channel estimation for diffusive molecular communications · 2018-03-07 · arxiv:1608.03745v1...

arX

iv:1

608.

0374

5v1

[cs.

IT]

12 A

ug 2

016

1

Channel Estimation for Diffusive MolecularCommunications

Vahid Jamali,Student Member, IEEE, Arman Ahmadzadeh,Student Member, IEEE, Christophe Jardin,Heinrich Sticht, and Robert Schober,Fellow, IEEE

Abstract—In molecular communication (MC) systems, theexpected number of molecules observed at the receiver over timeafter the instantaneous release of molecules by the transmitteris referred to as the channel impulse response (CIR). Knowledgeof the CIR is needed for the design of detection and equalizationschemes. In this paper, we present a training-based CIR estima-tion framework for MC systems which aims at estimating theCIR based on theobserved number of molecules at the receiverdue to emission of asequence of known numbers of molecules bythe transmitter. Thereby, we distinguish two scenarios dependingon whether or not statistical channel knowledge is available. Inparticular, we derive maximum likelihood (ML) and least sumof square errors (LSSE) estimators which do not require anyknowledge of the channel statistics. For the case, when statisticalchannel knowledge is available, the corresponding maximuma posteriori (MAP) and linear minimum mean square error(LMMSE) estimators are provided. As performance bound, wederive the classical Cramer Rao (CR) lower bound, valid for anyunbiased estimator, which does not exploit statistical channelknowledge, and the Bayesian CR lower bound, valid for anyunbiased estimator, which exploits statistical channel knowledge.Finally, we propose optimal and suboptimal training sequencedesigns for the considered MC system. Simulation results confirmthe analysis and compare the performance of the proposedestimation techniques with the respective CR lower bounds.

Index Terms—Molecular communications, channel impulseresponse estimation, Cramer Rao lower bound, and trainingsequence design.

I. I NTRODUCTION

Recent advances in biology, nanotechnology, andmedicine have enabled the possibility of communicationin nano/micrometer scale environments [1]–[3]. Thereby,employing molecules as information carriers, molecularcommunication (MC) has quickly emerged as a bio-inspiredapproach for man-made communication systems in suchenvironments. In fact, calcium signaling among neighboringcells, the use of neurotransmitters for communication acrossthe synaptic cleft of neurons, and the exchange of autoinducersas signaling molecules in bacteria for quorum sensing areamong the many examples of MC in nature [3], [4].

This paper was presented in part at the International Conference onCommunications (ICC) 2016, Kuala Lumpur, Malaysia [31].

V. Jamali, A. Ahmadzadeh, and R. Schober are with the Institute forDigital Communications at the Friedrich-Alexander University (FAU), Erlan-gen, Germany (email: [email protected], [email protected], [email protected]).

C. Jardin and H. Sticht are with the Institute for Biochemistry atthe Friedrich-Alexander University (FAU), Erlangen, Germany (email:[email protected] and [email protected]).

A. Motivation

The design of any communication system crucially dependson the characteristics of the channel under consideration.InMC systems, the impact of the channel on the number ofobserved molecules can be captured by the channel impulseresponse (CIR) which is defined as theexpected numberof molecules counted at the receiver at timet after theinstantaneous release of a known number of molecules by thetransmitter at timet = 0. The CIR, denoted byc(t), can beused as the basis for the design of equalization and detectionschemes for MC systems [5], [6]. For diffusion-based MC,the released molecules move randomly according to Brownianmotion which is caused by thermal vibration and collisionswith other molecules in the fluid environment. Thereby, theaverage concentration of the molecules at a given coordinatea = [ax, ay, az] and at time t after their release by thetransmitter, denoted byC(a, t), is governed by Fick’s secondlaw of diffusion [5]. Finding C(a, t) analytically involvessolving partial differential equations and depends on initialand boundary conditions. Therefore, one possible approachfor determining the CIR, which is widely employed in theliterature [6], is to first derive a sufficiently accurate analyticalexpression forC(a, t) for the considered MC channel fromFick’s second law, and to subsequently integrate it over thereceiver volume,V rec, i.e.,

c(t) =

∫∫∫

a∈V rec

C(a, t)daxdaydaz. (1)

However, this approach may not be applicable in many prac-tical scenarios as discussed in the following.

• The CIR can be obtained based on (1) only for thespecial case of a fullytransparent receiver where it isassumed that the molecules move through the receiver asif it was not present in the environment. The assumptionof a fully transparent receiver is a valid approximationonly for some particular scenarios where the interactionof the receiver with the molecules can be neglected.However, for general receivers, the relationship betweenthe concentrationC(a, t) and the number of observedmoleculesc(t) may not be as straightforward [7]–[9].

• Solving the differential equation associated with Fick’ssecond law is possible only for simple and idealisticenvironments. For example, assuming apoint sourcelocated at the origin of anunbounded environment andimpulsive molecule release,C(a, t) is obtained as [6]

C(a, t) =N Tx

(4πDt)3/2

exp

(−|a|2

4Dt

) [molecules

m3

],(2)

http://arxiv.org/abs/1608.03745v1

2

whereN Tx is the number of molecules released by thetransmitter att = 0 andD is the diffusion coefficient ofthe signaling molecule. However,C(a, t) cannot be ob-tained in closed form for most practical MC environmentswhich may involve difficult boundary conditions, non-instantaneous molecule release, flow, etc. Additionally,as has been shown in [10], the classical Fick’s diffusionequation might even not be applicable in complex MCenvironments as physico-chemical interactions of themolecules with other objects in the channel, such as othermolecules, cells, and microvessels, are not accounted for.

• Even if an expression forC(a, t) can be obtained for aparticular MC system, e.g. (2), it will be a function ofseveral channel parameters such as the distance betweenthe transmitter and the receiver and the diffusion coeffi-cient. However, in practice, these parameters may not beknown a priori and also have to be estimated [11], [12].This complicates finding the CIR based onC(a, t).

Fortunately, for most communication problems, includingequalization and detection, only theexpected number ofmolecules that the receiver observes at the sampling timesis needed [5], [6]. Hence, the difficulties associated withderivingC(a, t) can be avoided by directly estimating the CIR.Motivated by the above discussion, our goal in this paper is todevelop a general CIR estimation framework for MC systemswhich is not limited to a particular MC channel model or aspecific receiver type and does not require knowledge of thechannel parameters.

B. Related Work

In most existing works on MC, the CIR is assumed tobe perfectly known for receiver design [5], [6], [13]. In thefollowing, we review the relevant MC literature that focused onchannel characterization. Estimation of the distance betweena transmitter and a receiver was studied in [11], [12], [14]for diffusive MC. In [15], an end-to-end mathematical model,including transmitter, channel, and receiver, was presented, andin [16], a stochastic channel model was proposed for flow-based and diffusion-based MC. For active transport MC, aMarkov chain channel model was derived in [17]. Additionally,a unifying model including the effects of external noise sourcesand inter-symbol interference (ISI) was proposed for diffusiveMC in [18]. In [19], the authors analyzed a microfluidic MCchannel, propagation noise, and channel memory. For neuro-spike communications, a physical model for the channel be-tween two neurons was developed in [20]. In addition to thesetheoretical works, experimental results for the characterizationof MC channels were reported in [21], [22]. However, thefocus of [11], [12], [15]–[22] is either channel modeling orthe estimation of channel parameters, i.e., the obtained resultsare not directly applicable to CIR acquisition. In contrasttoMC, for conventional wireless communication, there is a richliterature on channel estimation, mainly for linear channelmodels and impairment by additive white Gaussian noise(AWGN), see [23]–[25], and the references therein. Channelestimation was also studied for non-linear and/or non-AWGNchannels especially in optical communication. For example, a

linear time-invariant system with chi-squared noise was usedin [26] to model the optical fiber channel. However, the MCchannel model considered in this paper is different from thechannel models considered in [23]–[26], and hence, theseresults are not directly applicable to MC.

In this paper, we consider the Poisson channel which wasintroduced in [27], [28] for molecule-counting receivers inMC systems. The Poisson channel was previously used tomodel optical communication channels with photon-countingreceivers, see [29] and the references therein. Moreover, theauthors of a recent paper [30] adopted the Poisson channelmodel for non-line-of-sight (NLOS) optical wireless com-munication and proposed two different channel estimators.However, to the best of the authors’ knowledge, channelestimation for the Poisson MC channel was first consideredin the conference version of this paper [31]. We note that thechannel estimators developed in this paper for MC systemsare different from the channel estimators presented in [30]for NLOS optical communications. Moreover, the estimatorsconsidered in [30] do not take into account the non-negativityof the CIR coefficients. In contrast, we have carefully con-sidered this non-negativity constraint on the CIR coefficientsin the proposed estimators. Finally, in contrast to [30], weassume that the mean of the noise is unknown and has to beestimated as well. Therefore, the results in [30] are not directlyapplicable to the MC channel considered in this paper.

C. Contributions

In this paper, we directly estimate the CIR of the MCchannel based on the channel output, i.e., the number ofmolecules observed at the receiver. In particular, we presenta training-based CIR estimation framework which aims atestimating the CIR based on the detected number of moleculesat the receiver due to the emission of a sequence of knownnumbers of molecules by the transmitter. Thereby, we considerthe following two scenarios:i) Non-Bayesian CIR estimationwhere statistical CIR knowledge is not available or not ex-ploited. For this case, we first derive the optimal maximumlikelihood (ML) CIR estimator. Subsequently, we obtain thesuboptimal least sum of square errors (LSSE) CIR estimatorwhich entails a lower computational complexity than the MLestimator. Additionally, we derive the classical Cramer Rao(CR) bound which constitutes a lower bound on the estimationerror variance of any unbiased estimator.ii) Bayesian CIRestimation where knowledge of the CIR statistics is availableand exploited. Here, we first investigate maximum a posteriori(MAP) estimation as the optimal approach which requiresfull statistical knowledge of the MC channel. Subsequently,we derive the linear minimum mean square error (LMMSE)estimator for the case when only the first and second orderstatistics of the MC channel are available. Additionally, wederive the Bayesian CR bound as a generalization of the clas-sical CR bound for estimators which exploit statistical channelknowledge. Finally, we study optimal and suboptimal trainingsequence designs for the considered MC system. Simulationresults confirm the analysis and evaluate the performanceof the proposed estimation techniques with respect to thecorresponding CR lower bound.

3

PSfrag replacements

TxRx

a)

PSfrag replacements

b)

Exp

ecte

dn

um

ber

of

ob

serv

edm

ole

cule

s

Time (ms)

DesiredComponent

ISIComponents

Fig. 1. a) An illustrative time snapshot of an MC system during the fourth symbol interval where the transmitted molecules in the first, second, third, andfourth symbol intervals are shown in blue, red, green, and orange colors, respectively, and noise molecules are shown inblack color. b) Illustration of ISI: theexpected number of observed molecules at the receiver if thetransmitter emitsNTx molecules at the beginning of each symbol interval. The vertical dottedlines indicate the beginning of a new symbol interval.

We note that in contrast to the conference version [31],which studies only the case when statistical channel knowl-edge is not exploited, this paper also provides optimal andsuboptimal CIR estimators, training sequence designs, andperformance bounds for the case when statistical channelknowledge is exploited. Moreover, many of the extensivediscussions, simulation results, and proofs provided in thispaper are not included in [31].

D. Organization and Notation

The remainder of this paper is organized as follows. InSection II, some preliminaries and assumptions are presentedand the classical and Bayesian CR bounds are derived. CIRestimators which do not require statistical channel knowledgeare proposed in Section III, and CIR estimators which dotake advantage of statistical channel knowledge are given inSection IV. In Section V, several different training sequencedesigns are presented. Numerical results are provided in Sec-tion VI, and conclusions are drawn in Section VII.

Notations: We use the following notations throughout thispaper: Ex{·} denotes expectation with respect to randomvariable (RV)x, | · | represents the cardinality of a set, and[x]+ = max{0, x}. Bold capital and small letters are usedto denote matrices and vectors, respectively.1 and 0 arevectors whose elements are all ones and zeros, respectively,I denotes the identity matrix,AT denotes the transpose ofA, ‖a‖ represents the norm of vectora, [A]mn denotes theelement in them-th row andn-th column of matrixA, tr{A}is the trace of matrixA, diag{a} denotes a diagonal matrixwith the elements of vectora on its main diagonal,vdiag{A}is a vector which contains the diagonal entries of matrixA,eig{A} is the set of eigen-values of matrixA, A � 0 denotesa positive semidefinite matrixA, anda ≥ 0 specifies that allelements of vectora are non-negative. Additionally,∇2

aaf(a)denotes the Hessian matrix of functionf(a) with respect tovector a. Furthermore,Poiss(λ) denotes a Poisson RV withmeanλ, Bin(n, p) denotes a binomial RV forn trials and

success probabilityp, and N (a,A) denotes a multivariatenormal RV with mean vectora and covariance matrixA.

II. PRELIMINARIES, ASSUMPTIONS, AND PERFORMANCE

BOUNDS

In this section, we present the considered MC channelmodel, formulate the CIR estimation problem, and derive thecorresponding classical and Bayesian CR bounds.

A. System Model

We consider an MC system consisting of a transmitter, achannel, and a receiver, see Fig. 1 a). At the beginning ofeach symbol interval, the transmitter releases a fraction of N Tx

molecules whereN Tx is the maximum number of moleculesthat the transmitter can release at once, i.e., concentration shiftkeying (CSK) is performed [2]. In this paper, we assumethat the transmitter emits only one type of molecule. Thereleased molecules propagate through the medium between thetransmitter and the receiver. We assume that the movementsof individual molecules are independent from each other. Thereceiver counts the number of observed molecules in eachsymbol interval. We note that this is a rather general modelfor the MC receiver which includes well-known receivers suchas the transparent receiver [6], the absorbing receiver [7], andthe reactive receiver [8].

Due to the memory of the MC channel, inter-symbol in-terference (ISI) occurs [18], [19], see Fig. 1 b). In particular,ISI-free communication is only possible if the symbol intervalsare chosen sufficiently large such that the CIR fully decaysto zero within one symbol interval which severely limits thetransmission rate and results in an inefficient MC systemdesign. Therefore, taking into account the effect of ISI, weassume the following input-output relation for the MC system

r[k] =L∑

l=1

cl[k] + cn[k], (3)

4

wherer[k] is the number of molecules detected at the receiverin symbol intervalk, L is the number of memory taps ofthe channel, andcl[k] is the number of molecules observedat the receiver in symbol intervalk due to the release ofs[k − l + 1]N Tx molecules by the transmitter in symbolinterval k − l + 1, where s[k] ∈ [0, 1] is the transmittedsymbol in symbol intervalk. Thereby, cl[k] can be wellapproximated by a Poisson RV with meancls[k − l + 1],i.e., cl[k] ∼ Poiss (cls[k − l+ 1]), see [5], [27]. Moreover,cn[k] is the number of external noise molecules detected bythe receiver in symbol intervalk but not released by thetransmitter. Noise molecules may originate from interferingsources which employ the same type of molecule as theconsidered MC system. Hence,cn[k] can also be modeled asa Poisson RV, i.e.,cn[k] ∼ Poiss (cn), wherecn = E {cn[k]}.

Remark 1: From a probabilistic point of view, we canassume that each molecule released by the transmitter insymbol intervalk− l+1 is observed at the receiver in symbolinterval k with a certain probability, denoted bypl. Thereby,the probability thatn molecules are observed at the receiverin symbol intervalk due to the emission ofN Tx moleculesin symbol intervalk − l + 1 follows a binomial distribution,i.e., n ∼ Bin(N Tx, pl). Moreover, assumingN Tx → ∞ whileN Txpl , cl is fixed, the binomial distributionBin(N Tx, pl)converges to the Poisson distributionPoiss(cl) [32]. Thisis a valid assumption in MC since the number of releasedmolecules is often very large to ensure that a sufficient numberof molecules reaches the receiver. The same reasoning appliesto the noise molecules. �

Unlike the conventional linear input-output model for chan-nels with memory in wireless communication systems [23],[24], the channel model in (3) is not linear sinces[k − l+ 1]does not affect the observationr[k] directly but via PoissonRV cl[k]. However, theexpectation of the received signal islinearly dependent on the transmitted signal, i.e.,

r[k] = E {r[k]} =

L∑

l=1

cls[k − l + 1] + cn. (4)

We note that for a givens[k], in general, the actual num-ber of molecules observed at the receiver,r[k], will differfrom the expected number of observed molecules,r[k], dueto the intrinsic noisiness of diffusion. Here, we emphasizethat the considered MC channel model is characterized bycl, l = 1, . . . , L, and cn, see Fig. 2, which we refer to as theCIR of the MC channel throughout the the remainder of thepaper.

B. CIR Estimation Problem

Let s = [s[1], s[2], . . . , s[K]]T be a training sequence oflengthK. Here, we assume continuous transmission. There-fore, in order to ensure that the received signal is only affectedby training sequences and not by the transmissions in previoussymbol intervals, we only employr[k], k ≥ L, for CIRestimation. Thereby, theK − L + 1 samples used for CIRestimation are given by

r[L] =Poiss (c1s[L]) + Poiss (c2s[L− 1]) + · · ·

PSfrag replacements

z−1z−1z−1

c1 c2 c3 cL−1 cLcn

c1[k] c2[k] c3[k] cL−1[k] cL[k]

s[k]

cn[k]r[k]∑

PoissPoissPoissPoissPoissPoiss

Fig. 2. Input-output signal model for an MC system withL memory taps.Here, block “z−1" denotes one symbol interval delay and block “Poiss"generates a Poisson RV with a mean equal to the input [5], [27]. Theconsidered MC channel is characterized bycl, l = 1, . . . , L, i.e., the CIR.The goal of this paper is to estimate both the CIR and the mean of the noise,cn, based on RVr[k].

+Poiss (cLs[1]) + Poiss (cn) (5a)

r[L+ 1] =Poiss (c1s[L+ 1]) + Poiss (c2s[L]) + · · ·

+Poiss (cLs[2]) + Poiss (cn) (5b)...

...

r[K] =Poiss (c1s[K]) + Poiss (c2s[K − 1]) + · · ·

+Poiss (cLs[K − L+ 1]) + Poiss (cn) . (5c)

For convenience of notation, we definer = [r[L], r[L +1], . . . , r[K]]T and c = [c1, c2, . . . , cL, cn]

T , and fr(r|c, s)is the probability density function (PDF) of observationrconditioned on a given channelc and a given training sequences. We assume that the CIR1, c, remains unchanged for asufficiently large block of symbol intervals during which CIRestimation and data transmission are performed. However, theCIR may change from one block to the next due to e.g. achange of the distance between transmitter and receiver. Tomodel this, we assume that CIRc is a RV which takes itsvalues in each block according to a PDFfc(c). To summarize,in each block, the stochastic model in (3) is characterized by c

and our goal in this paper is to estimatec based on the vectorof random observationsr.

C. CR Lower Bound

The classical CR bound is a lower bound on the variance ofany unbiased estimator of adeterministic parameter. However,this bound can be also generalized to stochastic parameterswhich leads to the Bayesian CR bound [32], [33]. In particular,under some regularity conditions2, the covariance matrix ofany unbiased estimate,c, of parameterc, denoted byC(ˆc),satisfies

C(ˆc)− I

−1 (c) � 0, (6)

1With a slight abuse of notation, we refer to vectorc as the CIR vectorthroughout the remainder of the paper althoughc also contains the mean ofthe noisecn.

2These conditions ensure that the Fisher information matrixexists. In par-ticular, the continuous differentiablity of the PDFsfr(r|c, s) andfr,c(r, c|s)is a sufficient condition [32].

5

where I (c) is the Fisher information matrix of parametervector c and is given by [33]

I (c) =

Er|c

{−∇2

ccln{fr(r|c, s)}

},

if c is a deterministic parameter

Er,c

{−∇2

ccln{fr,c(r, c|s)}

},

if c is a stochastic parameter

(7)

We assume that conditioned onc and s, the observationsin different symbol intervals are independent, i.e.,r[k] isindependent ofr[k′] for k 6= k′. This assumption is validin practice if the time interval between two consecutivesamples is sufficiently large, see [5] for a detailed discussion.Moreover, from (3), we observe thatr[k] is a sum of PoissonRVs. Hence,r[k] is also a Poisson RV with its mean equal tothe sum of the means of the summands, i.e.,r[k] ∼ Poiss(r[k])with r[k] = cn +

∑Ll=1 cls[k − l + 1] = c

Tsk and sk =

[s[k], s[k − 1], . . . , s[k − L + 1], 1]T . Therefore,fr(r|c, s) isgiven by

fr(r|c, s) =K∏

k=L

(cTsk

)r[k]exp

(−c

Tsk

)

r[k]!. (8)

For a positive semidefinite matrix, the diagonal elements arenon-negative, i.e.,

[C(ˆc)− I

−1 (c)]i,i

≥ 0. Therefore, for anunbiased estimator, i.e.,E

{ˆc}= c holds, with the estimation

error vector defined ase = c − ˆc, the classical CR boundfor deterministicc provides the following lower bound on thesum of the expected square errors

Er|c

{‖e‖2

}(9)

≥ tr{I−1 (c)

}= tr

[K∑

k=L

sksTk

cT sk

]−1

, CCRB(c),

whereCCRB(c) denotes the classical CR bound for a givenfixed/deterministic parameterc.

On the other hand, the Bayesian CR bound for stochasticc

is given by

Er,c

{‖e‖2

}(10)

≥ tr

[Ec

{−∇2

ccln{fc(c)} +K∑

k=L

sksTk

cT sk

}]−1

, BCRB.

The expectation in (10) is taken over RVc which takes itsvalues according to PDFfc(c). Note thatfc(c) depends onthe MC environment and a general analytical expression forfc(c) is not yet available in the literature. In practice,fc(c)can be obtained using historical measurements of the CIR, seeRemark 8 for further discussion.

Remark 2: By comparing (9) and (10), we observe thatEc {CCRB(c)} ≥ BCRB holds as a result of Jensen’s inequality[34]. In particular, for unbiased estimators which do not exploitany statistical knowledge of the MC channel, both lowerboundsEc {CCRB(c)} and BCRB are valid butEc {CCRB(c)}is a tighter bound. On the other hand, for unbiased estimatorswhich exploit some statistical knowledge of the MC channel,Ec {CCRB(c)} is not a valid lower bound whereasBCRB is avalid lower bound. �

III. CIR ESTIMATION WITHOUT STATISTICAL CHANNEL

KNOWLEDGE

In this section, we consider estimators which do not exploitstatistical knowledge of the MC channel. In particular, wederive the ML and the LSSE CIR estimators.

A. ML CIR Estimation

The ML CIR estimator finds the CIR estimate which maxi-mizes the likelihood of observation vectorr [32]. In particular,the ML estimator is given by

ˆcML = argmaxc≥0

fr(r|c, s), (11)

where fr(r|c, s) is given in (8). Maximizingfr(r|c, s) isequivalent to maximizingln(fr(r|c, s)) sinceln(·) is a mono-tonically increasing function. Hence, the ML estimate can berewritten as

ˆcML = argmaxc≥0

g(c) where (12)

g(c) ,

K∑

k=L

[− c

Tsk + r[k]ln

(cTsk

) ].

To present the solution of the above optimization problemrigorously, we first define some auxiliary variables. LetA ={A1,A2, . . . ,AN} denote a set which contains all possibleN = 2L+1 − 1 subsets of setF = {1, 2, · · · , L, n} exceptfor the empty set. Here,An, n = 1, 2, . . . , N , denotes then-th subset ofA. Moreover, letcAn andsAn

k denote reduced-dimension versions ofc and sk, respectively, which containonly those elements ofc and sk whose indices are elementsof setAn, respectively.

Lemma 1: The ML estimator of the CIR for the consideredMC channel is given by Algorithm 1 where the following non-linear system of equations is solved3 for differentAn

K∑

k=L

[r[k]

(cAn)T sAn

k

− 1

]sAn

k = 0. (13)

Proof: Please refer to Appendix A.Remark 3: Recall that in the system model, we assumed a

priori L non-zero taps and a noise with non-zero mean, i.e.,c > 0. Moreover, the consistency property of ML estimation[32, Chapter 4] implies that under some regularity conditions,notably that the likelihood is a continuous function ofc andthat c is not on the boundary of the parameter setc ≥ 0, weobtain ˆcML → c asK → ∞. As a result, the ML estimator isasymptotically unbiased. Therefore, for large values ofK, theML estimator becomes sufficiently accurate such that none ofthe elements ofcML is zero. In this case, Algorithm 1 reducesto directly solving (13) forAn = F , where the left-hand sideof (13) is the derivative ofg(c) given in (12) with respect toc. �

Motivated by the results in Lemma 1 and the discussion inRemark 3, we propose the following simpler but suboptimalestimate based on the ML estimate in (13).

3A system of nonlinear equations can be solved using mathematicalsoftware packages, e.g. Mathematica.

6

Algorithm 1 ML/LSSECIR EstimatecML/ˆcLSSE

initialize An = F and solve(13)/(17) to find cF

if cF ≥ 0 then

Set ˆcML = cF /ˆcLSSE = c

F

elsefor ∀An 6= F do

Solve(13)/(17) to find cAn

if cAn ≥ 0 holds then

Set the values of the elements ofˆcCAN, whose indicesare inAn, equal to the values of the correspondingelements incAn and the remainingL + 1 − |An|elements equal to zero;SaveˆcCAN in the candidate setC

elseDiscardcAn

end ifend forChooseˆcML/ˆcLSSE equal to thatcCAN in the candidate setC which maximizesg(c)/minimizes‖ǫ‖2

end if

Suboptimal ML-Based CIR Estimation: For a given ob-servation vectorr, the suboptimal estimatecMLsub is given byˆcMLsub = [c]+ where c is the solution of the following systemof equations

K∑

k=L

[r[k]

cT sk− 1

]sk = 0. (14)

Note that the above estimate is simpler than the optimal MLestimate in Lemma 1 since in the case that thec found from(14) does not satisfyc ≥ 0, we do not search for the optimalboundary points and instead just set the negative elements of cto zero. Moreover, considering Remark 3, this simpler estimatebecomes asymptotically identical to the ML estimate, i.e.,asK → ∞, we obtainˆcMLsub → ˆcML. As we will see in Section VI,cf. Fig. 4, ˆcMLsub approaches the performance ofˆcML even forsmallK.

Remark 4: We note that obtaining the ML estimate ofthe CIR might be computationally challenging for practicalMC systems as nanonodes have limited computational power.Nevertheless, ML estimation can serve as a benchmark forlow-complexity estimators such as the LSSE estimator. More-over, for applications where the nanoreceiver only collectsobservations, i.e., vectorr, and forwards them to an externalprocessing unit outside the MC environment, the computa-tional complexity of ML estimation may be affordable. Thiscase may apply e.g. in health monitoring when a computeroutside the body may be available for offline processing.�

B. LSSE CIR Estimation

The LSSE CIR estimator chooses thatc which minimizesthe sum of the square errors for observation vectorr. Thereby,the error vector is defined asǫ = r−E {r} = r−Sc whereS =[sL, sL+1, . . . , sK ]T . In particular, the LSSE CIR estimate can

be formally written as

ˆcLSSE = argminc≥0

‖ǫ‖2. (15)

Here, the square norm of the error vector is obtained as

‖ǫ‖2= tr{ǫǫT

}= tr

{(r− Sc)(r− Sc)T

}

= tr{STScc

T}− 2tr

{rTSc

}+ tr

{rr

T}, (16)

where we used the following properties of the trace:tr {A} =tr{A

T}

and tr {AB} = tr {BA} [35]. The LSSE estimateis given in the following lemma where we use the auxiliarymatrix S

An = [sAn

L , sAn

L+1, . . . , sAn

K ]T .Lemma 2: The LSSE estimator of the CIR for the considered

MC channel is given by Algorithm 1 where for a given setAn, cAn is obtained as

cAn =

((SAn)TSAn

)−1(SAn)T r. (17)

Proof: Please refer to Appendix B.Remark 5: The LSSE estimator employs in fact a linear

matrix multiplication to computecAn , i.e., cAn = FAnr

whereFAn =((SAn)TSAn

)−1(SAn)T . Moreover, since the

training sequences is fixed, matrixFAn can be calculatedoffline and then be used for online CIR estimation. Therefore,the calculation ofcAn for the LSSE estimator in (17) is con-siderably less computationally complex than the computationof ˆcAn for the ML estimator in (13) which requires solving asystem of nonlinear equations. �

Corollary 1: The LSSE estimator in Lemma 2 is biasedin general but asymptotically, asK → ∞, becomes unbiased.More precisely, the LSSE estimator in Lemma 2 is a consistentestimator, i.e.,cLSSE → c asK → ∞.

Proof: Please refer to Appendix C.The above corollary reveals that similar to the ML estimate,

the LSSE estimate is biased for short length sequences butbecomes asymptotically unbiased as the sequence length growsto infinity. Moreover, motivated by the results in Corollary1and similar to the sub-optimal estimator proposed based on theML estimate, we propose the following suboptimal estimatorbased on the LSSE estimate in (17).

Suboptimal LSSE-Based CIR Estimation: For a given obser-vation vectorr, the suboptimal estimatecLSSEsub is given by

ˆcLSSEsub =[(STS)−1

STr

]+. (18)

Note that the above suboptimal estimate asymptotically, asK → ∞, converges to the LSSE estimate, i.e.,ˆcLSSEsub → ˆcLSSE.Moreover, as we will see in Section VI, cf. Fig. 4,ˆcLSSEsub

approaches the performance ofˆcLSSE even for smallK.Remark 6: We note that for finiteK, the ML and LSSE

estimators in Algorithm 1 are biased in general. The pro-posed estimators are biased because of the non-negativity ofthe unknown parameters. Exploiting this information in theproposed estimators causes the biasedness for smallK butimproves the estimation performance by reducing the varianceof the estimation error. Hence, the error variances of the MLand LSSE estimates may fall below the classical CR bound.However, asK → ∞, the ML and LSSE estimators becomeasymptotically unbiased, cf. Remark 3 and Corollary 1, and

7

the classical CR bound becomes a valid lower bound. Theasymptotic unbiasedness of the proposed estimators is alsonumerically verified in Section VI, cf. Fig. 3. �

Remark 7: The appropriate length of the training sequencedepends on the required CIR accuracy and the coherence timeof the MC channel. In particular, shorter training sequencescan be used if higher estimation errors can be tolerated.Furthermore, since the coherence time of the MC channeldetermines how fast the MC channel changes, the length ofthe training sequence should be chosen small compared to thecoherence time such that the CIR acquisition overhead is low.�

IV. CIR ESTIMATION WITH STATISTICAL CHANNEL

KNOWLEDGE

In this section, we first present the MAP estimator assumingthat full statistical knowledge of the MC channel is available.Subsequently, we derive the LMMSE estimator for the casewhen only the first and second order statistics of the MCchannel are available.

A. MAP CIR Estimation

The MAP CIR estimator is the optimal estimator assumingthat full statistical knowledge of the MC channel is available,i.e., fc(c). In particular, the MAP CIR estimator choosesthat CIR c which maximizes the posterior expected valueof observation vectorr, i.e., fc(c|r, s). Thereby, the MAPestimate is given by

ˆcMAP = argmaxc≥0

fc(c|r, s)(a)=

fr(r|c, s)fc(c)

fr(r), (19)

where (a) follows the Bayes’ Theorem [32]. Note that theterm fr(r) does not depend on the estimatec. Moreover,considering thatln(·) is a monotonically increasing function,the MAP estimate can be rewritten in the following convenientform

ˆcMAP = argmaxc≥0

[g(c) + ln (fc(c))] , (20)

whereg(c) is given in (12).Remark 8: The PDFfc(c) depends on the considered MC

environment and a general analytical expression forfc(c)is not yet available in the literature. In practice,fc(c) fora particular MC channel can be obtained using historicalmeasurements ofc. Another convenient approximation formathematical derivations is to assume a particular shape forfc(c) and adjust the parameters of the PDF to match theexperimental data. For the simulation results presented inSection VI, we assume thatfc(c) is a multivariant GaussianPDF4 with a certain mean vectorµc and covariance matrixΦcc which are chosen based on our simulation parameters.However, we emphasize that the choice of the CIR distribution

4We note thatc ≥ 0 has to hold, whereas a Gaussian PDF may lead tonegative-valued realizations for the elements ofc. To resolve this issue, inSection VI, we usec = [cGaus]+, where c

Gaus is a Gaussian distributedvector. Thereby, for mathematical tractability of computing ˆcMAP, we adoptthe Gaussian PDFf

cGaus(cGaus) as an approximation forfc(c).

depends on the specific MC channel. Here, as an example, weadopt the Gaussian distribution for the PDF of the CIR.�

Similar to the ML and LSSE estimates provided in the pre-vious section, the optimal MAP estimate is either a stationarypoint or a boundary point, see Algorithm 1. Here, we do notprovide a detailed solution for the MAP estimate due to spaceconstraints, instead, we highlight the main steps for the casewhen the solution is a stationary point. For the case whenthe solution is a boundary point, we can employ the samemethodology as in Algorithm 1 to findcMAP. In particular, thestationary points of the cost function in (20) are obtained byequating its derivative to zero, i.e.,∂g(c)∂c + ∂ln(fc(c))

∂c = 0.Note that the first term∂g(c)

∂c is given on the left-hand sideof (13) for An = F and the second term is given by∂ln(fc(c))

∂c = − 12

(Φ

−1cc +Φ

−Tcc

)(c − µc) for the considered

Gaussian PDFfc(c). We note that although the complexityof the MAP estimator may not be affordable for practical MCsystems, we can still use it as a benchmark scheme to evaluatethe performance of less complex estimators, e.g., the LMMSEestimator.

B. LMMSE CIR Estimation

The MAP estimator in (20) requires full knowledge ofthe channel statistics, i.e.,fc(c), which is difficult to obtainfor practical MC channels, cf Remark 8. Therefore, in thefollowing, we consider an LMMSE-based estimator whichrequires only knowledge of the first and second order statisticsof the MC channel. In particular, in estimation theory, theaverage square norm of the estimation errore = c − ˆc isoften adopted as performance metric, i.e.,E

{‖e‖2

}. There-

fore, one can design an LMMSE estimator which directlyminimizesE

{‖e‖2

}. We note that the estimation error has

to be averaged over bothr and c, i.e., Er,c

{‖e‖2

}, since

both are randomly changing, and hence, have to be modeledas RVs. In this paper, we consider an LMMSE-based estimatorgiven byˆcMMSE = [FMMSE

r]+ whereFMMSE is the MMSE matrix.

Moreover, for mathematical tractability, we optimizeFMMSE

such that an upper bound on the estimation error is minimized,i.e.,

FMMSE = argmin

F

Er,c

{‖eMMSEup ‖2

}, (21)

whereeMMSEup = c − ˆcMMSEup and ˆcMMSEup = FMMSE

r. Note thatˆcMMSEup

yields an upper bound on the estimation error ofˆcMMSE becausewe neglect the side information that the CIR coefficients haveto be non-negative by dropping[·]+. Hereby,Er,c

{‖eMMSEup ‖2

}

is obtained as

Er,c

{‖eMMSEup ‖2

}= Er,c

{tr{eMMSEup

(eMMSEup

)T}}

= Er,c

{tr{(Fr− c)(Fr − c)T

}}

= tr{FΦrrF

T − FΦrc −ΦcrFT +Φcc

}, (22)

where

Φrr= EcEr|c

{rr

T}= Ec

{Scc

TST + diag {Sc}

}

= SΦccST + diag {Sµc} (23a)

Φrc= ΦTcr = EcEr|c

{rc

T}= Ec

{Scc

T}= SΦcc, (23b)

8

andµc = Ec{c} andΦcc = Ec

{cc

T}

. In addition, in (23a)and (23b), we useEX {tr {AXB}} = tr {AEX {X}B},which holds for general matricesA, B, andX. Furthermore, in(23a), we useEx

{xx

T}= λλT+diag{λ}, which is valid for

multivariate Poisson random vectorsx with covariance matrixC(x) = diag{λ}. The optimal MMSE matrix is given in thefollowing lemma.

Lemma 3: For the LMMSE-based estimatorcMMSE =[FMMSE

r]+, the optimal matrixFMMSE, i.e., the solution of the

optimization problem in (21), is given by

FMMSE = Φ

TccST(SΦccS

T + diag {Sµc})−1

. (24)

Proof: Please refer to Appendix D.Remark 9: Note that the optimal MMSE matrix depends

only on the training sequence and the first and second orderchannel statistics. Thus, the MMSE matrix can be computedoffline once and then be used for online CIR acquisitionas long as the channel statistics remain constant. Since thestatistics of the CIR,c, change much less frequently thanthe CIR itself, an infrequent update of the MMSE matrix issufficient. As we will see in Section VI, the LMMSE estimatorprovides a considerable performance gain compared to theML and LSSE estimators at the cost of acquiring the requiredknowledge of the MC channel statistics. �

Remark 10: Similar to the discussion regarding the MLand LSSE estimators in Remark 6, we note that the pro-posed MAP and LMMSE estimators are in general biased forshort sequence lengths and become asymptotically unbiasedas K → ∞. Therefore, the Bayesian CR bound in (10)may not be a valid lower bound for the MAP and LMMSEestimators for small values ofK, however, forK → ∞, itbecomes asymptotically a valid lower bound. Due to spaceconstraints, we skip the detailed proofs for the biasednessandasymptotic unbiasedness of the proposed MAP and LMMSEestimators. Instead, we numerically illustrate these propertiesin Section VI, cf. Fig. 3. �

V. TRAINING SEQUENCEDESIGN

In this section, we first present two optimal training se-quence designs for CIR estimation in MC systems based onthe LSSE and LMMSE estimators. Subsequently, we proposea suboptimal and simple training sequence which is suitablefor practical applications where the involved nanomachineshave limited computational processing capability. For futurereference, letS andSK denote the sets of all CSK symbolsand all possible training sequences, respectively.

A. LSSE-Based Training Sequence Design

We first consider a training sequence design which min-imizes anupper bound on the average estimation error ofthe LSSE estimator. First, we note that for training sequencedesign, the estimation error has to be averaged over bothr

and c since both are a priori unknown, and hence, have to bemodeled as RVs. Again, we recall that in the system model,we assumed a prioriL non-zero taps and a noise with non-zero mean. Therefore, neglecting the information thatc ≥ 0

has to hold in (15) yields an upper bound on the estimation

error for the LSSE estimator. This upper bound is adoptedhere for the problem of sequence design since the solutionof (15) after dropping constraintc ≥ 0 leads to the simpleand elegant closed-form solutioncLSSEup =

(STS)−1

Sr, whichwe can employ to find optimal training sequences. Moreover,this upper bound is tight asK → ∞ since ˆcLSSE > 0 holdsand we obtaincLSSE = ˆcLSSEup . In Section VI, Fig. 7, we shownumerically that even for short sequence lengths, this upperbound is not loose.

Defining the estimation error aseLSSEup = c − ˆcLSSEup , theexpected square error norm is obtained as

Er,c

{‖eLSSEup ‖2

}

= Er,c

{tr

{(c−

(STS)−1

STr

)(c−

(STS)−1

STr

)T}}

= Er,c

{tr{(

STS)−1

STrr

TS(STS)−1

}

−2tr{cr

TS(STS)−1

}+ tr

{cc

T}}

. (25)

Next, we calculate the expectation with respect to(r, c)in (25) in two steps, namely first with respect tor con-ditioned on c and then with respect toc. ExploitingEX {tr {AXB}} = tr {AEX {X}B} and Ex

{xx

T}

=λλT + diag{λ}, E

{‖eLSSEup ‖2

}can be calculated as

EcEr|c

{‖eLSSEup ‖2

}

= Ec

{tr{(

STS)−1

ST(Scc

TST)S(STS)−1

}

−2tr{cc

TSTS(STS)−1

}+ tr

{cc

T}

+tr{(

STS)−1

ST diag {Sc}S

(STS)−1

}}

= Ec

{tr{S(STS)−2

ST diag {Sc}

}}

= tr{STvdiag

{S(STS)−2

ST}µT

c

}. (26)

Remark 11: From (26), we observe that the square errornorm of c for the LSSE estimator can be factorized into twoterms, a first termST vdiag

{S(STS)−2

ST}

, which solelydepends on the training sequence, multiplied by a second termµc, which depends on the MC channel. �

The evaluation of the expression in (26) can be numericallychallenging due to the required inversion of matrixST

S,especially when one of the eigen-values ofS

TS is close to

zero. One way to cope with this problem is to eliminate allsequences resulting in close-to-zero eigen-values for matrixSTS during the search. Formally, we can adopt the following

search criterion for LSSE-based training sequence design

sLSSE = argmin

s∈S

tr{ST vdiag

{S(STS)−2

ST}µT

c

}, (27)

where S ={s ∈ SK

∣∣|x| > ε, ∀x ∈ eig{STS}}

and ε is asmall number which guarantees that the eigen-values of matrixSTS are not close to zero, e.g., in Section VI, we choose

ε = 10−9.

9

B. LMMSE-Based Training Sequence Design

Similar to the LSSE-based sequence design, the LMMSE es-timateˆcMMSE in Lemma 3 can be used as the basis for sequencedesign. In this paper, we choose the optimal LMMSE-basedtraining sequence such that the upper bound on the MMSEestimation error is minimized, i.e.,Er,c

{‖eMMSEup ‖2

}in (22).

In particular, substitutingFMMSE in (24) into (22), simplifyingthe resulting expression forEr,c

{‖eMMSEup ‖2

}, and removing

the terms which do not depend on the sequence, we obtainthe following criterion for the optimal LMMSE-based trainingsequence

sMMSE = argmax

∀s∈SK

tr{Φ

TccST(SΦccS

T + diag {Sµc})−1

SΦcc

}.(28)

We note that from (23a), we can conclude that matrixSΦccS

T + diag {Sµc} is a positive definite matrix, i.e., alleigen-values are positive.

Remark 12: We base our training sequence designs on theLSSE and LMMSE estimators (and not the ML and MAPestimators) because these two estimators lend themselves toelegant closed-form solutions for the estimated CIR whichleads to relatively simpler criteria for training sequencedesign,cf. (27) and (28). In Section VI, we employ an exhaustivecomputer-based search to find the optimal LSSE-based andLMMSE-based training sequences by evaluating (27) and (28),respectively. We note that the complexity of the exhaustivesearch can be accommodated since the optimal training se-quence is obtained offline and used for online CIR estimation.Nevertheless, to reduce the complexity of the exhaustivesearch, one can develop systematic approaches to solve theoptimization problems in (27) and (28). This constitutes aninteresting research problem which is beyond the scope ofthis paper and left for future work. �

C. ISI-Free Training Sequence Design

One simple approach to estimate the CIR is to construct atraining sequence such that ISI is avoided during estimation.In this case, in each symbol interval, the receiver will observemolecules which have been released by the transmitter in onlyone symbol interval and not in multiple symbol intervals.To this end, the transmitter releasesN Tx molecules everyL + 1 symbol intervals and remains silent for the rest of thesymbol intervals. This corresponds to the ON-OFF keying,i.e., s[k] ∈ {0, 1}, which is a special case of the general CSKmodulation adopted in this paper. In particular, the sequences is constructed as follows:

s[k] =

{1, if k−k0

L+1 ∈ Z

0, otherwise(29)

wherek ∈ {1, . . . ,K}, andk0 is the index of the first symbolinterval in which the transmitter releases molecules. Moreover,for this training sequence, the CIR can be straightforwardlyestimated as

ˆcISIFl =1

|Kl|

[ ∑

k∈Kl

[r[k]− ˆcISIFn

]]+, l = 1, . . . , L (30a)

ˆcISIFn =1

|Kn|

∑

k∈Kn

r[k], (30b)

whereKl ={k|k−k0−l+1

L+1 ∈ Z ∧ k ∈ {1, . . . ,K}}

, Kn ={k|k−k0−L

L+1 ∈ Z ∧ k ∈ {1, . . . ,K}}

, and [·]+ is needed toensure that all estimated channel taps are non-negative, i.e.,ˆcISIF ≥ 0 holds.

Remark 13: The ISI-free training sequence in (29) permitsthe use of the simple estimator in (30). Thereby, the mathemat-ical calculation forˆcISIFn boils down to averaging over thoseelements of the observation vectorr which are only affectedby the external noise. Similarly, the computation ofˆcISIFl

basically reduces to averaging over those elements ofr whichare affected bycl and the impact of the estimated mean of theexternal noisecISIFn is removed. The ISI-free training sequencein (29) and the corresponding simple estimator in (30) aresuitable options for applications where nanoreceivers withlow computational capability have to be employed for CIRestimation. We note that, for the ISI-free training sequence in(29), the asymptotic solutions of the ML, LSSE, MAP, andLMMSE estimates asK → ∞ reduce to the estimate givenin (30), i.e.,ˆcML = ˆcLSSE = ˆcMAP = ˆcMMSE = ˆcISIF. �

VI. PERFORMANCEEVALUATION

In this section, we evaluate the performances of the differentestimation techniques and training sequence designs developedin this paper for ON-OFF keying signaling, i.e.,s[k] ∈ {0, 1}.For simplicity, for the results provided in this section, wegenerate CIRc based on (1) and (2). However, we emphasizethat the proposed estimation framework is not limited to theparticular channel and receiver models assumed in (1) and (2).We use (1) and (2) only to obtain a CIRc which is representa-tive of a typical CIR in MC. In particular, we assume a pointsource with impulsive molecule release andN Tx = 105, afully transparent spherical receiver with radius50 nm, and anunbounded environment withD = 4.365×10−10 m2

s [13]. Thedistance between the transmitter and the receiver is assumed tobe|a| = 400 nm. The receiver counts the number of moleculesonce per symbol interval at timeTsmp = argmax t C(a, t)after the beginning of the symbol interval whereC(a, t) iscomputed based on (2). The noise mean is chosen ascn =0.2max t C(a, t). Furthermore, the symbol duration and thenumber of tapsL are chosen such thatcL+1 < 0.05c1. Basedon the aforementioned assumptions, we obtain a typical/defaultvalue for the CIR vector which is denoted bycdef, e.g.,cdef = [60.22, 11.76, 5.13, 12.04]T for L = 3. In practice,

however, there may be random variations in the underlyingchannel parameters, e.g.,D, |a|, N Tx, which leads to randomvariations in the CIR. To take this into account, we assume thatthe CIR vector in each estimation block is a realization of a RVaccording toc = [cGaus]

+= [cdef + σdiag{cdef}N (0, I)]

+,whereσ is the standard deviation of RVcGaus, see Remark 8for further discussion. We note that asσ → 0, the MC channelbecomes deterministic, and for largeσ, the MC channel ishighly stochastic.

For the results reported in Figs. 3-6, the training sequencesare constructed by concatenatingn copies of the binarysequence[1100100101] of length10, i.e.,K = 10n. Moreover,the results reported in Figs. 3-6 are Monte Carlo simulationswhere each point of the curves is obtained by averaging over

10

PSfrag replacements

LMMSE EstimationMAP Estimation

Suboptimal LSSE EstimationSuboptimal ML EstimationLSSE EstimationML Estimation

Est

imat

ion

Err

or

Mea

nin

dB

Sequence Length,K10 20 30 40 50 60 70 80 90 100

−16

−14

−12

−10

−8

−6

−4

−2

0

Fig. 3. Norm of the estimation error mean,‖E {e}‖, in dB vs. the trainingsequence length,K, for L = 3 andσ2 = 0.1.

PSfrag replacements

LMMSE EstimationMAP EstimationBayesian CR Bound

Suboptimal LSSE EstimationSuboptimal ML EstimationLSSE EstimationML EstimationClassical CR Bound

Est

imat

ion

Err

or

Var

ian

cein

dB


8

10

12

14

16

18

20

22

Fig. 4. Estimation error variance,E{

‖e− E {e}‖2}

, in dB vs. the

training sequence length,K, for L = 3 andσ2 = 0.1.

105 random realizations of observation vectorr and c. In thefollowing, in Figs. 3 and 4, we present results forL = 3 andσ2 = 0.1. In particular, in Fig. 3, we show the norm of themean of the estimation error,‖E {e}‖, wheree = ˆc − c, indB vs. the training sequence length,K. Since for a givenc,Er|c {e} = Er|c

{ˆc}− c holds, ‖E {e}‖ =

∥∥EcEr|c {e}

∥∥is a measure for theaverage bias of the estimatec. Weobserve from Fig. 3 that the mean of the estimation errordecreases as the sequence length increases. Therefore, theproposed estimators are biased for short sequence lengths,butas the sequence length increases, they become asymptoticallyunbiased, i.e.,E

{ˆc}

→ c as K → ∞, cf. Corollary 1 andRemarks 3, 6, and 10.

In Fig. 4, we show the estimation error variance5,E

{‖e− E {e}‖2

}, in dB vs. the training sequence length,

K. As expected, the variance of the estimation error decreaseswith increasing training sequence length. For CIR estimationwithout statistical channel knowledge, we observe in Fig. 4that the gap between the error variances of the suboptimalML-based (LSSE-based) estimate and the optimal ML (LSSE)estimate is small for short sequence lengths and vanishes asK → ∞. Furthermore, for the considered set of parameters,the ML estimator outperforms the LSSE estimator by a marginof less than1 dB. These results suggest that the simpleLSSE estimator provides a favorable complexity-performancetradeoff for CIR estimation in the considered MC system. ForCIR estimation with statistical channel knowledge, we observefrom Fig. 4 that the error variance of the LMMSE estimate isslightly higher than that of the MAP estimate which reveals theeffectiveness of the proposed LMMSE estimator. Additionally,for short sequence lengths, a reduction of up to5 dB in theestimation error variance is attainable with the MAP/LMMSE

5We note that the estimation error variancenormalized to ‖E {c} ‖2 yieldsvery small values. For instance, in Fig. 4, the value of the normalizedestimation error variance is approximately36 dB less than the value of theestimation error variance.

estimators as compared to the ML/LSSE estimators at the costof the required acquisition of statistical channel knowledge.However, this gain vanishes asK → ∞. Finally, we notethat for short sequence lengths, the variances of the proposedestimators can even be lower than the corresponding classi-cal/Bayesian CR bound as these estimators are biased and theCR bound is a valid lower bound only for unbiased estimators,see Fig. 3. However, asK increases, all estimators becomeasymptotically unbiased, see Fig. 3. Fig. 4 shows that, forlarge K, the error variance of the ML estimator coincideswith the classical CR bound and the error variance of theMAP estimator is very close to the Bayesian CR bound.

To further study the performance of the proposed estimators,in Figs. 5 and 6, we show results only for the suboptimalLSSE estimator and the LMMSE estimator. Note that theselinear estimators are easier to implement compared to the otherproposed estimators. In Fig. 5, we show the estimation errorvariance,E

{‖e− E {e}‖2

}, in dB vs. the training sequence

length, K for L = 3 and σ2 ∈ {1, 0.5, 0.1, 0.05, 0.01}.As expected, the error variance of the LMMSE estimatordecreases asσ2 decreases. In particular, whenσ2 → ∞,the statistical channel knowledge becomes useless and whenσ2 → 0, the first order statisticsµc become a perfectly preciseestimate ofc. Thereby, depending on the values ofK andσ2,we observe from Fig. 5 that the LMMSE estimator achievesgains of1-9 dB compared to the LSSE estimator at the expenseof requiring knowledge of the first and second order statisticsof the CIR. Moreover, as suggested by (26), the LSSE errorvariance depends only onµc which is approximately constantfor all curves in Fig. 5. In particular, the small change in theLSSE error variance in Fig. 5 is due to the fact that althoughby increasingσ2, the mean ofcGaus remains unchanged, themean ofc = [cGaus]

+ changes since the instances, in whichthe elements ofcGaus are negative, occur more frequently.

In order to compare the performances of the consideredestimators for different numbers of channel memory taps,L,

11

PSfrag replacements

LMMSE EstimationLSSE Estimation

σ2 = [0.01, 0.05, 0.1, 0.5, 1]

Est

imat

ion

Err

or

Var

ian

cein

dB


6

8

10

12

14

16

18

20

22

Fig. 5. Estimation error variance,E{

‖e− E {e}‖2}

, in dB vs. the

training sequence length,K, for L = 3 andσ2 = [0.01, 0.05, 0.1, 0.5, 1].

PSfrag replacements

LMMSE EstimationLSSE Estimation

L = [1, 2, 3, 4, 5]

No

rmal

ized

Est

imat

ion

Err

or

Var

ian

ce,

Var e

,in

dB


−35

−30

−25

−20

−15

−10

−5

Fig. 6. Normalized estimation error variance,Vare, in dB vs. the trainingsequence length,K, for L ∈ {1, 2, 3, 4, 5} andσ2 = 0.1.

we define the normalized variance of the estimation error as

Vare =E

{‖e‖2

}− ‖E {e} ‖2

‖E {c} ‖2. (31)

In Fig. 6, we show the normalized estimation error vari-ance,Vare, in dB vs. the training sequence length,K, forL ∈ {1, 2, 3, 4, 5} and σ2 = 0.1. Thereby, we observe thatthe error variance increases as the number of memory tapsincreases. This is due to the fact that asL increases, thenumber of unkown parameters which have to be estimatedincreases. Furthermore, Fig. 6 shows that the gap between theerror variances of the LMMSE and LSSE estimators increasesasL increases.

Next, we investigate the performances of the optimal LSSE-/LMMSE-based and ISI-free training sequence designs de-veloped in Section V. Here, we employ a computer-basedsearch to find the optimal sequences based on the LSSE-based criterion in (27) whereε = 10−9 and the LMMSE-based criterion in (28). We consider short sequence lengths,i.e., K ≤ 20, due to the exponential increase of the com-putational complexity of the exhaustive search with respectto the sequence length. Moreover, since there areL + 1unknown parameters, we require at leastL+1 observations forestimation, i.e.,K − L+ 1 ≥ L+ 1 or equivalentlyK ≥ 2L.In Table I, we present the optimal sequences obtained forL ∈ {1, 3, 5}, K ∈ {10, 20}, and σ2 = 0.1. We note thatthe optimal sequences which are obtained from (27) and (28)are not unique and only one of the optimal sequences is shownin Table I for each value ofK andL. The optimal sequenceshown in red font in Table I is an ISI-free sequence as definedin (29).

In Fig. 7, we show the normalized LSSE estimation errorvariance,Vare, in dB vs. the training sequence length,K,for L ∈ {1, 2, 3, 4, 5} and σ2 = 0.1. Thereby, we reportthe analytical results for the upper bound in (26) and MonteCarlo simulation results for105 random realizations ofr and

c. For the ISI-free sequence, we also report the performanceof the simple estimation method in (30) in addition to theperformance of LSSE estimation. Fig. 7 confirms that (26) isa tight upper bound even for short sequence lengths. Moreover,we observe from Fig. 7 that the difference between the errorvariances of the ISI-free sequence and the optimal sequenceincreases asL increases. This result suggests that for MCchannels with small numbers of taps, a simple ISI-free trainingsequence is a suitable option. Furthermore, as expected, theestimation error decreases with increasing training sequencelength and increases with increasing number of CIR coeffi-cients.

In Fig. 8, we show the normalized LMMSE estimationerror, Vare, in dB vs. the training sequence length,K, forL ∈ {1, 3, 5} and σ2 = 0.1. In general, we can observe asimilar behavior as in Fig. 7, hence, we highlight only thedifferences. In particular, the simulation results for LMMSEestimation and the corresponding upper bound in (22) coincidewhich reveals that the effect of the instances for whichthe elements ofFMMSE

r are negative is negligible and henceˆcMMSE = [FMMSE

r]+

≈ FMMSE

r. Moreover, forL = 3 andL = 5, there is a considerable gap between the variances ofthe MMSE estimators for the optimal LMMSE-based trainingsequence and the ISI-free sequence.

VII. C ONCLUSIONS

In this paper, we developed a training-based CIR estimationframework which enables the acquisition of the CIR basedon the observed number of molecules at the receiver due toemission of a sequence of known numbers of molecules by thetransmitter. In particular, for the case when statistical channelknowledge is not available, we derived the optimal ML esti-mator, the suboptimal LSSE estimator, and the classical CRlower bound. On the other hand, for the case when knowledgeof the channel statistics is available, we provided the optimalMAP estimator, the suboptimal LMMSE estimator, and the

12

TABLE IEXAMPLES OF OPTIMAL LSSE/LMMSE SEQUENCESOBTAINED BY A COMPUTER-BASEDSEARCH

FORL ∈ {1, 3, 5}, K ∈ {10, 20}, AND σ2 = 0.1.

Criterion K = 10 K = 20

L = 1LSSE-based s

∗ = [1001101110]T s∗ = [00101110111010101111]T

LMMSE-based s∗ = [1111000111]T s

∗ = [11011110101100010111]T

L = 3LSSE-based s

∗ = [1100001001]T s∗ = [10101101101101100000]T


∗ = [00110101100000111011]T

L = 5LSSE-based s

∗ = [1000001000]T s∗ = [11111110000100001000]T


∗ = [00000011001010100111]T

PSfrag replacements

ISI-Free Seq., Simp. Est., Sim. ResultISI-Free Seq., LSSE Est., Sim. ResultOpt. Seq., LSSE Est., Sim. ResultISI-Free Seq., LSSE Est., Analyt. Upp. BoundOpt. Seq., LSSE Est., Analyt. Upp. Bound

L = 5L = 4L = 3

L = 2

L = 1

No

rmal

ized

Est

imat

ion

Err

or

Var

ian

ce,

Var e

,in

dB

Sequence LengthK2 4 6 8 10 12 14 16 18 20

−25

−20

−15

−10

−5

Fig. 7. Normalized LSSE estimation error variance,Vare, in dB vs. thetraining sequence length,K, for L ∈ {1, 2, 3, 4, 5} andσ2 = 0.1.

PSfrag replacements

ISI-Free Seq., Simp. Est., Sim. ResultISI-Free Seq., LMMSE Est., Sim. ResultOpt. Seq., LMMSE Est., Sim. ResultISI-Free Seq., LMMSE Est., Analyt. Upp. BoundOpt. Seq., LMMSE Est., Analyt. Upp. Bound

L = 5L = 3

L = 1

No

rmal

ized

Est

imat

ion

Err

or

Var

ian

ce,

Var e

,in

dB

Sequence LengthK2 4 6 8 10 12 14 16 18 20

−28

−26

−24

−22

−20

−18

−16

−14

−12

−10

−8

−6

Fig. 8. Normalized LMMSE estimation error variance,Vare, in dB vs.the training sequence length,K, for L ∈ {1, 3, 5} andσ2 = 0.1.

Bayesian CR lower bound. Furthermore, we studied optimalLSSE-/LMMSE-based and suboptimal training sequence de-signs for the considered MC system. Our simulation resultsrevealed that the simple LSSE and LMMSE estimators pro-vide favorable tradeoffs between complexity and performance.Additionally, for LSSE estimation, the proposed simple ISI-free sequences achieve a similar performance as the optimalsequences whereas, for LMMSE estimation, the performancegap between the ISI-free and the optimal sequences is notnegligible specially for larger numbers of channel memorytaps.

In this paper, we investigated the performance of the pro-posed CIR estimators and training sequences in terms of theestimation error variance. Studying the effect of the quality ofthe CIR estimates on the end-to-end bit error rate performanceof MC systems is an interesting and important direction forfuture research.

APPENDIX APROOF OFLEMMA 1

The problem in (12) is a convex optimization problem invariable c becauseg(c) is a concave function inc and thefeasible setc ≥ 0 is linear in c. In particular,ln

(cTsk

)is

concave sincecT sk is affine and the log-function is concave[36, Chapter 3]. Therefore,g(c) is a sum of weighted concave

termsr[k]ln(cTsk

)and affine termscT sk which in turn yields

a concave function [36, Chapter 3]. For the constrained convexproblem in (12), the optimal solution falls into one of thefollowing two categories:

Stationary Point: In this case, the optimal solution is foundby taking the derivative ofg(c) with respect toc which isgiven in (13) forAn = F where cF = c and s

Fk = sk hold.

Note that this stationary point is the global optimal solution ofthe unconstrained version of the problem in (12), i.e., whenconstraintc ≥ 0 is dropped. Therefore, ifcF falls into thefeasible set, i.e.,cF ≥ 0 holds, it is also the optimal solutionof the constrained problem in (12) and hence, we obtainˆcML =cF .

Boundary Point: In this case, for the optimal solution, someof the elements ofc are zero. Since it is not a priori knownwhich elements are zero, we have to consider all possiblecases. To do so, we use auxiliary variablesc

An andsAn

k wheresetAn specifies the indices of the non-zero elements ofc. Fora givenAn, we formulate a new problem by substitutingcAn

andsAn

k for c andsk in (12), respectively. The stationary pointof the new problem can be found by taking the derivative ofg(cAn) with respect tocAn which is given in (13). Here, ifcAn ≥ 0 does not hold, we discardcAn , otherwise, it is a

candidate for the optimal solution. Therefore, we constructthe candidate ML CIR estimate, denoted byˆcCAN, such that

13

the elements ofcCAN whose indices are inAn are equal to thevalues of the corresponding elements inc

An and the remainingelements are equal to zero. The resultingˆcCAN is saved in thecandidate setC. Finally, the ML estimate,cML, is given by thatˆcCAN in setC which maximizesg(c).

The above results are concisely summarized in Algorithm 1which concludes the proof.

APPENDIX BPROOF OFLEMMA 2

The optimization problem in (15) is convex since‖ǫ‖2 isquadratic in variablec, ST

S � 0 holds, and the feasible setc ≥ 0 is linear in c [36, Chapter 4]. The constrained convexproblem in (15) can be solved using a similar methodologyas was used for finding the ML estimate in Lemma 1. Hence,in order to avoid repetition and due to the space constraint,we provide only a sketch of the proof for Lemma 2 in thefollowing. In particular, the optimal LSSE estimate is eithera stationary point or a boundary point for the problem in(15). Moreover, if the solution is a boundary point for theoriginal problem in (15), we can formulate a new problem bysubstitutingcAn and s

An

k for c and sk in (15), respectively,where the solution of the new problem has to be a stationarypoint, see the proof of Lemma 1. For a givenAn, the derivativeof ‖ǫAn‖2, whereǫAn = r−S

An cAn , with respect tocAn is

obtained as

∂‖ǫAn‖2

∂cAn

= 2(SAn)TSAn cAn − 2(SAn)T r

!= 0, (32)

where we employed the vector differentiation rules∂tr{Axx

T}∂x =

(A+A

T)x and ∂tr{Ax}

∂x = AT [35,

Table I]. Solving (32) leads to (17). Now, we first findthe stationary point of the original problem by substitutingAn = F in (17). If cF ≥ 0 holds, cF is the optimal LSSEestimate, otherwise, we have to calculatec

An for all possibleboundary points, i.e., all elements ofAn. Among all cAn forwhich c

An ≥ 0 holds, the LSSE estimate is the one whichminimizes‖ǫ‖2. These results are summarized in Algorithm 1which concludes the proof.

APPENDIX CPROOF OFCOROLLARY 1

To show that the LSSE estimator is biased in general, weneglect the constraint thatc ≥ 0 has to hold in (15) whichyields an upper bound on the estimation error for the LSSEestimator, denoted bycLSSEup =

(STS)−1

STr. Thereby, we

first show thatcLSSEup is unbiased. The unbiasedness ofˆcLSSEup iseasily shown as

Er|c

{ˆcLSSEup

}= Er|c

{(STS)−1

STr

}=

(STS)−1

STEr|c {r}

=(STS)−1

STSc = c. (33)

For the case whencLSSEup ≥ 0 holds, the originalcLSSE is identi-cal to ˆcLSSEup . Otherwise, for the case when some of the elementsof ˆcLSSEup are negative,cLSSE deviates fromcLSSEup to ensure thatˆcLSSE ≥ 0 holds. Therefore, the elements ofEr|c

{ˆcLSSE

}

have a positive bias compared to their respective elements in

Er|c

{ˆcLSSEup

}which indicates that the LSSE estimate,ˆcLSSE, is

biased.To show that the LSSE estimator becomes asymptotically

unbiased, asK → ∞, we first prove thatcLSSEup is a consistentestimator, i.e.,cLSSEup → c as K → ∞. Here, sinceˆcLSSEup

provides an upper bound on the estimation error ofˆcLSSE,we can conclude that ifcLSSEup is consistent, thencLSSE isalso consistent. In other words, asK → ∞, we obtainˆcLSSE → ˆcLSSEup → c and hence, the asymptotic unbiasnessof the LSSE estimator.

In the following, we employ a well-known approach whichis based on the law of large numbers (LLN) to show theconsistency ofcLSSEup [37], [38]. To apply the LLN, we rewriteS = [l1, l2, · · · , lL, ln] where ll, l ∈ {1, 2, . . . , L, n}, arethe columns ofS. We note that for each element of vectorSTr = [lT1 r, l

T2 r, · · · , l

TLr, l

Tn r], we have a deterministic term

ll, l ∈ {1, 2, . . . , L, n} and a stochastic termr. Therefore,

using the LLN, we obtain 1K−L+1 l

Tl r

K→∞−→ 1

K−L+1 lTl Sc.

Here, we emphasize that 1K−L+1r

K→∞−→ 1

K−L+1Sc does nothold. However, 1

K−L+1 lTl r is indeed an averaging over all

observations in which the elementcl is present based on (3).Substituting these results intocLSSEup , we obtain

ˆcLSSEup =(STS)−1

STr

K→∞−→

(STS)−1

STSc = c. (34)

SinceˆcLSSEup is a consistent estimator and also provides an upperbound on the estimation error forˆcLSSE, we can conclude thatˆcLSSE is a consistent estimator, too. This completes the proof.

APPENDIX DPROOF OFLEMMA 3

The optimization problem in (21) is a convex optimizationproblem since the cost function is quadratic inF [36]. More-over, since the feasible set ofF is not constrained, the globaloptimalF has to be a stationary point which can be found byequating the derivative ofEr,c

{‖eMMSEup ‖2

}with respect toF

to zero, i.e.,

∂Er,c

{‖eMMSEup ‖2

}

∂F

= F(Φrr +Φ

Trr

)− 2ΦT

rc

= 2F(SΦccS

T + diag {Sµc})− 2ΦT

ccST != 0. (35)

Solving the above linear equation leads to (24) and concludesthe proof.

REFERENCES

[1] T. Nakano, M. Moore, F. Wei, A. Vasilakos, and J. Shuai, “MolecularCommunication and Networking: Opportunities and Challenges,” IEEETrans. NanoBiosci, vol. 11, no. 2, pp. 135–148, Jun. 2012.

[2] N. Farsad, H. Yilmaz, A. Eckford, C. Chae, and W. Guo, “A Compre-hensive Survey of Recent Advancements in Molecular Communication,”IEEE Commun. Surveys Tutorials, vol. PP, no. 99, pp. 1–1, 2016.

[3] I. Akyildiz, F. Brunetti, and C. Blazquez, “Nanonetworks: A NewCommunication Paradigm,”Comput. Net., vol. 52, pp. 2260–2279, Apr.2008.

[4] B. Alberts, D. Bray, K. Hopkin, A. Johnson, J. Lewis, M. Raff,K. Roberts, and P. Walter,Essential Cell Biology. Garland Science,4th ed., 2014.

14

[5] A. Noel, K. Cheung, and R. Schober, “Optimal Receiver Design forDiffusive Molecular Communication with Flow and Additive Noise,”IEEE Trans. NanoBiosci., vol. 13, no. 3, pp. 350–362, Sept. 2014.

[6] M. Mahfuz, D. Makrakis, and H. Mouftah, “A ComprehensiveStudy ofSampling-Based Optimum Signal Detection in Concentration-EncodedMolecular Communication,”IEEE Trans. NanoBiosci., vol. 13, no. 3,pp. 208–222, Sept. 2014.

[7] A. Akkaya, H. Yilmaz, C. Chae, and T. Tugcu, “Effect of ReceptorDensity and Size on Signal Reception in Molecular Communication viaDiffusion With an Absorbing Receiver,”IEEE Commun. Lett., vol. 19,no. 2, pp. 155–158, Feb. 2015.

[8] A. Ahmadzadeh, H. Arjmandi, A. Burkovski, and R. Schober, “ReactiveReceiver Modeling for Diffusive Molecular Communication Systemswith Molecule Degradation,” inProc. IEEE ICC, Kuala Lumpur, May2016, pp. 1–7.

[9] H. Yilmaz, A. Heren, T. Tugcu, and C. B. Chae, “Three-DimensionalChannel Characteristics for Molecular Communications With an Ab-sorbing Receiver,”IEEE Commun. Lett., vol. 18, no. 6, pp. 929–932,Jun. 2014.

[10] G. Wei and R. Marculescu, “Miniature Devices in the Wild: ModelingMolecular Communication in Complex Extracellular Spaces,” IEEE J.Select. Areas Commun., vol. 32, no. 12, pp. 2344–2353, Dec. 2014.

[11] M. Moore, T. Nakano, A. Enomoto, and T. Suda, “MeasuringDistanceFrom Single Spike Feedback Signals in Molecular Communication,”IEEE Trans. Sig. Process., vol. 60, no. 7, pp. 3576–3587, Jul. 2012.

[12] A. Noel, K. Cheung, and R. Schober, “Bounds on Distance Estimationvia Diffusive Molecular Communication,” inIEEE Globecom, Dec.2014, pp. 2813–2819.

[13] A. Ahmadzadeh, A. Noel, A. Burkovski, and R. Schober, “Amplify-and-Forward Relaying in Two-Hop Diffusion-Based Molecular Communica-tion Networks,” in Proc. IEEE Globecom, San Diego, Dec 2015, pp.1–7.

[14] X. Wang, M. Higgins, and M. Leeson, “Distance Estimation Schemes forDiffusion Based Molecular Communication Systems,”IEEE Commun.Lett., vol. 19, no. 3, pp. 399–402, Mar. 2015.

[15] M. Pierobon and I. Akyildiz, “A Physical End-to-End Model for Molec-ular Communication in Nanonetworks,”IEEE J. Sel. Areas Commun.,vol. 28, no. 4, pp. 602–611, May 2010.

[16] D. Miorandi, “A Stochastic Model for Molecular Communications,”Nano Commun. Netw., vol. 2, no. 4, pp. 205–212, 2011.

[17] N. Farsad, A. Eckford, and S. Hiyama, “A Markov Chain Channel Modelfor Active Transport Molecular Communication,”IEEE Trans. Signal.Process., vol. 62, no. 9, pp. 2424–2436, May 2014.

[18] A. Noel, K. Cheung, and R. Schober, “A Unifying Model forExternalNoise Sources and ISI in Diffusive Molecular Communication,” IEEEJ. Sel. Areas Commun., vol. 32, no. 12, pp. 2330–2343, Dec. 2014.

[19] A. Bicen and I. Akyildiz, “End-to-End Propagation Noise and MemoryAnalysis for Molecular Communication over Microfluidic Channels,”IEEE Trans. Commun., vol. 62, no. 7, pp. 2432–2443, Jul. 2014.

[20] E. Balevi and O. Akan, “A Physical Channel Model for NanoscaleNeuro-Spike Communications,”IEEE Trans. Commun., vol. 61, no. 3,pp. 1178–1187, Mar. 2013.

[21] N. Farsad, N.-R. Kim, A. Eckford, and C. B. Chae, “Channel and NoiseModels for Nonlinear Molecular Communication Systems,”IEEE J.Select. Areas Commun., vol. 32, no. 12, pp. 2392–2401, Dec. 2014.

[22] A. Biral, D. Zordan, and A. Zanella, “Modeling, Simulation and Ex-perimentation of Droplet-Based Microfluidic Networks,”IEEE Trans.Molecular, Biol., and Multi-Scale Commun., vol. 1, no. 2, pp. 122–134,Jun. 2015.

[23] S. Crozier, D. Falconer, and S. Mahmoud, “Least Sum of Squared Errors(LSSE) Channel Estimation,”IEE Proc. F Radar Sig. Process., vol. 138,no. 4, pp. 371–378, Aug. 1991.

[24] M. Ozdemir and H. Arslan, “Channel Estimation for Wireless OFDMSystems,”IEEE Commun. Surveys Tutorials, vol. 9, no. 2, pp. 18–48,2007.

[25] Y. Liu, Z. Tan, H. Hu, L. Cimini, and G. Li, “Channel Estimation forOFDM,” IEEE Commun. Surveys Tutorials, vol. 16, no. 4, pp. 1891–1908, 2014.

[26] C. Yang, F. Yang, and Z. Wang, “Clustering-Based Adaptive ChannelEstimator for Optical Fiber Communication Systems,”IEEE PhotonicsTechnol. Lett., vol. 20, no. 20, pp. 1670–1672, Oct. 2008.

[27] H. Arjmandi, A. Gohari, M. Kenari, and F. Bateni, “Diffusion-BasedNanonetworking: A New Modulation Technique and Performance Anal-ysis,” IEEE Commun. Lett., vol. 17, no. 4, pp. 645–648, Apr. 2013.

[28] R. Mosayebi, H. Arjmandi, A. Gohari, M. Nasiri-Kenari,and U. Mitra,“Receivers for Diffusion-Based Molecular Communication:Exploiting

Memory and Sampling Rate,”IEEE J. Sel. Areas Commun., vol. 32,no. 12, pp. 2368–2380, Dec. 2014.

[29] D. Guo, S. Shamai, and S. Verdu, “Mutual Information andConditionalMean Estimation in Poisson Channels,”IEEE Trans. Inf. Theory, vol. 54,no. 5, pp. 1837–1849, May 2008.

[30] C. Gong and Z. Xu, “Channel Estimation and Signal Detection for Opti-cal Wireless Scattering Communication with Inter-Symbol Interference,”IEEE Trans. Wireless Commun., vol. 14, no. 10, pp. 5326–5337, Oct.2015.

[31] V. Jamali, A. Ahmadzadeh, C. Jardin, C. Sticht, and R. Schober, “Chan-nel Estimation Techniques for Diffusive Molecular Communications,” inProc. IEEE ICC, Kuala Lumpur, May 2016, pp. 1–7.

[32] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin,Bayesian DataAnalysis. Taylor & Francis, 2014, vol. 2.

[33] H. L. Van Trees,Detection, Estimation, and Modulation Theory. JohnWiley & Sons, 2004.

[34] M. D. Perlman, “Jensen’s Inequality for a Convex Vector-Valued Func-tion on an Infinite-Dimensional Space,”J. Multivariate Anal., vol. 4,no. 1, pp. 52–65, 1974.

[35] P. H. Schonemann, “On the Formal Differentiation of Traces andDeterminants,”Multivariate Behavioral Research, vol. 20, no. 2, pp.113–139, 1985.

[36] S. Boyd and L. Vandenberghe,Convex Optimization. Cambridge, U.K.:Cambridge Univ. Press, 2004.

[37] H. White, “A Heteroskedasticity-Consistent Covariance MatrixEstimator and a Direct Test for Heteroskedasticity,”Econometrica,vol. 48, no. 4, pp. 817–838, 1980. [Online]. Available:http://www.jstor.org/stable/1912934

[38] G. Milne, “Response of the Ordinary Least Squares Estimator (OLS) toDeterministic and Stochastic Noise,” inIEEE AFRICON, vol. 2, Sept.1996, pp. 1099–1104.

http://www.jstor.org/stable/1912934

channel estimation for diffusive molecular communications · 2018-03-07 · arxiv:1608.03745v1...

Documents