Diffusion Channel with Poisson Reception Process: Capacity Results and Applications

Transcription

1 Diffusion Channel with Poisson Reception Process: Capacity Results an Applications Hessam Mahavifar an Ahma Beirami Department of Electrical an Computer Engineering, University of California San Diego, USA Department of Electrical an Computer Engineering, Duke University, USA Abstract We consier a channel moel base on the iffusion of particles in the meium which is motivate by the natural communication mechanisms between biological cells base on exchange of molecules. In this moel, the transmitter secretes particles into the meium via a particle issemination rate. The concentration of particles at any point in the meium is a function of its istance from the transmitter an the particle issemination rate. The reception process is a oubly stochastic Poisson process whose rate is a function of the concentration of the particles in the vicinity of the receiver. We erive a close-form for the mutual information between the input an output processes in this communication scenario an establish useful properties about the mutual information. We also provie a signaling strategy using which we erive a lower boun on the capacity of the iffusion channel with Poisson reception process uner average an peak power constraints. Furthermore, it is shown that the capacity of iscretize iffusion channel can be a negligible factor of the capacity of continuous time iffusion channel. Finally, the application of the consiere moel to the molecular communication systems is iscusse. I. INTRODUCTION Diffusion channel with molecular reception process has emerge as an abstraction of a natural communication mechanism using signaling molecules between living organisms an nano-evices. This is a potential means of information transmission wherever conventional electromagnetic waves are prohibite, such as within human boy [], []. In the iffusion channel moel, the transmitter secretes particles in the meium that are subject to Brownian motion. In the large system limit, where the number of secrete particles in the meium is relatively large, we can assume that the number of particles in any ifferential volume element woul be concentrate aroun its mean, which is governe by the eterministic Fick s law of iffusion [3]. The information is conveye to the receiver by means of alteration of the concentration of particles in its vicinity. The iffusion channel has been moele by Pierobon an Akyiliz in [], where they efine the channel capacity problem in a general setting an aime at characterizing the capacity uner iffusion noise. Diffusion noise at a certain point is efine as the variation of particle concentration aroun its mean. In the abstraction consiere in this paper, the iffusion noise can be neglecte, as we assume that the reception process introuces a noise that is much larger than the iffusion noise. Hence, the limiting factor in etermining the capacity woul be the reception process. The Poisson reception process has been extensively stuie in the literature in the context of photon communication channel. In particular, the capacity of the Poisson reception channel was exactly erive by Kabanov [4]. Later, Wyner erive the error exponent for this channel [5], [6]. Therefore, the Poisson reception channel secons the well known Gaussian channel in that both its capacity an error exponents are completely characterize. In this paper we characterize the mutual information between the input an output stochastic processes of a iffusion channel with Poisson reception noise. The result is then use to show that the mutual information is a convex an ecreasing function of an aitive interference term, a property that will be exploite in orer to establish a lower boun on the capacity of the iffusion channel uner average an peak power constraints. This is accomplishe by proviing a signaling strategy base on the pulse amplitue moulation. We will prove that the capacity scales super-logarithmically with the peak power constraint. This result is then use to show that iscretization of the iffusion channel oes not suffice to stuy the capacity of the continuous time iffusion channel. A. Transmitter II. END-TO-END CHANNEL MODEL We assume that the transmitter is locate at the origin of the -imensional coorinate system where {,, 3} enotes the number of imensions. Let the particle concentration in the meium be enote by ρ(x, t), where x R enotes the coorinate an t [, ) is the time. The transmitter releases parcels in the meium at rate λ(t), for t [, ), which is also calle particle issemination rate. Further, let Λ enote the space of trajectories of λ. Observe that the transmitter power is irectly proportional to λ(t) which is subject to constraints. We assume that the transmitter is subject to a peak power constraint given by λ(t) A. () We further assume that the average power for all t [, ) is given by E{λ(t)} = µa, () for some µ [, ], where E enotes the expectation operator with respect to probability measure P λ over the space Λ of all trajectories of λ /5/$3. 5 IEEE 956 ISIT 5

2 B. Diffusion Channel The particle concentration is subject to the iffusion process governe by the following iffusion equation: { t ρ(x, t) = D ρ(x, t) + λ(t)δ( x ), (3) ρ(x, ) = where D is the iffusion constant, is the Laplacian operator, an δ is Dirac s elta function. The intereste reaer is referre to [] for further information about the physical interpretation of this equation. The solution to the above ifferential equation is the convolution with respect to t of λ(t) an the funamental solution ) Γ(x, t) = exp ( x. (4) / (4πDt) 4Dt In other wors, ρ(x, t) = Γ(x, t) λ(t) = t C. Receiver (Poisson Reception Process) Γ(x, s)λ(t s)s. (5) The receiver is assume to be at coorinate r, where particle concentration is given by ρ(r, t). The receiver is equippe with a number of receptors to which the particles may bin. The probability of bining of a particle in the vicinity of the receiver to a receptor epens on the particle concentration. The receiver then senses the particle concentration base on the number of particles boun to its receptors. Therefore, it is natural to assume that the receiver senses the particle concentration through a Poisson reception process whose rate is a function (enote by f) of the particle concentration ρ(r, t) at the receiver noe. Let N(t), t [, ) enote the receive signal (number of receive particles) which is governe by the Poisson reception process with rate γ(t) that is given by γ(t) = f(ρ(r, t)). (6) Observe that the receive signal N(t) is a oubly stochastic Poisson process (as γ(t) is itself a stochastic process). Let N enote the space of all trajectories of N(t). Therefore, γ(t) = f(γ(r, t) λ(t)) ef = Dλ(t), (7) where D is the iffusion channel operator. We stress that f epens on the physical properties of the receiver, such as the size an the number of receptors. It is natural to assume that f is a strictly increasing function as when more particles are in the vicinity the probability of bining with receptors increases. We further assume that f is a concave function. This is ue to the fact that when the particle concentration is lower, a tiny increase in the particle concentration increases more the probability of bining with the receptors. Further, we expect that f woul be boune as the the particle concentration increases. Hence, f(x) woul behave linearly with x for small x whereas it woul get saturate to a constant as x becomes large. III. END-TO-END MUTUAL INFORMATION To calculate the capacity of the iffusion channel moele in the previous section we woul nee to calculate the mutual information between the input process λ(t) an the 957 output process N(t) at interval [, T ], which is enote by I(λ t [,T ] ; N t [,T ] ). Let I T (λ, N) ef = I(λ t [,T ] ; N t [,T ] ). Let P λ,n enote the joint probability measure on λ an N on the space Λ N. Further, let P N λ enote the conitional probability measure. Hence, [ ( )] PN λ I T (λ; N) = P λ (λ) log (N) P N λ (N), Λ N P N (8) an the goal is to maximize T I T (λ; N) (9) with respect to P λ subject to input power constraints to erive the channel capacity as shall be escribe in Section IV. There has been a long history on stuying the relationship between the mutual information an estimation error [7], [8]. In particular, the mutual information between the input an output processes of the Poisson reception channel was given in terms of an estimator error [4]. This result is use in the following theorem in orer to characterize the mutual information I T (λ, N). Theorem Let γ(t) be given by (7) an φ( ) be the Raon- Nikoym erivative of the Poisson process as given by Then, I T (λ; N) = φ(x) = x log x. () T E {φ(γ(t)) φ(ˆγ(t)} t, () where E enotes the expectation operator with respect to the joint probability measure P λ,n, an ˆγ(t) is the minimum mean-square error (mmse) estimator of γ(t) given the process N τ [,t] as given by ˆγ(t) = E { γ(t) N τ [,T ] }. () Next, we establish a property of this mutual information that will be use in eriving the bouns in Section IV. Theorem If Pr{γ < } =, then I T (γ+γ ; N) is a convex an ecreasing function of γ in [, ). In other wors, for all γ, we have I T (γ + γ ; N) γ Proof. Observe that I T (γ + γ ; N) = γ I T (γ + γ ; N) T. (3) E {φ(γ(t) + γ ) φ(ˆγ(t) + γ )} t. (4) We shall show that { } E {(γ + γ ) ln(γ + γ ) (ˆγ + γ ) ln(ˆγ + γ )}. γ Note that γ {(γ + γ ) ln(γ + γ ) (ˆγ + γ ) ln(ˆγ + γ )} (5)

3 = γ + γ ˆγ + γ. (6) By using the fact that γ+γ is a convex function of γ an observing that ˆγ = E{γ N} we have { E }, (7) γ + γ ˆγ + γ which results in the first part of the claim. Also, note that {(γ + γ ) ln(γ + γ ) (ˆγ + γ ) ln(ˆγ + γ )} γ ( ) γ + γ = ln(γ + γ ) ln(ˆγ + γ ) = ln, (8) ˆγ + γ Since we have Pr{γ < } =, then ( ) γ + lim ln γ =. (9) γ ˆγ + γ The first part of theorem together with (9) completes the proof of the secon part. IV. LOWER BOUND ON THE CAPACITY We aapt a pulse amplitue moulation (PAM) signaling at the input λ(t). The PAM signaling has been consiere also in the context of photon communication with the Poisson reception process an the capacity uner certain constraints is erive [9], []. However, these results can not be irectly applie to our moel mainly ue to the iffusion channel which introuces memory to the system an results in inter-symbol interference for the pulses at the input of the channel. Let > enote the symbol uration. In the PAM signaling, for t an n N {}, we have λ(t) = λ n for t < (n + ). Let P(A, µ) enote the set of all istributions P λ of the ranom variables λ with λ [, A] an mean µa. We assume that λ n s are i.i. with the istribution P λ P(A, µ). Let also y n = N((n + ) ) N(). Then h j ef = Pr {y n = j} = e Λn Λ j n, () j! where Λ n = (n+) γ(t)t. For simplicity, we assume that f(.) which is a characteristic of the reception process is simply a linear function in the working regime, i.e. f(x) = gx. Let g (j+) Γ(r, s)s, for j j Then Λ n can be written as Λ n = g = g (n+) (n+) t g ( s)γ(r, s)s, for j =. ρ(r, t)t () Γ(r, t s)λ(s)s t = n h j λ n j j= () Remark. Since λ n is the transmitte signal at the n-th interval, one can interpret the term h λ n in () as the signal part an the rest of the terms as the inter-symbol interference (ISI) part. It is expecte that the ISI is comparable to 958 the signal an hence, can not be ignore. In fact, the ISI term resembles the ark current notion in classical photon communication channels. For a fixe j, let Λ an L enote the signal an the ISI part, i.e., Λ ef = h λ j an L ef = j i= h iλ j i. The ISI term can be upper boune as L ga Γ(r, s)s ef = L. (3) The integral in (3) is finite for the imensions 3, by using the efinition of Γ(r, s) in (4). For any >, let W : R + N {} enote the Poisson channel with the constant input intensity γ R + for a uration of. In other wors, the probability of observing y N {} at the output of W with the input γ is given by Pr {y = i} = e γ (γ ) i, (4) i! Lemma 3. Let γ be a ranom variable such that γ /h is istribute accoring to P λ. Consier the Poisson channel W with the input γ + L/ an the output y. Then I(λ j ; y j λ j ) I(γ; y). (5) This lemma is a straightforwar consequence of Theorem an it basically shows that increasing the ISI term will ecrease the mutual information between the input an the output of W at the j-th interval. Therefore, in orer to erive a lower boun on the capacity, one may assume the upper boun L on the ISI term. Lemma 4. The capacity C of the iffusion channel uner the PAM moulation with symbol uration an peak an average power constraints A an µa is lower boune as C max I(γ; y), (6) P λ P(A,µ) where γ /h is istribute as P λ P(A, µ) an y is the output of W with input γ + L/, where L is the maximum ISI term given by (3). The proof is remove ue to space constraints. The next step to lower boun the capacity is to fin the istribution P λ that maximizes (6) which seems to be an intractable task. In the next theorem, a lower boun on the capacity of iffusion channel is erive by picking a certain iscrete input istribution P λ. It has been pointe to us uring the review process that the results of [] can be invoke to this en. In fact the result of Theorem 5 bears a great resemblance with the lower bouns of [] on the capacity of iscrete-time Poisson channel. However, the input istribution introuce in the next theorem is iscrete an we provie a ecoing metho which leas to a signaling strategy while the input istribution in the similar result of [] is continuous. Theorem 5. For any ɛ, >, there exists A R + such that the capacity C of the iffusion channel with peak an average power constraints of A an µa, respectively, is lower boune as C ( ɛ)log A (7)

4 for A A. Furthermore, the lower boun can be achieve by eploying a PAM moulation scheme with symbol uration at the input. Proof. Throughout the proof, by approximation of some parameter with another parameter we mean that their ratio approaches as A grows large. Let k = c A, for a constant c, an m = A k. For the PAM scheme, we consier a iscrete istribution for P λ as follows: e (i )k ik µa e µa for i =,,..., m Pr {λ = ik} = e (i )k for i = m. The mean of P λ is well approximate by µa, since the quantization level k µa is O( A ). Also, the entropy of λ can be approximate as H(λ) log( µa k ) + ln µ + (8) = + log A + log µ( + ln ) log c, where ln µ + is the ifferential entropy of the exponential istribution with mean µ an log( µa k ) is the quantization term. Given the output y, the ecoer estimates λ as ˆλ where y ˆλ = k L h m + (9) Basically, after scaling the output y properly, the closet moulation point ik is chosen as ˆλ. Then we have } { P e = Pr {ˆλ λ Pr y N c } N erf(c ), (3) where N = h λ + L is the mean of y, an the constant c is relate to the constant c through other parameters as c = c h h + L/A. (3) Notice that from the efinition of L in (3) L/A only epens on an not on A. Then using Fano s inequality, H(λ y) h(p e ) + P e log(m ). (3) An by combining (8) an (3) we get I(λ; y) =H(λ) H(λ y) ( P e ) log A + log µ( + ln ) ( + P e ) log c h(p e ). (33) For the given ɛ, we choose constant c such that erf(c ) < ɛ, where c is given by (3). Then A is chosen such that (ɛ erf(c ) ) log A > ( + erf(c )) log c + h(erf(c )) (34) Combining (34) with (33) an using P e erf(c ) from (3) imply that I(λ; y) > ( ɛ) log A (35) Lemma 4 together with (35) complete the proof of theorem. 959 Corollary 6. The ratio C/ log A is unboune as A, i.e., C = ω(log A). The iscrete Poisson reception process characterize as W is equivalent to the iscrete-time Poisson channel which is a well-stuie subject. In particular, lower bouns an upper bouns on the capacity of the iscrete-time Poisson channel is provie in []. For A, the upper boun on the capacity is approximately given as log A. Using this together with Corollary 6 we arrive at the following corollary. For a fixe >, let C enote the capacity of the iscrete iffusion channel with iscretization. Corollary 7. The ratio C/C is unboune as the input power constraint A grows large. This corollary suggests that while stuying the iffusion channels we shoul keep in min that the iscretization oes not always preserve the essential aspects of the system an in particular the capacity of the channel. More precisely, a iscrete moel woul only provie a lower boun on the continuous time capacity an the upper boun on the capacity of the iscrete moel will be a negligible factor of the continuous time capacity when the input power constraint is large. V. APPLICATION TO MOLECULAR COMMUNICATION The iffusion channel moel consiere in this paper is motivate by recent stuies of molecular communication systems. In such a system, the transmitter can be a biological cell that uses molecular signaling for action coorination with other cells. As an example, pathogenic bacteria use molecular signaling to orchestrate their actions in the course of attacking their victim. This phenomenon is known as quorum sensing, which occurs among various species of bacteria an has been uner extensive scrutiny by biologists (cf. [] [4] an the references therein as a few examples). Molecular communication also provies a potential means of communication that suits biological meia such as human boy. The receiver in a molecular communication system, which can be also another biological cell, senses the concentration of signaling molecules using ligan receptors, as iscusse in Section II. The probability of bining of a molecule to a ligan receptor is stuie using a Markov chain moel in [5]. The problem of characterizing the channel capacity in a molecular communication system has been stuie uner various moels. In [6], Einolghozati et al. consiere a iscrete version of the molecular iffusion channel (where only two levels of molecular concentration can be communicate) an characterize the achievable transmission rates in this scenario using a telegraph moel. Recently, in [7], Arjmani et al. viewe molecular communication channel as the concatenation of a iscrete linear filter an a Poission reception process. Some bouns on the capacity of this iscrete moel were erive in [8]. However, as shown in the previous section, iscretization can hurt the throughput significantly an the continuous time capacity can not be achieve. In a ifferent line of work, Eckfor consiere the capacity of the molecular communication with Brownian motion, where the signal is encoe in timing between the molecular transmission [9]. This line of work becomes relevant when

5 the receiver is chosen to be much smarter than a Poisson reception process which requires the esign of nanomachines smarter than existing biological entities. Several other papers have appeare in the literature exploiting this moel for eriving capacity bouns [] [3]. VI. DISCUSSIONS AND CONCLUSION In this paper, we consiere a iffusion channel moel where the communication is via exchange of particles in the meium. The transmitter an the receiver together with the en-to-en channel are abstracte an the mutual information between the input an output processes is characterize in terms of the estimation error. A signaling strategy is propose base on the pulse amplitue moulation an a lower boun on the capacity of the iffusion channel is erive. There are several irections to exten the moel an the results presente in this paper. In our moel, the receiver is assume to have an infinite memory, i.e., the estimate ˆγ(t) is presente by assuming observation of N τ [,T ]. However, a more realistic scenario woul be to assume that the receiver has a finite memory, as the number of receptors is finite, an hence, one may assume that the receiver only observes N τ [T t,t ], where t is enforce by physical limitations of the receiver. It is worth mentioning that the analysis for eriving the lower boun in Section IV will be still vali if the memory of the receiver is at least, the uration of input pulses. The relation between the en-to-en mutual information an the mmse error is erive in Theorem assuming a non-causal estimator, i.e., ˆγ(t) epens on N τ [,T ] for t [, T ]. A practical approach woul be to assume a causal estimator an to try to boun the capacity uner this scenario. The results of [4] which establishes the relationship between the mutual information an error of a causal estimator for Poisson reception channels can be potentially useful for this approach. In orer to lower boun the capacity of the iffusion channel, we have assume that the peak power an average power of the input signal are subject to some constraints. These assumptions are mae accoring to the conventional communication scenarios. However, in a biological communication system, there might be some other natural constraints that shoul be taken into account. A more in-epth stuy of the behavior of these systems is neee in orer to establish a more comprehensive set of assumptions for this channel moel. It is straightforwar to erive upper bouns on the capacity of the iffusion channel moel using the well-establishe results on the capacity of Poisson channels [4] [6]. However, we believe that such bouns will not be very relevant for our moel. For instance, the capacity achieving schemes of [5], [6] for the Poisson channel are erive by assuming a pulse amplitue moulation whose uration goes to zero exponentially fast with the total signaling time of T. However, the Fourier transform of the iffusion channel characteristic of Γ(x, t) is almost zero for high frequencies, i.e., it behaves as a low pass filter. Hence, one can argue that the iffusion channel can not capture the rate of change of the input signal, if the uration of the input pulses is very small. Hence, it is a challenging problem to erive upper bouns that capture the effect of both the iffusion channel an the Poisson reception process. 96 ACKNOWLEDGEMENT The authors are grateful to the anonymous reviewer who brought to their attention the similar results erive in [] in the context of photon channels. REFERENCES [] I. F. Akyiliz, F. Fekri, R. Sivakumar, C. R. Forest, an B. K. Hammer, Monaco: funamentals of molecular nano-communication networks, IEEE Wireless Commun, vol. 9, no. 5, pp. 8,. [] M. Pierobon an I. F. Akyiliz, Capacity of a iffusion-base molecular communication system with channel memory an molecular noise, IEEE Trans. Inf. Theory, vol. 59, no., pp , Feb. 3. [3] E. L. Cussler, Diffusion: mass transfer in flui systems. Cambrige university press, 9. [4] Y. M. Kabanov, The capacity of a channel of the poisson type, Theory of Probability & Its Applications, vol. 3, no., pp , 978. [5] A. D. Wyner, Capacity an error exponent for the irect etection photon channel. I, IEEE Trans. Inf. Theory, vol. 34, no. 6, pp , Nov [6], Capacity an error exponent for the irect etection photon channel. II, IEEE Trans. Inf. Theory, vol. 34, no. 6, pp , Nov [7] T. E. Duncan, On the calculation of mutual information, SIAM Journal on Applie Mathematics, vol. 9, no., pp. 5, 97. [8] R. S. Liptser an A. N. Shiryaev, Statistics of Ranom Processes: II. Applications. Springer,, vol.. [9] S. Shamai, Capacity of a pulse amplitue moulate irect etection photon channel, in Communications, Speech an Vision, IEE Proceeings I, vol. 37, no. 6. IET, 99, pp [], On the capacity of a irect-etection photon channel with intertransition-constraine binary input, IEEE Trans. Inf. Theory, vol. 37, no. 6, pp , Nov. 99. [] A. Lapioth an S. M. Moser, On the capacity of the iscrete-time poisson channel, IEEE Trans. Inf. Theory, vol. 55, no., pp. 33 3, 9. [] M. B. Miller an B. L. Bassler, Quorum sensing in bacteria, Annual Reviews in Microbiology, vol. 55, no., pp ,. [3] P. K. Singh, A. L. Schaefer, M. R. Parsek, T. O. Moninger, M. J. Welsh, an E. Greenberg, Quorum-sensing signals inicate that cystic fibrosis lungs are infecte with bacterial biofilms, Nature, vol. 47, no. 685, pp ,. [4] X. Chen, S. Schauer, N. Potier, A. Van Dorsselaer, I. Pelczer, B. L. Bassler, an F. M. Hughson, Structural ientification of a bacterial quorum-sensing signal containing boron, Nature, vol. 45, no. 687, pp ,. [5] A. Einolghozati, M. Sarari, an F. Fekri, Capacity of iffusion-base molecular communication with ligan receptors, in Information Theory Workshop (ITW), IEEE. IEEE,, pp [6] A. Einolghozati, M. Sarari, A. Beirami, an F. Fekri, Capacity of iscrete molecular iffusion channels, in Information Theory Proceeings (ISIT), IEEE International Symposium on. IEEE,, pp [7] H. Arjmani, A. Gohari, M. N. Kenari, an F. Bateni, Diffusion-base nanonetworking: A new moulation technique an performance analysis, IEEE Communications Letters, vol. 7, no. 4, pp , 3. [8] H. Arjmani, G. Aminian, A. Gohari, M. N. Kenari, an U. Mitra, Capacity of iffusion base molecular communication networks in the LTI-Poisson moel, arxiv preprint arxiv:4.3988, 4. [9] A. W. Eckfor, Nanoscale communication with brownian motion, in Information Sciences an Systems, 7. CISS 7. 4st Annual Conference on. IEEE, 7, pp [] K. Srinivas, A. W. Eckfor, an R. S. Ave, Molecular communication in flui meia: The aitive inverse Gaussian noise channel, IEEE Trans. Inf. Theory, vol. 58, no. 7, pp ,. [] H. Li, S. M. Moser, an D. Guo, Capacity of the memoryless aitive inverse Gaussian noise channel, IEEE J. Selecte Areas Commun., vol. 3, no., pp , December 4. [] C. Rose an I. S. Mian, Signaling with ientical tokens: Lower bouns with energy constraints, in Information Theory Proceeings (ISIT), 3 IEEE International Symposium on. IEEE, 3, pp [3], Signaling with ientical tokens: Upper bouns with energy constraints, in Information Theory (ISIT), 4 IEEE International Symposium on. IEEE, 4, pp [4] D. Guo, S. Shamai, an S. Verú, Mutual information an conitional mean estimation in Poisson channels, Information Theory, IEEE Transactions on, vol. 54, no. 5, pp , 8.