Spatial Coding Techniques for Molecular MIMO

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1 Spatial Coding Techniques fo Molecula MIMO Matin Damath, H. Bikan Yilmaz, Chan-Byoung Chae, and Pete Adam Hoehe Faculty of Engineeing, Kiel Univesity, Gemany, {md, School of Integated Technology, Yonsei Univesity, Koea, {bikan.yilmaz, axiv: v1 [cs.et 15 Jun 2017 Abstact This pape studies spatial divesity techniques applied to multiple-input multiple-output (MIMO) diffusion-based molecula communications (DBMC). Two types of spatial coding techniques, namely Alamouti-type coding and epetition MIMO coding ae suggested and analyzed. In addition, we conside eceive-side equal-gain combining, which is equivalent to maximum-atio combining in symmetical scenaios. Fo numeical analysis, the channel impulse esponses of a symmetical 2 2 MIMO-DBMC system ae acquied by a tained atificial neual netwok. It is demonstated that spatial divesity has the potential to impove the system pefomance and that epetition MIMO coding outpefoms Alamouti-type coding. I. INTRODUCTION Molecula communication (MC) is a biologically inspied communication paadigm, whee molecules ae the infomation caies [1. MC is claimed to be a key technology in ealizing autonomous nanomachines (NMs), which size anges fom seveal nanometes up to a few micometes [2. The capability of NMs can be enhanced by woking in a coopeative manne [3, [4. Theefoe, communication at small scales is at a cucial point. The main application is anticipated to be in the medical secto, whee NMs can be used fo applications like tageted dug delivey, tissue engineeing, o health monitoing [4. In diffusion-based molecula communication (DBMC) [5, messenge molecules popagate, accoding to the law of diffusion, fom a souce to the sink. While this popagation is enegy efficient, this communication channel is fundamentally diffeent fom the classical adio-based wieless communication channel. Fo DBMC, the channel impulse esponse is slowly deceasing, which causes intesymbol intefeence (ISI) and uneliable tansmission [6. In ode to impove this uneliable tansmission, spatial divesity can be exploited in multiple-input multiple-output (MIMO) scenaios with multiple antennas at the tansmitte and/o eceive side. MIMO is a familia topic in classical wieless communication. In molecula communication, howeve, it has aely been consideed. The authos in [7 wee the fist to study molecula communication in conjunction with MIMO. They poposed diffeent techniques fo tansmitte divesity, divesity combining at the eceive side, and spatial multiplexing. Thei focus, howeve, was on multi-use intefeence, thus neglecting ISI thoughout the wok. In [8, the authos modeled a MIMO channel taking into account ISI and intelink intefeence (ILI). They focused then on spatial multiplexing and poposed diffeent detection algoithms. The authos applied thei algoithms to a tabletop molecula MIMO testbed and demonstated an Tx 1 Tx Tx 2 d Rx 1 a Rx 2 Fig. 1. Model of the diffusion-based molecula 2 2 MIMO system [8. impovement in the data ate compaed to thei single-input single-output (SISO) case. In [9, the authos analyzed the influence of a second absobing eceive on bit eo atio () and capacity fo a boadcast MC system. In this wok, the focus is on a MIMO channel consideing ISI and ILI. The main contibution is the analysis of diffeent spatial divesity algoithms at the tansmitte side. Fo the tansmitte side, we popose Alamouti-type coding and epetition MIMO coding; fo the eceive side, we popose equalgain combining that is equivalent to maximum-atio combining in symmetical scenaios. The divesity gain compaed to a SISO scenaio is investigated by means of a simulation, whee the influence of the system paametes is shown. The MIMO channel impulse esponses ae acquied by a tained atificial neual netwok (ANN). II. SYSTEM MODEL A. Topology and Popagation Model The system model unde investigation is simila to the system model intoduced in [8 and [10. As shown in Fig. 1, it consists of a tansmitte Tx and a eceive Rx in an infinite thee-dimensional homogeneous fluid medium without dift. In the sense of a 2 2 MIMO system, the Tx includes two point antennas Tx 1 and Tx 2, while the Rx includes two spheical eceive antennas Rx 1 and Rx 2 with adius that ae attached to the eflecting body of the Rx. Thoughout this wok, a symmetical scenaio is assumed whee Tx 1 is aligned to Rx 1 and Tx 2 is aligned to Rx 2. As a esult, the distance between Tx 1 and Rx 1, as well as between Tx 2 and Rx 2, is given as d. Futhemoe, the sepaation distance between Tx 1 and Tx 2, as well as between Rx 1 and Rx 2, is given as a. The fluid medium is descibed by the diffusion coefficient D. Rx

2 The molecules emitted by Tx 1 and Tx 2 popagate by Bownian motion, which is descibed by the Wiene pocess [11. Wheneve a diffusing molecule hits Rx 1 o Rx 2, it will be counted and pefectly absobed, i.e., it will be emoved fom the envionment. As a esult, the time histogam of absobed molecules at Rx 1 and Rx 2 follow the fist passage time concept. Fo a SISO scenaio in a 3-dimensional (3- D) envionment, thee exists a closed-fom fomula which descibes the pobability that a molecule hits Rx until time t afte its elease [12: F(t) = ( ) d d efc, (1) 4Dt whee efc( ) is the complementay eo function. Howeve, fo multiple absobing sphees inside the medium, no such closed-fom expession exists. Thus, a coesponding expession to (1) fo a given MIMO scenaio has to be obtained by eithe andom-walk-based simulations o by using a tained ANN as pesented in Section III. B. Communication Channel The modulation scheme unde investigation is on-off keying (OOK) [13 [15. Tx i emits eithe no molecules o N messenge molecules at the beginning of a symbol peiod of length T s to epesent bit u i [k = 0 o u i [k = 1, espectively. Molecules emitted by Tx 1 and Tx 2 ae of the same type. Futhemoe, Rx is assumed to be synchonized with Tx in time domain as suggested in [16. In addition, Rx 1 and Rx 2 pefom stength/ enegy detection at each symbol duation [17, [18. The MIMO channel can be sepaated into subchannels fom each tansmit antenna Tx i to each eceive antenna Rx j. Each subchannel is theeby chaacteized by the coesponding channel coefficients h ji [l (0 l L), which descibe the pobability that a molecule hits Rx j duing the lth time slot afte its emission at Tx i. All subchannels can be epesented by an equivalent discete-time channel model with effective channel memoy length L [19, [20. As a esult, the numbe of eceived molecules at Rx j can be descibed by the summation ove all subchannels elated to Rx j as y j [k = N Tx i=1 l=0 h ji [lx i [k l + n j [k, (2) whee N Tx is the numbe of tansmittes, n j [k descibes the amplitude dependent noise caused by the diffusive popagation of the molecules, and x i [k is the discete-time epesentation of the modulated data symbol tansmitted by Tx i at the stat of the kth tansmission inteval. Fo OOK it is defined as { N if u i [k = 1 x i [k = (3) 0 if u i [k = 0. Since the hitting pocess of molecules duing a bit peiod can be descibed by a binomial distibution [21, y j [k is epesented by the sum ove binomial distibutions y j [k N Tx i=1 l=0 B ( x i [k l, h ji [l ), (4) whee B (M, p) descibe a binomial distibution with M numbe of tials and success pobability p. With the help of (1), the channel coefficients fo a SISO scenaio can be easily calculated by h[l = F((l + 1)T s ) F(lT s ). (5) Howeve, fo multiple absobing sphees inside the medium, the channel coefficients h ji [l fo a given MIMO scenaio has to be obtained by eithe andom-walk-based simulations o by using a tained ANN as pesented in Sec. III. III. ANN FOR CHANNEL MODELING Fo modeling a molecula MIMO channel, we utilized the tained ANN of ou pevious wok [22. A tained ANN is able to estimate the channel coefficients h ji [l fo a given MIMO scenaio without unning simulations. In a fist step, we defined an expected analytical channel esponse function by intoducing fitting paametes into (1). The analytical channel esponse function at Rx 1 is defined as follows: ( F 11 (t, b 1, b 2, b 3 ) = b 1 d efc d (4D) b 2 t b 3 ), (6) whee b 1, b 2, and b 3 epesent the model fitting paametes. Similaly we define the esponse at Rx 2 (due to the coss link intefeence) as follows: ( ) d F 21 (t, b 4, b 5, b 6 ) = b 4 d 2 +a efc 2 +a 2 2 (4D) b, (7) 5 t b 6 whee b 4, b 5, and b 6 ae also model fitting paametes. In a second step, we fitted the expected analytical channel esponse functions to data obtained in extensive simulations. To detemine the b i values, we use a nonlinea least squaes cuve-fitting technique. These values, in conjunction with selected efeence system paametes d, a,, and D, ae the basis of taining and test datasets. Hence, the output of the cuve-fitting pocess consists of the model paametes b i fo each selected simulation scenaio. In a thid step, afte foming the taining and test datasets, the taining data is fed to the ANN taining pocess. Note that the tained ANN does not equie any simulation data. That is, the equied inputs ae abitay system paametes d, a,, and D. Afte taining, the ANN is able to pedict the fitting paametes b i fo these abitay system paametes. In Fig. 2, we pesent the channel coefficients that ae acquied fom extensive simulations and the tained ANN. We plot the h 11 [k and h 21 [k values by utilizing F 11, F 21, and the symbol duation. Ou esults validate and suppot the using of ANN to obtain the channel coefficients. IV. SPATIAL DIVERSITY Usually, spatial coding is pefomed along multiple tansmit antennas, wheeas combining stategies ae applied to multiple eceive antennas. In ode to achieve a spatial divesity gain at the Tx side, the same infomation is tansmitted ove multiple antennas. Theefoe, befoe spatial coding ove multiple Tx

3 h 21 [k h 11 [k Desied Signal ISI fom Tx1 d=20 7m: ANN d=20 7m: Simulation d=25 7m: ANN d=25 7m: Simulation Symbol Slot - k ILI Region ISI fom Tx1 d=20 7m: ANN d=20 7m: Simulation d=25 7m: ANN d=25 7m: Simulation Symbol Slot - k Fig. 2. Compaison of channel coefficients fom ANN and simulation data fo diffeent distances (a =13 µm, =5 µm, D =200 µm 2 /s, T s = 0.4 s) antennas is applied, the binay data sequence u is mapped onto a sequence of data symbols s. Theeby, s k denotes the kth data symbol of s. In what follows, two spatial coding techniques ae pesented - Alamouti-type coding and epetition MIMO coding, espectively. In addition, equal-gain combining, which is equivalent to maximum-atio combining (MRC) fo symmetical scenaios, is suggested as combining stategy. A. Alamouti-type Coding The Alamouti scheme [23 is a space-time block code widely used in adio-based communication systems fo spatial divesity. In its oigin, the Alamouti code can be epesented by the 2 2 tansmission matix [ G = s k s k+1 sk+1 sk, (8) whee the columns of the matix coespond to the tansmit antennas and the ows coesponds to two consecutive tansmission intevals [kt s (k +1)T s and [(k +1)T s (k +2)T s, espectively. As a esult, in the fist time slot, x 1 [k = s k is tansmitted via the fist tansmit antenna and x 2 [k = s k+1 is tansmitted simultaneously via the second tansmit antenna. In the second time slot, x 1 [k +1 = sk+1 is tansmitted via Tx 1 and x 2 [k +1 = sk is tansmitted simultaneously via Tx 2. Due to the fact that both tansmit antennas emit the same infomation, a spatial divesity gain can be achieved. The main advantage of the Alamouti scheme is that it is an othogonal space-time block code, i.e. G H G = 2I, whee G H is the Hemitian of matix G and I denotes the identity matix. Othogonality simplifies the implementation of a maximumlikelihood detecto, because ILI can be canceled completely. In the case of ISI, howeve, othogonality is getting lost and moe complex detection algoithms must be applied such as maximum-likelihood sequence estimation. In the case of molecula communication, the data symbols (amount of emitted molecules) ae non-negative and ealvalued athe than complex-valued. Fo OOK which is consideed thoughout this wok, data bits ae mapped onto data symbols s k {0, N} following the pinciple of (3). Hence, the classical Alamouti code has to be adapted to an Alamouti-type code that avoids the minus sign and the complex conjugate. As suggested in [24, the adaptation can be done by discading the complex conjugate opeation and eplacing negative symbols by s k := N s k. As a esult, the tansmission matix of the Alamouti-type code is given by [ s G = k s k+1. (9) N s k+1 s k B. Repetition MIMO Coding A simple altenative to othogonal Alamouti codes is offeed by epetition MIMO [25. In epetition MIMO the infomation is distibuted ove all tansmit antennas, whee each antenna tansmit exactly the same data symbol at the same time. As a esult, the tansmission matix fo a 2 2 MIMO scenaio is epesented by G = [ s k s k. (10) A big advantage of epetition MIMO is that, even in the pesence of ISI, single antenna detection algoithms can be applied. In addition, the ILI is constuctive and contibutes to the signal stength. C. Equal-gain Combining Befoe detection can be pefomed at Rx, the eceived signals of each eceive antenna have to be combined/selected in a cetain way. Following the equal-gain combining (EGC) algoithm, the signals of all eceive antennas ae equally weighted and combined: y[k = y 1 [k + y 2 [k. (11) Note that, fo symmetical scenaios, EGC is equivalent to MRC. In MRC, the combined signals ae weighted by a facto that is popotional to the coesponding channel quality. Fo symmetical scenaios, howeve, the channels at both eceive antennas ae the same. As a esult, the channel desciption fo the investigated scenaio, can be futhe simplified. Consideing that h 11 [l = h 22 [l and h 12 [l = h 21 [l, (11) can be estated as y[k = h[l (x 1 [k l + x 2 [k l) + n[k, (12) l=0 whee h[l h 11 [l + h 12 [l and n[k n 1 [k + n 2 [k.

4 TABLE I SIMULATION PARAMETERS USED FOR ANALYSIS. THE DEFAULT PARAMETERS ARE IN BOLD FACE. Paamete Value D [µm 2 /s {50, 100, 150, 200} [µm 5 d [µm 20 a [µm {11, 13, 15, 17} T s [s 0.6 L 3 N {500, 1000, 1500, 2000} K 10 6 R 1000 V. DETECTION ALGORITHMS Fo the bit eo analysis thoughout this pape, two diffeent detection algoithms ae consideed and adopted fom [26. Fist of all, the low-complexity adaptive theshold detecto (ATD) is applied, which woks independent of explicit channel knowledge: { 1 if y[k > y[k 1 û[k = 0 if y[k y[k 1. The second algoithm is maximum-likelihood sequence estimation (MLSE) based on the suboptimal squaed Euclidean distance banch metic ( 2 γ(y[k [ũ[k,..., ũ[k L) = y[k Nĥ[lũ[k l). Fo a single antenna system ĥ[l is assumed to be equal to the channel coefficients fom (5). Fo epetition coding with EGC ĥ[l = 2h[l, whee h[l is defined as in (12). In the case of Alamouti-type coding, the banch metic can be futhe adapted. Since the infomation of two bits is spead ove two consecutive time slots, the banch metic can be evaluated jointly fo both eceived numbes of molecules. l=0 VI. NUMERICAL RESULTS Fo simulative analysis, diffeent paametes settings ae consideed. These ae summaized in Table I, whee K is the bit sequence length fo one channel ealization and R is the total numbe of channel ealizations. It is assumed that afte (L + 1)T s = 2.4 s the emaining ISI is negligible. In ode to guaantee a fai compaison between the SISO and the 2 2 MIMO system by means of tansmitting enegy, the numbe of eleased molecules N in the case of the SISO scenaio is set twice as lage as fo the MIMO case. Futhemoe fo the SISO scenaio, thee is just a single tansmit and a single eceive antenna pesent in the envionment. The effect of spatial divesity is examined by means of a analysis. We investigate how the is impacted by the numbe of molecules N, sepaation distance a, and diffusion coefficient D. This is done by vaying one system paamete, while fixing the othe ones to the bold-faced values in Table I. Effect of Numbe of Emitted Molecules: In Fig. 3a, the effect of vaiation on the numbe of emitted molecules is shown. The numbe of molecules is popotional to the signal stength. In fact, if N is inceased, the vaiance lessens aound the expected channel impulse esponse and moe molecules hit the Rx sphee. Consequently, when N is inceased, all detection algoithms achieve bette pefomance when N is inceased. ATD benefits fom the ISI in the system [26. The best pefomance, howeve, is achieved by MLSE. This is due to the fact that MLSE implies channel equalization, which counteacts ISI. Repetition MIMO with ATD slightly outpefoms SISO tansmission in a egion with few molecules. Howeve, the spatial divesity gain fo ATD is not significant. Fo MLSE, the spatial divesity gain can be moe clealy obseved. The maximum impovement of epetition MIMO ove the SISO case is by a facto of almost 10. A facto of almost 400 is achieved, when powe nomalization (the emitted numbe of molecules in the SISO case is twice as lage as in the 2 2 MIMO case) is neglected. Fo the system consideed hee, Alamouti-type coding bings no impovement. The eason is that in case of epetition MIMO the ILI constuctively contibutes to the signal stength, wheeas in Alamouti-type coding ILI is competitive and as a esult moe destuctive. Effect of Antenna Sepaation: Fig. 3b shows the effect of the antenna sepaation distance on the MIMO system pefomance. Theeby, all MIMO systems pefom wose if the antenna sepaation distance is inceased, because the spatial gain fom ILI is deceased. Fo epetition MIMO with MLSE, a spatial divesity gain is still pesent fo a = 17 µm. Effect of Diffusion Coefficient: Fig 3c shows the effect of vaiation in the diffusion coefficient on the system pefomance. A lage diffusion coefficient leads to a moe spiky channel impulse esponse. As a esult, all detection algoithms unde consideation achieve a smalle if the diffusion coefficient is inceased. The spatial divesity gain fom epetition MIMO with ATD inceases with D. Howeve, the spatial divesity gain is not significant. The gap between epetition MIMO with MLSE and the SISO MLSE case is constant by a facto of aound 10 fo the analyzed paametes. As in Fig. 3a-3b, Alamouti-type coding does not show any spatial divesity gain compaed to a SISO system. VII. CONCLUSION In this pape, a diffusion-based molecula 2 2 MIMO communication system in a 3-D envionment was pesented. Channel coefficients wee obtained fom a tained ANN and incopoated into pefomance evaluations. Motivated fom the potential of spatial divesity in classical wieless communication, diffeent spatial divesity algoithms wee intoduced to the aea of MC and thei pefomances wee analyzed. In detail, Alamouti-type coding and epetition MIMO coding wee poposed at the tansmitte side. At the eceive side, equal-gain combining that is equivalent to MRC in symmetical scenaios, was pesented as a eceive combining stategy. In addition, adaptive theshold detection and maximumlikelihood sequence estimation wee adapted to the 2 2 MIMO scenaio. The divesity gain was analyzed by numeical

5 Adaptive theshold detecto (ATD) MLSE SISO Repetition MIMO Alamouti-type N a in µm D in µm 2 /s (a) Vaiation of numbes of molecules. (b) Vaiation of sepaation distance. (c) Vaiation of diffusion coefficient. Fig. 3. Bit eo ate pefomance as a function of the numbe of molecules (a), sepaation distance (b), and diffusion coefficient (c). If the coesponding paamete is not vaying, it is fixed to N = 1000, a = 11 µm, and D = 100 µm 2 /s. simulations. We leave pactical algoithms fo unsymmetical cases fo ou futue wok. ACKNOWLEDGMENT The wok of H. B. Yilmaz and C.-B. was in pat suppoted by the Basic Science Reseach Pogam (2017R1A1A1A ) though the NRF of Koea. REFERENCES [1 N. Fasad, H. B. Yilmaz, A. Eckfod, C.-B. Chae, and W. Guo, A compehensive suvey of ecent advancements in molecula communication, IEEE Commun. Suveys Tuts., vol. 18, no. 3, pp , [2 Y. Xia, P. Yang, Y. Sun, Y. Wu, B. Mayes, B. Gates, Y. Yin, F. Kim, and H. 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