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 T_s 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 T_s = 10 sec and L₁ = 70. Note that when T_s is small, say T_s = 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 T_s 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

⁻⁴

10

⁻³

10

⁻²

10

⁻¹

10

⁰

L

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 T_s.

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 T_s 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 arriving unintended molecules as a Poisson random process {N (t) : t ≥ 0} and N (t + τ ) − N (t) follows Poisson(ατ ) with noise rate α.

3.3 Detection Algorithms

In this section, we first describe the majority-vote detection algorithm and then elaborate the proposed threshold-based detection algorithm.

3.3. DETECTION ALGORITHMS 21

Figure 11: Block diagram of the threshold-based detection algorithm.

3.3.1 Majority-vote Detection

An intuitive detection method of the quantity-type-modulated molecular com-munication system is the majority vote, which counts the majority type of molecules once n molecules are gathered. We use an example to explain the majority-vote detection in Fig. 10. To illustrate that different types of molecules are released by the transmitter at different time slots, we use (U₁, U₂, U₃, · · · ) to represent that the transmitter releases a set of molecules U₁ at t = t₀, molecules U₂ at t = t₀+T_s, and molecules U₃at t = t₀+2T_s, and so on. Let us consider the example in Fig. 10, the transmitter releases (AAA, BBB, AAA, AAA, BBB) to convey the information sequence ‘1, 0, 1, 1, 0’. Due to the diffusion channel, the arriving molecules may be out of order. In this example, the receiver captures the molecules in the order of

‘A, A, B, A, B, A, B, A, A, A, A, B, B, B, A’. (3.1)

Note that in both (3.1) and “Received pattern” in Fig. 10, we do not show the time difference between two adjacently captured molecules, which may not be identical.

The majority-vote algorithm groups the molecules as ‘A, A, B’, ‘A, B, A’, ‘B, A,

3.3. DETECTION ALGORITHMS 22

A’, ‘A, A, B’, and ‘B, B, A’, and determines the majority type in each group as

‘A’, ‘A’, ‘A’, ‘A’, and ‘B’. Therefore, the detected bits are ‘1, 1, 1, 1, 0’, which has one bit error. This error results from the effect of ISI as described in Sec. 3.2.2.

3.3.2 Threshold-based Detection

To develop a detection algorithm that is suitable for operating under the diffu-sion channel, let us first gain some insights about how the diffudiffu-sion-based molec-ular communication system is performed. Consider a case that the transmitter releases (A, B), i.e., the transmitter releases a molecule of type A and then a molecule of type B after a time duration T_s. It is likely that the receiver receives ‘B, A’, which results in bit errors. However, if the transmitter releases (AAA , BBB), i.e., the transmitter releases three molecules of type A and then three molecules of type B after a time duration T_s, the probability that the receiver receives ‘B, B, B, A, A, A’ is small since it requires many crossovers happening to obtain this pattern. It is more likely that at least one type-A molecule arrives earlier than all the type-B molecules. Therefore, a reasonable principle for designing detection algorithm would be: once the receiver captures a type-A molecule before capturing a type-B molecule, the receiver infers that the transmitter releases (AAA) before (BBB). The remaining two type-A molecules that arrive later provide relatively little information, and they may even introduce ISI to other molecules; hence the receiver should ignore those two late-arriving molecules.

Based on the above observation, we propose the threshold-based detection al-gorithm. As shown in Fig. 11, the detector is composed of a classifier, two counters, and two threshold comparators. The classifier recognizes the type of the arriv-ing molecules. If the arrivarriv-ing molecule belongs to type A, the classifier generates a signal to counter A; if the arriving molecule belongs to type B, the classifier generates a signal to counter B. Counter A and counter B are used to count the number of arriving molecules of type A and B, respectively. Whenever a signal is sensed by the counter, it increases its stored value by one. The values stored in the counters are denoted by V_A and V_B respectively. When the value stored in

3.3. DETECTION ALGORITHMS 23

the counter reaches a predetermined threshold λ, bit ‘1’ or ‘0’ is generated. Once the stored value reaches n, the stored value is reset to zero. Note that when V_Aor V_B is in the region (λ, n], no output is generated by the detector. That is, n − λ molecules with the same type are ignored by the receiver.

Fig. 10 shows a simple example that does not consider the noise effect (or the noise rate α is negligibly small). The detection result in this example has no bit error. An example of the threshold-based detection algorithm considering the background noise is shown in Fig. 12. The transmitter releases three molecules at a time and the receiver applies the threshold-based detection with threshold λ = 3. In this example, only the background noise caused by type-A molecules is considered. The unintended molecules are equivalently inserted into the receiving molecule pattern. The inserted molecules cause wrong increment of the value stored in the counter, making the receiver output an erroneously detected bit.

This is termed bit insertion. When this happens, the detected bit sequence is right-shifted compared with the actual bit sequence.

Since n molecules are released in each symbol duration, the total number of arriving molecules at the receiver should be nL for each block. However, due to the background noise, the total number of molecules may be greater than nL. The proposed scheme in Sec. 3.2.1 suggests that after nL molecules are received, the receiver ignores the late-arriving molecules for a period of T . We set T = 2T_s since the probability that a molecule arrives late for more than 2T_s is small.

3.3.3 Trade-offs when Combating ISI and Noise Effect on Quantity-type Modulation

ISI comes from the crossover effect of the released molecules. Intuitively, adopt-ing a larger signaladopt-ing interval T_s at the transmitter will decrease the probability of crossovers. However, as T_s grows larger, the number of unintended molecules captured by the receiver during each symbol detection will increase. Thus, we expect that there should be an optimal T_s which minimizes the detection error caused by both ISI and background noise. The optimal T_s for threshold-based

3.3. DETECTION ALGORITHMS 24

Figure 12: A demonstrative example for the detection result under background noise using the threshold-based detection algorithm. λ = 3. In the example, the crossover effect is not taken into account.

detection will be derived in Sec. 3.4.

It can be observed that detection errors occur when unintended molecules are captured by the receiver, which causes the value stored in the counter to reach the threshold λ too early. Therefore, to combat the background noise, we may increase the value of λ. However, as will be shown later, a small λ is better for combating ISI. Details in determining λ is discussed in Sec. 3.5. Moreover, the value of n also affects the system performance since λ ≤ n, that is, n limits the possible choices of λ.

To deal with the background noise, it is also possible to modify the block size L. The effects of background noise can be mitigated by using a smaller L since the number of unintended molecules in a block can be reduced. Nevertheless, when a smaller block size is used, the total throughput decreases due to the longer duration between blocks.

In the next section, we analyze the system performance in the sense of bit

3.4. BER ANALYSIS 25

error rate (BER). Based on the BER performance, principles of choosing the above-mentioned system parameters will be discussed in Sec. 3.5.

3.4 BER Analysis

In this section, we mathematically compare the BER performance of the majority-vote detection algorithm and the proposed threshold-based detection algorithm.

In the following, we first derive BER under the case that the background noise is negligible, i.e. α approaches zero. We then analyze the case when α is not negligible such that increasing T_s would introduce more unintended molecules at the receiver.

3.4.1 Preliminaries

For the i-th information bit s_i ∈ {0, 1}, n molecules of type A or B are released.

By assuming that the prior probabilities Pr{s_i = 0} = Pr{s_i = 1} = 1/2, the BER can be written as

Pr{bs_i 6= s_i} = 1

2Pr{bs_i = 1|s_i = 0} +1

2Pr{bs_i = 0|s_i = 1}

= Pr{bs_i = 0|s_i = 1}, (3.2)

where bs_i is the information bit detected by the receiver. Equivalently, we can compute (3.2) by

Pr{bs_i 6= s_i} = 1

4Pr{bs_i = 0|s_i−1= 0, s_i = 1, s_i+1 = 0} +1

4Pr{bs_i = 0|s_i−1= 0, s_i = 1, s_i+1= 1}

+ 1

4Pr{bs_i = 0|s_i−1= 1, s_i = 1, s_i+1 = 0} +1

4Pr{bs_i = 0|s_i−1= 1, s_i = 1, s_i+1= 1}.

(3.3) Each term in (3.3) is affected by the ISI from {s_i−2, s_i−3, · · · } and {s_i+2, s_i+3, · · · }.

Here, we assume that the errors due to crossover of molecules happening to neigh-boring information bits are dominant compared with the error due to crossover of molecules happening to information bits that are more than two time intervals apart. Therefore, we compute (3.3) by considering the relations of s_i−1, s_i, and s_i+1 only.

3.4. BER ANALYSIS 26

Denote the first hitting time of the n molecules released for representing s_i as independent and identically distributed (i.i.d.) random samples X₁(i), · · · , X_n(i).

From the theorem of order statistics, if X₁(i), · · · , X_n(i) are arranged in increasing order as X₍₁₎(i) ≤ · · · ≤ X_(n)(i), the PDF and the cumulative density function (CDF) for the k-th smallest one are, respectively,

fX_(k)(y) = nn − 1

Now we define the switching between two symbols s_i and s_j as the event bsi = sj,bsj = si for si 6= sj. For convenience, we define the events Epqr and bEpqr

respectively as

E_pqr = {s_i−1= p, s_i = q, s_i+1= r},

Eb_pqr = {bs_i−1= p,bs_i = q,bs_i+1= r}. (3.6) We use the superscript to distinguish the majority-vote detection and the threshold-based detection forbs_i and bE_pqr, i.e.,bs^MVD_i and bE_pqr^MVD for the majority-vote detec-tion, and bs^TD_i and bE_pqr^TD(λ) for the threshold-based detection with threshold λ.

3.4.2 Analysis when Background Noise is Negligible

Let us begin by analyzing the majority-vote detection. To calculate the BER of the majority-vote detection, denoted by P_c^MVD, (3.3) is rewritten as

P_c^MVD = Pr{bs^MVD_i 6= s_i}

3.4. BER ANALYSIS 27

1. Molecules representing s_i cross over molecules representing s_i+1and then the event bE₀₀₁^MVD happens.

2. Molecules representing s_i−1cross over molecules representing s_i and then the event bE₁₀₀^MVD happens.

3. Molecules representing s_i−1, s_i, and s_i+1cross over in the way that the type-B molecules outnumber type-A molecules for all three detected symbols, and then the event bE₀₀₀^MVD happens.

Note that the above observation is made with the assumption in Sec. 3.4.1 that detection errors due to ISI is dominated by the crossovers of molecules from neigh-boring information bits, namely, s_i−1 and s_i+1. Fig. 13 shows example outcomes given the event E₀₁₀ which corresponds to the conditions mentioned above: the first outcome results from two crossovers between s_i and s_i+1, the second outcome results from two crossovers between s_i and s_i−1, and the third outcome results from one crossover between s_i and s_i+1 and one between s_i and s_i−1. The second and third terms in (3.7) follow similar arguments. Given E₀₁₁ or E₁₁₀, the event bs_i = 0 is equivalent to bE₁₀₁^MVD. The final term in (3.7) equals 0 since given event E₁₁₁, bs_i = 0 is impossible. Therefore, by the assumption in Sec. 3.4.1, we have P_c^MVD≈ 1

4Pr{ bE₀₀₁^MVD∪ bE₁₀₀^MVD∪ bE₀₀₀^MVD|E₀₁₀} +1

4Pr{ bE₁₀₁^MVD|E₀₁₁} +1

4Pr{ bE₁₀₁^MVD|E₁₁₀}.

(3.8) To furthur compute (3.8), we need the following lemma.

Lemma. Assume that the switching other than s_i and s_i+1is negligible. Under the quantity-type modulation and majority-vote detection without background noise, two transmitted symbols s_i and s_i+1 switch if and only if s_i 6= s_i+1, and the order statistics X₍ⁿ⁺¹

2 )(i) and X₍ⁿ⁺¹

2 )(i + 1) satisfy X₍ⁿ⁺¹

2 )(i) > X₍ⁿ⁺¹

2 )(i + 1) + T_s. (3.9)

Proof. Without loss of generality, we assume that s_i = 1 and s_i+1 = 0. For the

“if” part, the condition (3.9) is equivalent to having the crossover between the

3.4. BER ANALYSIS 28

n+1

2 -th molecule in s_i and the ⁿ⁺¹₂ -th molecule in s_i+1. Therefore, (3.9) guarantees that there are at most ⁿ⁻¹₂ molecules of type A arriving earlier than X₍ⁿ⁺¹

2 )(i), and there are at least ⁿ⁺¹₂ molecules of type B arriving earlier than X₍ⁿ⁺¹

2 )(i). Since

n−1

2 + ⁿ⁺¹₂ = n, we conclude that for the majority-vote detection, the detection results are bs_i = 0 and bs_i+1= 1, i.e., s_i and s_i+1 switch.

For the “only if” part, we assume that either s_i = s_i+1or X₍ⁿ⁺¹

2 )(i) < X₍ⁿ⁺¹

2 )(i+

1) + T_s, and we aim to prove s_i and s_i+1 do not switch. The first case s_i = s_i+1 is trivial since no switching happens between two identical symbols. For the other case X₍ⁿ⁺¹

2 )(i) < X₍ⁿ⁺¹

2 )(i + 1) + T_s, there are exactly ⁿ⁺¹₂ molecules of type A arriving no later than X₍ⁿ⁺¹

2 )(i), and there are at most ⁿ⁻¹₂ molecules of type B arriving no later than X₍ⁿ⁺¹

2 )(i). Since ⁿ⁺¹₂ + ⁿ⁻¹₂ = n, for the majority-vote detection, the detection results are bs_i = 1 and bs_i+1 = 0, i.e., s_i and s_i+1 do not switch.

After applying the lemma to (3.8), the BER under the majority-vote detection without background noise is given in (3.10).

Pr{bs^MVD_i 6= s_i} ≈ 1

Now we analyze the threshold-based detection algorithm. Similarly, by the assumption in Sec. 3.4.1, the BER of the threshold-based detection algorithm, denoted by P_c^TD(λ), can be derived as

3.4. BER ANALYSIS 29

Figure 13: Example of the events bE₀₀₁^MVD, bE₁₀₀^MVD, and bE₀₀₀^MVD given that the infor-mation sequence ‘0, 1, 0’ is transmitted.

If the threshold is λ, it can be observed that when s_i−16= s_i, the switching of s_i−1 and s_i occurs when

X_(λ)(i − 1) > X_(λ)(i) + T_s. (3.12) Therefore, (3.11) can be approximated by (3.13).

Pr{bs^TD_i 6= s_i} ≈ 1

4PrX_(λ)(i − 1) > X_(λ)(i) + T_s or X_(λ)(i) > X_(λ)(i + 1) + T_s + 1

4PrX_(λ)(i − 1) > X_(λ)(i) + T_s + 1

4PrX_(λ)(i) > X_(λ)(i + 1) + T_s (a)= 1

2PrX(λ)(i − 1) > X(λ)(i) + Ts + 1

2PrX(λ)(i) > X(λ)(i + 1) + Ts

− 1

4PrX(λ)(i − 1) > X(λ)(i) + Ts > X(λ)(i + 1) + 2Ts . (3.13)

The equality in (3.13) (denoted by ‘(a)’) results from the property Pr{A∪B} = Pr{A} + Pr{B} − Pr{A ∩ B}. Again, we utilize the fact that the probability of switching happening to the information bits that are more than two intervals apart is small. Then the negative term in (3.13) is negligible. Moreover, since the probability that s_i switches with s_i+1 and the probability that s_i−1 switches with

3.4. BER ANALYSIS 30

which can be evaluated numerically.

To compare the majority-vote detection algorithm and the threshold-based detection algorithm, we start from (3.8) and the lemma.

P_c^MVD≈ 1

where (3.15) comes from the union bound. Therefore, we conclude that the threshold-based detection with λ = ⁿ⁺¹₂ outperforms the majority-vote detection due to the extra term bE₀₀₀^MVD in (3.16).

3.4.3 Analysis when Background Noise is not Negligible

To approximate the BER when background noise is not negligible, we compute the BER resulted from the crossovers and the background noise respectively, and then discuss their joint effect. Denote P_cas the BER caused by the crossover effect (P_c = P_c^MVD or P_c = P_c^TD), P_n as the BER caused by the background noise, and

3.4. BER ANALYSIS 31

P_e as the aggregated BER. From the union bound, we have P_e ≤ P_c+ P_n. Note that we have already calculated P_c of the two algorithms in Sec. 3.4.2. The main problem of computing P_n is due to the difficulty of enumerating all the possible patterns of the arriving molecules when background noise is taken into account.

However, we can approximate P_n by computing the average number of errors in a block and dividing it by the block size L.

Although the detection performed by the receiver is asynchronous, the receiver would spend LT_s (or L time slots) on average to capture all nL molecules. For both detection algorithms, observe that whenever an unintended molecule arrives at the receiver, the following received molecule pattern in the block will be right-shifted by one. When the number of unintended molecules exceeds a certain value (which will be discussed in the following paragraphs), the molecules representing the ith bit will be right shifted such that the ith detection bit will not be performed on the corresponding molecules, and thus causes a bit insertion. An example for noise effect on the threshold-based detection is shown in Fig. 12. We assume the transmitter releases molecules (AAA, AAA, BBB) to convey information bits

‘1, 1, 0’. When background noise is negligible, the received molecule pattern will follow the order as in the example. On the other hand, when background noise is not negligible, a bit insertion is produced when the receiver receives λ = 3 unintended molecules. The subsequent bits will be right shifted and the detections

‘1, 1, 0’. When background noise is negligible, the received molecule pattern will follow the order as in the example. On the other hand, when background noise is not negligible, a bit insertion is produced when the receiver receives λ = 3 unintended molecules. The subsequent bits will be right shifted and the detections

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