7
Multi-Chemical Transmission

7.1 Introduction

M olecular communication is an alternative to the present transmission techniques, using particles instead of waves. This change in the propagation methods opens novel modulation methods not present in EM communication such as the characteristics of the particle being transmitted. Every chemical, molecule and compound have specific properties that make them unique and by exploiting this, the throughput and the channel capacity of the molecular communication can be increased. However, to test multiple chemical transmission, a sensor that can distinguish chemicals by a specific property must be used . As described in Chapter 3, a Membrane-Inlet Mass Spectrometer (MIMS) with a Quadrupole Mass Analyser is used as the receiver. Mass spectrometers (MS) have the ability to analy s e and distinguish multiple chemicals simultaneously which makes it a suitable detector for use in this application [251].

The novel contributions of the Chapter are as follows;

Multi-Chemical Transmission: An experimental proof-of-transmission along with theoretical comparison was conducted with additional analysis of multiple-chemical noise.
Molecular Quadrature Amplitude Modulation (MQAM): Influenced by its EM counterpart, theoretical analysis was done on this modulation scheme with analysis on the parameters of the communication for SER and MI .
Chemical Time Offset Keying (ChToK): A modulation method is proposed based on the relative position of signals in a given time-frame.
Chemical Ratio Keying (ChRK): A modulation method is proposed based on the amplitude of the chemicals relative to each other in a given symbol period.

In the simulations conducted in the chapter, it is assumed that the transmission possesses Additive White Gaussian Noise (AWGN) ( $N (μ_{N}, σ_{N}^{2})$ ) based on the single chemical noise analysis in [3] and the multi noise analysis conducted in this study. Finally, it is assumed that there are no interferences between carriers.

7.2 Multi Chemical Transmission

An experiment was undertaken to test the possibility of sending two chemical species simultaneously. To detect and analyse multiple chemicals simultaneously, a mass spectrometer (MS) was used which its inner workings were described in detail in Chapter 3. The signal chemicals were chosen as a cetone with tracked ion of $m ∕ z$ 43 and m ethanol with tracked ion of $m ∕ z$ 31 . The experimental parameters used in the experiment can be seen in Table 7.1 .

Table 7.1: Parameters for multi-chemical transmission study .

Experimental Parameter	Symbol	Value	Unit
Tracked signal flow ion	$m ∕ z$	31 ¹ , 43 ²	$D a$
Signal flow	$q$	8	$m l ∕ m i n$
Carrier flow	$Q$	750	$m l ∕ m i n$
Transmission distance	$x_{d}$	2.5 $\pm$ 0.1	$c m$
Acetone Detection delay [25]	$t_{d}$	20	$s$
Environment pressure	$P_{E}$	1 $\pm$ 0.003	$b a r$
Carrier flow pressure	$P_{F}$	1	$b a r$
Vacuum pump pressure	$P_{V}$	$1.95 \times 1 0^{- 6}$	$t o r r$
Environment temperature	$T_{E}$	297.35 $\pm$ 1.5	$K$
Diffusivity in air ³	$D$	0.15 ¹ , 0,124 ²	${c m}^{2} ∕ s$
Theoretical Parameter	Symbol	Value	Unit
Injected mass	$M_{0}$	1 ¹ , 3.6 ²	ng
Advective flow in $x$ -axis	$u_{x}$	0.04	cm/s
Advective flow in $y$ -axis & $z$ -axis	$u_{y}$ & $u_{z}$	0	cm/s
Transmission distance	$x_{d}$	2.5	cm
Detector radius	$R_{D}$	0.3	cm
Longitudinal d iffusivity	$D_{L}$	0.15 ¹ , 0,124 ²	$c m^{2} ∕ s$
Radial d iffusivity	$D_{T}$	0.005 ¹ , 0.003 ²	$c m^{2} ∕ s$
Noise mean & variance	$μ_{N}, σ_{N}^{2}$	1.21, 0.096	$p A, W ∕ i o n$

¹ Methanol ² Acetone ³ Normal air and Temperature

The experimental result of the transmission along with the theoretical comparison can be observed in Figure 7.0a . As can be seen , two signal chemicals ( a cetone & m ethanol) were transmitted simultaneously. However it must be noted that there is a difference in the amplitude from the captured particle of two chemical species. This can be caused by numerous effects such as the particles interaction with the detector inlet (i.e., PDMS membrane), the presence of another messenger particles interaction, the ioni s ation efficiency of the samples inside the detector. In this transmission, acetone produced a higher signal current than methanol. However, the retrieved signal of m ethanol (31 $D a$ ) has experienced considerable amount of distortion compared to a cetone (43 $D a$ ).

(a)

(b)

Figure 7.1: (A) Experimental results along with theoretical results of multi-chemical transmission . (Equations used in modelling: Eq. 4.11c and Eq. 4.14c) (B) Experimental along with theoretical fitting of multi-chemical noise .

The mathematical model presented in Chapter 4 , Section 4.2.1 and its comparison to experimental data can be seen in Figure 7.0a . As can be seen, the model shows agreement with the experimental data. To quantify the agreement with experimental data to the theoretical model Pearson correlation is used [352]:

ρ_{E, T} = \frac{cov (E, T)}{σ_{E} σ_{T}},

(7.1)

where $E$ is the experimental data and $T$ is the theoretical data. Based on the equation, Acetone produced a correlation coefficient of $ρ = 0.96$ and Methanol produced a correlation coefficient of $ρ = 0.94$ showing that the model shows validity in analysing and predicting multiple chemical transmission. A final comment needs to be made in the initial pulsation of the methanol signal. This “transient” behaviour can be caused by the complex interactions of the membrane and the chemicals. After the initial interaction the signal becomes easily predictable by the model developed and presented in Chapter 4.

7.3 Multi Chemical Noise

One of the important limitations in a communication system is the disturbances in the environment which influences the transmission. The noise in molecular communications can be caused by numerous effects. These include particles present in the environment, minute pressure differences in the inlet of the detector, temperature and pressure of the environment etc.. A study was done in [3] that shows experimentally that the noise of a single chemical in the environment effecting the transmission is AWGN . In this experiment, the presence of multiple chemicals noise is conducted to analyse the noise in a multi-channel transmission. The experimental parameters can be seen in Table 7.2 .

Table 7.2: p arameters for m ulti-chemical n oise study.

Experimental Parameter	Symbol	Value	Unit
Tracked a cetone ion	$m ∕ z$	43	$D a$
Tracked m ethanol ion	$m ∕ z$	38	$D a$
Environment pressure	$P_{E}$	1.005	bar
Carrier flow pressure	$P_{F}$	1	bar
Vacuum pump pressure	$P_{V}$	2.85 $\times 1 0^{- 6}$	torr
Environment temperature	$T_{E}$	291.85	K
Diffusivity of a cetone in a ir ¹	$D$	0.124	$c m^{2} ∕ s$
Diffusivity of m ethanol in a ir ¹	$D$	0.15	$c m^{2} ∕ s$

¹ Normal air and t emperature

The results can be seen in Figure 7.0b . As can be seen the results show a strong relation to Gaussian distribution for both mass-to-charge values. To quantify the goodness-of-fit, Kolmogorov-Smirnov test is used [351]. The equation of this test is given below :

D_{n} = \sup_{x} | F_{n} (x) - F (x) |,

(7.2)

Based on the test, the result for acetone is $D_{n}$ = 0.0267 and for methanol is $D_{n}$ = 0.0448 , which show s that the noise data is drawn from a Gaussian distribution with a significance level of $α$ = 0.05. From these results it can be seen that each chemical possesses a different noise mean ( $μ_{N}$ ) and variance ( $σ_{N}^{2}$ ). In addition, from Figure 7.0a it can be seen that the amplitude of signal reaches different values for different chemicals show that each chemical has a different signal-to-noise ratio (SNR). The multi-chemical noise measured can be given as :

\begin{align} μ_{A} = 1.45 \times 1 0^{- 3}, & σ_{A} = 0.33 \times 1 0^{- 3}, & (7.3a) \\ μ_{M} = 5.23 \times 1 0^{- 3}, & σ_{M} = 0.57 \times 1 0^{- 3} . & (7.3b) \end{align}

7.4 Molecular Quadrature Amplitude Modulation (MQAM)

By using more than a single chemical species to transmit information, the amount of bits encoded in a given symbol can be increased, which in turn increases the channel capacity ( $C$ ) of the transmission.

In this section a molecular analogy of quadrature amplitude modulation (QAM) is developed, whereby coding a bit value into the amplitude value of more than one chemical, increases the number of bits in a symbol.

The amount of elements that can be coded to a molecular amplitude can be expressed by the cardinality (i.e., number of elements in a set) of the input set with following equation :

c a r d (M) = ϕ^{C h},

(7.4)

where $ϕ$ is the number of modulated levels on the chemicals and $C h$ is the number of distinguishable chemicals used in the transmission of information. Based on this, the channel capacity of an $n$ block transmission with memory effects can be written using the following generali s ed equation [86] :

C ≜ \lim_{x \to \infty} s u p \frac{1}{n} I (X^{n}; Y^{n}) .

(7.5)

In the above equation $I (X^{n}; Y^{n})$ represents the mutual information between the input alphabet $X$ and the output $Y$ with the following identity [82] :

I (X^{n}; Y^{n}) = H (Y^{n}) - H (Y^{n} | X^{n}),

(7.6)

where $H (Y^{n})$ is the Shannon entropy of the probability vector $Y$ and $H (Y^{n} | X^{n})$ is the conditional entropy of $Y$ given $X$ .

As mentioned previously, an advantage of molecular communication over conventional EM communications is the ability to increase the amount of carrier signals in a given channel. For example, in the experimental analysis shown in Figure 7.0a , it was shown that it is possible to transmit multiple chemicals at the same time. These chemicals have the possibility of transmitting information with negligible interference.

Figure 7.2: A constellation diagram for use in molecular quadrature amplitude modulation (MQAM) .

Table 7.3: Theoretical p arameters of m olecular q uadrature a mplitude m odulation (MQAM) .

Theoretical Parameter	Symbol	Value	Unit
Injected mass	$M_{0}$	1	$n g$
Transmission distance	$x_{d}$	2.5	$c m$
Advective flow in $x$ -axis	$u_{x}$	0.1	$c m ∕ s$
Advective flow in $y$ -axis & $z$ -axis	$u_{y}$ & $u_{z}$	0	$c m ∕ s$
Longitudinal d iffusivity	$D_{L}$	0.1	${c m}^{2} ∕ s$
Radial diffusivity	$D_{T}$	$1 \times 1 0^{- 3}$	${c m}^{2} ∕ s$
Symbol duration	$T_{S}$	20	$s$
Detector radius	$R_{D}$	0.3	cm

Based on these experimental results shown in Figure 7.0a and Figure 7.0b a theoretical analysis was carried out on multiple chemical transmission. In this analysis, two chemicals were chosen as information carriers and assumed to cause no interference to each other. One chemicals is given the designation $𝜃_{Q}$ and the second chemical is defined as $𝜃_{I}$ . Two levels of mass were chosen ( $𝜃_{Q}$ = $𝜃_{I}$ = 1, $𝜃_{Q}$ = $𝜃_{I}$ = 2) for encoding information. The constellation diagram used in the simulation can be seen in Figure 7.2 . Based on these descriptions the SER and the MI can be seen in Figure 7.2a and Figure 7.2b respectively.

(a)

(b)

Figure 7.3: (A) Simulation results of the SER of MQAM t ransmission . (B) Simulation results of the MI of MQAM t ransmission .

As can be seen, the maximum information (compared to Shannon’s Law [85]) sees a considerable increase with each chemical species introduced, however the errors produced from the constellation experiences a considerable increase as well. This is due to the increase of symbols that can interfere with their neighbouring symbol values.

The upper bound symbol-error rate (SER) for the MQAM method with AWGN present in the transmission can be given as :

P_{s} \leq 1 - (1 - (1 - \frac{1}{\sqrt{M_{C}}}) e r f c (\sqrt{\frac{3 \log_{2} M_{C}}{2 (M_{C} - 1)} (\frac{E_{b}}{N_{0}})})),

(7.7)

where $M_{C}$ is the number of constellations in the modulation and $E_{b}$ is the energy per bit ( $W$ ) . The following subsection will focus on the effect of the three parameters ( longitudinal diffusion, advective flow and transmission distance) on the MI and SER of the MQAM. The values used in the theoretical study can be seen in Table 7.3 .

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7.4: Simulation results of molecular communication using m olecular q uadrature a mplitude m odulation (MQAM): (A) Constellation diagram of MQAM with different advective flow rates . (B) Constellation diagram of MQAM with longitudinal diffusivity rates . (C) Constellation diagram of MQAM with different transmission distance rates . (D) Distribution of received

𝜃_{I}

and

𝜃_{Q}

masses with different advective flow rates . (E) Distribution of received

𝜃_{I}

and

𝜃_{Q}

masses with different longitudinal diffusivity rates . (F) Distribution of received

𝜃_{I}

and

𝜃_{Q}

masses with different transmission distance rates . [Ideal

R_{X}

value (

∙

)]

7.4.1 Advective Flow

The effect of the advective flow ( $u_{x}$ ) on the modulation can be seen in Figure 7.3a . As it can be seen, the increase of flow has a profound effect on the position of the detected mass values ( $𝜃_{I}, 𝜃_{Q}$ ). Lower velocity values have shown that the detected bit values are scattered away from the ideal $R_{X}$ value. Increasing the flow rate, decreases the variance of the detected mass values and begins concentrating the values closer to the ideal $R_{X}$ value. The distribution of masses with different advective flow can be seen in Figure 7.3d .

(a)

(b)

Figure 7.5: (A) Simulation results of the SER of MQAM t ransmission for different advective flows . (B) Simulation results of the MI of MQAM t ransmission for different advective flows .

The theoretical analysis of $M_{C}$ = 4 analysis for different advective flows can be seen in Figure 7.4a for SER and Figure 7.4b for MI . As mentioned, advective flow plays a positive role in increasing the MI and decreasing the SER encountered during transmission.

7.4.2 Coefficient of Diffusivity

(a)

(b)

Figure 7.6: (A) Simulation results of the SER of MQAM t ransmission for different longitudinal diffusion . (B) Simulation results of the MI of MQAM t ransmission for different longitudinal diffusion .

Diffusivity plays a negligible effect on the transmission as a whole which can be observed in Figure 7.3b and the received mass distribution can be seen in Figure 7.3e . While the effect is necessary for transmission without external energy, when an active transport is present, the effect of diffusivity does not play a primary role on the clarity of the detected bits. This is likely due to the dominance of the advective flow over the diffusivity of the transmitted chemicals. This effect can also be observed in SER and MI in Figure 7.5b and Figure 7.5a respectively.

7.4.3 Transmission Distance

The effect of the transmission distance on the bit detection can be seen in Figure 7.3c and on received mass distribution can be seen in Figure 7.3e . As can be seen when transmission distance ( $x_{d}$ ) is increased, the divergence of the received bits from the ideal $R_{X}$ value also increases . In addition, as distance is decreased, the concentration of received bits approach closer to the ideal $R_{x}$ value. Unlike the advective flow this behaviour is detrimental to the communication. This effect can also be observed in SER and MI in Figure 7.6a and Figure 7.6b respectively.

(a)

(b)

Figure 7.7: (A) Simulation results of the SER of MQAM t ransmission for different transmission distance . (B) Simulation results of the MI of MQAM t ransmission for different transmission distance .

7.5 Chemical Time Offset Keying (ChToK)

An alternative way of modulating the signal is to add relative relations between each signal. As it is in EM communication, an implementation of Phase Shift Keying (PSK) is implemented into molecular communication in this study along with experimental test. The diagram of this modulation can be seen in Figure 7.8 where $T$ is the period of the pulse of a chemical (s), $δ T$ is the time frame which the chemical can be shifted (s) and $T_{F}$ is the duration of the frame (s). Therefore the cardinality of the modulation can be represented as :

Figure 7.8: Diagram representing the c hemical s hift k eying (ChSK) .

c a r d (M) = {(\frac{T_{F}}{δ T})}^{C h} .

(7.8)

If number of modulated levels ( $ϕ$ ) are also implemented into the modulation, the equation can be rewritten as :

c a r d (M) = {(ϕ (\frac{L}{δ T}))}^{C h} .

(7.9)

This modulation, as can be seen has four parameters that can be used to increase the number of bits transmitted. By introducing more chemicals to the modulation scheme, more information can be encoded to a single pulse of chemical mixture. This modulation could also be used for secure communication as the modulation can be established to be time sensitive which can be used to transmit information in a secure manner . This, however opens up additional problems that are needed to be overcome , the primary one being the synchroni s ation between the transmitter and the receiver. In addition due to the chemicals ’ diffusivity, physical limits are imposed on the time unit $δ T$ (i.e., Limit of Detection, Limit of Quantification, Sampling Rate).

The sampling rate is determined by the sensor being used. For the QMA used in the study, the sampling rate is determined by the mass range being scanned. For long mass ranges (50-1050) the maximum scan rate is 8.3 Hz however, as the MIMS can only analyse the samples that have a lower mass than 200 Da, the scan rate can be increased up to 28.14 Hz for use in multi-chemical communication.

7.5.1 Experimental Analysis

An experimental transmission of ChToK was done with the experimental parameters are shown in Table 7.4 . The experimental results of the ChToK can be seen in Figure 7.9 . The ASCII letter “ $x$ ” was transmitted by using Acetone and Methanol as signal chemicals, where in a given time frame, if the peak of the acetone appears first compared to methanol, it is given the symbol 0, and symbol 1 is given for the inverse case.

Table 7.4: Parameters for c hemical t ime o ffset k eying transmission study .

Experimental Parameter	Symbol	Value	Unit
Tracked signal flow ion	$m ∕ z$	31 ¹ , 43 ²	$D a$
Signal flow	$q$	8	$m l ∕ m i n$
Carrier flow	$Q$	750	$m l ∕ m i n$
Acetone detection delay [25]	$t_{d}$	20	$s$
Transmission distance	$x_{d}$	2.5 $\pm$ 0.1	$c m$
Detector radius	$R_{D}$	0.3	$c m$
Environment pressure	$P_{E}$	1 $\pm$ 0.003	$b a r$
Carrier flow pressure	$P_{F}$	1	$b a r$
Vacuum pump pressure	$P_{V}$	$1.95 \times 1 0^{- 6}$	$t o r r$
Environment temperature	$T_{E}$	297.35 $\pm$ 1.5	$K$
Diffusivity in air ³	$D$	0.15 ¹ , 0.124 ²	${c m}^{2} ∕ s$
Theoretical Parameter	Symbol	Value	Unit
Advective flow in $x$ -axis	$u_{x}$	0.02	cm/s
Advective flow in $y$ -axis & $z$ -axis	$u_{y}$ & $u_{z}$	0	cm/s
Transmission distance	$x_{d}$	2.5	cm
Detector radius	$R_{D}$	0.3	cm
Longitudinal d iffusivity	$D_{L}$	0.15 ¹ , 0.124 ²	$c m^{2} ∕ s$
Radial d iffusivity	$D_{T}$	0.001 ¹ $^{,}$ ¹	$c m^{2} ∕ s$
Noise mean & variance	$μ_{N}, σ_{N}^{2}$	1.21, 0.096	$p A, W ∕ i o n$

¹ Methanol ² Acetone ³ Normal air and t emperature

Figure 7.9: Experimental transmission of a cetone and m ethanol using ChToK .

Figure 7.10: Experimental transmission of the a cetone ( — ) along with the theoretical comparison ( — ) . (Equations used in modelling: Eq. 4.11c and Eq. 4.14c)

It can be observed that the signal amplitude of each chemical reach different peaks. This can be caused by various effects, such as inlet interactivity with chemicals, chemicals own interactivity or chemicals inherent properties. In each chemical period $T$ , there are two distinguishable peaks, leading or lagging relative to Acetone. Figure 7.10 is shown for the validation of the mathematical model in predicting the modulated signal for the transmission of acetone chemical . As can be seen the mathematical model, described in Chapter 4, shows agreement ( $ρ = 0.96$ ) with the experimentally transmitted signal.

7.6 Chemical Ratio Modulation (CRM)

One of the defining aspects of molecular communication is, as mentioned previously , the possibility of transmitting multiple chemicals simultaneously. This approach allows implementation of novel modulation approaches that may not be common in EM communications. One such being the use of multiple chemical ratios to transmit information. Here information is encoded based on the ratio between two or more chemicals transmitting simultaneously. A diagram that represents the chemical ratio keying can be seen in Figure 7.11 .

Figure 7.11: Diagram representing the c hemical r atio k eying (CRK) .

7.6.1 Calculation of the modulation matrix

To mathematically define the modulation, two chemicals ( $ϕ_{A}, ϕ_{B}$ ) are used. Let $ϕ_{A}$ and $ϕ_{B}$ be the chemicals with the following set :

\begin{align} ϕ_{A} = {{ϕ_{A}}_{0}, {ϕ_{A}}_{1}, {ϕ_{A}}_{2}, \dots, {ϕ_{A}}_{i}}, ϕ_{A} \subseteq ℕ, & (7.10a) \\ ϕ_{B} = {{ϕ_{B}}_{0}, {ϕ_{B}}_{1}, {ϕ_{B}}_{2}, \dots, {ϕ_{B}}_{j}}, ϕ_{B} \subseteq ℕ . & (7.10b) \end{align}

In this context the ratio is defined as the division of the chemicals:

M (i, j) = \frac{{ϕ_{A}}_{i}}{{ϕ_{B}}_{j}} .

(7.11)

The matrix that gives the possible combinations of modulations is given as:

M_{i, j} = [\begin{matrix} \frac{{ϕ_{A}}_{1}}{{ϕ_{B}}_{1}} & \frac{{ϕ_{A}}_{1}}{{ϕ_{B}}_{2}} & \frac{{ϕ_{A}}_{1}}{{ϕ_{B}}_{3}} & \dots & \frac{{ϕ_{A}}_{1}}{{ϕ_{B}}_{j}} \\ \frac{{ϕ_{A}}_{2}}{{ϕ_{B}}_{1}} & \frac{{ϕ_{A}}_{2}}{{ϕ_{B}}_{2}} & \frac{{ϕ_{A}}_{2}}{{ϕ_{B}}_{3}} & \dots & \frac{{ϕ_{A}}_{2}}{{ϕ_{B}}_{j}} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{{ϕ_{A}}_{i}}{{ϕ_{B}}_{1}} & \frac{{ϕ_{i}}_{1}}{{ϕ_{B}}_{2}} & \frac{{ϕ_{A}}_{i}}{{ϕ_{B}}_{3}} & \dots & \frac{{ϕ_{A}}_{i}}{{ϕ_{B}}_{j}} \end{matrix}] .

(7.12)

However, this matrix will possess values that are equal in terms of ratio (i.e., 4/2, 6/3, 8/4, ...) which are not suitable for use in modulation and may cause confusion in decoding the transmitted messages. The individuality of the ratios lies in the numbers that are co-primes. Two integers $ϕ_{A}$ , $ϕ_{B}$ are co-prime if the only positive integer (factor) that divides both of them is 1. Any ratio between two co-primes will produce a unique value in which information can be encoded given enough precision by the detector and the transmitter. To calculate the amount of combinations that can be generated, the Riemann-Zeta function is used with the following identity :

ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}} .

(7.13)

The relation of the above function to the problem in hand is as follows. Let the sum definition of $ζ (s)$ be the following expression :

ζ (s) = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + \frac{1}{5^{s}} + \dots, R (s) > 1,

(7.14)

From this series, to remove any number that is divisible by 2 (2| $n$ ) Eq. ( 7.15 ) is subtracted from ( 7.14 ) :

\frac{1}{2^{s}} ζ (s) = \frac{1}{2^{s}} + \frac{1}{4^{s}} + \frac{1}{6^{s}} + \frac{1}{8^{s}} + \frac{1}{1 0^{s}} + \dots .

(7.15)

The below equation represents the reciprocal of probability of $s$ amount of numbers not having a common divisor of 2 :

(1 - \frac{1}{2^{s}}) ζ (s) = 1 + \frac{1}{3^{s}} + \frac{1}{5^{s}} + \frac{1}{7^{s}} + \frac{1}{9^{s}} + \dots .

(7.16)

This action is carried out to remove the common divisor of 3 (3| $n$ ) :

\frac{1}{3^{s}} (1 - \frac{1}{2^{s}}) ζ (s) = \frac{1}{3^{s}} + \frac{1}{9^{s}} + \frac{1}{1 5^{s}} + \frac{1}{2 1^{s}} + \frac{1}{2 7^{s}} + \dots .

(7.17)

If this process is continued ad infinitum the following expression is obtained :

\dots (1 - \frac{1}{1 1^{s}}) (1 - \frac{1}{7^{s}}) (1 - \frac{1}{5^{s}}) (1 - \frac{1}{3^{s}}) (1 - \frac{1}{2^{s}}) ζ (s) = 1 .

(7.18)

The above expression gives the reciprocal probability of $s$ amount of numbers having no common divisors aside from 1 (i.e., co-prime) :

ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}} = \prod_{p p r i m e} \frac{1}{1 - p^{- s}} .

(7.19)

The right hand expression given in Eq. ( 7.19 ) is described as the Euler product which is an infinite product indexed by prime numbers. This connection allows the quantification and the calculation of the unique values of the modulation with high accuracy. While there are no closed form for odd values for Riemann-Zeta function, the closed form function for the even numbers can be expressed as :

ζ (2 n) = {(- 1)}^{n + 1} \frac{B_{2 n} {(2 π)}^{2 n}}{2 (2 n)!},

(7.20)

where $B_{2 n}$ is the Benoulli number. Particular values of the Riemann-Zeta function for $s$ = 2 $\dots$ 6 can be seen in Table 7.5 .

Table 7.5: Particular values of the Riemann-Zeta f unction .

Parameter	Function	Closed Form	Value	OEIS
$ζ (2)$ ¹	$1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \dots$	$\frac{π^{2}}{6}$	$1.6449 \dots$	A013661
$ζ (3)$ ²	$1 + \frac{1}{2^{3}} + \frac{1}{3^{3}} + \dots$	—	$1.2020 \dots$	A002117
$ζ (4)$	$1 + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \dots$	$\frac{π^{4}}{90}$	$1.0823 \dots$	A013662
$ζ (5)$	$1 + \frac{1}{2^{5}} + \frac{1}{3^{5}} + \dots$	—	$1.0369 \dots$	A013663
$ζ (6)$	$1 + \frac{1}{2^{6}} + \frac{1}{3^{6}} + \dots$	$\frac{π^{6}}{945}$	$1.0173 \dots$	A013664

¹ Known in literature as the Basel problem ² Known in literature as the Ap é ry’s constant

Based on the aforementioned equations, the number of individual values in a modulation matrix used can be expressed as:

c a r d (M) = \frac{1}{ζ (s)} \prod_{i = 1}^{i = s} ϕ_{i} .

(7.21)

In the above-given example, two chemicals were used. The amount of modulated values are therefore can be approximated as:

ϕ_{A} ϕ_{B} ζ (2) = ϕ_{A} ϕ_{B} \frac{π^{2}}{6} .

(7.22)

Comparison of actual individual values to estimated ones along with estimation error can be seen in Figure 7.11a and Figure 7.11b respectively.

(a)

(b)

Figure 7.12: (A) Comparison of the unique values of the ratio modulation ( — ) comparison to the function

ϕ_{A} ϕ_{B} \frac{π^{2}}{6}

. ( — )(B) Relative error (

η

%) of the

ϕ_{A} ϕ_{B} \frac{π^{2}}{6}

approximation .

To test the proposed modulation scheme, an experiment was conducted with the parameters given in Table 7.6 .

Table 7.6: Experimental parameters for c hemical r atio k eying ( ChRK ) .

Experimental Parameter	Symbol	Value	Unit
Tracked ion	m/z	43 ¹ , 42 ² , 57 ³	Da
Carrier flow	$Q$	1000	$m l ∕ m i n$
Detection delay [25]	$t_{d}$	15 ¹ , 14 ² , 20 ¹	s
Diffusivity in a ir	$D$	0.124 ¹ ,0.07 ¹ , 0.11 ³	$c m^{2} ∕ s$
Inner t ube diameter	$ϕ_{i n}$	19.80	$m m$
Outer t ube diameter	$ϕ_{o u t}$	24.25	$m m$

¹ Acetone ² Cyclopentane ³ n-Hexane

7.6.2 Experimental Analysis

In this setup, a clear acrylic tube is used to initiate a long distance transmission to show the resilience of the modulation to signal attenuation.

(a)

(b)

(c)

Figure 7.13: (A) Experimental transmission with increasing distance of (A) a cetone (B) c yclopentane (C) n- h exane .

(a)

(b)

(c)

(d)

Figure 7.14: Experimental results of chemical ratio k eying (ChRK) by using three different chemicals: a cetone, c yclopentane and n-h exane. The figures above show the signal strength with increasing distance : (A) 0.5 m, (B) 1.0 m, (C) 2.0 m and (D) 4.0 m .

The experimental results can be seen collectively in Figure 7.14 . From (a) to (d) the effect of attenuation for three chemicals by increasing distance can be observed. Decrease of signal strength is to be expected. However, it must be noted that each chemical experiences attenuation differently making some chemicals better at transmitting long distances than others. This effect can be seen in Figure 7.14a for their measured values and in Figure 7.14b for their values relative to 0.5 m measurement.

(a)

(b)

Figure 7.15: Experimental ratio values from the ratio modulation study : (A) Signal a mplitudes of the three transmitted chemicals relative to distance . (B) Amplitude values of chemicals relative to 0.5 m signal current values .

As can be seen in Figure 7.14b , the chemicals experience different attenuation rates. While Cyclopentane and Acetone experience similar attenuation, n-Hexane experiences higher attenuation compared to the other two chemicals. This difference can be caused by the unique parameters of the chemical species used for sending the pulse. In addition each chemical achieves a different maximum amplitude which can also be caused by the ioni s ation of samples, and their weight. The ratio of the two chemicals (Cyclopentane/Acetone) can be observed in Figure 7.15a . Unlike the signal amplitude, the ratio value experiences less of a change, making it a better option to transmit information over long distances. However, the stability of the ratio value depends on the chosen chemical combinations. As can be seen from the Figure 7.15a and 7.15b , some combinations can produce ratios that are maintained over distance whereas some ratios, seen in Figure 7.15b , are not suitable for ratio modulation usage.

(a)

(b)

Figure 7.16: Experimental ratio values from the ratio modulation study : (A) c yclopentane/ a cetone , (B) a cetone/ n-h exane .

7.7 Conclusion

In this chapter, an experimental and analytical study was done on the possibility of transmitting multiple chemicals at the same time. To achieve this, an odour gas generator was used to generate and transmit chemicals. For detector, a quadrupole mass spectrometer was utili s ed as a detector. The ability of the mass spectrometer, to distinguish and analyse multiple chemicals concurrently, makes the MS an invaluable tool for multi-molecular communications. To mathematically validate the experiments, 3D Advection-Diffusion Equation is used. Experimentally a multi-chemical transmission was done and shown to have strong agreement with mathematical model developed in this study. Experimentally the noise of the multi-chemical system was analysed and shown to behave as an Additive White Gaussian Noise with distinct values for the mean and variance for each chemical. Based on the mathematical model, a molecular version of QAM is developed in this study for macro-scale molecular communication and its properties are analysed. This chapter also proposed and experimented two types of molecular communication for use in macro-scale ratio modulation and phase shift keying. These modulation methods and the use of multiple chemical show that it is possible to greatly increase the achievable mutual information rate of the system which can create new application purposes for molecular communication and shows that there is still a possibility of increasing the throughput of the communication.

7 Multi-Chemical Transmission