Strength-based Signal Detection in PAM OOK Concentration-Encoded Molecular Communication

In this paper, the expressions of achievable strength-based suboptimum detection probabilities of concentration-encoded molecular communication (CEMC) system have been derived based on pulse-amplitude modulated (PAM) on-off keying (OOK) modulation scheme. Results show that the detection performance of a PAM OOK CEMC system significantly depends on the statistical parameters of diffusion noise and intersymbol interference (ISI). Analytical performances of ISI-free and ISI-affected scenarios have been explained and compared based on receiver operating characteristics (ROC) curves for impulse (i.e. spike)-modulated (IM) and PAM schemes. It is shown that the effects of diffusion noise and ISI on ROC can be explained separately based on their communication range-dependent statistics.


INTRODUCTION
Concentration-encoded molecular communication (CEMC) is an information encoding technique that relies on transmitting a single type of information molecules and varying the number of transmitted molecules in order to materialize molecular communication (MC) among bio-nanomachines [1,2].Pulse amplitude modulated (PAM) on-off keying (OOK) CEMC system is a common and popular form of MC among natural bionanomachines where a transmitting nanomachine (TN) releases a pulse of molecules in the form of a fixed number of molecules per second (s) for a finite length pulse-width Tp > 0 when the TN wants to send bit 1 and it does not release any molecule at all when it wants to send bit 0 [1].Correspondingly, in strength-based detection, a receiving nanomachine (RN) observes signal strength [3], i.e. the number of molecules available to RN during the symbol duration and compares it to a threshold.In impulse modulation (IM), the TN releases all the molecules at a time and so ideally Tp = 0.However, in PAM, 0 < Tp ≤ Tsym where Tsym denotes the symbol interval.When Tp = Tsym, it means that TN remains "on" in the entire Tsym in order to send bit 1 and "off" otherwise, noting that 0 < Tp < Tsym indicates a pulse-transmitted signal where the TN remains "on" for a period less than the symbol interval of the system.The quantity α = ܶ /ܶ ௦௬ is considered as the duty cycle of the transmitter system: α = 0 (ideal case, e.g.IM) and α = 1 (practical case, e.g.PAM with ܶ = ܶ ௦௬ ).Here, α varying from 0 to 1 denotes the percentage of the symbol interval being used by the TN in releasing the information molecules to represent bit 1.Given a fixed number of molecules AMo to be released in order to represent bit 1, when 0 < Tp ≤ Tsym, the TN adjusts the number of released molecules to ‫ܣ‬ ெ /ܶ over the pulse width Tp.Although PAM OOK with 0 < α ≤ 1 has been identified as a popular scheme for CEMC so far [1], its detection capability is still unexplored.While several relevant works [4][5][6] have investigated the performance of PAM OOK CEMC system and its dependence on communication range and transmission data rate, to the best of the author's knowledge, none of the available open literature has addressed the issue of achievable detection capability of PAM OOK scheme in CEMC.In this paper, the following contributions have been made: • First, the analytical expressions of simplistic strength-based suboptimum detection probability PD in the presence of diffusion-noise with and without the effects of ISI have been derived.• Second, the statistics of diffusion-based noise and intersymbol interference (ISI) for PAM OOK CEMC system have been derived.It is found that the dependence of diffusion-noise and ISI separately on the desired signal makes it extremely challenging to detect the desired signal correctly.• Finally, the effects of diffusion noise and ISI on PD have been separated and the detection performance have been explained when communication range and transmission data rate vary.This paper is organized as follows: Section 2 presents the system model.Section 3 provides the detection model with and without the effects of ISI at the RN.Section 4 derives the statistics of diffusion noise and ISI in the case of PAM OOK receiver in CEMC.Key findings obtained through numerical results have been provided in Section 5. Finally, Section 6 summarizes the importance of the results and concludes the paper.

SYSTEM MODEL
We consider a pair of nanomachines, namely TN and RN, in an unbounded three-dimensional propagation medium [1].TN and RN are respectively located at the origin (0,0,0) and any location represented by the vector ( ) , ,  r x y z = where r 2 = x 2 + y 2 + z 2 .TN and RN are assumed to be synchronized in time, which can be made possible by using external signals [4].As shown in Fig. 1, the TN transmits molecules in a time-slotted manner, meaning that, when α = 1, to send a bit 1, TN transmits molecules at a fixed rate of ‫ܣ‬ = ‫ܣ‬ ெ /ܶ ௦௬ molecules/s during the entire Tsym and it does not send any molecules when it wants to send a bit 0. When α = 0, the TN transmits all the AMo molecules at a time in an impulsive (i.e.spike) fashion, e.g. in IM.Since information molecules undergo ideal diffusion-based propagation [5], the number of molecules available at the RN is a random variable due to effects of both diffusion-based noise and ISI [6].The RN senses the occupancy of its receptors at uniform sampling intervals of ts = 1 s.In order to ensure statistically independent, uncorrelated samples of concentration intensity, ts needs to be reasonably larger than the waiting time of a molecule in the receiver sensing volume [9].While uniform sampling is adopted in this paper, other sampling techniques, e.g.nonuniform sampling, may also be used in such cases.

ISI-free Detection
Consider that signal detection model can be expressed in terms of hypotheses H0 and H1 when the TN sends bits 0 and 1 respectively as shown next.First, we consider the absence of ISI.For example, for the i-th symbol, this can be realized in two ways, namely, either by assuming TGuard = ∞ between two pulses corresponding to the two symbols, or by assuming all the previous bits bj, j ∈ {1,2,…, (i−1)}, being 0s so that they produce zero ISI at the RN respectively.In reality, TGuard 10τDS, where τDS is the root mean square (RMS) delay spread of CEMC channel [7].The number of molecules available at the RN at a given time follows binomial distribution [6], which can be approximated to a normal distribution in CEMC [6].The strength-based signal detection model can be expressed as below [9].
This yields the probabilities of false alarm (PFA) and detection (PD) as: Here ED s denotes the mean number of desired molecular signal strength, ( ) s ED σ is the standard deviation of signal strength, and (.) Q denotes the right-tail probability and can be expressed as denotes the desired signal to diffusion noise strength ratio, or alternatively, signal to noise ratio (SNR) at the RN.

ISI-affected Detection
When either 10τDS ≥ TGuard ≥ 0 or at least one of the previously transmitted bits bj, j ∈ {1,2,…, (i−1)} is a bit 1, ISI-producing molecules will impact the signal strength of the i-th symbol at the RN.For example, when TGuard = 0, TN sends pulses at every Tsym seconds.With the effects of ISI, the PAM OOK signal detection model can be expressed as below.
Here γ denotes a threshold that can be found for a given PFA.Unlike Eq. ( 2), Eq. ( 4) shows that in addition to Λ, in ISI-affected scenario, the detection performance is also a function of PFA and Note that the ROC as shown in Eq. ( 4) does not depend on ( ) .

ISI ED µ
Note also that Eq. ( 4) shows the impact of ISI in PD such that plugging Rσ = 0 into Eq.( 4) yields the ISI-free detection as shown in Eq. ( 2).
Let us now determine the worst case ISI-affected scenario as follows.For the i -th symbol, the worst-case detection would take place when all of the previous (i − 1) bits are 1s, hence contributing maximum to the ISI at the detection of the current bit.Therefore, in the worst-case scenario, the respective ISI quantities can be expressed as where M is the total number of previous bits considered (i.e.memory).For instance, for the i -th bit at the worst case, ( 1) where the superscript (WC) denotes the worst-case scenario.Therefore, the worst-case ROC can be found as

DIFFUSION-BASED NOISE AND ISI STATISTICS
In the analysis to follow, we show that, in a PAM OOK CEMC system, the signal strength can be expressed in terms of the impulse response of the channel.When the TN transmits AMo 1 molecules in an impulsive manner, the mean number of molecules (i.e.concentration) at the RN can be expressed as ( ) ( ) where as shown below ( ) p t is the probability of finding one molecule at time instant t at the virtual receiving volume (VRV) surrounding the RN [6] and dV = dx dy dz is the differential volume in VRV [6].
( ) ( ) Mean concentration of information molecules at the RN due to PAM OOK system can be found by taking integral of the channel output due to impulsive transmission [5].Therefore, using Eq. ( 7), with ts as the sampling interval, yields the mean and the variance of the desired signal strength as below ( ) where , i n are temporal indices 1 , samp N is the total number of samples taken in each symbol, and s t = 1 s is the sampling interval.
Similarly, at the i -th symbol, the mean and the variance of ISIproducing molecules originated in the j -th symbol, j ∈ {1,2,…, (i−1)}, can be expressed as below, where , k n are temporal indices. ( )

RESULTS
In the absence of ISI, PD depends on Λ only; however, when ISI is present, PD degrades and significantly depends on PFA, Λ, and Rσ.Numerical simulations have been performed in order to find Λ and Rσ when TN and RN are spatially apart by r and transmission data rate is given by Ω = 1/Tsym.In this paper, we provide the results of PAM scheme with α = 1, meaning that Tp = Tsym in all the results reported in this paper.In the absence of ISI, Eq. ( 2) yields the BER performance as shown in Fig. 2 where PD and BER depends on Λ only.When SNR is low, BER remains constant at 0.25; however, as SNR increases, BER decreases.shows that the PAM scheme performs worse than the IM scheme in the last-bit ISI scenario.Comparing Figs.4(c) with 4(d) shows that both the IM and PAM schemes provide almost the same performance in the worst-case ISI scenario, the most effective reason behind this being the high number of ISI-producing molecules present in the worst-case scenario with both schemes.

CONCLUSION
In this paper, performance of a strength-based suboptimum receiver has been presented for the PAM scheme with α = 1 in CEMC system.In addition, the performance of the PAM scheme is further compared with that of the IM scheme.The analytical results and numerical simulations show that the PAM scheme with α = 1 is more affected by the ISI-producing molecules and hence the lower PD values when compared to the IM scheme under the same scenario.However, in the worst-case scenario when all the previous bits are 1s, both IM and PAM systems are equally affected by the ISI-producing molecules and so yield almost the same worst-case ISI performances.Future works of this research include an analysis of the PAM system when 0 < α ≤ 1 and transmission data rate
and variance of ISIproducing molecular strength respectively.Suboptimum receiver operating characteristics (ROC) curves can be obtained as below in terms of PFA and PD[8].
e. the ratio of standard deviations of ISI strength to desired signal strength.Here, Rσ bears the characteristics of the ISI that impact PD at the current symbol.e. the ratio of mean values of ISI strength to desired signal strength.
on r and Tsym and hence the necessity to find communication range-and rate-dependent characteristics of PD.For a given PFA, since a high PD provides a low BER, useful variations in the detection performance can be observed by obtaining the quantities R σ , ( ) WC R σ , and Λ numerically, and plugging them in Eqs.(2), (4), and (6) in order to find PD and( ) 10) As shown in Eq. (10), the quantities ( ) using ( ) s t and ( ) p t functions.In numerical simulations, Mo A = 1000 has been assumed.

Fig. 4 .
Fig. 4. Detection probability versus SNR characteristics in the presence of ISI.