On the Impact of Transposition Errors in Diffusion-Based Channels

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by  Werner Haselmayr

1
On the Impact of Transposition Errors in
Diffusion-Based Channels
Werner Haselmayr†, Member, IEEE, Syed Muhammad Haider Aejaz†, Member, IEEE,
A. Taufiq Asyhari, Member, IEEE, Andreas Springer, Member, IEEE, and Weisi Guo∗, Member, IEEE
Abstract—Transposition errors fundamentally undermine relia- split between: i) fundamental understanding of molecular sig-
bilityandcapacityinmolecularcommunication,whenindividual naling channels [9]–[11], ii) exploiting the benefits of molec-
particles are used for information encoding. Recently, several
ular signaling in nano-medicine applications [5], [12]–[14],
channelcodingtechniqueshavebeenproposedformitigatingthe
and iii) developing and testing physical layer transmission
transposition effect. The presented techniques show promising
results for diffusion-based channels with drift. However, so far schemes [15]–[18] and signaling protocols [19] for reliable
no performance evaluation has been carried out for diffusion- multi-scale molecular signaling.
based channels without drift, although in this case, transpo-
sitions are more prominent. In this work, we first derive an
7 A. Review on the Transposition Effect
analytical expression for the uncoded bit error probability due
1
totranspositions.Then,wecomparetheuncodedandcodederror In MCvD, with or without drift, the information can be
0
2 performance over diffusion-based channels with and without encodedinthenumberofparticles,thereleasetimeofparticles
drift by means of computer simulations. The results reveal that
and the type of particles [7]. Due to the stochastic nature
n for diffusion-based channels without drift a transmission with
of diffusion-based channels, it may occur that a sequence of
a reasonablereliabilitycanonlybeachievedbyintroducingaguard
J time between the codewords. This research lays the foundation transmitted particles will arrive out-of-order at the receiver,

1 forfuturedevelopmentofstrategiestomitigatetranspositionsin i.e. particles that are released earlier arrive late – yielding
1 pure diffusion-based channels. transpositions1 of bits or symbols [20]. Transpositions funda-
IndexTerms—Guardtime,Lévydistribution,molecularcommu- mentallylimittheperformanceoftime-andtype-basedmodu-
]
T nication, inverse Gaussian distribution, transposition/crossover lation schemes using individual particles. If the information is
I effect encodedinthenumberofparticles,transpositionshavealmost
.
s no effect, since the arrival time averaged over all particles is
c unlikely to vary significantly. Similarly, transmitting multiple
[ I. INTRODUCTION
particles instead of a single particle to convey time- or type-

1 Molecular communication broadly defines the transmission based information reduces the transposition effect [21], [22].
v of information using biochemical molecules over multiple However, if nano-machines are only able to release a small
1
distancescales[1].Withinmulti-cellularorganisms,molecular number of particles due to molecular reservoir capacity issues
7
communication within cells, between local cells, and across or when nano-machines are moving [23], the receiver cannot
9
2 thebodyoftheorganism(e.g.,hormones)isessentialforcoor- rely on the law of large numbers to average out stochastic
0 dinatedcellularaction-reaction.Betweenorganisms,molecular errors. Thus, in these scenarios, it is important to consider the
1. communicationtakesplaceoverseveralkilometersdistancein effect of transpositions.
0 air and under water (e.g., pheromones), and is used to signal Theimplementationofanoptimaldetectorfortime-andtype-
7 intent, assist navigation, and warn of impending dangers [2]. based modulation is infeasible, even for a small number of
1
The aforementioned molecular communication largely relies particles,sinceallpossiblepermutationsoftheparticlesarrival
:
v on messenger molecules to transverse channels using some must be taken into account [22], [24]. A sub-optimal detector
i formofnormaloranomalousdiffusionmechanism,potentially fortime-basedmodulationwasproposedin[24],whichunfor-
X
combining microscopic discrete random walk with macro- tunately is not applicable to type-based modulation. In recent
r
a scopic continuum fluid mechanics. years, several block coding techniques have been proposed
Due to the potential for ultra-high energy efficiency [3], for mitigating the transposition effect [15], [25], [26]. The
device dimension scalability, and bio-compatibility, molecular code design is no longer based on the Hamming distance,
communication via diffusion (MCvD) has gathered intense which is only useful if the bits are corrupted by noise and is
research interest in academia [4]–[7] and also industry (see likely to be ineffective in the case of transpositions. Hence,
IEEE standardization efforts under P1906.1-2015 [8]). The other attributes, like for example the Hamming weight [15]
majority of MCvD research and research projects have been or a molecular coding distance [25] were considered as
suitable coding design paradigms. However, so far the error
Manuscript received January 9, 2017; revised Month Day, 2017. †The performance of the codes was only evaluated for diffusion-
first two authors contributed equally to this work. ∗Corresponding author. based channels with drift, although transpositions are more
W. Haselmayr, S. M. H. Aejaz, A. Springer are with the Johannes Kepler
prominent in diffusion-based channels without drift. This is
University Linz, Austria, A. T. Asyhari is with the University of Bradford,
UnitedKingdomandW.GuoiswiththeSchoolofEngineering,University
ofWarwick,UnitedKingdom. 1In[20],transpositionsarereferredtoascrossovers.
2
because the tail of the first hitting time distribution for pure particlesarriveatthereceiveratrandomtime.Thearrivaltime
diffusion-based channels is heavier compared to flow-induced of the mth particle is given by [22], [27]
channels [22], [27].
Y =X +Z , m=1,...,M, (2)
m m m
B. Contributions & Organization where Zm denotes the random propagation time until the first
arrival, which is referred to as first hitting time.
Previous studies only provided analytical expressions for the
The first hitting time of a diffusion-based channel without
transposition probability of two particles [26], [28]. In this
drift follows a Lévy distribution, with its probability density
workweprovideananalyticalexpressionfortheprobabilityof
function (PDF) given by [27]
each permutation that can occur when a sequence of particles
(cid:114)
is released. Moreover, we derive the bit errors caused by each c (cid:16) c (cid:17)
fL(z)= exp − , z >0, (3)
permutation. Z 2πz3 2z
To the best of our knowledge, the block coding schemes
and its cumulative distribution function (CDF) given by
proposed for combating transposition errors have been only
(cid:18)(cid:114) (cid:19)
evaluated for diffusion-based channels with drift (e.g., [15] c
FL(z)=erfc , z >0, (4)
providesacomparativeperformanceanalysis).Sincethetrans- Z 2z
position effect is more prominent in diffusion-based channels
withthescaleparameterc=d2/(2D ).Thedistancebetween
withoutdrift,itisofhighinterestwhetherthecodingschemes p
transmitterandreceiverisdenotedbydandD isthediffusion
provide a reasonable error performance in such a scenario. p
coefficientofthetransmittedparticles.Thecomplementaryer-
Thus, in this work we provide an extensive error performance √
rorfunctionerfc(x)isdefinedbyerfc(x)=2/ π(cid:82)∞e−t2dt.
analysis for pure diffusion-based channels and a comparative x
The first hitting time for a diffusion-based channel with drift
study with flow-induced diffusion-based channels by means
follows an inverse Gaussian distribution with its PDF given
of computer simulations. The results show, that for pure
by [22]
diffusion-basedchannelsareasonableperformancecanonlybe
achieved by introducing a guard time between the codewords. (cid:114) λ (cid:18) (z−µ)2(cid:19)
The rest of this paper is organized as follows: The system fIG(z)= exp −λ , z >0, (5)
Z 2πz3 2µ2z
model is presented in Section II and the uncoded bit error
probability due to transpositions is derived in Section III. and its CDF given by
The most promising block coding techniques for mitigating (cid:32)(cid:114) (cid:18) (cid:19)(cid:33)
λ z
transposition errors are discussed in Section IV. Simulation FIG(z)=φ −1
Z z µ
results are presented in Section V and Section VI provides
concluding remarks. (cid:18) (cid:19) (cid:32) (cid:114) (cid:18) (cid:19)(cid:33)
2λ λ z
+exp φ − +1 , z >0, (6)
µ z µ
II. SYSTEMMODEL
with mean µ = d/v, the shape parameter λ=d2/(2D ) and
We consider the transmission of L information bits p√
the standard normal distribution φ(x) = exp(−x2/2)/ 2π.
b=[b ,...,b ]T over a diffusion-based channel with and
1 L The positive drift velocity from the transmitter to the receiver
without drift. The bit sequence is divided into Q blocks
isdenotedbyv.Notethatinthecaseofnodrift(v =0m/s)the
of length2 k = L/Q. If an (n,k) block code is applied,
mean µ tends to infinity and the inverse Gaussian distribution
each k-bit block is mapped to an n-bit codeword, i.e.
turns into a Lévy distribution, i.e. lim fIG(z)=fL(z).
c =[c ,...c ]T, q =1,...Q. Thus, in the coded case µ→∞ Z Z
q q,1 q,n The receiver collects the particles based on their arrival time
M = nQ coded bits are transmitted. In the uncoded case
and treats them as bit 0 and bit 1 if type-A and type-B are
the number of transmitted bits is M =kQ=L, since n=k.
received,respectively.Ifcodingisappliedatthetransmitter,a
As modulation scheme, we use binary molecule shift keying
proper decoding scheme is applied to obtain the information
(MoSK) [7], [29], which employs two different particles for
bits (cf. Section IV).
informationencoding.Weemploytype-Aandtype-B particles
to encode bit 0 and bit 1, respectively. The mth particle is
released by the transmitter into the medium at time III. TRANSPOSITIONEFFECT
Duetotherandomarrivaltimeoftheparticles,theymayarrive
X =(m−1)T , m=1,...,M, (1)
m b out-of-order and, thus, transpositions occur [20]. According
where T denotes the bit interval. The released particles to [26], we define that a level-l transposition occurs when
b
propagateaccordingtoBrownianmotion,withorwithoutdrift two particles are released exactly l bit intervals apart, but
and independently of each other, to the receiver. The receiver theirreceivingorderchanges.Forexample,supposethatthree
is capable of detecting the two types of particles and particles particlesarereleasedintheorder(p1,p2,p3)andarereceived
arriving at the receiver are absorbed and removed from the in the order (p2,p1,p3), which is illustrated in Fig. 1. In this
environment. Due to the stochastic nature of the channel, the case a level-1 transposition happened. If coding is applied
at the transmitter, we can distinguish between two cases of
2Withoutlossofgenerality,weassumethatk=L/Q∈N. transpositions:
3
and the ceiling function (cid:100)x(cid:101) denotes the mapping to the
smallest following integer number. Combining (7) and (8)
gives the uncoded bit error probability for a transmission of
M bits
(cid:88)
P = P (π)P (π). (10)
b perm b
π∈P
It is important to note that the calculation of the bit error
probability is only feasible if M is small, since P has M!
elements. In the following example, we derive the bit error
probability for the case when two particles are released and
Fig.1.Illustrationofthetranspositioneffect.Threeparticlesarereleasedin
theorder(p1,p2,p3)andarereceivedintheorder(p2,p1,p3). relatethisexpressiontootherdefinitionsusedintheliterature.
Example 1. We consider an uncoded transmission of
L=M =2 bits using a single block, i.e. Q = 1 and k = 2.
1) Intra-codeword transpositions refer to transpositions
Thus,thesetP includesM!=2permutationsπ andπ .The
within a codeword. 0 1
probability that the particles p and p arrive in the correct
2) Inter-codeword transpositions denote transpositions over 1 2
order, π =(1,2)(cid:44)(p ,p ), is given by (7)
different codewords. 0 1 2
Intheuncodedcase,onlyintra-codewordtranspositionsoccur, (cid:90)∞(cid:90)y2
P (p ,p )= f (y −x )f (y −x )dy dy
since only a single block/codeword is transmitted. In the perm 1 2 Z 2 2 Z 1 1 1 2
following,weprovideananalyticalexpressionfortheuncoded 0−∞
bit error probability due to transpositions. The derivation can (cid:90)∞ (cid:90)y2
be divided into two steps. First, we determine the probability = f (y −x ) f (y −x )dy dy . (11)
Z 2 2 Z 1 1 1 2
for each permutation that can occur when a sequence of M
0 −∞
particlesisreleased.Second,weprovideanexpressionforthe
Similarly, the probability that the particles p and p arrive
bit errors caused by each permutation. Let P represent the 1 2
out-of-order, π =(2,1)(cid:44)(p ,p ), can be computed by
set of all possible permutations on M released particles. The 1 2 1
probability that a certain permutation π ∈ P arrives at the (cid:90)∞ (cid:90)y1
receiver is given by Pperm(p2,p1)= fZ(y1−x1) fZ(y2−x2)dy2dy1. (12)
(cid:90)∞y(cid:90)π(M) y(cid:90)π(2) 0 −∞
By making the substitution y −x = z and y −x = z
P (π)= ··· 1 1 1 2 2 2
perm
and let x and x be 0 and T , respectively (cf. (1) and (2)),
1 2 b
0 −∞ −∞
we can rewrite (12) as follows
M−1
(cid:89)
fZ(yπ(M−i)−xπ(M−i))dyπ(1)···dyπ(M), (cid:90)∞ z1(cid:90)−Tb
i=0 Pperm(p2,p1)= fZ(z1) fZ(z2)dz2dz1 (13)
(7)
0 −∞
where fZ(z) denotes the PDF of the Lévy distribution or the (cid:90)∞
inverse Gaussian distribution given in (3) and (5), respec- = f (z )F (z −T )dz , (14)
Z 1 Z 1 b 1
tively. The release time of the mth particle is denoted by
0
x =(m−1)T (cf. (1)) and π(i) denotes the ith element
m b where F (z) denotes the CDF of the Lévy distribution or the
of the permutation π (e.g., π(1) = 2 for π =(2,1,3)). Z
inverseGaussiandistributiongivenin(4)and(6),respectively.
It is important to note that a permutation may not di-
It is important to note that the expression in (14) corresponds
rectly translate to an error. For example, consider three
to [26, Eq. (7)], which describes the probability that a trans-
particles, each of either type-A or type-B, that are re-
position between two particles happens. Moreover, through
leased in the order (1,2,3)(cid:44)(p ,p ,p ). If the permutation
1 2 3 substituting z by z +z in (13) and by changing the order
(2,1,3)(cid:44)(p ,p ,p ) arrives at the receiver, no error occurs 2 1
2 1 3 of the integrals results in
when (p = A,p = A,p = B), but an error occurs if
2 1 3
(p2 = B,p1 = A,p3 = B). The bit error probability for a −(cid:90)Tb(cid:90)∞
certain permutation can be obtained as follows Pperm(p2,p1)= fZ(z1)fZ(z+z1)dz1dz
disp(π) −∞ 0
Pb(π)= 2M , π ∈P, (8) −(cid:90)Tb
= (f ∗f−)(z)dz
where we denote the number of displacements of a permuta- Z Z
tion π by −∞
−(cid:90)Tb
M (cid:24) (cid:25)
(cid:88) |π(i)−i| = f dz =F (−T ), (15)
disp(π)= , (9) ∆Z ∆Z b
M
i=1 −∞
4
TABLEI TABLEII
CODEWORDSASSIGNMENTFORTHE(4,2,1)ISI-FREECODE. COMPARISONOFTHEBERWITHTHEBITERRORPROBABILITYPbINTHE
UNCODEDCASE(DP=600µm2/s,d=50µm,Tb=0.1s).
Assignmentifthepreviouscodewordhas
Informationbits
lastbit0 lastbit1 n Pb BER
00 0000 1111 2 0.2462 0.2461
01 0001 1000 3 0.3292 0.3297
10 0011 1100 4 0.3670 0.3711
11 0111 1110
C. ISI-free Codes
with f− = f (−z) and f (z) and F (z) denote the
Z Z ∆Z ∆Z A family of (n,k,l) ISI-free codes that address intra- and
PDF and the CDF of the random variable ∆Z = Z −Z ,
2 1 inter-codeword transpositions have been presented in [26],
respectively. Thus, the integral on the r.h.s of (15) defines
where l denotes the correctable transposition level. An
the probability Pr(∆Z < −T ) which is equivalent to the
b example of a (4,2,1) ISI-free code, with code rate 1/2, is
probability that the particles p and p arrive out-of-order. It
1 2 shown in Tab. I. It can be observed that, similar to DHW
isimportanttonotethatfortheLévydistributionaclosed-form
codes, intra-codeword transpositions are mitigated using
expression for the PDF f was derived in [27, Eq. (19)].
∆Z codewords with distinct Hamming weights. Moreover, level-l
However, no expression for the CDF F (−T ) was provided
Z b transpositions between adjacent codewords are suppressed by
in[27],butitcanbeeithercalculatednumericallyorbyusing
ensuring that for two consecutive codewords, the l last bits
tables of the standardized CDF.
of the first codeword are identical to the first l bits for the
According to (8), the bit error probability for the permutation
second codeword.
P (p ,p ) is given by P (p ,p )=1/2 and, thus, the total
perm 2 1 b 2 1
bit error probability results in P =1/2P (p ,p ).
b perm 2 1 The decoding complexity of the aforementioned codes is very
low,sinceitonlyinvolvescountingthenumberof1’s.Analyt-
ical expressions that approximate the bit error probability of
IV. TRANSPOSITIONERRORCORRECTIONCODES
repetition and ISI-free codes were presented in [26]. The ap-
In recent years, several block coding techniques for miti- proximationusesthepairwisetranspositionprobabilitydefined
gating the transposition effect have been introduced. In an in (12) and neglects transpositions that are larger than level-2.
(n,k) block code, a k-bit block is mapped to an n-bit code- A comparative performance analysis of the aforementioned
word. Most existing block codes have been designed based block codes was presented [15]. However, the performance
on maximizing the minimum pairwise Hamming distance be- was only evaluated for diffusion-based channels with drift.
tween the 2k codewords chosen from 2n possible codewords. Thus, in the next section we provide a comprehensive perfor-
However, this criterion is only useful if most of the bits are manceanalysisforpurediffusion-basedchannelsandcompare
corrupted by noise and is likely to be ineffective when bits the results with flow-induced channels.
exchange their position. Thus, for the design of codes that
mitigate transpositions, relevant attributes that are preserved
V. SIMULATIONRESULTS
afterthepermutationofthebitpositionshavetobeidentified.
For the sake of completeness, we review the most promising In this section, we present analytical and simulation results
block codes in the following. for the bit error probability in diffusion-based channels with
and without drift. We consider uncoded as well as coded
transmission and the simulation results were obtained using
A. Repetition Codes Monte-Carlo simulations.
Repetition codes repeat a particular bit n times. In the case
of bit transpositions, the structure of a codeword is preserved A. Uncoded Case
and, thus, the code is effective for mitigating intra-codeword
Figs. 2 – 4 show the impact of the diffusion coefficient D ,
p
transpositions. However, repetition codes have a code rate of
thedistanced,andthebitintervalT ontheuncodedbiterror
b
R =1/n which becomes very small for a large n.
rep rate(BER)fordifferentblocksizesn.Intheuncodedcase,we
considerthetransmissionofasingleblock,i.e.Q=L/n=1,
and, thus, only intra-block/codeword transpositions occur. If
B. Distinct Hamming Weight Codes
not otherwise stated, we use D = 600µm2/s, d = 50µm
p
Distinct Hamming weight (DHW) codes are based on the and T = 0.1s. Tab. II compares the BER results, obtained
b
observation that the Hamming weight of a codeword is pre- via simulations, with the bit error probability P , which was
b
served when bits exchange their position [15]. Thus, such obtained by evaluating the analytical expression in (10). It
codes are effective for combating intra-codeword transposi- shows that the simulation results match the analytical values
tions. The codebook of DHW codes consists of codewords very well.
with distinct Hamming weights. For example, the codebook To illustrate the impact of different diffusion coefficients, we
of a (4,2) DHW code, with code rate 1/2, is given by chose D ∈{600µm2/s,4500µm2/s}, which corresponds to
p
C ={0000,1000,1100,1110}. the diffusion coefficient of glucose and hydrogen in 25◦C
DHW
5
100 100

10−1 10−1
Rate Rate
Error Dp=600µm2/s, v=0m/s Error
Bit 10−2 DDpp==660000µµmm22//ss,, vv==240000µµmm//ss Bit 10−2 TTbb==00..11ss,, vv==02m00/µsm/s
Tb=0.1s, v=400µm/s
Dp=4500µm2/s, v=0m/s Tb =10s, v=0m/s
Dp=4500µm2/s, v=200µm/s Tb=10s, v=200µm/s
Dp=4500µm2/s, v=400µm/s Tb=10s, v=400µm/s
10−3 10−3
2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10
Block Size n Block Size n
Funigc.od2e.dImcapsaect(do=f th5e0µdmiffuasnidonTbco=effi0c.1iesn)t.Dp on the bit error rate in the F(Digp.4=.I6m0p0aµcmto2f/tshaenbditdin=ter5v0alµTmb).onthebiterrorrateintheuncodedcase
100 distance is reduced and the bit interval is increased.
It can be seen from Figs. 2 – 4 that an uncoded transmission
withreasonablereliability(e.g.,BER<10−4)isonlypossible
10−1 for large bit intervals and high drift velocities. Moreover, the
reliabilitycanbefurtherimprovedbyreducingthedistanced.
ate10−2 Unfortunately, changing the distance is hardly possible and a
R high drift is often not available in the medium. Furthermore,
Bit Error 10−3 d=50µm, v=0m/s idnicffreeraesnintgbltohcekbciotdiinntgertveaclhrneidquuceesshtahveebrietcreanttel.yFboerenthpisrorpeaossoend,,
d=50µm, v=200µm/s which we evaluate in the next section.
d=50µm, v=400µm/s
10−4 d=10µm, v=0m/s
B. Coded Case
d=10µm, v=200µm/s
d=10µm, v=400µm/s In this section, we evaluate different block coding techniques,
namely repetition codes, ISI-free codes and DHW codes (cf.
2 3 4 5 6 7 8 9 10
Block Size n Section IV), to mitigate the transposition effect and, thus, to
enable a reliable transmission even for low drift scenarios. In
the coded case, we consider the transmission of L bits which
Fig. 3. Impact of the distance d on the bit error rate in the uncoded case
(Dp=600µm2/sandTb=0.1s). are divided into Q = L/n > 1 blocks of size n and, thus,
intra- and inter-codeword transpositions occur.
Fig. 5 shows the BER versus guard time T , which cor-
G
water, respectively3 [30]. It can be observed from Fig. 2 that responds to the time between consecutive codewords. The
theimpactofdifferentdiffusioncoefficientsisnegligibleinthe aim for introducing a guard time is to reduce inter-codeword
caseofdiffusion-basedchannelswithoutdrift.Ifthereisadrift transpositions, since currently proposed codes can only tackle
in the medium, particles with lower diffusion coefficient have a limited number of transpositions between codewords. It
a better error performance than particles with higher diffusion can be observed that when no guard time is introduced, i.e.
coefficients, since particles with lower diffusion coefficient TG = Tb = 0.1s, coding does not significantly improve the
experience a higher support due to drift. BER performance. However, for a guard time of TG > Tb,
Figs. 3 and 4 show the impact of different distances d and a substantial BER improvement can be achieved. The perfor-
different bit intervals4 T . We chose d ∈ {10µm,50µm} mance improvement for diffusion-based channels with drift is
b
and T ∈ {0.1s,10s}. For diffusion-based channels without much higher than for diffusion-based channels without drift.
b
driftonlyasmallgaincanbeobserved,butfordiffusion-based This is because the tail of the Lévy distribution is heavier
channels with drift the BER improves significantly when the compared to the inverse Gaussian distribution, which results
in high level transpositions that are not limited to adjacent
3Note that for binary MoSK two particles with almost equal Dp are codewords. Thus, we expect that applying codes that can
required. combat high level transpositions between adjacent codewords
4We observed through simulations that the BER for Tb = 10s with will not significantly boost the performance.
v=200µm/s and v = 400µm/s lies far below 10−9. However, we did
Fig. 6 shows the BER versus the guard time, which is chosen
not determine the exact values due to the very long simulation times and,
thus,thecorrespondingBERcurvesarenotshowninFig.4. according to the percentile of particles arrivals, e.g., the 99th
6
100 100

10−1
10−2
Rate10−2 Rate
Bit Error 10−3 Rep. (5,1), v=0m/s Bit Error 10−4
ISI−free (4,2,1), v=0m/s
10−4 DRHepW. ((54,,12)),, vv==200m0µ/sm/s 10−6 Uncoded
ISI−free (4,2,1), v=200µm/s Rep. (5,1)
ISI−free (4,2,1)
DHW (4,2), v=200µm/s
DHW (4,2)
10 −1 100 101 102 103 104 10−8 5 10 15 20 25 30 35 40 45 50
Guard Time T (s) Block Size α⋅n
G
Fig. 5. Coded bit error rate versus guard time TG (Dp = 600µm2/s, Fig. 7. Coded bit error rate versus block size for diffusion-based channels
d=50µm,Tb=0.1s). with drift (Dp=600µm2/s, d=50µm, Tb =0.1s, v =200µm/s, 99th
percentile).
100
10−1 100
10−2 10−1
e
at
R
or 10−3 10−2
Bit Err10−4 Rep. (5,1), v=0m/s or Rate10−3
ISI−free (4,2,1), v=0m/s Err
10−5 DRHepW. ((54,,12)),, vv==200m0µ/sm/s Bit 10−4
ISI−free (4,2,1), v=200µm/s
DHW (4,2,1), v=200µm/s Uncoded
10−6 10−5 Rep. (5,1)
0 10 20 30 40 50 60 70 80 90 100 ISI−free (4,2,1)
Guard Time T (nth percentile)
G DHW (4,2)
10−6
5 10 15 20 25 30 35 40 45 50
Fig. 6. Coded bit error rate versus percentile of particle arrivals Block Size α⋅n
(Dp=600µm2/s,d=50µm,Tb=0.1s).
Fig. 8. Coded bit error rate versus block size for diffusion-based channels
withoutdrift(Dp=600µm2/s,d=50µm,Tb=0.1s,99thpercentile).
percentile corresponds to the guard time where 99% of the
released particles are expected to be observed at the receiver.
Fordiffusion-basedchannelswithoutdrift,aBERbelow10−5
can be achieved with a repetition code at the 99th percentile.
Figs. 7 and 8 show the BER when multiple codewords are
For flow-induced diffusion-based channels, a similar BER
transmitted consecutively, i.e. a guard time is only inserted
is achieved at the 75th percentile, but a BER below 10−7
betweentheconcatenatedcodewords.Thesizeoftheresulting
is achieved at the 99th percentile (cf. Fig. 7). Please note
blocks is given by αn, where α denotes the number of
that for pure diffusion-based channels, the 99th percentile
concatenated codewords. We can observe that concatenating
corresponds to the guard time T =13262s, whereas for
G codewords deteriorates the BER performance. A reasonable
diffusion-basedchannelswithdriftthe75thand99thpercentile
BER performance can only be achieved for diffusion-based
corresponds to the guard time T =0.3s and T =0.52s,
G G channels with drift, if a repetition code is applied.
respectively. Thus, to achieve a reasonable BER performance
for diffusion-based channels without drift, almost all particles It is important to note that the superior BER performance of
ofacodewordmustbereceivedbeforethenextistransmitted. the repetition code compared to the ISI-free and DHW code
The resulting guard time is very large due to the heavy tail is due to its low code rate. We included the results for the
of the Lévy distribution. For flow-induced diffusion-based repetition code to show that especially for diffusion-based
channel, a significantly lower guard time is required to obtain channels without drift, codes with low rates are required to
comparable results. enable a reliable transmission.
7
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