On Molecular Timing Channels with α-stable Noise

Transcription

1 On Moleular Timing Channels with α-stable Noise Yonathan Murin, Nariman Farsad, Mainak Chowdhury, and Andrea Goldsmith Department of Eletrial Engineering, Stanford University, USA Abstrat This work studies ommuniation over moleular timing MT hannels with α-stable noise. The transmitter simultaneously releases multiple small information partiles, where the information is enoded in the time of release. The reeiver deodes the transmitted information based on the random time of arrival of the information partiles, whih is represented as an additive noise hannel. For a diffusion-based MT hannel, without flow, this noise follows the Lévy distribution. For this ase, the maximum-likelihood ML detetor is derived and shown to have high omputational omplexity. It is further shown that for any additive hannel with α-stable noise, α < 1, suh as the DBMT hannel, a linear reeiver is not able to take advantage of the release of multiple information partiles. Thus, instead of the ommon low omplexity linear approah, a new detetor, whih is based on the first arrival FA among all the transmitted partiles, is derived. It is shown that for small number of partiles the performane of the FA detetor is very lose to that of the ML detetor. On the other hand, via error exponent analysis, it is shown that the performane of the two detetors differ when the number of sent partiles is large. Thus, in the regime of small to medium number of sent partiles, the FA detetor is an attrative alternative to the relatively ompliated ML detetor. 1 Introdution Moleular ommuniation MC is an emerging field in whih nano-sale devies ommuniate with eah other via hemial signaling, based on exhanging small information partiles [1, 2]. For instane, in biologial systems MC an take plae using hormones, pheromones, or ribonulei aid moleules. To embed information in these partiles one may use the partile s type [3], the partiles onentration [4, 5], the partile number [6], or the time of release [7, 8]. These partiles an be transported from the transmitter to the reeiver via diffusion, ative transport, bateria, and flow, see [2, Se. III.B] and referenes therein. Although this new field is still in its infany, several basi experimental systems serve as a proof of onept for transmitting short messages at low bit rate [9, 10]. There are several similarities between traditional eletromagneti EM ommuniation and MC. These similarities motivated using tools and algorithms developed for EM ommuniation in designing MC ommuniation systems. The work [11] studied on-off transmission via diffusion of information partiles, where the information is reovered at the reeiver based on the measured onentration. A hannel model with finite memory was proposed, whih involves additive Gaussian noise, along with several sequene detetion algorithms suh as maximum a-posteriori MAP detetion and maximum likelihood ML detetion. The work [12] studied a similar setup proposing a tehnique for inter-symbol interferene ISI mitigation and deriving a redued-state ML sequene detetion algorithm. Finally, [13] studied 1

2 on-off transmission over diffusive moleular hannel with flow, proposed ML sequene detetion algorithm, and designed a family of weighted sums detetors. While the above works build upon the similarities between EM ommuniation and MC, namely, linear hannel model with additive and in some ases Gaussian noise, there are aspets in whih MC is fundamentally different from traditional EM ommuniation. For instane, in EM ommuniation the symbol duration is fixed, while in MC the symbol duration is often a random variable RV. Therefore, information partiles may arrive out-of-order, whih plaes a big hallenge for the detetor, in partiular when the transmitted information partiles are indistinguishable [14]. In this work, we study reeiver design for MC systems where information is modulated theough the time of release of the information partiles, whih is reminisent of pulse positionmodulation [15]. A ommon assumption, whih is aurate for many sensors, is that the partile is absorbed by the reeiver and removed from the environment. In suh ase, the random delay until the partile arrives at the reeiver an be represented as an additive noise term. For diffusion-based hannels without flow, this additive noise is Lévy-distributed [16], while for diffusion-based hannels with flow, this additive noise follows an inverse Gaussian IG distribution [17]. Fig. 1 provides a shemati desription of suh hannels. At first glane, the ases of diffusion with and without flow may seem similar; however, a loser look reveals a fundamental differene whih stems from the different properties of the additive noise. The Lévy distribution has an algebrai tail 1 [18, 19], while the tail of the IG distribution, similarly to the standard Gaussian distribution, deays exponentially. Thus, traditional linear detetion and signal proessing tehniques whih work well in the presene of Gaussian or IG noise, may perform poorly in the presene of noise with an algebrai tail. The observation that hannels with additive noise, whih has an algebrai tail, require different detetion methods was already stated in [20] based on numerial simulations. Yet, in this work we provide a rigorous proof that linear proessing annot improve the detetion performane in hannels orrupted by additive α-stable noise [21, 18], with α < 1. It should be noted that the Lévy distribution belongs to the family of α-stable distributions. 2 These distributions are ommonly used in modeling impulsive noise [20, 22, 23]. Yet, to the best of our knowledge, the fous of all previous studies was on symmetri stable distributions, also referred to as SαS distributions. Our fous in this work is on timing hannels orrupted by additive α-stable noise, suh as the Lévy noise, whih is not symmetri. Apart from the fat that the tails of the additive noise deay slowly, in the onsidered diffusion-based timing hannel ordering in time is not preserved, namely, information partiles from onseutive hannel uses may arrive out of order. This gives rise to ISI. In this work, however, we fous on settings without ISI suh as ommuniation systems in whih onseutive transmissions are far enough apart,or a nano-sale sensor whih one-in-awhile sends a limited number of bits, modulated in a single hannel use, to a entralized moleular ontroller, and then remains silent for a long period. Hene, we study independent onseutive hannel uses whih an be analyzed separately. 1 An RV X has an algebrai tails if there exists, α > 0 suh that lim x x α Pr{ X > x} =. 2 The Lévy distribution is α-stable with α =

3 Main Contributions We study transmission over a diffusion-based moleular timing DBMT hannel without flow, assuming that onseutive hannel uses are independent and identially distributed i.i.d. We onsider an MC system in whih the information is enoded in the time of release of the information partiles, where this time is seleted out of a set with finite ardinality, namely, a finite onstellation is used. At eah transmission M information partiles are simultaneously released at the time orresponding to the urrent symbol, while the reeiver s objetive is to detet this transmission time. Note that M is onstant and does not hange from one transmission to the next, i.e., information is not enoded in the number of partiles. We derive the ML detetion rule whih, as expeted, requires relatively high omputational omplexity and therefore is not neessarily suitable for nano-sale MC systems. This motivates studying detetors with lower omplexity. A ommon approah in traditional EM ommuniation, whih was also proposed in [17], is to use a linear detetor. We show that for any α stable noise with α < 1, suh linear proessing inreases the noise dispersion, whih is reminisent of inreasing the variane in linear proessing of independent Gaussian noise signals. This inreased dispersion degrades the probability of orret detetion ompared to the ase of a single partile. To the best of our knowledge this is the first proof for the destrutive effet of linear proessing in ommuniation in the presene of noise with heavy algebrai tails and harateristi exponents smaller than unity. Sine linear detetors annot take advantage of the multiple transmitted partiles, we derive a new detetor whih is based on the first arrival FA among the M information partiles. We show that the onditional probability densities of the FA onentrate around the possible transmitted signal. This results in an effet reminisent of dereasing the variane in linear proessing of independent Gaussian noise signals. We further analytially show that the performane of the proposed FA detetor are very lose to those of the optimal ML detetor, for small to moderate values of M. On the other hand, we use error exponent analysis to show that for large values of M, ML signifiantly outperforms the FA detetor, whih agrees with the fat that the FA is not a suffiient statisti for the onsidered detetion problem. The rest of this paper is organized as follows. The problem formulation and some preliminaries are presented in Setion 2. The ML detetor and linear detetion are studied in Setion 3. The FA detetor is derived in Setion 4, while its performane is ompared to the performane of the ML detetor in Setion 5. Numerial results are presented in Setion 6, and onluding remarks are provided in Setion 7. 2 Preliminaries 2.1 Notation We denote the set of real numbers by R, the set of positive real numbers by R +, and the set of integers by N. Other than these sets, we denote sets with alligraphi letters, e.g., B. We denote RVs with upper ase letters, e.g., X, Y, and their realizations with lower ase letters, e.g., x, y. An RV takes values in the set X, and we use X to denote the ardinality of a finite set. We use f Y y to denote the probability density funtion PDF of a ontinuous RV 3

4 Figure 1: Diffusion-based moleular ommuniation timing hannel. X denotes the release time, Z denote the random propagation time, and Y denotes the arrival time. Y on R, f Y X y x to denote the onditional PDF of Y given X, and F Y X y x to denote the onditional umulative distribution funtion CDF. We denote vetors with boldfae letters, e.g., x, y, where the k th element of a vetor x is denoted by x k. Finally, we use erf to denote the omplementary error funtion given by erfx = 2 π x e u2 du, log to denote the natural logarithm, and E{ } to denote stohasti expetation. 2.2 System Model Fig. 1 illustrates a moleular ommuniation hannel in whih information is modulated on the time of release of the information partiles. We assume that the information partiles themselves are idential and indistinguishable at the reeiver. Therefore, the reeiver an only use the time of arrival to deode the intended message. The information partiles propagate from the transmitter to the reeiver through some random propagation mehanism e.g. diffusion. We make the following assumptions about the system: A1 The transmitter perfetly ontrols the release time of eah information partile, and the reeiver perfetly measures the arrival times of the information partiles. Moreover, the transmitter and the reeiver are perfetly synhronized in time. A2 An information partile whih arrives at the reeiver is absorbed and removed from the propagation medium. A3 All information partiles propagate independently of eah other, and their trajetories are random aording to an i.i.d. random proess. This is a reasonable assumption for many different propagation shemes in moleular ommuniation suh as diffusion in dilute solutions, i.e., when the number of partiles released is muh smaller than the number of moleules of the solutions. 4

5 Note that these assumptions have been traditionally onsidered in all previous works, e.g. [5, 8, 24, 25, 26], in order to make the models tratable. Let X be a finite set of onstellation points on the real line: X {ξ 0, ξ 1,..., ξ L 1 }, 0 ξ 0 ξ L 1, and let ξ L 1 < T s < denote the symbol duration. The k th transmission takes plae at time K 1T s + X k, X k X, k = 1, 2,..., K. At this time, M N information partiles are simultaneously released into the medium by the transmitter. Note that we assume that at eah transmission the same number of information partiles is released. The transmitted information is enoded in the sequene {K 1T s + X k } K k=1, whih is assumed to be independent of the random propagation time of eah of the information partiles. Let Y k denote an M-length vetor onsisting of the times of arrival of eah of the information partiles released at time k 1T s + X k. It follows that Y k,m > X k, m = 1, 2,..., M. Thus, we obtain the following additive noise hannel model: Y k,m = k 1T s + X k + Z k,m, k = 1, 2,..., K, m = 1, 2,..., M, 1 where Z k,m, m = 1, 2,..., M, is a random noise term representing the propagation time of the m th partile of the k th transmission. Note that Assumption A3 implies that all the RVs Z k,m are independent. In the hannel model 1, partiles may arrive out of order, whih results in a hannel with memory. In this work, however, we assume that eah information partile arrives before the next transmission takes plae. This assumption an be formally stated as: A4 T s is a fixed onstant hosen to be large enough suh that the transmission times X k obey Y k,m kt s with high probability. 3 With this assumption, we obtain an i.i.d. memoryless hannel model whih an be written as: Y m = X + Z m, m = 1, 2,..., M. 2 In the rest of this work we fous on this memoryless hannel model. Assumption A4 implies that T s is hosen suh that onseutive transmissions are suffiiently separated in time. We further note that the model 2 also represents well the setting of a nano-sale sensor whih infrequently sends a symbol whih onveys a limited number of bits to a entralized moleular ontroller, and then remains silent for a long period. Thus, the effetive ommuniation hannel is memoryless. To simplify the presentation, in most of the work we restrit our attention to the ase of binary modulations, i.e., X = {ξ 0, ξ 1 }. We note that all the results and tehniques derived in this work an be easily extended to more then two elements in the set X. Let S {0, 1}, be an equiprobable bit to be sent over the hannel 2 to the reeiver, and denote the estimate of S at the reeiver by Ŝ. We note that our results an be easily extended to the ase of different a-priori probabilities. Our objetive is to design a simple reeiver whih minimizes the probability of error P ε = Pr{S Ŝ}. In order to minimize P ε we maximize the spaing between 3 Formally, let η be arbitrarily high probability, then we hoose T s suh that Pr{Y k,m < kt s} > η, k = 1, 2,..., K, m = 1, 2,..., M. 5

6 ξ 0 and ξ 1, and without loss of generality we use the following mapping for transmission: { 0, s = 0 XS =, s = 1. 3 Note that the above desription of ommuniation over an MT hannel is fairly general and an be applied to different propagation mehanisms as long as Assumptions A1 A4 are not violated. Before presenting the DBMT hannel, we briefly disuss a lass of distributions with algebrai tail: the stable distributions. For a detail desription we refer the reader to [18, 27]. 2.3 Stable Distributions Definition 1 Stable Distribution. An RV X has a stable distribution if for two independent opies of X, X 1 and X 2, and onstants {a j } 3 j=1, a j R + and a 4 R, the following holds: a 1 X 1 + a 2 X 2 d = a3 X + a 4, 4 where d = denotes equality in distribution, i.e., both expressions follow the same probability law. Stable distributions an also be defined via their harateristi funtion. Definition 2 Charateristi Funtion of a Stable Distribution. Let < µ <, 0, 0 < α 2, and 1 β 1. Further define: { tan πα Φt, α 2, α 1 2 π log t, α = 1. Then, the harateristi funtion of a stable RV X, with loation parameter µ, sale or dispersion parameter, harateristi exponent α, and skewness parameter β, is given by: ϕt; µ,, α, β = exp {jµt t α 1 jβsgntφt, α}. 5 In the following, we use the notation S µ,, α, β to represent a stable distribution with the parameters µ,, α, and β. Apart from several speial ases, stable distributions do not have losed-form PDFs. The exeptional ases are the Gaussian distribution α = 2, the Cauhy distribution α = 1, and the ase of α = 1 2 whih was very reently derived in [28, Theorem 2]. Note that the Lévy distribution is a speial ase of the results of [28] with β = 1. Finally, we note that all stable distributions, apart from the ase α = 2, have infinite variane, and all stable distributions with α 1 also have infinite mean. In fat, this statement an be generalized to moments of order p α, see [19]. Next, we desribe the DBMT hannel. 6

7 2.4 The DBMT Channel In diffusion-based propagation, the released partiles follow a random Brownian path from the transmitter to the reeiver. In this ase, to speify the random additive noise term Z m in 2, we define a Lévy-distributed RV as follows: Definition 3 Lévy Distribution. Let Z be Lévy-distributed with loation parameter µ and sale parameter [18]. Then, its PDF given by: exp 2πz µ f Z z = 3 2z µ, z > µ, 6 0, z µ and its CDF given by: erf 2z µ, z > µ F Z z =. 7 0, z µ The Lévy distribution belongs to the lass of stable distributions with the parameters S µ,, 1 2, 1. Thus, its harateristi funtion is given by: { ϕt = exp jµt } 2jt. Let d denote the distane between the transmitter and the reeiver, and D denote the diffusion oeffiient of the information partiles in the propagation medium. Following along the lines of the derivations in [17, Se. II], and using the results of [29, Se. 2.6.A], it an be shown that for 1-dimensional pure diffusion, the propagation time of eah of the information partiles follows a Lévy distribution, denoted in this work by L µ, with = d2 2D and µ = 0. Thus, Z m L 0,, m = 1, 2,..., M. Remark 1. The work [16] showed that a saled Lévy distribution an also model the first arrival time in the ase of an infinite, three-dimensional homogeneous medium without flow. Hene, our results an be extended to 3-D spae by simply introduing a salar fator. 2.5 Transmission over the Single-Partile DBMT Channel We onlude this setion of preliminaries with the study of the relatively simple ase in whih a single information partile is released, i.e., M = 1. For this setup, the deision rule whih minimizes the probability of error, and the minimal probability of error, are given in the following proposition: Proposition 1. The deision rule whih minimizes the probability of error when M = 1, is given by: { 0, y 1 < θ Ŝ ML y 1 = 8 1, y 1 θ, 7

8 where θ is the unique solution, in the interval [, + 3 ], of the following equation in y 1: y1 y 1 y 1 log = y 1 3, y 1 > > 0. 9 Furthermore, the probability of error of this deision rule is given by: P ε = erf + erf. 10 2θ 2θ Remark 2. The first term on the right-hand-side RHS of 10 orresponds to the probability of error when X = 0 is transmitted, while the seond term orresponds to the ase of X =. As we onsider a non-negative and heavy-tailed distribution, 1 erf 2θ erf 2θ, whih implies that the hannel is asymmetri, and the probabilities of error in sending S = 0 or S = 1 are different. This an be takled by alternating the assignments of bits in 3 over time, or by applying oding dediated to asymmetri hannels, see [30, 31, 32] and referenes therein. Proof of Proposition 1. The optimal symbol-by-symbol deision rule is the MAP rule [33, Ch. 4.1]. As we onsider a binary detetion problem with equiprobable onstellation points, the MAP rule speializes to the ML rule, whih using the mapping 3 is written as: f Y X y x = 0 f Y X y 1 x = Ŝ = 0 Ŝ = 1 1, y 1 >. 11 Plugging the density in 6 with µ = x into the left hand side LHS of 11, and applying log on both sides, we obtain 9. The uniqueness of the threshold θ follows from the fat that the PDFs for both hypotheses are shifted versions of the Lévy PDF, whih is unimodal. A formal and rigorous proof for this uniqueness is provided in Appendix A. Regarding the probability of error, for the ase of y 1 <, we note that due to ausality s must be equal to 0. For y 1 we write: P ε a = 0.5 Pr{y > θ s = 0} + {y θ s = 1} b = erf + erf, 2θ 2θ where a follows from the assumption that the symbols are equiprobable, and b follows from 7. We emphasize that this proposition an be easily extended to the ase of unequal a-priori symbol probabilities. Next, we study ML and linear detetion for transmission over the DBMT hannel with M > 1. 8

9 3 Transmission over the DBMT Channel for M > 1: ML and Linear Detetion The probability of error in moleular ommuniation an be redued by transmitting multiple information partiles for eah symbol [17], [34], namely, using M > 1 partiles for eah transmission. 4 In fat, in [8] we showed that the apaity of the DBMT hannel sales linearly with M. Yet, in [8] we did not provide analysis of the probability of error, nor an effiient deoding method. In this setion we first present the ML detetor for the DBMT hannel, and then disuss more pratial detetion approahes. 3.1 ML Detetion Let y = {y m } M. hannel outputs y: The following proposition haraterizes the ML detetor based on the Proposition 2. The deision rule whih minimizes the probability of error for M 1, is given by: Ŝ ML y = { 0, M log ym y m + 3 1, otherwise, 1 y my m > 0 12 Proof. The proof follows the same lines as the proof of Prop. 1. More preisely, as the a- priori probabilities are equal, the optimal detetion rule is ML. Using Assumption A3 the joint onditional density of y is a produt of the individual onditional densities, and applying log results in 12. Although the above ML detetor minimizes the probability of error, it laks an exat performane analysis and is relatively ompliated to ompute in nano-sale devies; this in partiular holds for the log operation [35, 36, 37]. In the following we denote the probability of error of the ML detetor by P ε,ml. In general, the asymptoti performane of the ML detetor an be derived using the approah of [20, Ch ], yet this does not seem to lead to losed-form expressions. In traditional wireless ommuniation, the ommon approah is to apply a linear signal proessing based on the sequene y. The omplexity of suh a reeiver is signifiantly lower ompared to that of the ML detetor, and for an AWGN hannel this approah is known to be optimal [38, Ch. 3.3]. In fat, even in non-gaussian problems suh as transmission over a timing hannel with drift [17, Se. IV.C.2], modeled by the additive IG noise AIGN hannel, the linear approah yields signifiant performane gains ompared to the transmission of a single partile. We note that the linear approah is motivated by the fat that the mean of the RV y m minimizes the power of the shifted variable Y X over all possible shifts [19, eq. 25]. This favors using linear tehniques in signal proessing problems involving finite seond-order moments. 5 However, the signals in our problem have infinite 4 As we assume that the transmitter and the reeiver are perfetly synhronized, the best strategy is to simultaneously release M moleules. Releasing the M moleules in different times an only inrease the ambiguity at the reeiver and therefore inrease the probability of error [17, Se IV.C]. 5 Note that the IGN distribution onsidered in [17] has a finite variane. 9

10 seond moments, whih renders the linear approah sub-optimal. In the next subsetion we argue that when the transport mehanism is based only on diffusion, then suh a linear reeiver in fat degrades the performane ompared to the transmission of a single partile. The sub-optimality of linear signal proessing of signals orrupted by α-stable additive noise was already observed in [20, Ch ], yet, to the best of our knowledge, the analysis in the next sub-setion is the first to rigorously show this sub-optimality. 3.2 Linear Detetion In this subsetion we onsider linear detetion of signals transmitted over an additive hannel orrupted by a stable noise with harateristi exponent smaller than unity, namely, we use the hannel model 2, with the minor hange that Z m S 0,, β, α, α < 1. Thus, the results presented in this subsetion also hold for the Lévy-distributed noise. Let {w m } M, w m R +, M w m = 1 be a set of oeffiients, and onsider ML detetion based on Y LIN M w my m : ˆX LIN = argmax f YLIN Xy LIN X = x. 13 x {0, } Let P ε,lin denote the probability of error of the detetor ˆX LIN. We now have the following theorem: Theorem 1. The probability of error of the linear detetor is higher than the probability of error of the detetor in 8, namely, P ε,lin P ε, where P ε is given in 10. Proof. We show that given X = x, Y LIN S x, LIN, α, β, with LIN. Note that when X = x is given, then the Y m s are independent. Therefore, the harateristi funtion of Y LIN, given X = x, is given by: ϕ YLIN X=xt = a = M exp {jxw m t w m t α 1 jβsgnw m tφw m t, α} M exp {jxw m t w m t α 1 jβsgntφt, α} { M } = exp {jxw m t w m t α 1 jβsgntφt, α} { b = exp jxt M w α m t α 1 jβsgntφt, α = exp {jxt LIN t α 1 jβsgntφt, α}, where a follows from the fat that w m > 0 and from the fat that Φt, α is independent of t, for α < 1 ; b follows from the fat that M w m = 1; and follows by defining LIN = M wα m 1 α. Therefore, given X = x, we have Y LIN S x, LIN, α, β. Sine } 10

11 1 M w m 1, m = 1, 2,..., M, we have m wα α 1, and therefore LIN. Finally, as is the dispersion of the distribution, and sine stable distributions are unimodal [21, Ch. 2.7], it follows that the probability of error inreases with. Therefore, we onlude that P ε,lin P ε. As the Lévy distribution is a speial ase of the family S 0,, β, α, α < 1, it follows that the linear detetor degrades the performane ompared to the detetor in 8. This is numerially demonstrated in Setion 6. Remark 3. The differene between the AIGN hannel or the AWGN hannel and the hannel onsidered in this paper stems from the fat that for the AIGN, weighted averaging an derease the noise variane, namely, the tails of the noise. On the other hand, in the ase of the Lévy distribution, averaging leads to heavier tail, and therefore to higher probability of error. Remark 4. In order to implement the ML detetor 12, the reeiver must wait for all partiles to arrive. However, as the Lévy distribution has heavy tails, this may result in very long reeption intervals. In fat, the average waiting time of suh a reeiver will be infinite. The detetor in 13 an be implemented with shorter waiting intervals, yet, these intervals are signifiantly longer then the ones required in the single-partile detetion problem. In the next setion we present a simple detetor, whih requires a short reeption interval in the order of the single-partile ase, and ahieves performane very lose to that ahieved by the ML detetor. 4 Transmission over the DBMT Channel for M > 1: FA Detetion The detetor proposed in this setion detets the transmitted symbol based only on the FA among the M partiles, namely, it waits for the first partile to arrive and then applies ML detetion based on this arrival. In terms of omplexity, the FA detetor simply ompares the first arrival to a threshold; this is in ontrast to the ompliated ML detetor in 12. Let y FA = min{y 1, y 2,..., y M }. In the sequel we show that the PDF of Y FA is more onentrated around the transmitted symbol than the original Lévy distribution, whih is reminisent of the lower variane ahieved by averaging in AIGN and AWGN hannels. The FA detetor is presented in the following theorem: Theorem 2. The deision rule whih minimizes the probability of error, based on y FA, is given by: { 0, y FA < θ M Ŝ FA y FA = 14 1, y FA θ M, 11

12 where θ M θ M 1, θ 1 = θ, is the solution of the following equation in y FA : 2M 1 3 y FA y FA log yfa 1 erf 2y FA + log y FA 1 erf = 3, 15 2y FA for y FA > 0. The probability of error of the FA detetor is given by: M M P ε,fa = erf erf. 16 2θ M 2θ M Proof. The detetion rule that minimizes the probability of error is the ML detetor based on Y FA. This requires the PDF and CDF of Y FA given X. Let F Y X y x denote the CDF of y m given X. Assumption A3 implies that given X, the hannel outputs Y 1, Y 2,..., Y M are independent. Hene, using basi results from order statistis [39, Ch. 2.1], we write: F YFA Xy FA x = 1 Pr{Y > y X = x} M a M = 1 1 erf 2y x Ψ, M, y x, 17 where a follows from 7. Next, to obtain the PDF of Y FA given X, we write: f YFA Xy FA x = F Y FA Xy FA x y FA 1 erf = M f Y X y x 2y x = M 2πy x 3 exp 2y x M 1 1 erf Hene, the ML deision rule based on the measurement y FA is given by: f YFA Xy FA x = 0 f YFA Xy FA x = Ŝ = 0 Ŝ = 1 M y x 1, y FA >. 19 Plugging the density in 18 into the LHS of 19, and applying some algebrai manipulations we obtain 15. To show that θ M θ M 1 we first note that by plugging 18 into 19 it follows that θ M is the solution of the following equation: f Y X y FA x = 0 f Y X y FA x = = 1 erf 2y FA 1 erf 2y FA M

13 x =0,M =1 x = Δ,M =1 x =0,M =3 x = Δ,M =3 f Y X y x θ 3 θ y Figure 2: The onditional probability densities f Y X y x = 0 and f Y X y x =, for = 2, and = 1. Now, for M = 1, the RHS of 20 equals 1, and θ 1 [, + 3 ]. Thus, in this interval, the LHS of 20 ahieves the value 1. An expliit evaluation of the derivative of the LHS of 20 shows that in this range the derivative is negative, and therefore the LHS of 20 dereases with y FA, independently of M. On the other hand, the RHS of 20 inreases with M for all y FA. Therefore, we onlude that the solution of 20 dereases with M. Regarding the probability of error, we first note that for y FA <, due to the ausality of the arrival time, S must be equal to 0, and therefore the probability of error is zero. For y FA we write: P ε,fa = F YFA Xy FA x = θ M + F YFA Xy FA x = θ M. By plugging the CDF in 17 into this expression we obtain 16. Finally, we note that this theorem an also be easily extended to the ase of unequal a-priori symbol probabilities. Example 1. Consider sending information partiles with diffusion oeffiient D = 10µm 2 /s, see [40], and let the distane between the transmitter and the reeiver be d = 4 10µm. This implies that = 2s. We further set = 1, and using Prop. 1, for M = 1, we obtain the optimal deision threshold θ = The onditional probability densities f Y X y x = 0 and f Y X y x = are illustrated in Fig. 2. Fig. 2 also depits the onditional probability distributions for M = 3. For this ase the optimal deision threshold is θ 3 = It an be observed that for M = 3 the onditional PDFs are more onentrated near X = 0 and X =, respetively, ompared to the ase of M = 1. Furthermore, the tails of the onditional PDFs for M = 3 are smaller than those for M = 1. Finally, note that while the tail dereases exponentially in M, the PDF around X = 0 or X = inreases linearly with M, see 18. Remark 5. The FA detetion framework an be diretly extended to the ase of L > 2. In suh ases the detetion will be based on L 1 thresholds, eah separates between two neighbor transmission points. Furthermore, as the onditional PDFs onentrate near x when M 13

14 inreases, we onlude that by inreasing M one an support larger L for a given target probability of error. 6 This is demonstrated in Setion 6. 5 Performane Comparison of the ML and FA Detetors It is natural to wonder how lose is the probability of error of the FA detetor to the probability of error of the ML detetor? Clearly, Y FA is not a suffiient statisti for deoding based on y, yet, our numerial simulations indiate that for low to moderate values of M, these detetors are almost equivalent. On the other hand, when M is large, we use error exponent analysis to show the superiority of the ML detetor over the FA detetor. 5.1 Small M To study the performane gap between the two detetors, when M is relatively small, we derive an upper bound on the probability that there is a mismath between the deisions of the two detetors. More preisely, let P mm Pr{ŜMLy ŜFAy}. The following theorem upper bounds P mm : Theorem 3. Let gx log x x + 3xx, x >. The equation gx = 0 has a unique solution x, where gx > 0 for < x < x, and gx < 0 for x < x. Furthermore, the mismath probability is upper bounded by: where 0 = 1, and 1 = 0. P mm P ub mm = i=0 Proof. The proof is provided in Appendix B. { } Ψ, M, x i Ψ, M, ī, 21 Remark 6. Reall that f YFA Xy x = x 0 onentrates around x = x 0 when M inreases. On the other hand, x and are independent of M and depend on the propagation of a single partile. Therefore, when M inreases, the upper bound in 21 beomes loose. We further note that the upper bound in 21 is tightened when inreasing. For instane, let M = 2, = 1, and = 1. For this setting P ε,ml = , P ε,fa = , and P mm ub = If we inrease to be equal to 5 we obtain: P ε,ml = , P ε,fa = , and P mm ub = On the other hand, for larger values of M, e.g., M = 5, we have P ε,ml = , P ε,fa = , P mm ub = , for = 1, and P ε,ml = , P ε,fa = , P mm ub = 0.001, for = 5. For large values of M, we next analyze the error exponents of the FA and ML detetors, and show that in this regime the ML detetor signifiantly outperforms the FA detetor. 6 Note that when L > 2, then Pr{S Ŝ} refers to the symbol error probability. 14

15 5.2 Large M Let P ε M denote the probability of error of a given detetor, as a funtion of M. The error exponent is then given by: E = lim M M P ε log M. In the following we first derive the error exponent of the FA detetor, and then numerially ompare it to the exponent of the ML detetor. This numerial omparison indiates that the error exponent of the ML detetor is higher than that of the FA detetor. This implies that the two detetors are not equivalent, even though for low values of M they ahieve very similar performane. This observation is supported by the fat that the first arrival is not a suffiient statisti for deoding based on y. The following theorem presents the error exponent of the FA detetor: Theorem 4. The error exponent of the FA detetor is given by: E FA = log 1 erf Proof Outline. Reall the probability of error of the FA detetor in 16, repeated here for ease of referene: M M P ε,fa = erf erf 2θ M 2θ M Based on the observations in Remark 2, and noting that both PDFs are right-sided, namely, different than zero only for y > x, we intuitively expet the first term on the RHS of 16 to be larger than the seond term. In suh ase, the error exponent of the FA detetor is governed by the first term, and as θ M when M, we obtain 22. In Appendix C we rigorously analyze the saling behavior of the seond term in 22 and show that it yields the same error exponent as the first term, thus, leading to 22. Next, we disuss the error exponent of the ML detetor. Deriving a losed form expression for this error exponent seems intratable, therefore, we present an impliit expression and evaluate it numerially. The problem of reovering x based on the M i.i.d. realizations {y m } M belongs to the lass of binary hypothesis problems, whih are studied in [41, Ch. 11]. In partiular, the error exponent for the probability of error is exatly the Chernoff information [41, Theorem ]. We emphasize that this optimal error exponent is independent of the prior probabilities assoiated with the two values of the transmitted symbol x, see the disussion in [41, pg. 388]. Thus, the assumption of equiprobable bits plaes no limitation on the error exponent of the ML detetor. Next, for ontinuity of presentation, we briefly present the appliation of the results of [41, Ch. 11.9] to our problem. Let π 0 > 0 and π > 0 denote a-prior probabilities for sending x = 0 and x =, respetively, for a fixed M. Furthermore, let g 0 y and g y denote the likelihood funtions. 15

16 orresponding to x = 0 and x =, respetively. Finally, let I ondition denote the indiator funtion whih take the value 1 if the ondition is satisfied and zero otherwise. Sine given x the {y m } M are independent, it follows that the probability of error of the ML detetor, as a funtion of M, an be written as: M M P M ε,ml = π M 0 g 0 y m I g 0 y m < g y m dy y + π y M M g y m I g 0 y m > Next, we define J M y min { M g 0y m, M g y m M g y m dy. 23 } dy, and note that 23 satisfies: min {π 0, π } J M P M ε,ml max {π 0, π } J M. 24 logp M ε,ml Observe that for fixed π 0 and π, the error exponent lim M M, equals the error exponent of J M, namely, logp M ε,ml lim logj M = lim M M M M, whih is exatly the Chernoff information, see [41, pg. 387]. We further write: { M } M J M = min g 0 y m, g y m dy a y min s:0 s 1 b e ME ML, y M s M 1 s g 0 y m g y m dy where a follows from the fat that for any positive numbers a, b and a real number s [0, 1] we have mina, b a s b 1 s, and b follows from defining: E ML min g0yg s 1 s ydy. 25 s:0 s 1 log logj The above argument establishes an upper bound on the error exponent lim M M M. A lower bound follows diretly from the ahievable distribution in [41, Theorem ]. The two bounds oinide as M. Thus, we onlude that the error exponent of the ML detetor is given in 25. Example 2. In ontrast to 22, deriving a losed form expression for the error exponent 25 seems intratable. Hene, we numerially evaluated E ML using a PYTHON sript based on the SCIPY library. Table 1 details both E ML and E FA for {0.1, 0.2, 0.3, 0.4}, and {0.5, 1, 2}. 16 y

17 Pε ML, M =1 FA, M =2 FA, M =3 ML, M =2 ML, M =3 LIN, M =2 LIN, M = [s] Figure 3: P ε vs., for = 1 [s] and M = 1, 2, 3. Note that when M inreases, very small values of an be used. For instane, for M = , = 2 and = 0.1, we obtain P ε,fa = It an be observed that for small values of, and large values of, the relative differene between the two values is larger. This implies that in severe propagation onditions, the ML detetor has a signifiant advantage upon the FA detetor. On the other hand, when the propagation onditions improve, the relative differene is small. This observation is also refleted in Figure 4, see Setion 6. = 0.1 = 0.2 = 0.3 = 0.4 = 0.5, E ML = 0.5, E FA = 1, E ML = 1, E FA = 2, E ML = 2, E FA Table 1: E ML and E FA for different values of and. Next, we present numerial evaluations of our results. 6 Numerial Results We begin our numerial evaluations with the probability of error for the different detetors. Fig. 3 depits the probability of error versus different values of, for M = 1, 2, 3, for the ML, FA, and linear detetors. Throughout this setion 10 6 trials were arried out for eah point. When M = 1 all the detetors are idential. For larger values of M, the probability of error of the ML detetor was evaluated numerially, while the probability of error of the FA detetor was alulated using 16. For the linear detetor we assumed w m = 1 M whih leads 17

18 10 0 = =0.5 Pε FA, =0.2 ML, =0.2 FA, =0.5 ML, = M Figure 4: P ε vs. M, for = 2 [s] and = 0.2, 0.5 [s]. to LIN = M. It an be observed that the probability of error dereases with and with M. Moreover, as stated in Setion 5, Fig. 3 shows that the ML and FA detetors are pratially indistinguishable for small values of M. Finally, note that the linear detetor indeed degrades performane as M inreases. Fig. 4 depits the probability of error versus the number of released partiles M, for the ML and FA detetors, = 0.2, 0.5, and = 2. Here, 10 6 trials were arried out for eah M point. It an be observed that for small values of M, as indiated by Fig. 3, the FA and ML are indistinguishable. On the other hand, when M inreases, the superiority of the ML detetor is revealed, e.g., M 50. This supports the results and onlusions stated in Setion 5.2. Note that Fig. 4 also indiates that for large enough M, the probability of error deays exponentially with M. This implies that if hanges, e.g., the distane between the transmitter and reeiver inreases, one an ahieve the same P ε,s by inreasing M. This is demonstrated in Fig. 5, where P ε,s is fixed to 0.01, and the required M is presented as a funtion of, for different values of. Finally, we note that one an trade probability of error with number of bits onveyed in eah transmitted symbol, namely, log 2 L. More preisely, for a given and L, by using M large enough, one an ahieve any target probability of error. This is demonstrated in Fig. 6, whih shows that the symbol probability of error P ε,s of 10 3, an be ahieved when is about 7 seonds, for different pairs of L and M. This implies that by using large values of M the transmitter an send short messages using a single-shot transmission with relatively small values of. 7 Conlusion We studied ommuniation over MT hannels with α-stable noise assuming that multiple information partiles are simultaneously released at eah transmission. We first derived the 18

19 10 3 Δ =0.5 Δ =1 Δ =2 Δ =5 M [s] Figure 5: The number of partiles M required to ahieve P ε = 0.01, as a funtion of, for the FA detetor Pε,s M =25,L=8 M =50,L=16 M = 100,L= Δ [s] Figure 6: P ε,s vs., for = 1 [s] and the M, L pairs: 25, 8, 50, 16, 100,

20 ML detetor and argued that it is impratial for nano-sale devies due to its high omplexity. Then, we onsidered the linear detetion framework, whih was shown to be effetive in Gaussian hannels or in MT hannels in the presene of drift. We showed that when the noise is stable with harateristi exponent smaller then unity, then linear proessing inreases the noise dispersion, whih results in higher probability of error. This is supported by the fat that stable signals with harateristi exponent smaller than unity have an infinite mean, and therefore linear proessing is not favorable. To take advantage of the multiple transmitted partiles, we then derived the FA detetor and showed that for low to medium values of M it ahieves a probability of error very lose to that of the ML detetor. On the other hand, sine the first arrival is not a suffiient statisti for the detetion problem, it is not expeted that the FA and the ML detetors will be equivalent for any value of M. To rigorously prove this statement we derived the error exponent of both detetors and showed that indeed, in this aspet, ML is superior. Our derivations indiate that the FA detetor has a nie property that the onditional densities onentrate around the transmitted values. This implies that by using M large enough one an use large onstellations, thus, onveying several bits in eah transmission. This property is very attrative for moleular nano-sale sensors that are required to send a limited number of bits and then remain quiet for a long period of time. A Proof for the Uniqueness of θ in 9 First we note that the mode of a standard Lévy-distributed RV is 3, and therefore, the deision threshold must lie in the interval [, + 3 ]. The uniqueness of the solution stems from the fat that the PDF of the Lévy distribution is unimodal, and from the fat that the PDFs in the two hypotheses are shifted version of the Lévy distribution. More preisely, note that for y 1, the LHS of 9 tends to zero, while for y the LHS of 9 is larger than 3. We now show that the derivative of the LHS of 9, whih is given by 2y 1 log y1 y 1, is positive. This implies that 9 has a unique solution. We write: y1 2y 1 log a = log w y 1 1 w b 1 1 w = 0. W + 1 W w 1 Here, a follows by setting w = 1 y 1. Note that sine y 1, then w [0, 1]. For step b we use the inequality 1 1 w logw. Thus, as the derivative is positive, we onlude that 9 has a unique solution in the desired range. 20

21 B Proofs for Theorem 3 First we prove that the equation gx = 0 has a unique solution. Then we derive the properties of gx, and finally, we derive the upper bound on the mismath probability. Let α = 3. Thus, we an write gx as gx = log x x + α xx, x >. First, we show that gx has a single extreme point whih is larger than. Writing the derivative of gx we have: gx x = α 2x + xx x 2 x 2. Thus, the extrema points of gx are the roots of the polynomial x 2 2α+ 2 x + α. Plugging α = 3 and using the expressions for roots of a quadrati equation we obtain that the extreme points are given by: x 1 = , x 2 = Now, it an be observed that x 1 > 3 + > whih proves the existene of an extrema point larger than. For x 2 we write: x 2 = = < 0. Hene, in the range x >, the funtion gx has a single extreme point. Next, we note that lim x gx =, while lim x gx = 0. Therefore, x 1 is a minimum point. Thus, the equation gx = 0 has a single solution in the range x >. Next, we upper bound the mismath probability. Let mismath denote the event of Ŝ ML y ŜFAy. We write: Pr{mismath} = 0.5 Pr{mismath x = 0} + Pr{mismath x = }. B.1 B.1 Upper Bounding Pr{mismath x = 0} We begin with upper bounding Pr{mismath x = 0}. Note that if y FA then ŜMLy = Ŝ FA y = 0, and therefore we analyze only the ase of < y FA. Reall that for < y FA < θ M the FA detetor deides ŜFAy = 0. Hene, a mismath event ours when the ML detetor delares ŜMLy = 1, whih ours if see 12: M ym log + 3 y m 1 y m y m < 0. B.2 21

22 The LHS of B.2 an be written as: M ym log + y m 3 yfa = log + 3 y FA 1 y m y m 1 M y FA y FA + m=2 ym log y m y m y m. Therefore, B.2 an be written as: yfa log + 3 Let φy = log y FA B 1 y y y + 3 { 1 M y FA y FA < m=2 ym log y m 3 1 yy. Next, we define the set: y 2, y 3,..., y M : φy < M m=2 ym log y m 3 1 y m y m. } 1. y m y m B.3 Thus, Pr{mismath x = 0}, when < y FA < θ M, is given by: Pr{mismath x = 0, < y FA < θ M } θm = f YFA X y x = 0 f Y2,Y 3,...,Y M Xy 2, y 3,..., y M x = 0d 2 dy 3... dy M dy a θm f YFA X y x = 0dy B 1 y = Ψ, M, θ M Ψ, M,, B.4 where a follows from the fat that in the seond integrand is a joint PDF, and therefore it is upper bounded by 1. Following similar arguments, for Pr{mismath x = }, when < y FA < θ M, we obtain: Pr{mismath x =, < y FA < θ M } Ψ, M, θ M Ψ, M, 0. B.5 B.2 Upper Bounding Pr{mismath x = 0} for θ M < y FA First, we reall that when θ M < y FA then ŜFAy FA = 1. Hene, a mismath event takes plae if the ML detetor delares ŜMLy = 0, whih ours if see 12: M ym log + 3 y m 1 y m y m > 0. B.6 We showed that if y FA > x then gy m < 0, m = 1, 2,..., M. In suh ase the LHS of B.6 is negative and ŜMLy = 1, thus, there is no mismath. Therefore, Pr{mismath x = 0, θ M < 22

23 y FA } = Pr{mismath x = 0, θ M < y FA < x }. Now, we define the set: B 2 y { y 2, y 3,..., y M : φy > M m=2 and write Pr{mismath x = 0, θ M < y FA < x } as: ym log y m 3 } 1, y m y m Pr{mismath x = 0, θ M < y FA < x } x = f YFA X y x = 0 f Y2,Y 3,...,Y M Xy 2, y 3,..., y M x = 0d 2 dy 3... dy M dy θ M B 2 y x θ M f YFA X y x = 0dy = Ψ, M, x Ψ, M, θ M. B.7 Following similar arguments, Pr{mismath x =, θ M < y FA } is upper bounded by: Pr{mismath x = 0, θ M < y FA } Ψ, M, x Ψ, M, θ M. B.8 Combining B.4, B.5, B.7, and B.8 we onlude the proof. C Proof of Theorem 4 Let a M M M 1 erf 2θ M and bm 1 1 erf 2θ M. Then, expliitly writing the probability of error in 16, the error exponent of the FA detetor is given by: M ε,fa E FA = lim log P M M = lim log 0.5 a M + b M M { M = min lim log a M M M Sine θ M, when M, we write: lim log a M M M = lim M log 1 erf 2θ M M M, lim log b } M. C.1 M M = log 1 erf. 2 C.2 Next, we analyze the seond term in C.1, and note that this term depend on the rate of onvergene of θ M to. Again, we use the fat that θ M as M, and write θ M = + δ M, where δ M 0. We then haraterize the saling behavior of δ M to zero, for large 23

24 values of M. As θ M is the deision threshold, by equating the two PDFs in 18, we have the following equality in terms of δ M : δm + δ M 3 2 e 2 δ M +δ M = 1 erf 2 +δ M 1 erf 2δ M M 1. C.3 logerfx Let M be suffiiently large, and reall the equality lim x = 1. Furthermore, for x 2 M large enough we an write + δ M. Thus, we write C.3 as: δm + δ M 3 2 e 2 δ M +δ M 3 δm 2 e 1 β 2 δ M 1 e 2δ M where we let β 1 erf 2 1, and note that 1 erf 2 +δ m M 1, C.4 β. We now assume that δ M sales as d 1 M, for some onstant d 1, and show that for large enough M the LHS and RHS of C.4 have the same saling. This also enables finding the onstant d 1, and alulating the error exponent of the seond term in C.1. We write C.4 as: 3 d1 2 3 M 2 e M 2d 1 = β M e M 1 βm 1 + M 1e M 2d 1. M 2d 1 C.5 Thus, by noting that the two sides of C.5 must sale to zero at the same rate, we write: 7 e 2d 1 M = e M1+log β e d 1 = Having the saling law of δ M, we now write the seond term in C.1 as: lim log b log M M M = lim M log = lim M = 1 log β e log β C.6 e. M 1 erf 2θ M Me 2d 1 M M = log 1 erf Finally, by plugging C.2 and C.7 into C.1, we onlude the proof. M. C Note that as we are interested in the error exponent, we apply analysis whih fouses only on the saling law, and therefore we ignore terms whih sale slower, e.g., M

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