that the type of molecules could yield ad-ditional embedded information, and that the increase in the quantity of molecules per transmission could result in higher performance, we propose a new modula-tion scheme called quantity-type modulamodula-tion. Consider a type-based modulated system, the transmitter releases different types of molecules representing differ-ent information bits or symbols. When the molecules arrive at the receiver, the receiver captures those molecules and attains information based on their types.
Nevertheless, the arrival times of the molecules at the receiver are random due to the diffusion process. This results in the phenomenon that the molecules released earlier may arrive late, leading to wrong information detection. A way to remedy this is to release a group of molecules of the same type at a time to improve the system reliability.
An intuitive method to detect the quantity-type modulated molecular commu-nication system is using majority vote—information bits are detected according to the type of molecules that outnumber another. However, as it is shown in this paper, the performance of the majority vote detection algorithm is disappointing.
Hence, in this chapter, we introduce a novel detection algorithm called threshold-based detection by exploiting the characteristics of the diffusion channel. Theo-retical approximations of the bit error rate (BER) performance of the proposed threshold-based detection algorithm for the quantity-type-modulated molecular
18
3.2. SYSTEM MODEL 19
communication system are derived. Both the simulation and theoretical results confirm the significant performance improvement over the majority vote detection, either without or with background noise.
3.2 System Model
In this section, based on the diffusion channel modeled by Brownian motion as in 2.2, we introduce the communication scheme adopted in this chapter.
3.2.1 Quantity-type Modulation
We assume that two types of information molecules , ‘A’ and ‘B’, have the same radius r and are distinguishable for both transmitter and receiver. This can be achieved by using isomers as described in [26]. The transmission is assumed to be time-slotted with interval Ts and the transmitter releases n molecules of one type at the beginning of each time slot to represent an information bit, where n is an odd number. That is, n type-A molecules represent bit ‘1’ and n type-B molecules represent bit ‘0’. In addition, the information bits are firstly partitioned into blocks with L bits. When nL molecules are released, the transmitter should wait for a period of time T before the next transmission. The receiver always gathers molecules on arrival, that is, there is no fixed time slots at the receiver and the asynchronous detection, i.e., majority-vote detection or threshold-based detection, is performed. Moreover, whenever nL molecules are captured, the receiver should also wait for a period of time T before the next detection. The waiting time between blocks is set to avoid detection errors caused by background noise (which will be mentioned in the next subsection) affecting the next block.
3.2.2 ISI and Noise Effect on Quantity-type Modulation
Due to the randomness of the first hitting times of transmitted molecules, a molecule may arrive at the receiver in advance of the molecule(s) released earlier . We use the term crossover to describe this phenomenon. Crossovers result in intersymbol interference (ISI) and may lead to detection errors.
3.3. DETECTION ALGORITHMS 20
Figure 10: A demonstrative example for the majority-vote detection algorithm and the threshold-based detection algorithm with λ = 1 and λ = 2. The boxed numbers represent the value stored in the counter that reaches the threshold λ.
In diffusion-based molecular communications, it is likely for the receiver to capture molecules that are not released by the corresponding transmitter. Those unintended molecules, which we call background noise in the rest of this study, may come from the environment or other transmitters. We model the number of
In diffusion-based molecular communications, it is likely for the receiver to capture molecules that are not released by the corresponding transmitter. Those unintended molecules, which we call background noise in the rest of this study, may come from the environment or other transmitters. We model the number of