I need a summery for this paper

User Generated

onyxuhmnvz2017

Engineering

Description

.

Unformatted Attachment Preview

60 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017 Equivalent Discrete-Time Channel Modeling for Molecular Communication With Emphasize on an Absorbing Receiver Martin Damrath,∗ Student Member, IEEE , Sebastian Korte, and Peter Adam Hoeher, Fellow, IEEE Abstract — This paper introduces the equivalent discretetime channel model (EDTCM) to the area of diffusionbased molecular communication (DBMC). Emphasis is on an absorbing receiver, which is based on the so-called first passage time concept. In the wireless communications community the EDTCM is well known. Therefore, it is anticipated that the EDTCM improves the accessibility of DBMC and supports the adaptation of classical wireless communication algorithms to the area of DBMC. Furthermore, the EDTCM has the capability to provide a remarkable reduction of computational complexity compared to random walk based DBMC simulators. Besides the exact EDTCM, three approximations thereof based on binomial, Gaussian, and Poisson approximation are proposed and analyzed in order to further reduce computational complexity. In addition, the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm is adapted to all four channel models. Numerical results show the performance of the exact EDTCM, illustrate the performance of the adapted BCJR algorithm, and demonstrate the accuracy of the approximations. Index Terms — Equivalent discrete-time channel model, molecular communication, nanomachines. I. I NTRODUCTION ECENTLY, a remarkable progress in the field of nanotechnology has been achieved, providing new opportunities in many different research areas [1]. One important area is dealing with the realization of so called nanomachines (NMs), which are autonomous machines in the size of several nanometers up to a few micrometers [2]. NMs are likely to enable new application scenarios in the future. Besides industrial and consumer usage like food and water quality control or intelligent textile fabrics, and the environmental usage like biodegradation or air pollution control, the main application is anticipated to be in the medical sector, where NMs can be used for applications like targeted drug delivery, tissue engineering, or health monitoring [1], [3]. However, NMs are very limited with respect to complexity and energy due to their size [3], [4]. In fact, it is assumed that a single NM can perform only simple tasks [5]. Thus, a swarm of NMs has to be realized to provide even more complex functions [1], [6]. The key technology for enabling an autonomous R Manuscript received September 20, 2016; revised December 2, 2016; accepted January 2, 2017. Date of publication January 10, 2017; date of current version March 2, 2017. Asterisk indicates corresponding author. ∗ M. Damrath is with the Faculty of Engineering, University of Kiel, Kiel, Germany (e-mail: md@tf.uni-kiel.de). S. Korte and P. A. Hoeher are with the Faculty of Engineering, University of Kiel, Kiel, Germany. Digital Object Identifier 10.1109/TNB.2017.2648042 swarm is communication between several NMs. Inspired by nature, molecular communication (MC) [7]–[10] seems to be the most promising communication technique. Like in the case of calcium signaling in epithelial cells [8], [11], quorum sensing between bacteria [12], [13], or neurotransmitters in the synaptic cleft [3], molecules can be used as information carrier which passively diffuse through the media. MC comes with the benefit that it is inherently biocompatible, energy efficient, and has been evolved over the last billions of years in nature [1]. The diffusion-based MC (DBMC) channel, e.g. studied in [14]–[17], is fundamentally different to a classical wireless communication channel. Radio waves propagate deterministically in a given environment, whereas molecules perform a random walk. Due to the diffusive random propagation of the molecules, the channel impulse response is slowly decreasing, which causes intersymbol interference (ISI) and unreliable transmission [18]. Thus, it is of great importance to invent new or to adapt existing communication algorithms to the DBMC channel. However, existing practical implementations and numerical simulations are often time-consuming and abstract, which complicates an exhaustive analysis of the impact of DBMC model on transmitter and/or receiver algorithms. In [19] a first initiative towards a practical implementation is given, by introducing a macroscopic MC testbed based on an electrical sprayer and an alcohol sensor. While it provides a proof of principle, it has to deal with hardware challenges, like the non-linearity of the sensor [20]. Furthermore, it is difficult to control environmental parameters. In [10] a layered architecture of MC is presented. Furthermore, the relation between a transmitters’ input and a receivers’ output is generally formulated and specified for a targeted drug delivery scenario. In [21] a system model including multiple transmitters and multiple receivers is described, assuming detection by Michaelis-Menten enzymatic kinetics. In [22] the communication channel model is described by a binomial distribution, which is approximated in [15], [23] as a Gaussian and a Poisson distribution, respectively. In [24] a simulation environment for three-dimensional MC based on Brownian motion is proposed with focus on parallelization to enhance execution time. Therefore, a high-level architecture is designed to distribute the molecular simulation scenario load. Another method to decrease the simulation time of a 3D Brownian motion system is given in [25], where the authors propose a dual time-step approach which uses larger simulation time steps for particles that are far away from the target. In [26] a framework for molecular nanonetworks 1536-1241 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. DAMRATH et al.: EQUIVALENT DISCRETE-TIME CHANNEL MODELING FOR MOLECULAR COMMUNICATION called NanoNS is build on top of the well-known network simulator NS-2. NanoNS is able to simulate the diffusive molecular communication channel, where the diffusion is based on the multiparticle lattice gas automata algorithm. The medium is divided into lattice sides, which reduces simulation complexity. In [27] the so-called N3Sim simulation framework for nanonetworks in two- and three-dimensional bounded and unbounded environments is introduced. While it simulates the diffusion-based MC by Brownian motion, it offers different simulation settings like multiple transmitters with different transmit pulses, absorbing or transparent receivers or molecule harvesters. Furthermore, the inertia and interaction of molecules can be simulated. However, it computes the position and velocity of each molecule for each time step, which makes simulations with a very high number of molecules impossible. In [28] BiNS, a simulation platform designed for information exchange at nanoscale, is presented. Thereby, propagation of molecules is based on a Brownian molecular diffusion process in a three-dimensional environment. Elastic collision between molecules is taken into account. Furthermore, nanomachines can be simulated with a specific mass, size, lifetime and mobility model. The signal transduction is performed by receptors placed on the outer surface of a nanomachine. In [29] BiNS is extended to BiNS2 which is used to simulate diffusion-based molecular communication with drift in blood vessels. In addition, a grid approach is given to enable distributed computations, which improves simulation time. In [30] a three-dimensional end-to-end simulator for diffusionbased molecular communication called MUCIN is introduced. Besides the ability of sending consecutive symbols, frequency-based and concentration-based modulation, imperfect reception, and ISI mitigation function, it also offers a fast simulation based on the characteristic function and binomial distributed random variables. In [31] the authors analyze the effect of approximating the binomial distribution inside the channel model by Gaussian and Poisson distributions. In this contribution the characteristic function is used to derive an equivalent discrete-time channel model (EDTCM) for DBMC with emphasize on an absorbing receiver. The EDTCM is familiar in the wireless communications community to model noisy channels with memory in the discrete-time domain. Therefore, it supports the understanding of the DBMC channel and makes it easier to adapt classical wireless communication algorithms to the area of MC. Furthermore, it offers a simple implementation for numerical simulations, which provides a faster tool for analyzing system performances when applying long bit sequences or large number of molecules. In addition, three approximations are presented, which further improve simulation time. Moreover, the Bahl-Cocke-JelinekRaviv (BCJR) algorithm is presented for each channel model. In contrast to [23], the channel model in this contribution is extended to arbitrary memory lengths and visualized as an equivalent discrete-time channel model. Furthermore, it is directly applied by means of numerical simulations to give an alternative to random walk based simulations. In addition, the accuracy and computational complexity of each channel approximation is analyzed. 61 Fig. 1. Model of the diffusion-based molecular communication system under investigation. The remainder of this article is organized as follows: In Section II the system model that is assumed throughout this contribution is summarized. Based on the system model the exact EDTCM and three approximations thereof are derived in Section III. Additionally, the accuracy of the approximations is analyzed and the computational complexity of all system models under investigation is discussed. In Section IV the detection algorithms that are applied in the numerical analysis in Section V are briefly presented. Thereby, the focus is on adapting the transition probability of the BCJR algorithm to the channel models under investigation. Section VI finally summarizes the work and gives an outlook for future work. II. S YSTEM M ODEL The system model under investigation is similar to the system model introduced in [32]. Thus, in the following, only a brief description is given. For more detailed information the reader is referred to [32]. As shown in Fig. 1, the system model consists of a single point source transmitter (Tx) and a single spherical receiver (Rx) with radius r in an infinite three-dimensional homogeneous fluid medium without drift, where d > r is the distance between Tx and Rx. The fluid medium is described by the diffusion coefficient D, which is assumed to be constant. The modulation scheme under investigation is on-off keying (OOK). The Tx emits either no molecules or N molecules at the beginning of a bit period of length Tb to represent bit u[k] = 0 or u[k] = 1, respectively:  if u[k] = 1, N δ(t − kTb ) x(t) = (1) 0 otherwise, where δ(.) is the Dirac delta function. Molecules emitted by Tx are the only molecules in the medium and propagate by Brownian motion. Rx is assumed to be synchronized with Tx in time domain [33], [34] and behaves perfectly absorbing, while performing strength/energy detection [35]–[37]. Consequently, a molecule is detected and removed from the environment whenever it hits the surface of the receiver. By means of strength/energy detection, the number of hitting molecules is accumulated for each bit period separately. As a result Rx follows the first passage time concept. The probability per unit of time that a molecule hits Rx at time instant t after being released at t = 0 can be described as an inverse 62 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017 Fig. 2. Block diagrams for exact and approximated EDTCM. Gaussian distribution [38]: r (d − r ) f hit (t) = √ e d 4π D t 3 2 − (d−r) 4D t . (2) Consequently, the probability that a molecule hits Rx until t is given by    t d −r r . (3) f hit (τ )dτ = erfc √ Fhit (t) = d 4D t 0 III. E QUIVALENT D ISCRETE -T IME C HANNEL M ODELS Numerical simulation of the random propagation of information molecules is of significant computational complexity, especially for a large number of molecules and for long channel memories. Those simulations lead to a trade-off between simulation step size and simulation precision. For those cases an EDTCM based on stochastic distributions can improve simulation time and precision. Furthermore, an EDTCM provides an easier access to the main properties of the channel and makes it easier to adapt classical wireless communication algorithms to MC channels. In the following the exact EDTCM is derived and three approximations thereof are proposed to further reduce complexity. Subsequently, the accuracy of the approximations are analyzed and the computational complexity of all channel models is discussed. Furthermore, the signal to interference plus noise ratio is defined and investigated. A. Exact EDTCM The basis for the EDTCM is given by (3) and the assumption of a perfectly absorbing and summing receiver. As a result, the probability for a molecule to reach Rx during the lth bit duration [lTb , (l + 1)Tb ] after release can be calculated by h l = Fhit ((l + 1)Tb ) − Fhit (lTb ), (4) where h l can be interpreted as the lth channel coefficient of the EDTCM. The scenario that a single molecule arrives the Rx during a certain bit duration can be represented by a Bernoulli experiment with two possible events of either hitting Rx or not. For N molecules released at the same time and under the assumption that molecules propagate independently and not change the hitting probability at Rx, the N Bernoulli experiments can be described by the binomial distribution z l ∝ B (N, h l ) , (5) where z l is the number of received molecules during the lth bit duration after their release. As depicted in Fig. 2(a), the number y[k] of received molecules at time instant k when transmitting a bit sequence is a summation over all z l [22]: y[k] = L  B (x[k − l], h l ) , (6) l=0 where L is the effective channel memory length and x[k] is the discrete representation of the OOK modulated symbol:  N if u[k] = 1, x[k] = (7) 0 otherwise. The summation over binomial distributions is exactly described by a Poisson binomial distribution. Notice that an infinite lifetime of molecules leads to L → ∞. However, in simulations L cannot exceed the length of the bit sequence when the receiving process is truncated with the last transmitted bit. Furthermore, the main energy is usually located in the first channel coefficients, thus L can be further reduced while still achieving meaningful results [23]. B. Binomial Approximation For a large memory length L it is time-consuming to create a single binomial random variable for each branch in Fig. 2(a). DAMRATH et al.: EQUIVALENT DISCRETE-TIME CHANNEL MODELING FOR MOLECULAR COMMUNICATION 63 However, there is no simple solution to describe the sum of binomial distributions with different probabilities exactly. Therefore, we apply different approximations to further reduce the complexity. The effect of these approximations is analyzed in Section III-E, while the effect on the error performance is analyzed in Section V. The sum of different binomial distributions can be approximated by a new binomial distribution. If h l is sufficient small y[k] can be approximated as   L  (8) y[k] ≈ B N, u[k − l]h l . l=0 The corresponding EDTCM is shown in Fig. 2(b). This kind of approximation is rather unusual, but as it will be shown in Section III-E, it is superior to other approximations under certain conditions. C. Gaussian Approximation Especially if the number of draws is high and the success probability is not close to zero or one, the binomial distribution can be well approximated by the Gaussian distribution [23]. The relation between the two distributions can be approximated as B (N, h l ) ≈ N μl = Nh l , σl2 = Nh l (1 − h l ) , (9) with mean μl and variance σl2 . Since the sum of independent Gaussian distributions can be described by a single Gaussian distribution, y[k] can be approximated as   L  2 y[k] ≈ N z[k], σ [k] = x[k − l]h l (1 − h l ) = z[k] + n[k], where z[k] = L l=0 (10) x[k − l]h l describes the expected number l=0 of molecules at Rx, and n[k] ∝ N (0, σ 2 [k]) represents an additive amplitude-dependent noise term, which models the uncertainty of the number of hitting molecules at Rx, which is known as Brownian noise [39]. This channel model visualized in Fig. 2(c) is similar to the additive white Gaussian noise (AWGN) model with memory, which is well-known in the wireless communications community [40]. The only difference lies in the amplitude-dependent white Gaussian noise. Please note that the EDTCM might be extended by a second noise process that models the background noise caused by molecules present in the propagation medium before transmission, which is related to the AWGN process frequently assumed in wireless communications. The big advantage of EDTCM with Gaussian approximation is the simplicity in implementation, because z[k] and σ 2 [k] can be efficiently calculated by two convolutions (see Fig. 3) and there is only one access to a Gaussian random generator per time index k (see Fig 2(c)). While the number of received molecules is an unsigned integer value, the Gaussian distribution is defined for real numbers. Therefore, y[k] needs to be postprocessed: y  [k] = max {0, y[k]} . (11) Fig. 3. Variance calculation for the EDTCM with Gaussian approximation. This postprocessing step causes an increasing approximation error if the mean of y[k] is close to zero. D. Poisson Approximation Especially for unlikely events the binomial distribution can be well approximated by the Poisson distribution. The relation between both distributions can be approximated as B (N, h l ) ≈ P (λl = Nh l ) , (12) where P(λ) describes a Poisson distribution with parameter λ which is equal to the mean and variance of the distribution [15]. Similar to the Gaussian distribution, the sum of several independent Poisson distributions can be described by a single Poisson distribution, thus y[k] can be approximated as   L  x[k − l]h l . (13) y[k] ≈ P λ[k] = z[k] = l=0 The resulting EDTCM is shown in Fig. 2(d). E. Accuracy of Approximations The channel models introduced in Sections III-B, III-C, and III-D approximate the sum of binomial random variables defined for the exact channel model introduced in Section III-A. The accuracy of approximation can be analyzed by investigating the mean Kullback-Leibler divergence (MKLD) between the exact Poisson binomial probability mass function PR and the proposed approximations thereof over all possible input sequences s:  1 D̄(R||B/N /P) = L+1 2 s∈{0,1} L+1 ⎡ ⎤   wH (s)N  (n) P R ⎦, ×⎣ PR (n) log2 PB/N /P (n) n=0 (14) where wH (·) is the Hamming weight. In Fig. 4(a)–4(c) the MKLD for the system parameters given in Table I is shown. For the binomial approximation, the MKLD is decreasing with an increasing distance, which is similar to a 64 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017 Fig. 4. Mean Kullback-Leibler divergence between Poisson binomial distribution and approximations (Tb = 0.5 s, L = 5). TABLE I S IMULATION PARAMETERS U SED FOR A NALYSIS ; THE D EFAULT PARAMETERS A RE IN B OLDFACE of the binomial Poisson distribution except for the Poisson distribution for small distances and the binomial distribution for a small number of molecules. A practical comparison by means of bit error simulations will confirm this result in Section V. F. Complexity Analysis decreasing hitting probability. Furthermore, it is decreasing with an increasing number of molecules, similar to the case of Gaussian approximation. An increasing number of molecules reduces the uncertainty of the channel output which is represented by a decreasing variance. For the Poisson approximation, the MKLD is decreasing when the distance is increased. This confirms the statement that the Poisson approximation fits well for unlikely events. In Fig. 4(e)–4(f) the differences between the MKLD of the approximations is shown. Therefore, it is possible to define specific regions where an approximation is superior with respect to another approximation. Often the difference between the MKLD is small in the scenario under investigation. In case of a small number of molecules it can be stated that the binomial approximation lead to a higher approximation error. Furthermore, the Poisson approximation should be avoided for small distances. Overall, all three approximations lead to a fair approximation The main parameters (number of molecules, channel coefficients, mean, variance) of the channel models introduced in Sections III-A, III-B, III-C, and III-D can be calculated efficiently by convolutions or can be stored in look-up tables. Therefore, the main computational complexity is due to the random number generators (RNGs). There exist efficient algorithms for Gaussian RNG, Poisson RNG [41] and binomial RNG [42] with constant complexity O(1) independent of N and h l . Thus, the complexity of all approximation models is O(1), while the complexity of the exact channel model is O(L). Therefore, the suggested approximations reduce the complexity by a factor of L. As a result, the complexity reduction of the proposed approximations improves with the channel memory length L. G. Signal to Interference Plus Noise Ratio Due to the fact that there is amplitude-dependent noise present in the MC system, it is useful to define an averaged signal to interference plus noise ratio (SINR). Because the mean and variance of the sum of binomial distributions are equal to the mean and variance of the Gaussian approximation, DAMRATH et al.: EQUIVALENT DISCRETE-TIME CHANNEL MODELING FOR MOLECULAR COMMUNICATION the averaged SINR is defined as μ2 SINR = 2 μISI + σ 2N =  N h0 l=1 l=0 h 20 2 L hl + N (h l − h l2 ) 1 p(u=1)N l=1 L l=0 , (15) (h l − h l2 ) where μ is the averaged number of molecules that reach Rx in their first bit interval after release, μISI is the averaged number of molecules that reach Rx after their first bit interval after release and causes ISI, σ 2N is the averaged diffusion noise variance, and N = p(u = 1)N is the averaged number of emitted molecules. The definition of SINR visualizes the fact that the diffusion noise can be reduced by increasing the number of molecules N, while the ISI is only dependent on the channel coefficients. For the bit error analysis throughout this paper four different detection algorithms are applied and compared. First of all, the common fixed threshold detector (FTD) is used, where the fixed threshold is chosen to be optimal in terms of minimizing the BER by means of an exhaustive search. Second, the low-complexity adaptive threshold detector (ATD) introduced in [32] is applied, where the threshold is determined by the previous number of received molecules. Third, the maximumlikelihood sequence estimator (MLSE) based on the squared Euclidean distance metric is applied [43]. As a fourth detector we adapt the a posteriori probability (APP) detector, which can be implemented by means of the BCJR algorithm [44]. While the latter two algorithms are too complex to be implemented inside a NM (at least in the near future), they serve as a benchmark in the simulations. The MLSE and the BCJR algorithm work on a trellis diagram with M L states [43], [44], where M is the symbol cardinality (in case of OOK M = 2). For adapting the BCJR algorithm to the EDTCM it is necessary to define the transition probability γ (si , s j , k) from state s i = [si,1 , · · · , si,L ] to state s j at time index k. For the EDTCM proposed in Section III-A, the probability mass function (PMF) of the sum over binomial distributions can be described as a Poisson binomial distribution, which can be calculated with help of the discrete Fourier transformation as ξs ,s N i j  1 C −my[k] γ si , s j , k = 1 + ξsi ,s j N   N  1 + C m − 1 sl h l . (18) According to the channel models introduced in Sections III-B, III-C, and III-D, the PMF can be approximated by binomial, Gaussian, and Poisson distributions, respectively. Therefore, we can approximate γ (si , s j , k) as follows:     N−y[k]  N y[k]  γB s i , s j , k = psi ,s j 1 − psi ,s j , (19) y[k]   2    y[k] − μsi ,s j 1 exp − , γN s i , s j , k =  2σs2i ,s j 2πσ 2 s i ,s j (20)   γP s i , s j , k = y[k] λsi ,s j y[k]!   exp −λsi ,s j , (21) where psi ,s j = ũh 0 + L  sl h l , (22) l=1 IV. D ETECTORS  L   l=1 2 L + N hl = N    (m) = 1 + C m − 1 ũh 0 · 2 L s i ,s j 65 s i ,s j (m) , (16) m=0 where ξsi ,s j = wH (si ) + wH (ũ) is the Hamming weight of the state si plus the Hamming weight of the bit hypothesis ũ from state s i to state s j , and   2π , (17) C = exp j 1 + ξsi ,s j N μsi ,s j = λsi ,s j = N psi ,s j ,  σs2i ,s j = μsi ,s j − N ũh 20 − L   sl h l2 . (23) (24) l=1 Note that it is straightforward to adapt the MLSE metric to the transition probabilities γ (si , s j , k). However, in this work the adaptation is applied only to the BCJR algorithm, which is optimal in the sense that it minimizes the bit error probability. V. S IMULATION R ESULTS For simulative analysis throughout this paper the scenario under investigation is inspired by [32] and summarized in Table I, where K is the bit sequence length for one channel realization and R is the total number of channel realizations. In contrast to [32] the denaturation of the molecules is fixed to the channel memory length L = 5. In order to verify the introduced EDTCM and approximations thereof, the results of [32] are reproduced by applying the detectors described in Section IV to the proposed channel models. During the simulations the impact of transmission distance d, number of molecules N, and bit duration Tb on the BER is investigated. This is done by varying one system parameter, while fixing the other ones to N = 1000, d = 30 µm, and Tb = 0.5 s. The simulation results based on the exact EDTCM can be seen in Fig. 5 and show similar findings as in [32]. All detection algorithms perform better if N is increased, d is decreased, and/or Tb is increased, which is equivalent to increase the signal to interference plus noise ratio in classical wireless communications. While an increasing number of molecules additionally attenuate the stochastic behavior of the diffusion process, a decreasing transmission distance and a longer bit duration weaken the effect of ISI. The only detection algorithm that is not improving the performance with an increase of Tb in Fig. 5(c) is ATD. In fact, the ATD algorithm benefits from ISI 66 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017 Fig. 5. Bit error rate performance as a function of the number of molecules (a), transmitting distance (b), and bit duration (c). If the corresponding parameter is not varying, it is fixed to N = 1000, d = 30 µm, and Tb = 0.5 s. Fig. 6. Normalized mean squared error between the bit error rate of the exact channel model (see Fig. 5) and approximations thereof (fixed threshold detection and adaptive threshold detection). and struggles if ISI is decreased. Furthermore, the results show that MLSE and BCJR outperforms simple detection algorithms like FTD and ATD, while ATD is superior to FTD for a certain range of Tb and d (for a more detailed analysis the reader is referred to [32]). The BCJR algorithm introduced in Section IV achieves a slightly better performance than the MLSE algorithm by adapting the transition probability to the channel statistics. However, the obtained BER in Fig. 5 are smaller and especially the MLSE detector performance is remarkably improved compared to [32]. This is due to two main reasons. Firstly, the simulation results using the exact channel model are equal to a random walk simulation with a sufficient small simulation time step, which provides more accurate results. In fact, the simulation time step has to be as small as possible, but much larger than τp = m/(6πηrs ) to model the propagation as Brownian motion [45]. Thereby m and rs are the mass and the Stokes’ radius of the propagating molecule, and η is the viscosity of the fluid medium. Secondly, the channel memory length in this work is set to L = 5, while there is an infinite channel memory length assumed in [32]. Nevertheless, Fig. 5 confirms the qualitative statements obtained in [32]. In order to analyze the impact of the approximations compared to the exact channel model, Figs. 6 and 7 show the normalized mean squared error (MSE) between the BER obtained by the exact channel model and the BER obtained by the approximated channel models, where the squared error is normalized to the BER of the exact channel model. The MSE is shown for the cases of FTD and ATD (Fig. 6) and for the case of BCJR detection (Fig. 7). Overall, for the system under consideration in Fig. 6, the approximations do not deviate remarkably from the exact channel model (for almost every point MSEBER < 10−3 ). In Fig. 6(a) the effect of the number of molecules on the MSE is shown. For N ≥ 5000, the MSE of the binomial and Poisson approximation is nearly constant, while the MSE of the Gaussian approximation is still DAMRATH et al.: EQUIVALENT DISCRETE-TIME CHANNEL MODELING FOR MOLECULAR COMMUNICATION 67 Fig. 7. Normalized mean squared error between the bit error rate for BCJR with Poisson binomial distributed transition probabilities and BCJR algorithm with approximated transition probabilities, respectively. Each BCJR algorithm is applied to the exact channel model and to the related approximated channel model. decreasing for the FTD. This result supports the fact that the binomial distribution converges to the Gaussian distribution for n → ∞. From the trend in Fig. 6(b) and Fig. 6(c) it can be concluded that the approximation accuracy gets better if the transmission distance is increased and/or the bit duration is decreased. An increasing transmission distance and/or decreasing bit duration leads to smaller channel coefficients. This is especially beneficial for the Poisson distribution, which is a good approximation for unlikely events. As a result, the performance of the Poisson approximated channel gets worse if the channel coefficients are increased. This can be observed in Fig. 6(b) and 6(c) at d = 20 µm and Tb = 1.5 s. In Fig. 7 the normalized MSE of the BCJR algorithm applied to the channel approximations and to the exact channel model is shown. Thereby, the transition probabilities for the BCJR algorithm are adapted to the various approximated channel models described in Section IV. Each of those approximated BCJR algorithms is applied to the exact channel model and to the approximated version of the channel model related to the applied transition probabilities. Furthermore, the abscissae are limited to the area of Fig. 5, where the BCJR algorithm produces reasonable BER for the simulation parameters under investigation. In general, the MSE follows the same qualitative behavior of ATD in Fig. 6, except for the performance of ATD applied to the Gaussian approximated channel model. If N or Tb is increased, the MSE increases as well, while an increasing d leads to a decreasing error. From Fig. 7 it can be observed that applying the binomial BCJR and the Poisson BCJR to the exact channel model give a better MSE performance than applying them to the related approximated channel models. In contrast, the Gaussian BCJR behaves differently. For N ≥ 1000, d ≤ 40 µm and over the whole range of simulated bit durations, applying the Gaussian BCJR to the Gaussian approximated channel model gives a lower MSE than applying the Gaussian BCJR to the exact channel model. As a result, Figs. 6 and 7 give an insight, which approximated channel model and which approximated BCJR detector can be applied. Thereby, the choice is affected by the acceptable MSE and the given simulation parameters. As an example, (i) assuming MSEBER ≤ 10−3 , (ii) assuming simulation parameters according to Table I with N = 1000, d = 30 µm and Tb = 0.5 s, and (iii) assuming that the BCJR algorithm should not run with the Poisson binomial distributed transition probability (in order to save computational complexity at the detection step). In that case the Gaussian channel model together with the Gaussian BCJR should be applied. VI. C ONCLUSION In this paper an equivalent discrete-time channel model for three-dimensional diffusion-based molecular communication using the first passage time concept is introduced. Due to the fact that the EDTCM is well-known in the wireless communications community, the proposed model facilitates the understanding of diffusion-based molecular communication and supports the adaptation of classical wireless communication algorithms to the area of MC. Furthermore, it implies a great potential for reducing simulation complexity compared to random walk based MC simulators. While the exact EDTCM is based on the summation of binomial distributions, three channel models are proposed which approximate the summation by binomial, Gaussian, and Poisson distributions, respectively, in order to reduce computational complexity. The approximation accuracy is analyzed by means of the KullbackLeibler divergence. Favored regions for each approximation are highlighted. In addition, the transition probability for the BCJR algorithm is adapted to each proposed channel model. In numerical BER simulations the exact EDTCM in conjunction with FTD, ATD, MLSE, and BCJR is studied. In addition, the improvement of the adapted BCJR algorithm in comparison to MLSE is shown. Furthermore, the difference between the exact EDTCM and the approximations thereof in conjunction with 68 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 16, NO. 1, JANUARY 2017 the proposed detection methods are analyzed. The results show an acceptable small error over a wide range of simulation parameters for the scenario under consideration. While OOK and energy detection is assumed throughout this work, the channel models can be extended easily to other modulation schemes like molecule shift keying or concentration shift keying, and different detection strategies like amplitude detection. Future work includes expanding the EDTCM to additional assumptions, e. g. including the process of ligand receptor binding, or to consider transmitters with non-zero spatial size. R EFERENCES [1] I. F. Akyildiz, F. Brunetti, and C. Blàzquez, “Nanonetworks: A new communication paradigm,” Comput. Netw. J., vol. 52, no. 12, pp. 2260–2279, 2008. [2] Y. Xia et al., “One-dimensional nanostructures: Synthesis, characterization, and applications,” Adv. Mater., vol. 15, no. 5, pp. 353–389, 2003. [3] T. Nakano, A. W. Eckford, and T. Haraguchi, Molecular Communication. Cambridge, U.K.: Cambridge Univ. Press, 2013. [4] T. Nakano, M. Moore, A. Enomoto, and T. Suda, Molecular Communication Technology as a Biological ICT. New York, NY, USA: Springer, 2011. [5] T. Suda, M. Moore, T. Nakano, R. Egashira, and A. Enomoto, “Exploratory research on molecular communication between nanomachines,” in Proc. Conf. Genetic Evol. Comput. Conf. (GECCO), vol. 25. 2005, p. 29. [6] B. Atakan, O. Akan, and S. Balasubramaniam, “Body area nanonetworks with molecular communications in nanomedicine,” IEEE Commun. Mag., vol. 50, no. 1, pp. 28–34, Jan. 2012. [7] S. Hiyama et al., “Molecular communication,” in Proc. NSTI Nanotechnol. Conf., 2005. [8] T. Nakano, T. Suda, M. Moore, R. Egashira, A. Enomoto, and K. Arima, “Molecular communication for nanomachines using intercellular calcium signaling,” in Proc. IEEE Conf. Nanotechnol., Jul. 2005, pp. 478–481. [9] T. Nakano, M. J. Moore, F. Wei, A. V. Vasilakos, and J. Shuai, “Molecular communication and networking: Opportunities and challenges,” IEEE Trans. NanoBiosci., vol. 11, no. 2, pp. 135–148, Jun. 2012. [10] T. Nakano, T. Suda, Y. Okaie, M. J. Moore, and A. V. Vasilakos, “Molecular communication among biological nanomachines: A layered architecture and research issues,” IEEE Trans. NanoBiosci., vol. 13, no. 3, pp. 169–197, Sep. 2014. [11] E. Carafoli, “Calcium signaling: A tale for all seasons,” Proc. Nat. Acad. Sci. USA, vol. 99, no. 3, pp. 1115–1122, 2002. [12] B. L. Bassler, “How bacteria talk to each other: Regulation of gene expression by quorum sensing,” Current Opinion Microbiol., vol. 2, no. 6, pp. 582–587, 1999. [13] M. B. Miller and B. L. Bassler, “Quorum sensing in bacteria,” Annu. Rev. Microbiol., vol. 55, no. 1, pp. 165–199, 2001. [14] M. Pierobon and I. F. Akyildiz, “A physical end-to-end model for molecular communication in nanonetworks,” IEEE J. Sel. Areas Commun., vol. 28, no. 4, pp. 602–611, May 2010. [15] M. Pierobon and I. F. Akyildiz, “Diffusion-based noise analysis for molecular communication in nanonetworks,” IEEE Trans. Signal Process., vol. 59, no. 6, pp. 2532–2547, Jun. 2011. [16] M. Pierobon and I. F. Akyildiz, “Noise analysis in ligand-binding reception for molecular communication in nanonetworks,” IEEE Trans. Signal Process., vol. 59, no. 9, pp. 4168–4182, Sep. 2011. [17] M. Pierobon and I. F. Akyildiz, “Capacity of a diffusion-based molecular communication system with channel memory and molecular noise,” IEEE Trans. Inf. Theory, vol. 59, no. 2, pp. 942–954, Feb. 2013. [18] M. U. Mahfuz, D. Makrakis, and H. T. Mouftah, “Characterization of molecular communication channel for nanoscale networks,” in Proc. BIOSIGNALS, Valencia, Spain, 2010, pp. 327–332. [19] N. Farsad, W. Guo, and A. W. Eckford, “Tabletop molecular communication: Text messages through chemical signals,” PLoS ONE, vol. 8, no. 12, p. e82935, 2013. [20] N. R. Kim, N. Farsad, C. B. Chae, and A. W. Eckford, “A universal channel model for molecular communication systems with metaloxide detectors,” in Proc. IEEE Int. Conf. Commun. (ICC), Jun. 2015, pp. 1054–1059. [21] T. Nakano, Y. Okaie, and A. V. Vasilakos, “Transmission rate control for molecular communication among biological nanomachines,” IEEE J. Sel. Areas Commun., vol. 31, no. 12, pp. 835–846, Dec. 2013. [22] M. J. Moore, T. Suda, and K. Oiwa, “Molecular communication: Modeling noise effects on information rate,” IEEE Trans. NanoBiosci., vol. 8, no. 2, pp. 169–180, Jun. 2009. [23] M. C. S. Kuran, H. B. Yilmaz, T. Tugcu, and B. Özerman, “Energy model for communication via diffusion in nanonetworks,” Nano Commun. Netw., vol. 1, no. 2, pp. 86–95, 2010. [24] A. Akkaya, G. Genc, and T. Tugcu, “HLA based architecture for molecular communication simulation,” Simul. Modell. Pract. Theory, vol. 42, pp. 163–177, Jan. 2014. [25] A. Toth, D. Banky, and V. Grolmusz, “3-D Brownian motion simulator for high-sensitivity nanobiotechnological applications,” IEEE Trans. NanoBiosci., vol. 10, no. 4, pp. 248–249, Dec. 2011. [26] E. Gul, B. Atakan, and O. B. Akan, “NanoNS: A nanoscale network simulator framework for molecular communications,” Nano Commun. Netw., vol. 1, no. 2, pp. 138–156, 2010. [27] I. Llatser, D. Demiray, A. Cabellos-Aparicio, D. T. Altilar, and E. Alarcón, “N3sim: Simulation framework for diffusion-based molecular communication nanonetworks,” Simul. Model. Pract. Theory, vol. 42, pp. 210–222, Mar. 2014. [28] L. Felicetti, M. Femminella, and G. Reali, “A simulation tool for nanoscale biological networks,” Nano Commun. Netw., vol. 3, no. 1, pp. 2–18, 2012. [29] L. Felicetti, M. Femminella, and G. Reali, “Simulation of molecular signaling in blood vessels: Software design and application to atherogenesis,” Nano Commun. Netw., vol. 4, no. 3, pp. 98–119, 2013. [30] H. B. Yilmaz and C.-B. Chae, “Simulation study of molecular communication systems with an absorbing receiver: Modulation and ISI mitigation techniques,” Simul. Model. Pract. Theory, vol. 49, pp. 136–150, Dec. 2014. [31] H. B. Yilmaz and C.-B. Chae, “Arrival modelling for molecular communication via diffusion,” Electron. Lett., vol. 50, no. 23, pp. 1667–1669, 2014. [32] M. Damrath and P. A. Hoeher, “Low-complexity adaptive threshold detection for molecular communication,” IEEE Trans. NanoBiosci., vol. 15, no. 3, pp. 200–208, Apr. 2016. [33] M. J. Moore and T. Nakano, “Oscillation and synchronization of molecular machines by the diffusion of inhibitory molecules,” IEEE Trans. Nanotechnol., vol. 12, no. 4, pp. 601–608, Jul. 2013. [34] L. Felicetti, M. Femminella, G. Reali, T. Nakano, and A. V. Vasilakos, “TCP-like molecular communications,” IEEE J. Sel. Areas Commun., vol. 32, no. 12, pp. 2354–2367, Dec. 2014. [35] M. Mahfuz, D. Makrakis, and H. Mouftah, “Spatiotemporal distribution and modulation schemes for concentration-encoded medium-tolong range molecular communication,” in Proc. 25th Biennial Symp. Commun. (QBSC), May 2010, pp. 100–105. [36] M. U. Mahfuz, D. Makrakis, and H. T. Mouftah, “On the characterization of binary concentration-encoded molecular communication in nanonetworks,” Nano Commun. Netw., vol. 1, no. 4, pp. 289–300, 2010. [37] I. Llatser, A. Cabellos-Aparicio, M. Pierobon, and E. Alarcon, “Detection techniques for diffusion-based molecular communication,” IEEE J. Sel. Areas Commun., vol. 31, no. 12, pp. 726–734, Dec. 2013. [38] H. B. Yilmaz, A. C. Heren, T. Tugcu, and C.-B. Chae, “Threedimensional channel characteristics for molecular communications with an absorbing receiver,” IEEE Commun. Lett., vol. 18, no. 6, pp. 929–932, Jun. 2014. [39] H. ShahMohammadian, G. G. Messier, and S. Magierowski, “Optimum receiver for molecule shift keying modulation in diffusion-based molecular communication channels,” Nano Commun. Netw., vol. 3, no. 3, pp. 183–195, 2012. [40] P. Hoeher, “A statistical discrete-time model for the WSSUS multipath channel,” IEEE Trans. Veh. Technol., vol. 41, no. 4, pp. 461–468, Nov. 1992. [41] J. H. Ahrens and U. Dieter, “Computer generation of Poisson deviates from modified normal distributions,” ACM Trans. Math. Softw., vol. 8, no. 2, pp. 163–179, 1982. [42] V. Kachitvichyanukul and B. W. Schmeiser, “Binomial random variate generation,” Commun. ACM, vol. 31, no. 2, pp. 216–222, 1988. [43] G. D. Forney, Jr., “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inf. Theory, vol. IT-18, no. 3, pp. 363–378, May 1972. [44] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, no. 2, pp. 284–287, Mar. 1974. [45] T. Li, S. Kheifets, D. Medellin, and M. G. Raizen, “Measurement of the instantaneous velocity of a Brownian particle,” Science, vol. 328, no. 5986, pp. 1673–1675, 2010.
Purchase answer to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Explanation & Answer

Attached.

Running head: EQUIVALENT DISCRETE-TIME CHANNEL MODELS

Equivalent Discrete-Time Channel Models
Institution Affiliation
Date

1

EQUIVALENT DISCRETE-TIME CHANNEL MODELS

2

Equivalent Discrete-Time Channel Models (EDTCM) has been widely recognized in the
wireless communication world (Damrath, Korte, & Hoeher, 2017). Technology has enabled
molecular communication (MC) highly achievable. MC has proved to be energy efficient and
biocompatible. Various channels such as diffusion-based molecular communication (DBMC)
uses radio waves which spread using natural laws. However, the molecules diffuse and make the
channel response decrease thus causing inter-symbol interference. Due to this, new
communication algorithms have been invented using the DBMC channel which unfortunately is
exhaustive and time-consuming.
The first step in achieving a practical methodology has been the introduction of a
macroscopic MC which has an alcohol sensor and an electric sprayer. Communication channel
models rely on Binomial distribution, ...


Anonymous
Great! 10/10 would recommend using Studypool to help you study.

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Related Tags