A molecular communication system operating in a pipe propagation channel with no induced flow is considered. Experimentally, it is shown that discrepancies in channel impulse response can be accurately modelled by an additive noise model. The noise amplitude is Nakagami distributed, and the shape and spread parameters of the distribution increase monotonically with propagation distance. Furthermore, demonstrated how the proposed noise model can be used to calculate the bit-error-rate and the capacity of a binary symmetric channel

Guo, Weisi

Qiu, Song

Wang, Siyi

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http://wrap.warwick.ac.uk           Original citation: Qiu, Song, Wang, Siyi and Guo, Weisi. (2015) An experimental Nakagami distributed noise model for molecular communication channels with no drift. Electronics Letters, Volume 51 (Number 8). pp. 611-613.  Permanent WRAP url: http://wrap.warwick.ac.uk/68732                      Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.  Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners.  To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.  Copies of full items can be used for personal research or study, educational, or not-for profit purposes without prior permission or charge.  Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.  Publisher’s statement: "This paper is a postprint of a paper submitted to and accepted for publication in Electronics Letters and is subject to Institution of Engineering and Technology Copyright. The copy of record is available at IET Digital Library"  A note on versions: The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher’s version.  Please see the ‘permanent WRAP url’ above for details on accessing the published version and note that access may require a subscription.  For more information, please contact the WRAP Team at: publications@warwick.ac.uk  An Experimental Nakagami Distributed NoiseModel for Molecular CommunicationChannels with No DriftS. Qiu and S. Wang and W. GuoWe consider a molecular communication system operating in apipe propagation channel with no induced flow. Experimentally, weshow that discrepancies channel impulse response can be accuratelymodelled by an additive noise model. The noise amplitude is Nakagamidistributed, and the shape and spread parameters of the distributionincrease monotonically with propagation distance. The paper furtherdemonstrates how the proposed noise model can be used to calculatebit-error-rate and capacity of a binary symmetric channel.Background: Molecular communications utilizes chemical moleculesas an alternative carrier for information [1, 2]. Compared to existingwave-based propagation, molecular diffusion has the potential topropagate more efficiently and reliably in certain confined and micro-scale environments, or when there is overwhelming electromagneticinterference [3]. Indeed, this is perhaps why molecular communicationsis prevalent in nature, both at the inter-organism and at the inter-cellscales [4].For the past few years, the vast majority of molecular communicationsresearch has been modeling based, utilizing classical diffusion equations[2, 5, 6, 7]. In a real pulse modulated molecular communication system,noise can unfortunately arise from a host of sources and it is difficult totake all of them into account or argue which may dominate. Examplesof noise include, but are not limited to: inconsistent mechanical pulseemission, atmospheric contamination, physical disturbances, andreceiver design [8]. The vast majority of literature assume a Gaussiandistributed additive noise [9], which is not unreasonable given thatrecent research has shown this to be accurate in a high velocity turbulentflow 3-dimensional channel [10]. The noise model for a random walkdiffusion channel with no induced drift velocity is lacking. Such achannel is common in low pressure industrial applications (i.e., pipenetworks) and enclosed environments (i.e., caves and tunnels). Utilizinga recently built molecular communication test-bed [11], we characterizethe additive noise model in a pipe channel for molecular communications.Experimental Setup: The pulse-modulated transceiver design used inthis paper can be found in [11], and the overall experimental setup isfound in [3]. Briefly, the system converts an alpha-numeric messageinto a binary code, and uses on-off-keying (OOK) line coding. Themodulation scheme employed is chemical concentration (amplitude)modulation (BASK), and the chemical used is ethanol. At the transmitter,an electronically controlled mechanical spray acts as a transmitteremitting a number of molecules (M). Each input is defined as pulse witha finite small duration. The chemical concentrations are sprayed intothe channel and received at a chemical sensor positioned at a distanced away. The chemical sensor converts concentration measurements intoelectrical signals. With the appropriate signal processing, the output canbe decoded back into the original message. An approximately constanthumidity and temperature is maintained. The propagation channel iscomposed of a straight iron pipe of up to 245cm long, with two sealediron chambers at each end containing the transmitter and receiversrespectively. The pipe diameter is 4cm with a thickness of 2mm. Thechemicals are initially emitted in a way that ensures that the overall driftvelocity in the channel is zero.Noise Model: Theoretically, the molecular concentration at any pointin space and time can be found by using hitting probability densityequation in 1-dimensional space and the molecules are constrained to betransmitted only in one direction [2]:φhit(x, t) =1√piDtexp[− x24Dt], (1)where D is the diffusivity, t is the transmission time, x is the distancebetween transmitter and receiver. The received pulse in reality will not0 0.5 1 1.5 2100  Error at d=60cmNakagami Fit0 1 2 3 4100  Nakagami FitError at d=127cm0 2 4 6100  Error at d=245cmNakagami FitN (mg/L) Fig. 1. Noise (error) PDF with fitted Nakagami distributions (Log Scale)behave precisely in accordance with the expression due to a number ofnoise sources in the system.Therefore, the realistic received signal pulse shape will be:ϑ(x, t) = φhit(x, t) + I +N, (2)where I is the inter-symbol-interference (ISI) from previous andsuccessive pulses (which we do not consider here); and N is the additivenoise, whose distribution is our chief concern.Additive noise is defined as the difference between the measuredreceived signal and the average received signal of the system. Given areliable number of experiments for each of the different channel lengths,we apply the Levenberg-Marquardt algorithm to curve fit a probabilitydistribution. It was found that the modified Nakagami distribution fittedthe additive noise data. The Nakagami distribution has two parameters: ashape parameter µ, and a spread parameter ω and the probability densityfunction (PDF) of the noise can be expressed as:fN (z;µ, ω) =1|z|Γ(µ)(µωz2)µexp(−µωz2). (3)The domain of the traditional Nakagami distribution is [0,+∞) whilein this case, the noise can be negative and the domain is extended to(−∞,+∞) to reflect this. By doing so, the pdf had to scale down by2 from general Nakagami distribution expression otherwise the integralof the unmodified pdf across the whole domain will exceed 1. This PDFcaptures the fact that the additive noise can be negative mathematically asopposed to the standard Nakagami distribution where the random variableis strictly positive.Table 1: Nakagami Distribution Fitting ParametersChannel Length, d Shape, µ Spread, ω Correlation, R260cm 0.122 0.312 0.59127cm 0.291 2.510 0.76245cm or greater 0.310 4.058 0.69The results in Fig. 1 show three examples of the noise (error)measurements given as symbols and the theoretical Nakagamidistribution given as the line. In order to examine the match clearly, wedisplay the noise on a log-scale. Over 10000 empirical data points wereused to generate each noise distribution. In terms of accuracy, the PDFregression fit has a correlation of R2 = 0.59–0.76. An example of theNakagami Distribution parameters are displayed in Table 1 for variouslengths. It can be seen that with increased pipe length, the shape andthe spread parameters of the Nakagami distribution increase. The resultssaturate at lengths significantly longer than 2.5m, and the parameters at2.5m can be used as a stable value for longer distance propagation.ELECTRONICS LETTERS 22nd January 2015 Vol. 00 No. 000 2 4 6 8 10 1200.20.40.60.81max  Capacity, bits/s/chemical  d=60cm Nakagami Noise Modeld=127cm Nakagami Noise Modeld=245cm Nakagami Noise ModelGaussian Noise Model1 2 3 4 5 6 7 8 9 10 11 1210-610-410-2100max  BER  d=60cm Nakagami Noise Modeld=127cm Nakagami Noise Modeld=245cm Nakagami Noise ModelGaussian Noise ModelFig. 2 Comparison of BER and Capacity performance for different noisemodels at different received peak concentration levels φmax..Performance: We primarily consider only the additive noise in thesystem. If the receiver samples at the peak of the captured moleculeconcentration φmax., the captured concentration including noise isϑ′ = φmax. +N . The distribution of the additive noise can follow theproposed Nakagami distribution (3) or a normal distribution N (0, 1) [9].The transmission system is a OOK modulation scheme with ρ probabilityof transmitting a 1. At the optimal sampling point, the minimum errorprobability (MEP) criterion of a standard Bayesian detection frameworkis given by the following with decision threshold η [9]:ϑ′ ≷10σ2ϑ′µϑ′1 − µϑ′0log(1− ρρ)+12(µϑ′1 + µϑ′0 )≡ η, (4)where µϑ′0 =E[ϑ′|χ= 0] = µN , µϑ′1 =E[ϑ′|χ= 1] = φmax. + µN ,and σ2ϑ′ = Var[ϑ′|χ= 0] = Var[ϑ′|χ= 1] = σ2N .The average error probability is:Pe = ρF(µϑ′1 − ησϑ′)+ (1− ρ)F(η − µϑ′0σϑ′), (5)where the F-function represents the complementary cumulativedistribution function (CCDF) of the noise distribution. For line-codingwith an equal probability of transmitting a 1 and 0 (ρ= 0.5), the BER isreduced to: Pe =F(φmax.2σN).For a Gaussian-distributed noise model, the F-function is wellestablished to be the Q-function. For the modified Nakagami-distributednoise model, the F-function is:F(x) = 1−Γ(µ, µωz2)2Γ(µ)z < 012− γ(µ,µωz2)2Γ(µ)z > 0,(6)where Γ(s, x) and γ(s, x) are the upper and lower incompletegamma function, respectively. The associated variance is σ2N = ω(1−1µ[Γ(µ+0.5)Γ(µ)]2).Note that the probability of error given a 1 is transmitted is equal tothe probability of error given a 0 is transmitted. Therefore, with ρ= 0.5,the system is a binary symmetric channel. The achievable throughput ofthe binary symmetric system is:C = 1−H(Pe) = 1 + Pe log2(Pe) + (1− Pe) log2(1− Pe). (7)The results in Figure 2 show that the specific parameters of the noisemodel can have a dramatic effect on the BER and capacity performance.This is a testament to the importance of employing accurate noisedistributions as well as relevant parameters for calculating BER andcapacity performances.Conclusions: This paper has developed a simple and effective Nakagamidistributed noise model for molecular communications in a zero velocitypipe channel. The shape and spread parameters of the model increasemonotonically with pipe length until the parameters converge at lengthsgreater than a certain value. The proposed noise model can be used as abasis for further molecular communications research and we demonstrateits application to bit-error-rate and capacity performance analysis.Song Qiu and Weisi Guo (School of Engineering, University of Warwick,United Kingdom)Siyi Wang (Department of Electrical and Electronic Engineering, Xi´ anJiaotong-Liverpool University, China)Corresponding Author E-mail: weisi.guo@warwick.ac.ukReferences1 T. Nakano, A. Eckford, and T. Haaguchi, Molecular Communication.Cambridge University Press, 2013.2 W. Guo, S. Wang, A. Eckford, and J. Wu, “Reliable communicationenvelopes of molecular diffusion channels,” Electronics Letters, vol. 49,no. 19, pp. 1248–1249, Sep. 2013.3 S. Qiu, W. Guo, S. Wang, N. Farsad, and A. Eckford, “A molecularcommunication link for monitoring in confined environments,” in IEEEInternational Conference on Communications (ICC), 2014, pp. 718–723.4 T. D. Wyatt, “Fifty years of pheromones,” Nature, vol. 457, no. 7227,pp. 262–263, Jan. 2009.5 C. Bai, M. Leeson, and M. Higgins, “Minimum energy channel codesfor molecular communications,” Electronics Letters, vol. 50, no. 23, pp.1669–1671, Nov. 2014.6 K. Srinivas, A. Eckford, and R. Adve, “Molecular Communication inFluid Media: The Additive Inverse Gaussian Noise Channel,” IEEETrans. on Information Theory, vol. 8, no. 7, pp. 4678–4692, Jul. 2012.7 I. Llaster, A. Cabellos-Aparicio, M. Pierobon, and E. Alarcon,“Detection Techniques for Diffusion-based MolecularCommunication,” IEEE Journal on Selected Areas in Communications(JSAC), vol. 31, no. 12, pp. 726–734, Jan. 2014.8 M. Pierobon and I. F. Akyildiz, “Diffusion-based noise analysis formolecular communication in nanonetworks,” IEEE Transactions onSignal Processing, vol. 59, no. 6, pp. 2532–2547, 2011.9 L. Meng, P. Yeh, K. Chen, and I. Akyildiz, “MIMO communicationsbased on molecular diffusion,” in IEEE Global CommunicationsConference (GLOBECOM), Dec. 2012, pp. 5380–5385.10 N. Farsad, N. Kim, A. Eckford, and C. Chae, “Channel and NoiseModels for Nonlinear Molecular Communication Systems,” IEEEJournal on Selected Areas in Communications (JSAC), pp. 1–14, Nov.2014.11 N. Farsad, W. Guo, and A. Eckford, “Tabletop molecularcommunication: Text messages through chemical signals,” PLOSONE, vol. 8, Dec. 2013.2

An experimental Nakagami distributed noise model for molecular communication channels with no drift

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An experimental Nakagami distributed noise model for molecular communication channels with no drift

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