1

Exploiting Diversity in Molecular Timing
Channels via Order Statistics

arXiv:1801.05567v1 [] 17 Jan 2018

Yonathan Murin, Nariman Farsad, Mainak Chowdhury, and Andrea Goldsmith
Department of Electrical Engineering, Stanford University, USA
Abstract—We study diversity in one-shot communication over
molecular timing channels. We consider a channel model where
the transmitter simultaneously releases a large number of information particles, while the information is encoded in the time
of release. The receiver decodes the information based on the
random time of arrival of the information particles. The random
propagation is characterized by the general class of right-sided
unimodal densities. We characterize the asymptotic exponential
decrease rate of the probability of error as a function of the
number of released particles, and denote this quantity as the
system diversity gain. Four types of detectors are considered: the
maximum-likelihood (ML) detector, a linear detector, a detector
that is based on the first arrival (FA) among all the transmitted
particles, and a detector based on the last arrival (LA). When the
density characterizing the random propagation is supported over
a large interval, we show that the simple FA detector achieves a
diversity gain very close to that of the ML detector. On the other
hand, when the density characterizing the random propagation
is supported over a small interval, we show that the simple LA
detector achieves a diversity gain very close to that of the ML
detector.

I. I NTRODUCTION
In many communication systems it is common to modulate
the information bits into the amplitude or into the phase of
the transmitted signal. In this work we consider a different
transmission approach in which the information is embedded
in the timing of the transmissions. The resulting channels
are commonly referred to as timing channels. Communication
over timing channels was studied in three main contexts: communication via queues, i.e., queuing timing channels [2]–[7],
molecular communications, i.e., molecular timing channels,
[8]–[15], and covert (secure) timing channels [16]–[18].
We study a model for molecular timing channels where
information is modulated through the time of release of information particles (see [19] for a detailed discussion regarding
applications of molecular communications). These information
particles represent molecules in the context of molecular
communications, or tokens using the terminology of [11], [12].
We focus on a one-shot communication scenario where the
transmitter simultaneously releases multiple identical information particles, and the time of release is selected out of a set
with finite cardinality. The receiver’s objective is to detect
this time of release. The released particles are assumed to
randomly and independently propagate to the receiver, where
upon their arrival they are absorbed and removed from the
environment. Hence, the random delay until a particle arrives
The authors are with the Department of Electrical Engineering, Stanford
University, Stanford, CA, 94305 USA. This work was presented in part at the
IEEE Global Communication Conference (GLOBECOME), December 2017,
Singapore, [1]. This research was supported in part by the NSF Center for
Science of Information (CSoI) under grant CCF-0939370, and the NSERC
Postdoctoral Fellowship fund PDF-471342-2015.

at the receiver can be represented as an additive noise term.
Our objective in this work is to characterize the asymptotic
exponential decrease rate of the probability of error, at the
receiver, as a function of the number of released particles. We
refer to this quantity as the system diversity gain. The formal
definition of diversity gain is given in Section II.
Comparing the diversity gains of different detection techniques indicates which method achieves a lower probability
of error when the number of particles used for communication is large, without the need for explicitly calculating the
probability of error. Thus, such a comparison can simplify
the system design. Note that in [20], [21] we also considered
a molecular timing channel with diversity; however, in these
works we derived upper and lower bounds on the capacity of
the molecular timing channel while in the current work we
study the diversity gain in the probability of error for oneshot communication. We believe that the results derived in
this paper also provide insights into achievable schemes that
will result in tighter lower bounds on the capacity.
Since we consider a causal molecular timing channel, we
focus on propagation models characterized by noise densities
with support on the positive real line. In particular, in molecular communications, the particles propagate to the receiver
following a random Brownian path. When the propagation is
based solely on diffusion, the additive noise associated with
random delay follows the Lévy distribution [14], [22]. When
the diffusion is accompanied by a drift, this additive noise
follows the inverse Gaussian (IG) distribution [9], [10]. In the
model studied in [11], [12], the additive noise representing the
propagation of the tokens follows an exponential distribution
(the exponential delay can represent systems with chemical
reactions that cause the particles to decay quickly [23]). Propagation based on diffusion without a drift, when the information
particles have finite life span, was studied in [13], [21],
while [24] considered diffusion-based communication when
the information particles experience exponential degradation.
Motivated by the above propagation models, in this work
we study the general class of propagation delays where the
associated noise density is continuous, differentiable, and
unimodal.1 We derive expressions for the system diversity gain
associated with four types of detectors: the optimal maximum
likelihood (ML) detector, a linear detector based on the mean
of the arrival times, a detector that is based on the first arrival
(FA) among the transmitted particles [14], and a detector that is
based on the last arrival (LA) among the transmitted particles.
1 When the maximum of a probability density function of a continuous
distribution is at a single value (or a continuous interval), the density is referred
to as unimodal (as opposed to the case of multiple maxima which is referred
to as multimodal). The local maximizing values are the modes of the density.

2

One of the main results presented in [14] is that in the case of a
Lévy-distributed additive noise, linear detection under multiple
particle release has worse performance than linear detection
based on a single particle release. This degradation is due to
the fact that the Lévy distribution has heavy tails that render
linear processing highly suboptimal. It was further shown in
[14] that for a small number of released particles, the probability of error achieved by the FA detector is indistinguishable
from that achieved by the ML detector; thus, this detector
provides a simple and attractive alternative to ML detection
for a small number of released particles.
In this work we consider the complementary setting where
the number of released particles is large. We show that if the
mode of the density of the noise is at zero, for example as
is the case for uniform or exponential distributions, then the
FA and ML detectors are equivalent. Moreover, even if the
mode is larger than zero, when the density of the noise is
supported over a large interval (e.g. the positive real line),
the FA detector can achieve a diversity gain very close to
the one achieved by the ML detector, and can significantly
outperform the linear detector. We emphasize that this holds
regardless of the tails of the noise, and contradicts the common
use of linear processing, known to maximize the signal-tonoise ratio (or minimize the bit error rate) in systems with
receive diversity and additive Gaussian noise (see [25] and [9,
Sec. IV.C.2]). Our results indicate that for detection of signals
transmitted over molecular timing channels (when the density
of the noise is supported over a large interval), the FA detector
is a much better alternative to the high-complexity ML detector
as compared to linear processing.
While the FA detector performs very well for noise densities
supported over the positive real line (e.g. the Lévy and IG
distributions), we further show that when the density of the
noise is supported over a short interval, the FA detector can
be significantly outperformed by linear detection. In this case,
we show that if the mode of the density of the noise is at
the maximum value of the support, then the LA and ML
detectors are equivalent. Moreover, even if this condition does
not hold, when the density of the noise is supported over a
small interval, the LA detector can achieve a diversity gain
very close to the one achieved by the ML detector, and can
significantly outperform the linear detector. Thus, our results
indicate that detection based on order statistics of the arrival
times, namely, based on the FA and LA, exploits the diversity
of the channel in a near-optimal manner, thereby establishing
a low-complexity near-ML detection framework for one-shot
communication over timing channels.
The rest of this paper is organized as follows. The problem
formulation is presented in Section II. The diversity gain
of the ML and linear detectors are derived in Section III.
The diversity gain of the FA and LA detectors is derived in
Section IV. Analysis of the diversity gain of specific densities
is provided in Section V, and the paper is concluded in
Section VI.
Notation: We denote sets with calligraphic letters, e.g., X ,
where R+ denotes the set of positive real numbers. We denote
RVs with upper case letters, e.g., X (except M and L which
are used to denote constants), and their realizations with lower

Fig. 1: Molecular communication timing channel. X denotes the release time,
Z denote the random propagation time, and Y denotes the arrival time.

case letters, e.g., x. An RV takes values in the set X , and
we use |X | to denote the cardinality of a finite set. We use
fZ (z) to denote the probability density function (PDF) of a
continuous RV Z on R+ and FZ (z) to denote its cumulative
distribution function (CDF). We denote vectors with boldface
letters, e.g., y, where the k th element of a vector y is denoted
by yk . Finally, we use log(·) to denote the natural logarithm.
II. P ROBLEM F ORMULATION
A. System Model
The molecular timing channel is illustrated in Fig. 1. In
this channel model the information is modulated on the time
of release of the information particles X, where Z denotes
the random time required for the information particle to
propagate from the transmitter to the receiver. We make the
following assumptions about the system (these assumptions
are consistent with those made in previous works, for instance
[8]–[11], [21], [22], [26]):
i) The information particles are assumed to be identical and
indistinguishable, thus, the information is encoded only
in the time of release of the particles. At the receiver, the
information is decoded based only on the time of arrival.
ii) The time-synchronization between the transmitter and
the receiver is perfect, the transmitter perfectly controls
the particles’ release time, and the receiver perfectly
measures their arrival time.
iii) Every information particle that arrives at the receiver is
absorbed and removed from the system.
iv) The information particles propagate independently of
each other, and their trajectories are random according
to an i.i.d. random process.
Let X be a finite set of constellation points on the real
line: X , {ξ0 , ξ1 , . . . , ξL−1 }, 0 ≤ ξ0 ≤ · · · ≤ ξL−1 = ∆.
Observing l ∈ {0, 1, . . . , L − 1} with equal probability, the
transmitter simultaneously releases M information particles
into the medium at time X ∈ X . The release time X is
assumed to be independent of the random propagation time
of each of the information particles. Let {Ym }M
m=1 denote the
M arrival times of each of the information particles released at
time X. Due to causality, we have Ym > X, m = 1, 2, . . . , M .
This leads to the following additive noise channel model:
Ym = X + Zm ,

m = 1, 2, . . . , M,

(1)

3

where Zm ∈ R+ is a random noise term representing the
propagation time of the mth particle. Note that Assumption
iv) implies that the RVs Zm are independent of each other.
The channel model (1) represents the setting of a transmitter
(e.g., a nano-scale sensor) that infrequently sends a symbol
(which conveys a limited number of bits) to a receiver (e.g., a
centralized molecular controller), and then remains silent for
a long period. Thus, the communication has a one-shot nature
and there is no inter-symbol interference.
In this paper we restrict our attention to the case of a binary
modulation, i.e., X = {0, ∆}. We note that the derived results
can be extended to more than two elements in the set X (see
the approach taken in [26, Sec. VI]). It can also be extended
to the case of unequal a-priori probabilities.
Let X̂ denote the estimation of X at the receiver. We
denote the probability of error, when M particles are used, by
(M)
Pε
, Pr{X 6= X̂}. As all the particles are simultaneously
(M)
released, Pε
can decrease when M is increased, [9], [14],
[26]. Specifically, in this paper we focus on the exponential
(M)
in the asymptotic limit of increasing M ,
decrease of Pε
defined by the quantity D given by:
(M)

− log Pε
.
(2)
M→∞
M
Remark 1 (System diversity gain). The channel (1) has a
single input and multiple outputs. Thus, by simultaneously
releasing M particles we achieve receive diversity. This is
the motivation for referring to D as the system diversity gain.
(M)
Clearly, if Pε
does not decrease at least exponentially with
M , then the system diversity gain is D = 0. As the propagation
of all particles is independent and identically distributed (see
Assumption iv)), the channel (1) can also be viewed as a
single-input-multiple-output channel where all the channel
outputs experience an independent and identical propagation
law.
D , lim

function fZ (z) supported over R+ .2 We further assume that
the density function fZ (z) is continuous, differentiable, and
unimodal.
Remark 2 (Generality of the studied model). Note that we do
not restrict fZ (z) to have finite first or second moments.
The above assumptions hold for the random propagation
models used to characterize many molecular timing channels.
For instance, in diffusive molecular communications the released particles follow a random Brownian path from the transmitter to the receiver. In the case of diffusion without a drift,
the RVs Zm are Lévy-distributed [26, Def. 1], while in the case
of diffusion with a drift, the RVs Zm follow the IG distribution
[9, eq. (3)]. Moreover, if the information particles have a finite
deterministic life time (see [21]), then the resulting densities
of the RVs Zm are the clipped-Lévy density (diffusion without
a drift) and the clipped-IG distribution (diffusion with drift).
Another example for a random propagation model that satisfies
these assumptions is the exponential noise that was considered
in the communication model of [11]. We emphasize that the
results derived in the following sections hold for any channel
model obeying (1) (not only in the case of molecular timing
channels), provided that the density of the noise is continuous,
differential, right-sided, and unimodal.
Next, we derive the diversity gain of the ML detector and
the linear (mean) detector.
III. S YSTEM D IVERSITY G AIN OF THE ML
D ETECTORS

AND

L INEAR

A. The ML Detector
Let y = {ym }M
m=1 . The ML detector is given by the
following decision rule:
(
PM
fZ (ym )
0,
m=1 log fZ (ym −∆) ≥ 0
(3)
X̂ML (y) =
∆, otherwise.
(M)

We emphasize that this description of communication over
a molecular timing channel is fairly general and can be applied
to different propagation mechanisms as long as Assumptions
ii)–iv) hold. Next, we discuss the random propagation model
for our channel.

For many types of fZ (z) an explicit expression for Pε,ML ,
the probability of error of the ML detector, is not available.
However, as recovering x from the M i.i.d. realizations
{ym }M
m=1 belongs to the class of binary hypothesis problems,
the diversity gain is exactly the Chernoff information. This is
formalized in the following proposition:

B. The Random Propagation Model

Proposition 1. The diversity gain for the ML detector in (3)
is given by:
Z ∞

s
1−s
DML = − min log
(fZ (y)) ·(fZ (y −∆)) dy . (4)

Our assumptions on the propagation model require the
following definition of the class of weakly unimodal (quasiconcave) functions [27, Sec. 3.4.1]:
Definition 1. A function f (z) is said to be weakly unimodal
if there exists a value ζ for which it is weakly monotonically
increasing for z ≤ ζ and weakly monotonically decreasing for
z ≥ ζ. Note that for a weakly unimodal function the maximum
value can be reached for a continuous range of values of z.
In the following we refer to the class of weakly unimodal
functions simply as unimodal functions.
We now define the proposed model associated with the
timing channel (1):
Definition 2. For the channel model of (1), we focus on propagation models characterized by noise Zm with a noise density

s:0≤s≤1

y=∆

Proof: The result follows from combining [28, Theorem
11.9.1] and [28, eq. (11.239)] for continuous distributions, see
the detailed discussion in [26, Sec. V.B].
As the ML detector minimizes the probability of error for
equiprobable signaling, it also maximizes the diversity gain D.
At the same time, the ML detector has two main drawbacks.
First, it is relatively complicated to compute in low-complexity
devices (e.g., nano-scale sensors) due to the logarithm of the
2 Note that [15] considered a molecular timing channel with differential
transmission where the noise density support is R. Yet, for the channel model
(1), the noise can have only positive values.

4

quotient of the densities. Second, the ML detector requires all
the particles to arrive. In some scenarios this may require very
long delays, in particular when fZ (z) has heavy tails (e.g., the
Lévy distribution). A detector that (partially) addresses the first
drawback is the linear detector discussed next.
B. The Linear Detector
If the additive noise is Gaussian, i.e., Zm ∼ N (0, σz2 ), then
the optimal detector is linear [25, Ch. 3.3]. Even when the
noise is not Gaussian, detection based on a linear combination
of the received signals {ym }M
m=1 can significantly improve the
probability of error, as observed in [9, Sec. IV.C.2] for the
case of additive IG noise. We note here that the main benefit
of the linear detector is its simplicity; yet, it may also require
long delays. Before formally defining the linear detector, we
comment on the destructive effect linear detection can have
on performance in the case of heavy tailed fZ (z).
Remark 3 (The destructive nature of linear detection for
heavy-tailed noise). In [26, Thm. 1] it is shown that, for the
case of Lévy-distributed propagation, a linear detector (e.g.,
applying ML detection based on the averaged arrival times)
increases the dispersion of the noise.3 Thus, the probability
of error of a linear detector for M > 1 is lower bounded by
the probability of error of an optimal detector for the case of
M = 1, which means that the linear detector has a diversity
gain of zero.
To derive the diversity gain of the linear detector we use
tools from large deviations
theory [29]. Consider detection
PM
1
based on YLIN , M
m=1 Ym , and let

(5)
ΛZ (ρ) , log EZ eρZ
denote the cumulant generating function of Z. Further, define
Λ∗Z (v) , sup {λv − ΛZ (λ)}

(6)

necessarily symmetric, to maximize DLIN we use the fact that
the two densities differ only in a shift and require the two
terms on the right-hand-side (RHS) of (8) to be the same.
Thus, we find the point at which the right tail (Λ∗Z (µ+α))
equals the left tail (Λ∗Z (µ−∆+α)). This leads to the decision
threshold µ + α and to (7). If such a point does not exist,
then the decision intervals do not overlap, which implies zero
probability of error and DLIN = ∞.
In the next section we discuss detection based on order
statistics, and show that for some noise distributions the
proposed detectors are equivalent to the optimal ML detector.
IV. D ETECTION BASED

ON

O RDER S TATISTICS

The detectors proposed in this section detect the transmitted
symbol based on either the first or the last arrivals among the
M particles. Specifically, the detector waits for the first (last)
particle to arrive and then applies ML detection based on this
arrival. It is shown that for M large enough this can be done
by simply comparing the first (last) arrival to a threshold.
A. The FA Detector
Let yFA , min {y1 , y2 , . . . , yM }. The FA detector is the
ML detector based on YFA . Before discussing the performance
of the ML detector, we note that for a fixed value of M ,
fYFA |X (yFA |x) is not necessarily unimodal. The following
lemma provides sufficient conditions for fYFA |X (yFA |x) to be
unimodal, for sufficiently large (yet finite) M .
Lemma 1. Let fZ (z) be a unimodal density supported on the
interval (0, τ ), τ ≤ ∞, and fZ′ (z) its derivative. If there exists
an ǫ > 0′ such that, for every 0 < z ≤ ǫ, the function
f (z)
g(z) = fZ2 (z) is monotonically decreasing, then there exists
Z
a sufficiently large and finite M0 for which the density of
ZFA , min{Zm }M
m=1 is unimodal for M > M0 .
Proof: The proof is provided in Appendix A.

λ

to be the rate (Cramér) function [29, Sec. 2.2]. The diversity
gain of X̂LIN (YLIN ), the ML detector based on YLIN , is stated
in the following theorem.

Remark 4 (Shifted support). In Lemma 1 it is assumed that
the support of fZ (z) is (a, b), a = 0, b ≤ ∞. The lemma can
be easily extended to the case of a > 0.

Theorem 1. Let fZ (z) be a density with E{Z} = µ < ∞, and
E{(Z − µ)2 } < ∞. Further assume that ΛZ (ρ) is finite over
some interval in R. If there exists α ∈ (max{∆ − µ, 0}, ∆),
such that Λ∗Z (µ+α) = Λ∗Z (µ−∆+α), then:

Remark 5 (Sufficient conditions for unimodality). Lemma
1 provides sufficient conditions for the density of the FA,
fYFA |X (yFA |x), to be unimodal when M is sufficiently large. It
is possible that fYFA |X (yFA |x) will be unimodal even if these
conditions do not hold. It is also possible that fYFA |X (yFA |x)
will be unimodal for values of M smaller than M0 . In
Appendix C we show that the conditions of Lemma 1 hold
for the IG and for the Lévy densities.

Otherwise, DLIN

DLIN = Λ∗Z (µ + α) .
= ∞.

(M)

(7)

(M)

Proof: Let Pε|0 and Pε|∆ denote the probabilities of error
given x = 0 and x = ∆, respectively. The diversity gain is
now given by:


(M)
(M)

− log Pε|∆ 
− log Pε|0
, lim
DLIN = min
. (8)
lim
M→∞

M→∞
M
M
From Cramér’s Theorem [29, Thm. 2.2.3] we have
limM→∞ log Pr{YLIN > y0 |X = x} = Λ∗Z (y0 ) , y0 > µ. Similarly, Cramér’s Theorem states that limM→∞ log Pr{YLIN <
y∆ |X = x} = Λ∗Z (y∆ ) , y∆ < µ. Recalling that fZ (z) is not
3 The

dispersion of the noise is also known as its scale.

Next, we assume that fYFA |X (yFA |x) is unimodal for a given
finite M , and provide the detection rule and probability of
error of the FA detector.
Proposition 2. Let fYFA |X (yFA |x) be unimodal for a given
value of M , and let mZ be the mode of fZ (z). Further, let
FZ (z) be the CDF of the noise. Then, the ML detector based
on yFA is given by:
(
0, yFA < θM
X̂FA (yFA ) =
(9)
∆, yFA ≥ θM ,

5

where θM is the solution of the following equation in yFA for
∆ ≤ yFA ≤ mZ :
M−1

fZ (yFA )
1 − Fz (yFA − ∆)
.
(10)
=
fZ (yFA − ∆)
1 − Fz (yFA )

fZ (y) ≤ fZ (y − ∆), y ≥ ∆. The ML detection rule can be
written as:

If (10) does not have a solution, then θM = ∆. Furthermore,
the probability of error of the FA detector is given by:


(M)
Pε,FA = 0.5 (1−Fz (θM ))M +1−(1−Fz (θM −∆))M . (11)

Therefore, the ML detector declares X̂ML (y)=0 only if there
exists Ym < ∆ (otherwise it declares X̂ML (y) = ∆). Since
testing if there exists Ym < ∆ can be implemented based on
yFA , we conclude that in this case the ML detector reduces to
the FA detector, and therefore the detectors are equivalent.

Proof Outline: As fYFA |X (yFA |x) is a PDF, the ML detector
amounts to comparing yFA to a threshold, which can be found
by equating the two densities. Since the two densities differ
only by a shift, if (10) does not have a solution, then θM = ∆.
Finally, (10)–(11) are obtained by noting that FYFA |X (yFA |x) =
1 − (1 − FZ (y − x))M and fYFA |X (yFA |x) = M ·fZ (y − x)·
(1 − FZ (y − x))M−1 .
The following theorem presents the diversity gain of the FA
detector.
Theorem 2. Let fYFA |X (yFA |x) be unimodal for all M > M0 .
Then, the diversity gain of the FA detector is given by:
DFA = − log (1 − FZ (∆)) .

(12)

Proof: Before proving (12) we note that if fYFA |X (yFA |x)
is unimodal for all M > M0 , then θM → ∆ when M → ∞.
This follows from the extreme value theorem [30, Thm. 1.8.4],
which implies that the limiting distribution of the considered
densities concentrates towards the release time x (namely, a
Dirac delta at x), thus θM → ∆.
Next, we recall that ZFA = min{Zm }M
m=1 , and let θM =
(M)
∆+ δM , δM → 0. Note that Pε,FA can be bounded as follows:
1
1
(M)
Pr{ZFA ≥ ∆ + δM } ≤ Pε,FA ≤ Pr{ZFA ≥ ∆}. (13)
2
2
For the RHS of (13), recalling that Pr{ZFA ≥ ∆} = (1 −
FZ (∆))M , we write
− log Pr{ZFA ≥ ∆}
= − log(1−FZ (∆)).
(14)
M→∞
M
For the left-hand-side (LHS) we have Pr{ZFA ≥ ∆ + δM } =
(1 − FZ (∆ + δM ))M . Using a Taylor expansion of log(1 −
FZ (∆ + δM ))M around ∆, we obtain:
lim

M

log(1−FZ (∆ + δM ))


fZ (∆)δM
2
+O(δM ) . (15)
= M log(1−FZ (∆))−
1 − FZ (∆)

Therefore, since δM → 0, we have:
− log Pr{ZFA ≥ ∆ + δM }
lim
= − log(1−FZ (∆)). (16)
M→∞
M
Combining (13)–(16) we conclude the proof.
We now consider the special case where the mode of fZ (z)
is zero, e.g., the uniform or exponential densities:
Theorem 3. Let fZ (z) be a continuous, differentiable, and
unimodal density with mode mZ ≥ 0 and fZ (z) = 0, z < mZ .
Then, the FA and ML detectors are equivalent, namely, they
have the same probability of error.
Proof: We focus on the case of mZ = 0. The proof for
mZ > 0 follows along similar lines. Since mZ = 0, then

X̂ML (y) = argmax
x

M
Y

fZ (ym |X = x).

(17)

m=1

The following corollary is a direct consequence of Thm. 3.
Corollary 1. Under the conditions of Thm. 3 DFA = DML .
As stated in Thm. 3, and as exemplified in Section V, in
some cases DFA is very close to the optimal diversity gain
DML . Unfortunately, as is also shown in Section V, there are
cases where there is a substantial gap between DFA and DML .
This motivates discussing a detection based on the complement
order statistics, namely the last arrival.
B. The LA Detector
Let yLA , max {y1 , y2 , . . . , yM }. Similarly to the FA
detector, the LA detector is the ML detector based on YLA .
Before discussing the performance of the ML detector, we
note that for a given M , fYLA |X (yLA |x) is not necessarily
unimodal. The following lemma provides sufficient conditions
for fYLA |X (yLA |x) to be unimodal, for sufficiently large (yet
finite) M .
Lemma 2. Let fZ (z) be a unimodal density supported on
(0, τ ), τ ≤ ∞, and fZ′ (z) its derivative. If there exists an ǫ > 0
f ′ (z)
such that, for every τ −ǫ < z ≤ τ , the function g(z) = fZ2 (z) is
Z
monotonically decreasing, then there exists a sufficiently large
and finite M0 for which the density of ZLA , max{Zm }M
m=1
is unimodal for M > M0 .
Proof: The proof is provided in Appendix B.
Next, we assume that fYLA |X (yLA |x) is unimodal for a given
finite M , and provide the detection rule and probability of
error of the LA detector.
Proposition 3. Let fZ (z) be a unimodal density supported on
(0, τ ), τ ≤ ∞, fYLA |X (yLA |x) be unimodal for a given value
of M , and mZ be the mode of fZ (z). Further, let FZ (z) be
the CDF of the noise Zm for all m. Then the ML detector
based on yLA is given by:
(
0, yLA < ϑM
(18)
X̂LA (yLA ) =
∆, yFA ≥ ϑM ,
where ϑM is the solution of the following equation in yLA for
∆ ≤ yLA ≤ τ :
M−1

fZ (yLA )
Fz (yLA − ∆)
.
(19)
=
fZ (yLA − ∆)
Fz (yLA )
If (19) does not have a solution, then ϑM = τ . Furthermore,
the probability of error of the LA detector is given by:

1
(M)
(20)
Pε,LA = Fz (ϑM − ∆)M +1−Fz (ϑM )M .
2

6

Proof: The proof follows along the same lines as the proof
of Proposition 2, and thus it is omitted.
The following theorem presents the diversity gain of the LA
detector.
Theorem 4. Let fYLA |X (yLA |x) be unimodal for all M > M0 .
Then, the diversity gain of the LA detector is given by:
DLA = − log (FZ (τ − ∆)) .

(21)

Proof: Similarly to the proof of Thm. 2, we note that if
fYLA |X (yLA |x) is unimodal for all M > M0 , then ϑM → τ
when M → ∞. This follows from the extreme value theorem
[30, Thm. 1.8.4], which implies that the limiting distribution
of the considered densities concentrates towards it maximal
value τ , thus θM → τ .
Next, we recall that ZLA = max{Zm }M
m=1 , and write ϑM =
(M)
τ − ηM , ηM → 0. Similarly to (13), Pε,LA can be bounded as
follows:
(M)

Pr{ZLA ≤ τ −ηM −∆} ≤ 2Pε,LA ≤ Pr{ZLA ≤ τ −∆}. (22)
For the RHS of (22), recalling that Pr{ZFA ≤ τ−∆} = FZ (τ−
∆)M , we write:

0. Therefore, the LA detector better be used only when τ is
finite and not too big (see the numerical results reported in
Section V).
Next we compare the system diversity gains achieved by the
two detectors as a function of τ . Let f˜Z (z) be a continuous,
differentiable, and unimodal density with support R+ . Further,
let the noise density fZ (z) be generated from f˜Z (z) via:
(˜
fZ (z)
, z≤τ
(25)
fZ (z) = F̃Z (τ )
0,
otherwise,
for 0 < τ ≤ ∞. Thus, fZ (z) is a clipped version of f˜Z (z)
(z)
at τ , and the CDF of Z is given by FZ (z) = F̃F̃Z (τ
. We now
)
Z
ask: What is the range of τ where DFA ≥ DLA ? To answer
this question we compare (12) and (21) and write:
log (1 − FZ (∆)) = log (FZ (τ − ∆))
!
!
F̃Z (τ − ∆)
F̃Z (∆)
= log
⇔ log 1 −
F̃Z (τ )
F̃Z (τ )
⇔ F̃Z (τ ) − F̃Z (∆) = F̃Z (τ − ∆).

(26)

− log Pr{ZLA ≤ τ −∆}
= − log(FZ (τ −∆)). (23)
M→∞
M
For the LHS we have Pr{ZLA ≤ τ − ηM − ∆} = FZ (τ −
ηM − ∆)M . Similarly to (15), using the Taylor expansion of
log FZ (τ − −ηM − ∆)M around τ −∆, we obtain:

Let τ ∗ be the solution of (26). Then, for a fixed ∆, for τ ≤
τ ∗ the LA detector achieves a higher system diversity gain,
while for for τ ≥ τ ∗ the FA detector achieves a higher system
diversity gain.
We conclude this section with two remarks discussing
extensions to the considered FA and LA detectors.

− log Pr{ZLA ≤ τ − ηM − ∆}
= − log(FZ (τ −∆)).
M→∞
M
(24)

Remark 6 (Other order statistics). While the ML detector (3)
is optimal for detection based on all the particle arrivals, the
FA and LA detectors, namely, (9) and (18), are optimal for
detection based only on the FA (LA) of a particle. One can
use order statistics theory to design optimal detectors based
on the first (last) M0 ≤ M particle arrivals. Yet, the analysis
of such detectors is significantly more involved. Moreover,
as indicated in the next section, combining the FA and LA
detectors based on τ ∗ can achieve system diversity gains very
close to those achieved by the ML detector.

lim

lim

Combining (22)–(24) concludes the proof.
We now consider the special case where the mode of fZ (z)
is at τ :
Theorem 5. Let fZ (z) be a continuous, differentiable, and
unimodal density with support (0, τ ), τ ≤ ∞ and mode mZ =
τ . Then, the LA and ML detectors are equivalent, namely, they
have the same probability of error.
Proof: Since mZ = τ , then fZ (y) ≥ fZ (y −∆), y ≤ τ . The
ML detection rule can be written as in (17). Therefore, the
ML detector declares X̂ML (y) = 0 only if there exists Ym > τ
(otherwise it declares X̂ML (y) = 0). Since testing if there exists
Ym > τ can be implemented based on yLA , we conclude that
in this case the ML detector reduces to the LA detector, and
therefore the detectors are equivalent.
The following corollary is a direct consequence of Thm. 5.
Corollary 2. Under the conditions of Thm. 5 DLA = DML .
C. Comparing the FA and LA Detectors
Thm. 3 and Thm. 5 imply that the FA and the LA detectors
are optimal for different types of noise densities. While the FA
detector is optimal when the mode is at zero, the LA detector
is optimal when the mode is at τ . Thus, FA is optimal for
monotonically decreasing densities, while LA is optimal for
monotonically increasing densities. It can further be observed
that for a finite ∆, if the support of fZ (z) is R+ , then DLA =

Remark 7 (Larger constellations). The FA, LA, linear, and ML
detectors can be extended to the case of larger constellations,
i.e., L > 2. In this case the ML detector requires comparing
all L hypotheses. On the other hand, as discussed in [26, Sec.
L−1
VI], given a simple choice of the constellation points {ξl }l=0
(see [26, Fig. 4]), optimal detection based on YFA (or YLA ) can
be implemented by comparing only two hypothesizes. These
two hypothesizes can be easily found based on their modes.
Since for the FA detector the conditional density concentrates
towards x (or for the LA detector towards τ ), for a fixed L one
can find large enough M such that any non-zero probability of
error can be achieved. This enables sending short messages of
several bits using a large number of particles. Note that when
L scales with M a more involved analysis is required. This
analysis involves the rate of convergence of δM to zero (or
ηM to τ ) for the specific noise density.
In the next section we explicitly evaluate the formulas
derived above and the resulting diversity gains for several
specific propagation densities: the uniform, exponential, IG,
and Lévy distributions. We also provide numerical analysis

7

of the system diversity gains for the clipped Lévy and IG
distributions, as a function of the clipping parameter τ .

DML
DFA
DLIN

V. N UMERICAL R ESULTS

∆ = 0.5
0.4766
0.4541
0.0308

∆=1
1.1070
1.1029
0.1180

∆ = 1.5
1.6657
1.6648
0.2499

TABLE II: DML , DFA and DLIN for IG(1, 1).

We begin our numerical study considering densities with
support R+ (i.e., the case of τ = ∞).
A. Densities With τ = ∞
Before discussing specific densities, we recall (21) which
implies that, for a finite ∆, if τ = ∞ then DLA = 0. Therefore,
in this subsection we focus on the diversity gain of the ML,
linear, and FA detector, and do not discuss the LA detector.
1) The Exponential Distribution: Let Z ∼ EXP(b), i.e.,
the exponential density with rate parameter b > 0. An explicit
evaluation of (4) results in DML = b∆, and following Corollary
1, we have DML = DFA . Considering the linear detector, for the
exponential density Λ∗Z (v) = bv−1−log(bv), v > 0. Moreover,
it can be shown that the α in (7) is given by:
α=

1 − eb∆ (1 − b∆)
.
(eb∆ − 1)b

(27)

3) The Lévy Distribution: We last consider the Lévy density, fZ (z) ∼ L(µ, b), with a location parameter4 µ and
a scale parameter b > 0. Similarly to the IG density, in
Appendix C it is shown that for fZ (z) ∼ L(µ, b), the conditions of Lemma 1 hold and therefore fZ (z) is unimodal
for sufficiently large M . Moreover, explicitly evaluating DML
seems intractable, yet, it can be evaluated numerically. For
the FA detector we use theq
CDF 
of the Lévy density to
b
, where erfc(·) is the
obtain DFA = − log 1−erfc
2∆

complementary error function. As for the linear detector we
recall Remark 3 which states that for Lévy-based propagation
DLIN = 0.
Table III details the system diversity gain for µ = 0, b = 1,
and ∆ ∈ {0.5, 1, 1.5}.

Plugging this value into (7) results in:
1 + eb∆ (b∆ − 1) − (eb∆ − 1) log
DLIN =

eb∆ − 1




b∆

b∆e
eb∆ −1

DML
DFA

.

(28)

As an example, let b = 1, and consider ∆ ∈ {0.5, 1.5, 2.5}.
Table I details the resulting diversity gains.
DML , DFA
DLIN

∆ = 0.5
0.5
0.0312

∆ = 1.5
1.5
0.2729

∆ = 2.5
2.5
0.7216

∆ = 0.5
0.1791
0.1711

∆=1
0.3828
0.3817

∆ = 1.5
0.5350
0.5348

TABLE III: DML and DFA for L(0,1).

It can be clearly observed that the gap in system diversity
gain between the FA and ML detectors is very small for both
the IG and Lévy densities.
Next, we consider noise densities with a finite support (i.e.,
τ < ∞).

TABLE I: DML , DFA and DLIN for EXP(1).

2) The Inverse-Gaussian Distribution: Let Z ∼ IG(µ, b),
i.e., the IG density with mean µ and shape parameter b > 0.
In Appendix C it is shown that for fZ (z) ∼ IG(µ, b), the
conditions of Lemma 1 hold and therefore fZ (z) is unimodal
for sufficiently large M . While explicitly evaluating DML
seems intractable, it can be evaluated numerically. For the FA
detector we use the CDF of the IG density to obtain:
r 
!
b ∆
DFA = − log 1−Φ
−1
∆ µ
r 
!!
2b
b
∆
+1
,
(29)
−e µ Φ −
∆ µ
where Φ(x) is the CDF of a standard
 qGaussianRV. Finally,
2
b
for the IG density ΛZ (τ ) = µ 1− 1− 2µb τ , τ ≤ 2µb 2 .

B. Densities With τ < ∞
We first consider the uniform distribution supported over
[0, τ ]. Since the uniform distribution is constant over its
support, according to Def. 1 the whole support constitutes the
mode of the density (in particular, both z = 0 and z = τ ).
1) The Uniform Distribution: Let fZ (z) ∼ U(0, τ ), τ > ∆,
i.e., the uniform density over [0, τ ]. Following Corollaries 1
τ
. For the linear detector
and 2, DML = DFA = DLA = log τ −∆
we do not have a closed form expression for Λ∗Z (v). However,
we can use the moment generating function (MGF) of the
uniform distribution and numerically find DLIN .
For instance, let τ = 1, and consider ∆ ∈ {0.25, 0.5, 0.75}.
Table IV details the resulting diversity gains. The table indicates large performance gains of the FA and LA detectors over
linear detection.

Explicitly calculating Λ∗Z (v) for the IG density we obtain:


b −µ2 + 2µ µv − 1 v + v 2
∗
.
(30)
ΛZ (v) =
2µ2 v

While for the IG density finding an explicit expression for
α seems intractable, it can be found numerically using (30).
Table II details the diversity gain for µ = 1, b = 1, and ∆ ∈
{0.5, 1, 1.5}.

DML , DFA , DLA
DLIN

∆ = 0.25
0.2879
0.0956

∆ = 0.5
0.6931
0.4086

∆ = 0.75
1.3863
1.0798

TABLE IV: DML , DFA and DLIN for U(0, 1).

Next, we consider the truncated (clipped) Lévy and IG
distributions.
4 Note

that µ is not the mean, as the mean of the Lévy density is ∞.

8

12

FA
LA
LIN
ML

IG, shape=1, mean=1

8

5

6
4
2
0
1.0

1.5

2.0

2.5
τ

3.0

3.5

1.1

8

1.2

1.3

1.4

1.5

τ

1.6

1.7

Fig. 3: Diversity gain versus τ for Z ∼ IG(1, 1, τ ) and ∆ = 1 (zoomed).

4.5
4.0

FA
LA
LIN
ML

Levy, dispersion=1

3.5
Diversity Gain

Diversity Gain

2

4.0

FA
LA
LIN
ML

Levy, dispersion=1

10

6
4

3.0
2.5
2.0
1.5

2
0
1.0

3

1

Fig. 2: Diversity gain versus τ for Z ∼ IG(1, 1, τ ) and ∆ = 1.
12

FA
LA
LIN
ML

IG, shape=1, mean=1

4
Diversity Gain

Diversity Gain

10

1.0
1.5

2.0

2.5
τ

3.0

3.5

4.0

Fig. 4: Diversity gain versus τ for Z ∼ L(0, 1, τ ) and ∆ = 1.

2) The Truncated Lévy and IG Distributions: Considering
the truncated Lévy and IG distributions is motivated by scenarios where the information particles degrade over time [21],
[24]. Specifically, these truncated distributions model a finite
lifespan of the particles, where the particles are dissipated
immediately after the time interval [0, τ ]. Such truncated noise
densities can be (approximately) achieved by using enzymes
or other chemicals to quickly degrade the particles [31]–[33].
Following the approach taken in Section IV-C, we let f˜Z (z)
be a continuous, differentiable, and unimodal density with
support R+ . The noise density fZ (z) is obtained by truncating
f˜Z (z) at τ . As stated in Section IV-C, the PDF and CDF are
˜
(z)
given by fZ (z) = F̃fZ (z)
and FZ (z) = F̃F̃Z (τ
, respectively.
)
Z (τ )
Z
Recall that the Lévy or IG densities correspond to diffusion
with and without a drift, respectively, and let IG(µ, b, τ ) denote
the IG distribution truncated at τ . Similarly, let L(µ, b, τ )
denote the Lévy distribution truncated at τ . Figs. 2 and 3
depict the diversity gain achieved by the ML, linear, FA, and
LA detectors, for Z ∼ IG(1, 1, τ ). It can be observed that for
large values of τ the FA detector achieves diversity gain very
close to the ML, while the LA and linear detectors perform
poorly. This extends the results reported in Table II. On the
other hand, when τ is small the diversity gain achieved by
the LA detector is very close to that achieved by the ML
detector; in fact, the curves are practically indistinguishable.
In this regime the FA detector performs poorly, while the linear
detector is superior to the FA yet inferior to the LA. Using
(26) we have that the curves corresponding to the FA and LA
detector intersect at τ ∗ = 1.208. Indeed, Fig. 2 indicate that

1.1

1.2

1.3

1.4

τ

1.5

1.6

1.7

1.8

Fig. 5: Diversity gain versus τ for Z ∼ L(0, 1, τ ) and ∆ = 1 (zoomed).

by using the LA detector for τ < 1.208 and the FA detector
for τ ≥ 1.208, one achieves diversity gain very close to the
diversity gain achieved by the ML detector.
Figs. 4 and 5 depict the diversity gain achieved by the ML,
linear, FA, and LA detectors, for Z ∼ L(0, 1, τ ). Similarly to
the truncated IG, the FA performs well for large values of τ
(which supports the results reported in Table III), while the
LA performs well for small values of τ . Here, (26) leads to
τ ∗ = 1.282.
VI. C ONCLUSION
We have studied one-shot communication over molecular
timing channels assuming that M information particles are
simultaneously released and that their propagation follows
a unimodal density. We defined the system diversity gain
D to be the exponential rate of decrease of the detection
(M)
probability of error Pε
when M grows asymptotically large.
We then derived closed form expressions for the D achievable
by four detectors: the optimal ML detector, a linear detector,
the FA detector, and the LA detector. We showed that the
FA detector achieves a diversity gain very close to that of
the ML detector, while being simpler and having significantly
shorter delays, when the density of the noise is supported
over a large interval (for instance R+ ). In particular, for
delay densities where the mode of the density is zero, the
FA detector is optimal. We also showed that when the density
of the noise is supported over a small interval, the LA detector
achieves a diversity gain very close to that of the ML detector.
Particularly, for delay densities where the mode of the density

9

′

is at the maximum of its support, the LA detector is optimal.
Our numerical evaluations show that by combining the FA
and LA detectors one can achieve performance very close to
that of the ML detector for all ranges of support intervals.
Specifically, for almost every support interval, this combined
detector outperforms the linear detector. We conclude that
this combined detector constitutes a low-complexity near-ML
detection framework for one-shot communication over timing
channels.

f (z)(1−F (z))

that Z f 2 (z)Z
will also be monotonically decreasing. In
Z
such case there is a z0 for which:
′
z < z0
fZ (z)(1−FZ (z))
≷ M − 1.
fZ2 (z)
z > z0

(A.4)

Hence, the density is unimodal
for all M > M0 , where
 ′
fZ (ξ)(1−FZ (ξ))
+ 1 and ξ =
M0 is given by M0 =
f 2 (ξ)
Z

′

f (z)(1−F (z))
argmaxz Z f 2 (z)Z
, z > ǫ.
Z

A PPENDIX A
P ROOF OF L EMMA 1
A unimodal density supported on R+ belongs to one of the
following classes of densities:
1) Unimodal densities with mode mZ = 0.
2) Unimodal densities with mZ > 0, and limz→0+ fZ (z) =
ν > 0.
3) Unimodal densities with mZ > 0, and limz→0+ fZ (z) = 0.
We next show that densities from the first two classes are unimodal for sufficiently large M regardless of the conditions of
Lemma 1. Then, we show that the conditions stated in Lemma
1 ensure that densities from the third class are unimodal for
sufficiently large M .
From Def. 1, if a unimodal density has more than a single
maximum, then the maximum must be a continuous interval.
Thus, the derivative of the density changes its sign at most
once. Next, recall that fZFA (z) = M ·fZ (z)·(1−FZ (z))M−1 .
Hence, the derivative of fZFA (z) is given by:
 ′
′
fZFA (z) = M fZ (z)(1−FZ (z))M−1

−fZ2 (z)(M − 1)(1−FZ (z))M−2 . (A.1)
′

Setting fZFA (z) = 0 we obtain conditions indicating when
fZFA (z) decreases:
′

′

fZFA (z) ≤ 0

fZ (z)(1−FZ (z))
≤ M − 1.
fZ2 (z)

⇔

(A.2)
′

For densities that belong to the first class we have fZ (z) ≤ 0.
′
Therefore, as (1−FZ (z)) and fZ2 (z) are positive, fZFA (z) is
non-increasing and unimodal for any M .
For the second class
we note that in the range 0 < z ≤ mZ ,
′
f (z)(1−F (z))

fZ2 (z) ≥ ν 2 . Thus, Z f 2 (z)Z
is positive and bounded, and
Z
by choosing M large enough fZFA (z) is decreasing for any z
and therefore unimodal.
Finally, for densities in the third class with
limz→0+ fZ (z) = 0, since fZ (z) is assumed to be
differentiable, it is possible that:5

A PPENDIX B
P ROOF OF L EMMA 2
Similarlly to the derivation in Appendix A, Def. 1 implies
that if a unimodal density has more than a single maximum,
then the maximum must be a continuous interval. Thus, the
derivative of the density changes its sign at most once. Next,
recall that fZLA (z) = M · fZ (z) · (FZ (z))M−1 . Hence, the
derivative of fZLA (z) is given by:

 ′
′
fZLA (z)=M FZ (z)M−2 fZ (z)FZ (z)+fZ2 (z)(M −1) . (B.1)
′

Setting fZLA (z) = 0 we obtain conditions indicating when
fZLA (z) decreases:
′

′

fZLA (z) ≤ 0

fZ (z)FZ (z)
≤ −(M − 1).
fZ2 (z)

⇔

(B.2)

Thus we need to show that if fZLA (z) starts decreasing it does
not increase again. To show this we recall that the support of
′
fZ (z) is (0, τ ) and that |fZ (z)| < ∞. We now consider two
cases:
1) limz→τ fZ (z) = ν > 0
2) limz→τ fZ (z) = 0
For the first case (B.2) implies that then there exist an M0
such that for M > M0 , fZLA (z) is increasing for all z in
(0, τ ). Thus, in this case fZLA (z) is clearly unimodal.
′
For the second case we note that (B.2) holds only if fZ (z) <
0 (since FZ (z) and fZ2 (z) are positive). Moreover, only when
z → τ it is possible that:
′

f (z)FZ (z)
= −∞,
lim Z 2
z→τ
fZ (z)

(B.3)

The condition of the lemma
ensures that there is an interval
′
fZ (z)FZ (z)
(τ − ǫ, τ ), ǫ > 0, where
monotonically decreases.
2 (z)
fZ
Therefore, by choosing M large enough it can be guaranteed
that once fZLA (z) starts decreasing it does not increase again,
thus, it is unimodal.

′

lim+

z→0

fZ (z)(1−FZ (z))
= ∞.
fZ2 (z)

(A.3)

Since (1−FZ (z)) is monotonically decreasing with z, requiring
′

that

fZ (z)
2 (z)
fZ

will decrease monotonically for 0 < z < ǫ ensures

5 Recall that since m > 0, then there exists an ǫ > 0 such that 0 ≤
Z
′
fZ (z), 0 < z < ǫ.

A PPENDIX C
T HE C ONDITIONS OF L EMMA 1 AND L EMMA 2 FOR THE
L ÉVY AND IG D ENSITIES
′

f (z)

In this section we evaluate the function g(z) = fZ2 (z) for the
Z
Lévy and IG Densities. We show that for z small enough the
derivative of g(z) is negative and therefore it monotonically

10

decreases as required. We begin with the Lévy distribution
where we write:
r
π c
fZ′ (z)
gLev (z) = 2
(C.1)
=
e 2z (c − 3z).
fZ (z)
2cz

Thus, g ′ (z) is given by:
r
c
π
′
e 2z (c2 − 2cz + 3z 2 ).
gLev (z) = −
5
8cz

(C.2)

′
Observe that gLev
(z) < 0 for any finite z and c > 0. Thus,
for the Lévy distribution, the conditions of both lemmas hold.
Next, we consider the IG distribution, where we have:
r
b(z−µ)2
π
2µ2 z (b(µ2 − z 2 ) − 3µ2 z),
(C.3)
gIG (z) =
e
2µ4 bz

and
b(z−µ)2
π
2µ2 z
e
8µ8 bz 5
× (3µ4 z 2 +b2 (µ2 −z 2 )2 +b(6µ2 z 3 −2µ4 z)).
(C.4)

′
gIG
(z) = −

r

′
Again, it can be shown that gIG
(z) < 0 for any finite z > 0
and µ, b > 0. Thus, for the IG distribution, the conditions of
both lemmas hold.

ACKNOWLEDGMENT
The authors would like to thank Andrea Montanari for
comments which greatly simplified the proof of Thm. 2.
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