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arXiv:2203.02784v1 [cs.IT] 5 Mar 2022
1
Deterministic Identification for Molecular
Communications over the Poisson Channel
Mohammad J. Salariseddigh, Uzi Pereg, Holger Boche, Christian Deppe,
Vahid Jamali, and Robert Schober
Abstract Various applications of molecular communications (MC) are event-triggered, and, as a consequence,
the prevalent Shannon capacity may not be the right measure for performance assessment. Thus, in this
paper, we motivate and establish the identification capacity as an alternative metric. In particular, we study
deterministic identification (DI) for the discrete-time Poisson channel (DTPC), subject to an average and a
peak power constraint, which serves as a model for MC systems employing molecule counting receivers. It is
established that the codebook size for this channel scales as 2(nlog n)R, where nand Rare the codeword length
and coding rate, respectively. Lower and upper bounds on the DI capacity of the DTPC are developed. The
obtained large capacity of the DI channel sheds light on the performance of natural DI systems such as natural
olfaction, which are known for their extremely large chemical discriminatory power in biology. Furthermore,
numerical simulations for the empirical miss-identification and false identification error rates are provided for
finite length codes. This allows us to quantify the scale of error reduction in terms of the codeword length.
Index Terms
Channel capacity, deterministic identification, molecular communication, and Poisson channel
I. Introduction
Molecular communication (MC) is a new paradigm in communication engineering where information
is transmitted via signaling molecules [2], [3]. Over the past decade, synthetic MC has been extensively
studied in the literature from different perspectives including channel modeling [4], modulation and
detection design [5], biological building blocks for transceiver design [6], and information theoretical
This paper was presented in part at the IEEE Global Communications Conference (GC 2021) [1].
M. J. Salariseddigh, U. Pereg, and C. Deppe are with the Institute for Communications Engineering, Technical University of
Munich, Germany (e-mail: {mjss, uzi.pereg, christian.deppe}@tum.de). H. Boche is with the Chair of Theoretical Information
Technology, Technical University of Munich, Germany (e-mail: boche@tum.de). V. Jamali is with the Department of Electrical
and Computer Engineering, Princeton University, USA (e-mail: jamali@princeton.edu). R. Schober is with the Institute for Digital
Communications, Friedrich-Alexander University Erlangen-Nürnberg (FAU), Germany (e-mail: robert.schober@fau.de).
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performance characterization [7], [8]. Moreover, several proof-of-concept implementations of synthetic
MC systems have been reported in the literature, see, e.g., [9]–[11].
Despite the recent theoretical and technological advancements in the field of MCs, the fundamental
channel capacity limits of most MC systems are still unknown. A mathematical foundation for information
theoretical analysis of diffusion-based MC is established in [12] where a channel coding theorem is
proved. The information rate capacity of diffusion-based MC was studied in [13] where both channel
memory and molecular noise are taken into account. For diffusion-based MC, the capacity limits of
molecular timing channels are investigated in [14] and lower and upper bounds on the corresponding
capacity are reported. In [15], a new characterization of capacity limits and capacity achieving distribu-
tions for the particle-intensity channel are studied. Capacity bounds on point-to-point communication are
studied in [8] and a corresponding mathematical framework is established. A comprehensive overview
of mathematical challenges and relevant mathematical tools for studying molecular channels is provided
in [7].
Various applications of MC within the framework of sixth generation wireless networks (6G) [16],
[17] are associated with event-triggered systems, where Shannon’s message transmission capacity, as
considered in [7], [8], [12]–[15], may not be the appropriate performance metric. In particular, in event-
detection scenarios, where the receiver wishes to decides about the occurrence of a specific event in terms
of a reliable Yes / No answer, the so-called identification capacity is the relevant performance measure
[18]. Specific examples of the identification problem in the context of MC can be found in targeted
drug delivery [19], [20] and cancer treatment [21]–[23], where, e.g., a nano-device’s objective may be to
identify whether or not a specific cancer biomarker exists around the target tissue; in health monitoring
[24], [25] where, e.g., one may be interested in whether or not the pH value of the blood exceeds a
critical threshold; in natural pheromone communications [26], [27] where, e.g., a male insect searches for
sex pheromones indicating the presence of a nearby female insect; etc. For these tasks, the identification
capacity is deemed to be a key quantitative measure [16], [17]. Motivated by this discussion, this paper
focuses on the problem of identification in the context of MC systems.
A. Related Work on Identification Capacity
In Shannon’s communication paradigm [28], a sender, Alice, encodes her message in a manner that
will allow the receiver, Bob, to reliably recover the message. In other words, the receiver’s task is to
determine which message was sent. In contrast, in the identification setting, the coding scheme is designed
to accomplish a different objective [18]. The decoder’s main task is to determine whether a particular
message was sent or not, while the transmitter does not know which message the decoder is interested
in. Ahlswede and Dueck [18] introduced a randomized-encoder identification (RI) scheme, in which the
codewords are tailored according to their corresponding random source (distributions). It is well-known
that such distributions do not increase the transmission capacity for Shannon’s message transmission
task [29]. On the other hand, Ahlswede and Dueck [18] established that given local randomness at the
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encoder, reliable identification is accomplished with a codebook size that is double-exponential in the
codeword length n, i.e., ∼22nR [18], where Ris the coding rate. This behavior differs radically from the
conventional message transmission setting, where the codebook size grows only exponentially, with the
codeword length, i.e., ∼2nR. Therefore, RI yields an exponential gain in the codebook size compared to
the transmission problem.
Other non-standard properties of the RI capacity compared to the message transmission capacity can be
observed for the discrete memoryless channel (DMC) with feedback, namely, strictly causal feedback from
the receiver to the transmitter can increase the identification capacity of a DMC [30], but not the message
transmission capacity [31]. Furthermore, for the compound wiretap channel [32], the secure RI capacity
fulfills a dichotomy theorem that is in strong contrast to the transmission capacity, namely, when the secure
RI capacity is greater than zero a price is not paid for secure identification, i.e., the secure RI capacity
coincides with the RI capacity. Other unusual behaviors of the RI capacity, in terms of computability and
continuity for a correlation-assisted DMC are reported in [33]. For instance, the identification capacity for
the DMC is not Turing computable. Also, it cannot be represented as the maximization of a continuous
function. The construction of RI codes is considered in [34]–[37]. Nevertheless, it can be difficult to
implement RI codes. Therefore, from a practical point of view, it is of interest to consider the case where
the codewords are not selected based on a distribution but rather by means of a deterministic mapping
from the message set to the input space. In the literature, this approach is also referred to as identification
without randomization [38] or deterministic identification (DI) [39]–[41].
In the deterministic coding setup for identification, for DMCs, the codebook size grows only exponen-
tially in the codeword length, similar to the conventional transmission problem [18], [38], [40], [42], [43].
However, the achievable identification rates are significantly higher compared to the transmission rates
[40], [41]. Deterministic codes often have the advantage of simpler implementation and simulation [44],
[45] and explicit construction [46]. In our recent works [39]–[41], we have considered DI for channels
with an average power constraint, including DMCs and Gaussian channels with fast and slow fading,
respectively. In the Gaussian case, we have shown that the codebook size scales as 2(nlog n)R, by deriving
bounds on the DI capacity. Furthermore, DI for Gaussian channels is also studied in [39], [47]–[49].
B. Contributions
In this work, we consider MC systems employing molecule counting receivers, where the received signal
has been shown to follow the Poisson distribution1when the number of released molecules is large, see
[4, Sec. IV], [53], [54] for details. To the best of the authors’ knowledge, the identification capacity of
the DTPC has not been studied so far. In particular, we consider DI over a DTPC under average and
peak power constraints that account for the limited molecule production / release rate of the transmitter.
1The discrete-time Poisson channel (DTPC) is also a useful model for optical communication systems with direct-detection
receivers [50]–[52].
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By deriving positive bounds on the DI capacity of the DTPC, we establish that the codebook size for
deterministic encoding scales as 2(nlog n)R.
The approach to derive the bounds on the capacity is similar to that for Gaussian channels [39], namely,
to obtain the lower bound we exploit the existence of an appropriate sphere packing within the input
space where the mutual distance between the centers of the spheres does not fall below a certain value,
and for the upper bound, we assert a certain minimum distance between the codewords of any given
sequence of codes with vanishing error probabilities. However, the analysis and the upper bound for
the DTPC are different. Here, in the achievability proof, we consider the packing of hyper spheres with
radius ∼n1
4inside a larger hyper cube. While the radius of the small spheres in the Gaussian case
[39] tends to zero, here the radius grows in the codeword length, n. Yet, we show that we can pack a
super-exponential number of spheres within the larger cube. In the converse part of the proof for the
DTPC, the derivation of the upper bound on the capacity is more involved compared to that for Gaussian
channels [39] and leads to a larger upper bound. Instead of establishing a minimum distance between
the codewords (codeword-wise distance), as in the Gaussian case [39], we use a criterion imposed on the
symbols of every two codewords, namely, we show that for each pair of different codewords, there exists
at least one index for which the ratio of the corresponding symbols is different from 1 (symbol-wise
distance).
The enlarged codebook size of the identification problem compared to the transmission problem may
have interesting implications for MC system design. To elaborate on this, we consider two general
categories of codes, where the channel uses within each codeword are realized in different manners,
namely spatial and temporal codes. For spatial codes, a different type of signaling molecule is used for
each channel use. Such a setup can be used to model molecule-mixture communications in mammalian
and insect olfactory systems, where a given mixture of different types of molecules represents a codeword
[55]–[57]. Thereby, the results presented in this paper may shed light on the extremely large identification
capability of natural olfactory systems. In contrast, for temporal codes, different channel uses are realized
by releasing the same type of molecules in different time instances. On the one hand, the hardware
complexity for temporal codes is lower since only one type of molecule is needed; on the other hand,
the receiver has to be equipped with memory to store and jointly process the observations of all channel
uses within one codeword. Temporal codes may find applications in synthetic MC identification systems
used in, e.g., targeted drug delivery and environmental monitoring [4], [6].
Although our theoretical results target the capacity of the DI channel in the standard asymptotic
definition, i.e., as n→ ∞, we also provide numerical simulations for finite size codes. In particular, our
numerical simulations evaluate the empirical miss-identification (type I) and false identification (type II)
error rates of the proposed achievable scheme. These results allow us to quantify the error reduction in
terms of the codeword length and shed light on the error behavior as a function of the codeword length.
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C. Organization
The remainder of this paper is structured as follows. In Section II, scenarios for application of DI in the
context of MC are discussed and the required preliminaries regarding DI codes are established. Section III
provides the main contributions and results on the message identification capacity of the DTPC. Section IV
presents simulation results for the empirical type I and type II error rates. Finally, Section V of the paper
concludes with a summary and directions for future research.
D. Notations
We use the following notations throughout this paper: Calligraphic letters X,Y,Z,...are used for finite
sets. Lower case letters x, y, z, . . . stand for constants and values of random variables, and upper case letters
X,Y,Z,... stand for random variables. Lower case bold symbol xindicates a row vector of size n, that is,
x= (x1,...,xn). Bold symbol 1nindicates the all-one row vector of size n. The distribution of a random
variable Xis specified by a probability mass function (pmf) pX(x)over a finite set X. All logarithms and
information quantities are for base 2. The set of consecutive natural numbers from 1through Mis denoted
by [[M]]. The set of whole numbers is denoted by N0,{0,1,2,...}. The gamma function for non-positive
integer xis denoted by Γ(x)and is defined as Γ(x) = (x−1)!, where (x−1)! ,(x−1) ×(x−2) ×···×1.
We use the small O notation, f(n) = o(g(n)), to indicate that f(n)is dominated by g(n)asymptotically,
that is, limn→∞ f(n)
g(n)= 0. The big O notation, f(n) = O(g(n)), is used to indicate that |f(n)|is bounded
above by g(n)(up to constant factor) asymptotically, that is, lim supn→∞ |f(n)|
g(n)<∞. We use the big
Omega notation, f(n) = Ω(g(n)), to indicate that f(n)is bounded below by g(n)asymptotically, that
is, g(n) = O(f(n)). The ℓ2-norm and ℓ∞-norm are denoted by kxkand kxk∞, respectively. Furthermore,
we denote the n-dimensional hyper sphere of radius rcentered at x0with respect to the ℓ2-norm by
Sx0(n, r) = {x∈Rn:kx−x0k ≤ r}. An n-dimensional cube with center (A
2,...,A
2)and a corner at the
origin, i.e., 0= (0,...,0), whose edges have length Ais denoted by Q0(n, A) = {x∈Rn:0≤xt≤
A, ∀t∈[[n]]}.
II. System Model, MC Scenarios for DI, and Preliminaries
In this section, we present the adopted system model, introduce MC scenarios for DI, and establish
some preliminaries regarding DI coding.
A. System Model
We focus on an identification setup, where the decoder wishes to reliably determine whether or not
a particular message was sent by the transmitter, while the transmitter does not know which message
the decoder is interested in, see Figure 1. To achieve this objective, we establish a coded communication
between the transmitter and the receiver over nchannel uses of an MC channel. We consider a stochastic
release model, see Figure 2, where for the t-th channel use, the transmitter releases molecules with
rate xt(molecules/second) over a time interval of Trls seconds into the channel [7]. These molecules
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Information
Source
Transmitter
Encoding Release
Mechanism Diffusion/Advection/Reaction
Processes
Physical Channel Receiver
Yes/No
Reception
Mechanism Decoding
Fig. 1. End-to-end chain of communication setup for DI in a generic MC system. Receiver declares a Yes/ No in a reliable manner.
propagate through the channel via diffusion and/or advection, and may even be degraded in the channel
via enzymatic reactions [4]. We assume a counting-type receiver which is able to count the number of
received molecules. Examples include the transparent (perfect monitoring or passive) receiver, which
counts the molecules at a given time within its sensing volume [11], the fully absorbing (perfect sink)
receiver, which absorbs and counts the molecules hitting its surface within a given time interval [58], and
the reactive (ligand-based) receiver which counts the number of molecules bound to the ligand proteins
on its sensing surface at a given time [59]. Assuming that the release, propagation, and reception of
individual molecules are statistically similar but independent of each other, the received signal follows
Poisson statistics when the number of released molecules is large, i.e., xtTrls ≫1[4, Sec. IV]. Let X∈R≥0
and Y∈N0denote random variables (RVs) modeling the rate of molecule release by the transmitter and
the number of molecules observed at the receiver, respectively. The input-output relation for the DTPC
is given as follows
Y=Pois (ρX +λ),(1)
where ρX is the mean number of observed molecules due to release by the transmitter, ρ=pchTrls,
and pch ∈(0,1] denotes the probability that a given molecule released by the transmitter is observed
at the receiver. The value of pch depends on the propagation environment (e.g., diffusion, advection,
and reaction processes) and the reception mechanism (e.g., transparent, absorbing, or reactive receiver)
as well as the distance between transmitter and receiver, see [4, Sec. III] for the characterization of pch
for various setups and Figure 3 for an example illustration of reception process via absorbing receiver.
Moreover, λ∈R>0is the mean number of observed interfering molecules originating from external noise
sources which employ the same type of molecule as the considered MC system.
The letter-wise conditional distribution of the DTPC output is given by
W(y|x) = e−(ρx+λ)(ρx +λ)y
y!.(2)
Standard transmission schemes employ strings of letters (symbols) of length n, referred to as codewords,
that is, the encoding schemes use the channel in nconsecutive times to transmit one message. As a
consequence, the receiver observes a string of length n, referred to as output vector (received signal). We
assume that different channel uses are orthogonal. This assumption is justified for different MC scenarios
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Encoder
Particle Storage
xt
Fig. 2. Transmission configuration in a Poisson concentration type. The value of the channel input, xtcontrols the outlet size
bio-chemically where it adjusts the width of the gate, that is, xt= 0 represents a closed gate and xt=Aaccounts to a maximal
opening. At every time instance t, the width of the open gate, marked in blue, increases with the input value xt. The exact number
of emitted particles is modelled by a Poisson distribution with mean of xtTrls for Trls to be the symbol interval.
in Section II-B. Therefore, for nchannel uses, the transition probability law reads
Wn(y|x) =
n
Y
t=1
W(yt|xt) =
n
Y
t=1
e−(ρxt+λ)(ρxt+λ)yt
yt!,(3)
where x= (x1,...,xn)and y= (y1,...,yn)denote the transmit codeword and the received signal,
respectively. The codewords are subject to peak and average power constraints as follows
0≤xt≤Pmax and 1
n
n
X
t=1
xt≤Pavg ,(4)
respectively, ∀t∈[[n]], where Pmax, P avg >0constrain the rate of molecule release per channel use and over
the entire nchannel uses in each codeword, respectively. We note that while the average power constraint
for the Gaussian channel is a non-linear (square) function of the symbols (signifying the signal energy),
here for the DTPC, it is a linear function (signifying the number of released molecules) [7].
Spherical Receiver
Information Molecule
Fig. 3. Trajectory of captured particles within the surface of an absorbing receiver. Each small propagation step is governed by
Brownian motion. Once the information particles hits the absorbing boundary, they will turn in-reversibly into another type of
molecule. As a result, such hitting particles will immediately disappear from the environment.
8
TX RX
Fig. 4. Illustration of an olfactory-inspired MC system, where three orthogonal molecule types, namely type square, triangle, and
circle, are shown. The transmitter secretes a mixture of these molecules corresponding to a particular message. Each receptor located
on the receiver surface is sensitive to only one type of molecule. The receiver’s task is to determine whether or not a desired message
(molecular mixture) has been sent by the transmitter.
B. Spatial vs. Temporal Channel Uses
The coding scheme proposed in this paper requires nindependent uses of the MC channel; however,
how the MC channel is accessed for each channel use may depend on the application of interest. In
the following, we introduce two application scenarios, which employ spatial and temporal channel uses,
respectively.
Spatial channel use: The mammalian olfactory system is believed to possess the capacity of discriminat-
ing many thousands of different chemical mixtures, where each chemical mixture is often associated with a
specific event or an external object [55], [60], [61]. Examples of such mixtures include mating pheromones,
social pheromones, food odorants, repellent odorants (e.g., implying toxic materials or predators), etc.
Hence, the Ahlswede-Dueck identification problem applies to natural olfaction since each molecular mixture
conveys a particular message that the receiver may be interested in. Here, each molecular mixture can be
seen as a codeword, where the channel is accessed spatially and simultaneously via the different types
of molecules in the mixture. The extraordinary discriminatory power of natural olfactory systems has
motivated researchers to develop biosensors that mimic the structural and functional features of natural
olfaction [62], [63]. Motivated by this, we consider a communication scenario, where the transmitter
releases a mixture of ndifferent types of molecules to convey a message to the receiver, see Figure 4.
The receiver is equipped with a dedicated type of receptor for each type of molecule, which ensures the
orthogonality of the nchannel uses. The receiver’s task is to determine whether or not a desired message
(molecular mixture) has been sent by the transmitter.
Temporal channel use: For spatial channel uses, the complexity of transmitter and receiver may be
high as they have to be able to generate and detect ndifferent types of molecules, respectively. To avoid
this complexity, one may employ only one type of molecule and access the MC channel at different
time instances. Thereby, transmitter and receiver have to be equipped with memory for generation and
9
processing of all nchannel uses, respectively. In addition, due to the dispersive nature of the diffusive
MC channel, the channel has memory and proper measures have to be taken to ensure the orthogonality
of different channel uses. An immediate approach is to make the symbol duration sufficiently large such
that the channel response (practically) decays to zero within each symbol interval. However, this may
lead to an inefficient design due to the reduction of the rate of channel access. More efficient approaches
proposed in the literature include the use of enzymes [64] and reactive cleaning molecules [65] to generate
a concentrated channel response for the desired signaling molecules, see, e.g., [4, Fig. 15].
C. DI Coding for the DTPC
The definition of a DI code for the DTPC is given below.
Definition 1 (Poisson DI Code).An L(n, R), n, λ1, λ2DI code for a DTPC Wunder average and peak
power constraints of Pave and Pmax, respectively, and for integer L(n, R), where nand Rare the codeword
length and coding rate, respectively, is defined as a system (U,D)which consists of a codebook U=
{ui}i∈[[L]] ⊂ Rn, such that
0≤ui,t ≤Pmax and 1
n
n
X
t=1
ui,t ≤Pavg ,(5)
∀i∈[[L]],∀t∈[[n]], and a collection of decoding regions D={Di}i∈[[L]] with
L(n,R)
[
i=1 Di⊂Nn
0.
Given a message i∈[[L]], the encoder transmits ui, see Figure 5, the decoder’s aim is to answer the
following question: Was a desired message jsent or not? There are two types of errors that may occur:
Rejection of the true message or acceptance of a false message. These errors are referred to as type I and
type II errors, respectively.
The corresponding error probabilities of the identification code (U,D)are given by
Pe,1(i) = 1 −X
y∈Di
Wnyui(miss-identification error) ,(6)
Pe,2(i, j) = X
y∈Dj
Wnyui(false identification error) .(7)
and satisfy the following bounds
Pe,1(i)≤λ1and Pe,2(i, j)≤λ2,(8)
∀i, j ∈
i6=j[[L]] and every λ1, λ2>0. A rate R > 0is called achievable if for every λ1, λ2>0and sufficiently
large n, there exists an (L(n, R), n, λ1, λ2)DI code. The operational DI capacity of the DTPC is defined
as the supremum of all achievable rates, and is denoted by CDI (W, L).
III. DI Capacity of the DTPC
In this section, we first present our main results, i.e., lower and upper bounds on the achievable
identification rates for the DTPC. Subsequently, we provide the detailed proofs of these bounds.
10
iEnc Molecular Channel Dec
j
Yes/No
uiY
Fig. 5. DI scheme for MC with under the DTPC model. Relevant processes in the molecular channel are diffusion, advection, and
reaction. Ytis Poisson distributed with rate ρui,t +λ. Given the observation Y, decoder asks whether jequals ior not.
A. Main Results
The DI capacity theorem for the DTPC is stated below.
Theorem 1.The DI capacity of the DTPC Wsubject to average and peak power constraints of the form
n−1Pn
t=1 ui,t ≤Pave and 0≤ui,t ≤Pmax, respectively, in the super-exponential scale, i.e., L(n, R) =
2(nlog n)R, is bounded by
1
4≤CDI (W, L)≤3
2.(9)
Proof: The proof of Theorem 1 consists of two parts, namely the achievability and the converse proofs,
which are provided in Sections III-B and III-C, respectively.
Before we provide the proof, we highlight some insights obtained from Theorem 1 and its proof.
Scale: Theorem 1 shows a different behavior compared to the traditional scaling of the codebook size
with respect to codeword length n. The bounds given in Theorem 1 are valid in the super-exponential
scale of L= 2(nlog n)Rwhich is in between the conventional exponential and double exponential codebook
sizes (see Figure 6). Given such a codebook size it follows [40, see Rem. 1] that the capacity values in
the standard codebook sizes, i.e., exponential and double exponential, are infinite and zero, respectively.
Budget for Molecule Release: The proposed capacity bounds in the super-exponential scale are inde-
pendent of the values of Pave and Pmax as long a the codeword length ngrows sufficiently large, i.e.,
n→ ∞. However, for finite n, the codebook size is indeed a function of Pave and Pmax. This can be readily
seen from the achievability proof, where the codebook size in its raw form (see (21)) before division by
the dominant term reads
L(n, R) = 2(nlog n)R+n(log A
e√a)+o(n).(10)
where A= min (Pave, P max )and a > 0is a parameter of the codebook construction, cf. (13). In other
words, the codebook size increases as Aincreases; however, since Aappears in a term that is exponential
in n, i.e., ∼2n(log A
e√a), the influence of Abecomes negligible compared to the dominant super-exponential
term, i.e., 2(nlog n)Ras n→ ∞2. While the proof of Theorem 1 mainly concerns the asymptotic regime of
n→ ∞, we are still able to get some insight for finite n, too. For instance, the error constraints in (8)
2It is interesting to recall that the codebook size for the transmission capacities of both the DTPC [66, see Eq. (5)] and the
Gaussian channel [67], [68] scale with 2nlog √Ain terms of A.
11
L(n, R)
2√nR
Covert Commun. (DMC)
2nR
Transmission (DMC & Poisson)
2(nlog n)R
DI (Gaussian & Poisson)
22√nR
Covert ID (BIDMC)
22nR
Randomized ID (DMC & Poisson)
222...2nR
DI with Feedback (Gaussian)
Fig. 6. Spectrum of codebook sizes for different transmission and identification setups. Apart from the conventional exponential
and double exponential codebook sizes for transmission [28] and RI [18], respectively, different non-standard codebook sizes are
observed for other communication tasks, such as covert communication [69] or covert identification [70] for the binary-input DMC
(BIDMC), where the codebook size scales as 2√n R and 22√nR , respectively. For the Gaussian DI channel with feedback [47], the
codebook size can be arbitrarily large. In [48] the result is generalized for channels with non-discrete additive white noise and
positive message transmission feedback capacity.
can be met by the proposed achievable scheme even for finite nif Ais sufficiently large and a=Ω(A2),
cf. (31), (40), and (42). A comprehensive study of the achievable DI rates for finite nconstitutes an
interesting research topic for future work, but is beyond the scope of this paper.
Adopted Decoder: For the achievability proof, we adopt a decoder that upon observing an output
sequence y, declares that the message jwas sent if the following condition is met
y−Eyuj
2
−E
y−Ey|uj
2uj≤nδn.(11)
where uj= [uj,1,...,uj,n]is the codeword associated with message jand δnis a decoding threshold. In
contrast to the popular distance decoder used for Gaussian channels [39] that includes only the distance
term ky−E(y|uj)k, the proposed decoder in (11) comprises the additional correction term Eky−
E(y|uj)k2|uj. This choice stems from the fact that the noise in the DTPC is signal dependent [4].
Therefore, the variance of ky−E(y|uj)kdepends on the adopted codeword uj, which implies that
unlike for the Gaussian channel, here the radius of the decoding region is not constant for all codewords.
B. Achievability
Consider DTPC W. We show achievability of (9) using a packing of hyper spheres and a distance
decoder. We pack hyper spheres with radius ∼n1
4inside a larger hyper cube. While the radius of the
spheres in a similar proof for Gaussian channels vanishes, as nincreases [39], the radius here diverges to
infinity. Yet, we can obtain a positive rate while packing a super-exponential number of spheres satisfying
the power and error constraints in (6)-(8). A DI code for the DTPC Wis constructed as follows.
Codebook construction: Let
A= min (Pave, P max ).(12)
12
A/2
√nǫn
A√n
Fig. 7. Illustration of a saturated sphere packing inside a cube, where small spheres of radius r0=√nǫncover a larger cube.
Yellow colored spheres are not entirely contained within the larger cube, and yet they contribute to the packing arrangement. As
we assign a codeword to each sphere center, the 1-norm and arithmetic mean of a codeword are bounded by Aas required.
In the following, we restrict ourselves to codewords that meet the condition 0≤xt≤A,∀t∈[[n]]. We
argue that this condition ensures both the average and the peak power constraints in (4). In particular,
when Pave ≥Pmax, then A=Pmax and the constraint 0≤xt≤Aautomatically implies that the constraint
1
nPxt≤Pave is met, hence, in this case, the setup with average and peak power constraints simplifies
to the case with only a peak power constraint. On the other hand, when Pave < P max , then A=Pave
and by 0≤xt≤A,∀t∈[[n]], both power constraints are met, namely 1
nPxt≤Pave and 0≤xt≤Pmax,
∀t∈[[n]]. Hence, in the following, we restrict our considerations to a hyper cube with edge length A.
We use a packing arrangement of non-overlapping hyper spheres of radius r0=√nǫnin a hyper cube
with edge length A, where
ǫn=a
n1
2(1−b),(13)
and a > 0is a non-vanishing fixed constant and 0< b < 1is an arbitrarily small constant.
Let Sdenote a sphere packing, i.e., an arrangement of Lnon-overlapping spheres Sui(n, r0),i∈[[L]],
that are packed inside the larger cube Q0(n, A)with an edge length A, see Figure 7. As opposed to
standard sphere packing coding techniques [71], the spheres are not necessarily entirely contained within
the cube. That is, we only require that the centers of the spheres are inside Q0(n, A)and are disjoint from
each other and have a non-empty intersection with Q0(n, A). The packing density ∆n(S)is defined as
the fraction of the cube volume Vol Q0(n, A)that is covered by the spheres (see [71, Ch. 1]), i.e.,
∆n(S),Vol Q0(n, A)∩SL
i=1 Sui(n, r0)
Vol Q0(n, A).(14)
Sphere packing Sis called saturated if no spheres can be added to the arrangement without overlap. In
particular, we use a packing argument that has a similar flavor as that observed in the Minkowski–Hlawka
theorem for saturated packing [71]. Specifically, consider a saturated packing arrangement of
L(n,R)
[
i=1 Sui(n, √nǫn)(15)
13
spheres with radius r0=√nǫnembedded within cube Q0(n, A). Then, for such an arrangement, we have
the following lower [72, Lem. 2.1] and upper bounds [71, Eq. 45] on the packing density
2−n≤∆n(S)≤2−0.599n.(16)
The volume of a hyper sphere of radius ris given by [71, Eq. (16)]
Vol Sx(n, r)=πn
2
Γ(n
2+ 1) ·rn.(17)
We assign a codeword to the center uiof each small sphere. The codewords satisfy the input constraint
as 0≤ui,t ≤A,∀t∈[[n]],∀i∈[[L]], which is equivalent to
kuik∞≤A . (18)
Since the volume of each sphere is equal to Vol(Su1(n, r0)) and the centers of all spheres lie inside the
cube, the total number of spheres is bounded from below by
L=
Vol SL
i=1 Sui(n, r0
Vol(Su1(n, r0))
≥
Vol Q0(n, A)∩SL
i=1 Sui(n, r0)
Vol(Su1(n, r0))
=∆(S)·Vol Q0(n, A)
Vol(Su1(n, r0))
≥2−n·An
Vol(Su1(n, r0)) ,(19)
where the first inequality holds by (14) and the second inequality holds by (16). The above bound can
be further simplified as follows
log L≥log An
Vol Su1(n, r0)!−n
(a)
=nlog A
√πr0+ log n
2!−n
(b)
=nlog A−nlog r0+1
2nlog n−nlog e+o(n),(20)
where (a)exploits (17) and (b)follows from Stirling’s approximation, see Appendix A. Now, for r0=
√nǫn=√an 1+b
4, we obtain
log L≥nlog A
√a−1
4(1 + b)nlog n+1
2nlog n−nlog e+o(n)
=1−b
4nlog n+n(log A
e√a) + o(n),(21)
where the dominant term is of order nlog n. Hence, for obtaining a finite value for the lower bound of
the rate, R, (21) induces the scaling law of Lto be 2(nlog n)R. Therefore, we obtain
R≥1
nlog n"1−b
4nlog n+nlog A
e√a+o(n)#,(22)
which tends to 1
4when n→ ∞ and b→0.
14
Encoding: Given message i∈[[L]], transmit x=ui.
Decoding: Let
δn=cρ2ǫn=cρ2an 1
2(b−1) ,(23)
where 0< b < 1is an arbitrarily small constant and 0< c < 2is a constant. To identify whether message
j∈ M was sent, the decoder checks whether the channel output ybelongs to the following decoding
set:
Dj=ny∈ Yn:D(y;uj)≤δno,(24)
where
D(y;uj) = 1
n
n
X
t=1 yt−ρuj,t +λ2−ρuj,t +λ(25)
is referred to as the decoding metric evaluated for observation vector yand codeword uj.
Error Analysis: Consider the type I errors, i.e., the transmitter sends ui, yet Y/∈ Di. For every i∈[[L]],
the type I error probability is bounded by
Pe,1(i) = Pr D(Y;ui)> δnui,(26)
where the condition means that x=uiwas sent. In order to bound Pe,1(i), we apply Chebyshev’s
inequality, namely
Pr D(Y;ui)−EnD(Y;ui)uio> δnui!≤
var nD(Y;ui)uio
δ2
n
.(27)
First, we derive the expectation of the decoding metric as follows
ED(Y;ui)|ui=1
n
n
X
t=1 hvar Yt|ui,t−ρui,t +λi= 0 .(28)
Now, since the channel is memoryless, we can compute the variance as follows
var D(Y;ui)|ui=1
n2
n
X
t=1
var Yt−ρui,t +λ2ui,t.(29)
Next, we present a useful lemma.
Lemma 2.Let Z∼Pois(λZ)be a Poisson RV with mean λZ. The following inequality holds
En(Z−λZ)4o≤7λ4
Z+λ3
Z+λ2
Z+λZ.
Proof: The proof is provided in Appendix B.
Using the above lemma, we bound the variance of the decoding metric as follows
EnD(Y;ui)2uio(a)
=var nD(Y;ui)uio
(b)
≤EnD(Y;ui)2uio
(c)
=1
nEYt−ρui,t +λ4ui,t
15
≤7
n(ρA +λ)4+ (ρA +λ)3+ (ρA +λ)2+ (ρA +λ).(30)
where (a)follows since ED(Y;ui)= 0,(b)follows since var{Z} ≤ EZ2, and (c)holds by letting
Z=Yt−ρui,t +λ2and exploiting an upper bound on the fourth non-central moment of a Poisson
random variable (see Appendix B). Therefore, exploiting (27), (28) and (30), we can bound the type I
error probability in (26) as follows
Pe,1(i) = Pr D(Y;ui)> δnui
≤
7(ρA +λ)4+ (ρA +λ)3+ (ρA +λ)2+ (ρA +λ)
nδ2
n
=
7(ρA +λ)4+ (ρA +λ)3+ (ρA +λ)2+ (ρA +λ)
c2ρ4a2nb
≤λ1,(31)
for sufficiently large nand arbitrarily small λ1>0.
Next, we address type II errors, i.e., when Y∈ Djwhile the transmitter sent ui. Then, for every
i, j ∈[[L]], where i6=j, the type II error probability is given by
Pe,2(i, j) = Pr D(Y;uj)≤δnui.(32)
Then, we have
D(Y;uj) = 1
n
n
X
t=1 Yt−ρui,t +λ+ρui,t −uj,t2
|{z }
B
−1
n
n
X
t=1 ρuj,t +λ.(33)
Observe that term Bcan be expressed as follows
B=1
n
Y−(ρui+λ1n)
2+
ρui−uj
2+ 2ρ
n
X
t=1 ui,t −uj,tYt−ρui,t +λ
.(34)
Then, define the following events
Hj
i=
1
n
n
X
t=1 Yt−ρui,t +λ+ρui,t −uj,t2−ρuj,t +λ≤δnui,t
E0=
2ρ
n
n
X
t=1 ui,t −uj,tYt−ρui,t +λ
> δnui
E1=
1
n
Y−(ρui+λ1n)
2+
ρui−uj
2−
n
X
t=1 ρuj,t +λ
≤2δnui
.
Exploiting the reverse triangle inequality, i.e., |β|−|α| ≤ |β−α|, and letting β=Band α=1
nPn
t=1 ρuj,t +λ,
we obtain the following upper bound on the type II error probability
Pe,2(i, j) = Pr Hj
i= Pr |β−α| ≤ δn
≤Pr |β| − |α| ≤ δn
16
= Pr
|B| − 1
n
n
X
t=1 ρuj,t +λ≤δn
≤Pr
B−1
n
n
X
t=1 ρuj,t +λ≤δn
.(35)
Now, applying the law of total probability to event
B=
B−1
n
n
X
t=1 ρuj,t +λ≤δn
,(36)
over E0and its complement Ec
0, we obtain
Pe,2(i, j)≤Pr (B ∩ E0) + Pr (B ∩ Ec
0)
≤Pr (E0) + Pr (E1),(37)
where the first expression in the last line follows since Hj
i∩ E0⊂ E0. The latter expression holds since,
given events Ec
0and B, we immediately get event E1.
We now proceed with bounding Pr (E0). By Chebyshev’s inequality, the probability of this event can
be bounded as follows
Pr(E0)≤
varnPn
t=1 ui,t −uj,tYt−ρui,t +λuio
n2δ2
n/(4ρ2)
=4ρ2Pn
t=1(ui,t −uj,t )2·var{Yt|ui,t}
n2δ2
n
=4ρ2Pn
t=1(ui,t −uj,t )2·(ρui,t +λ)
n2δ2
n
≤4ρ2(ρA +λ)Pn
t=1(ui,t −uj,t )2
n2δ2
n
=4ρ2(ρA +λ)
ui−uj
2
n2δ2
n
.(38)
Observe that
ui−uj
2(a)
≤kuik+
uj
2
(b)
≤√nkuik∞+√n
uj
∞2
(c)
≤√nA +√nA2
= 4nA2,(39)
where (a)holds by the triangle inequality, (b)follows sincek·k ≤ √nk·k∞, and (c)is valid by (18). Hence,
we obtain
Pr(E0)≤4ρ2(ρA +λ)
ui−uj
2
n2δ2
n
≤16nρ2(ρA +λ)A2
n2δ2
n
17
=16ρ2(ρA +λ)A2
nδ2
n
=16(ρA +λ)A2
c2ρ2a2nb
≤ζ0,(40)
for sufficiently large n, where ζ0>0is an arbitrarily small constant.
We now proceed with bounding Pr (E1)as follows. Based on the codebook construction, each codeword
is surrounded by a sphere of radius √nǫn, that is
ui−uj
2≥4nǫn.(41)
Thus, we can establish the following upper bound for event E1:
Pr(E1) = Pr
1
n
Y−(ρui+λ1n)
2+
ρui−uj
2−
n
X
t=1 ρuj,t +λ
≤2δnui
(a)
≤Pr
1
n
Y−(ρui+λ1n)
2−
n
X
t=1 ρuj,t +λ
≤2(c−2)ρ2ǫnui
= Pr
1
n
n
X
t=1 Yt−ρui,t +λ2−ρuj,t +λ
≤2(c−2)ρ2ǫnui
(b)
≤
var 1
nPn
t=1 Yt−ρui,t +λ2
2(c−2)ρ2ǫn2
(c)
≤
7(ρA +λ)4+ (ρA +λ)3+ (ρA +λ)2+ (ρA +λ)
n2(c−2)ρ2ǫn2
=
7(ρA +λ)4+ (ρA +λ)3+ (ρA +λ)2+ (ρA +λ)
4(c−2)2ρ4a2nb
≤ζ1,(42)
for sufficiently large n, where ζ1>0is an arbitrarily small constant. Here, (a)follows from (41) and
(23), (b)holds by Chebyshev’s inequality as given in (27), and (c)follows by Lemma 2. Therefore,
Pe,2(i, j)≤Pr(E0) + Pr(E1)
≤ζ0+ζ1
≤λ2,(43)
We have thus shown that for every λ1, λ2>0and sufficiently large n, there exists an (L(n, R), n, λ1, λ2)
code.
18
C. Converse Proof
We show that the capacity is bounded by CDI (W, L)≤3
2. The derivation of this upper bound for
the achievable rate of the DTPC is more involved than the derivation in the Gaussian case [39]. In our
previous work on Gaussian channels with fading [39], the converse proof was based on establishing
a minimum distance between each pair of codewords. Here, on the other hand, we use the stronger
requirement that the ratio of the letters of every two different codewords is different from 1for at least
one index.
We begin with the following lemma on the ratio of the letters of every pair of codewords.
Lemma 3.Suppose that Ris an achievable rate for the DTPC. Consider a sequence of (L(n, R), n, λ(n)
1,
λ(n)
2)codes (U(n),D(n))such that λ(n)
1and λ(n)
2tend to zero as n→ ∞. Then, given a sufficiently large n,
the codebook U(n)satisfies the following property. For every pair of codewords, ui1and ui2, there exists
at least one letter t∈[[n]] such that
1−ρui2,t +λ
ρui1,t +λ
> ǫ′
n,(44)
for all i1, i2∈[[L]], such that i16=i2, with
ǫ′
n=Pmax
n1+b,(45)
where b > 0is an arbitrarily small constant.
Proof: The proof is given in Appendix C.
Next, we use Lemma 3 to prove the upper bound on the DI capacity. Observe that since
vi,t =ρui,t +λ > λ ,
Lemma 3 implies
ρui1,t −ui2,t=vi1,t −vi2,t
(a)
> ǫ′
nvi1,t
(b)
> λǫ′
n,(46)
where (a)follows by (44) and (b)holds by (46). Now, since kui1−ui2k ≥ ui1,t −ui2,t , we deduce that
the distance between every pair of codewords satisfies
kui1−ui2k>λǫ′
n
ρ.(47)
Thus, we can define an arrangement of non-overlapping spheres Sui(n, λǫ′
n), i.e., spheres of radius λǫ′
n
that are centered at the codewords ui. Since the codewords all belong to a hyper cube Q0(n, P max )with
edge length Pmax, it follows that the number of packed small spheres, i.e., the number of codewords L,
is bounded by
L=
Vol SL
i=1 Sui(n, r0
Vol(Su1(n, r0))
19
≤∆(S)·Vol Q0(n, P max )
Vol(Su1(n, r0))
≤2−0.599n·Pn
max
Vol(Su1(n, r0)) ,(48)
where the last inequality follows from inequality (16). Thereby,
log L≤log Pn
max
Vol Su1(n, r0)!−0.599n
=nlog Pmax −nlog r0−nlog √π+1
2nlog n
2−n
2log e+o(n)−0.599n , (49)
where the dominant term is again of order nlog n. Hence, for obtaining a finite value for the upper bound
of the rate, R, (49) induces the scaling law of Lto be 2(nlog n)R. Hence, by setting L(n, R) = 2(nlog n)R
and r0=λǫ′
n
2ρ=λP max
2ρn1+b, we obtain
R≤1
nlog nnlog Pmax −nlog r0−nlog √π+1
2nlog n
2−n
2log e+o(n)−0.599n
=1
nlog n
1
2+ (1 + b)nlog n−n log λ√πe
2ρ+ 1.0599!+o(n)
,(50)
which tends to 3
2as n→ ∞ and b→0. This completes the proof of Theorem 1.
IV. Simulation Results
We emphasize that the main result of this paper is the characterization of the DI capacity for the
DTPC (cf. Theorem 1), which by definition holds for asymptotically large codewords, i.e., as n→ ∞.
Nevertheless, the proposed achievability scheme presented in Section III-B is based on a constructive
proof, which allows us to generate a practical code even for finite n. Therefore, in the following, we
evaluate the performance of an explicitly constructed codebook in terms of empirical type I and type II
error rates. The values of the parameters used in the proposed simulation setup and codebook construc-
tion are summarized in Table I. The codebook construction is briefly sketched in the following. At first,
codewords are generated uniformly, that is, the value of each symbol is chosen uniformly distributed
between 0and A. Next, in order to realize the minimum distance property of the codebook, once a
codeword is created, before adding it to the codebook, it is verified whether it has at least a minimum
Euclidean distance of 2√nǫnfrom all previously generated codewords or not. In the course of codeword
generation, if a codeword violates the minimum distance property, it is discarded and a new codeword
is generated and the procedure is repeated until the desired codebook size is obtained. To simulate the
receiver’s task, the distance decoder in (11) is implemented and the empirical type I and type II error
rates for finite codeword lengths are obtained via Monte Carlo simulation. In practical MC systems, very
large codeword lengths might not be feasible due to restrictions on time, energy, etc. Therefore, we focus
on a range of small codeword lengths, i.e., 19 ≤n≤28. Moreover, since rates R≥1
4are achievable by
the proposed scheme only as n→ ∞, we choose a smaller rate, i.e., R= 0.1, for codebook generation for
finite n. However, we study a codebook with super-exponential size in n, i.e., L= 2(nlog n)R, which is the
20
TABLE I
Parameters of The Simulations
Description Notation Value
Minimum of power constraints A= mi n (Pave, P max)1000 molecules/s
Release time Trls 1 s
Prob. molecules reaching the receiver pch 0.01
Expected number of interfering molecules λ0.2
Code rate R0.1
Codeword length n[19 - 28]
Codebook size L= 2(nlog n)R[268 - 11273]
Codebook parameters a, b, c 105, 0.99, 1
3
Codebook precision ǫn=an 1
2(b−1) [9.853 - 9.834]
Decoding threshold δn=cρ2ǫn[3.284 - 3.278]
Codebook minimum distance 2√nǫn[27.36 - 33.18]
Number of iterations - 7×105
key claim of Theorem 1. Without loss of generality, we assume that the transmitter sends message i= 1
and denote the empirical type I and type II error rates (average and maximum) by ¯e1(i), and ¯eave
2,¯emax
2,
respectively.
In Figure 8a, we observe that the empirical type I error rates decrease when the codeword length
increases. Figure 8b shows the empirical type II error rate, where a similar phenomenon as for the
type I error rate is observed for the average and maximum error rates. The results in Figure 8a and
8b suggest that any arbitrarily small values for the error probabilities can be achieved if the codeword
length is sufficiently increased, which is consistent with the DI capacity result reported in Theorem 1.
These empirical observations are supported by the theoretical error analysis provided in the proof of
Theorem 1, where increasing the codeword length can reduce type I and II error probabilities below any
arbitrarily small values. Furthermore, the simulation results in Figure 8a and 8b show that the achieved
error rates for the constructed code with R= 0.1, decay faster than the theoretical upper bounds provided
in (31), (40) and (42) evaluated for b= 0.99. Nevertheless, the general trend of the empirical error rates
as functions of the codeword length is well captured by the analytical upper bounds.
We note that to fully characterize the asymptotic behavior of the decoding errors as a function of the
codeword length for every value of the rate 0< R < C, knowledge of the corresponding channel reliability
function is required [73]. To the best of the authors’ knowledge, the channel reliability function for DI has
not been studied in the literature so far, neither for the Gaussian channel [41] nor the Poisson channel
[1], [74]. We note that even for the conventional message transmission problem, the characterization of
the channel reliability function is difficult, as the corresponding channel reliability function is not Turing
21
19 20 21 22 23 24 25 26 27 28
4
5
6
7
8
·10−2
Codeword length
Empirical Type I Error Rate
¯e1(1) −simulation
O(1
n0.99 )−analytical (31)
(a)
19 20 21 22 23 24 25 26 27 28
1
2
3
4
5
6
·10−3
Codeword length
Empirical Type II Error Rate
¯emax
2−simulation
¯eave
2−simulation
O(1
n0.99 )−analytical (40) & (42)
(b)
Fig. 8. Impact of codeword length on the empirical type I and type II error rates. Larger lengths decrease the empirical rates.
computable [73].
V. Summary and Future Directions
In this paper, we studied the DI problem over the DTPC, which may serve as a model for event-
triggered based tasks in the context of MC for applications such as targeted drug delivery, health condition
monitoring, olfactory systems, etc. In particular, we derived lower and upper bounds on the DI capacity
of the DTPC subject to average and peak power constraints in the codebook size of L(n, R) = 2(nlog n)R=
nnR. Our results revealed that the super-exponential scale of nnR is the appropriate scale for the DI
capacity of the DTPC, which was proved by finding a suitable sphere packing arrangement embedded in
a hyper cube. We emphasize that this scale is sharply different from the ordinary scales in transmission
and RI settings, where the codebook size grows exponentially and double exponentially, respectively.
The results presented in this paper can be extended in several directions, some of which are listed in
the following as potential topics for future research works:
•Our observations for the codebook size of the DTPC and Gaussian channels [39] lead us to conjecture
that the codebook size for any continuous alphabet channel would be a super-exponential function,
i.e., 2(nlog n)R. However, a formal proof of this conjecture remains unknown.
•We assumed that the channel uses are orthogonal, which implies a memoryless channel for temporal
codes and independent molecule reception for spatial codes. In practice, however, the DTPC may ex-
hibit memory [7] and non-orthogonal molecule reception [55], the investigation of which constitutes
an interesting research problem.
•This study has focused on a point-to-point system and may be extended to multi-user scenarios (e.g.,
broadcast and multiple access channels) or multiple-input multiple-output channels which become
relevant in complex MC nano-networks.
22
•Another interesting research topic is to investigate the behavior of the DI capacity in the sense
of Fekete’s Lemma [75], that is, to verify whether the pessimistic (C= lim inf n→∞ log L(n,R)
nlog n) and
optimistic (C= lim supn→∞
log L(n,R)
nlog n) capacities [76] are equal or not.
Acknowledgments
Salariseddigh was supported by the German Research Foundation (DFG) under grant DE 1915/2-1.
Pereg and Deppe were supported by the German Federal Ministry of Education and Research (BMBF)
under Grants 16KIS1005 (LNT, NEWCOM) and 16KISQ028. Boche was supported by the BMBF un-
der Grant 16KIS1003K (LTI, NEWCOM), and the national initiative for “Molecular Communications"
(MAMOKO) under Grant 16KIS0914. Jamali was supported by the DFG under Grant JA 3104/1-1. Schober
was supported by MAMOKO under Grant 16KIS0913.
Appendix A
Volume of a Hyper Sphere With Growing Radius
To solidify the idea of packing spheres within a hyper cube, we reveal and explain a counter-intuitive
phenomenon regarding the packing of hyper spheres with growing radius in the codeword length inside
a hyper cube. We observe that despite the fact that the hyper sphere’s radius tends to infinity as the
codeword length goes to infinity ∼n1
4its volume tends to zero. In fact, the volume of the sphere vanishes
super-exponentially inverse, i.e., ∼n−n
4, such that in the asymptotic analysis, we can accommodate a
super-exponential number of such hyper spheres inside the hyper cube. We note that the ratio of the
spheres in our construction grows with n, as ∼n1
4. It is well-known that the volume of an n-dimensional
unit-hyper sphere, i.e., with a radius of r0= 1, tends to zero, as n→ ∞ [71, Ch. 1, Eq. (18)]. Nonetheless,
we observe that the volume still tends to zero for a radius of r0=nc, where 0< c < 1
2. More precisely,
lim
n→∞ Vol Su1(n, r0)= lim
n→∞
πn
2
Γ(n
2+ 1) ·rn
0
= lim
n→∞
πn
2
n
2!·rn
0
= lim
n→∞ r2π
nr0!n
,(51)
where the last equality follows by Stirling’s approximation [77, P. 52], that is,
log n! = nlog n−nlog e+o(n).(52)
The last expression in (51) tends to zero for all r0=ncwith c∈(0,1
2). On the other hand when n→ ∞,
the volume of a hyper cube Q0(n, A)with edge length Ais given by
lim
n→∞ Vol Q0(n, A)= lim
n→∞ An=
0A < 1,
1A= 1 ,
∞A > 1.
(53)
23
Now, to derive how many spheres can be packed inside the hyper cube Q0(n, A)we derive the log-ratio
of the volumes as follows
log Vol Q0(n, A)
Vol Su1(n, r0)!= log An
πn
2rn
0·n
2!!
=nlog A
√πr0+ log n
2!
=nlog A−nlog r0−nlog √π+1
2nlog n
2−n
2log e+o(n)
=1
2−cnlog n+nlog A
√πe −3
2+o(n),(54)
where the last equality follows from r0=nc. Now, since the dominant term in (54) involves nlog n, we
deduce that codebook size should be L(n, R) = 2(nlog n)R, thereby by (19) we obtain
R≥1
nlog n
log Vol Q0(n, A)
Vol Su1(n, r0)!−n
=1
nlog n"1
2−cnlog n+nlog A
√πe −3
2+o(n)#,(55)
which tends to 1
2−cwhen n→ ∞.
Appendix B
Moment Generating Function of Poisson Random Variable
The moment-generating function (MGF) of a Poisson variable Z∼Pois(λZ)is given by
GZ(α) = eλZ(eα−1) .(56)
Hence, for X=Z−λZ, the MGF is given by
GX(α) = eλZ(eα−1−α).(57)
Since the fourth non-central moment equals the fourth order derivative of the MFG at α= 0, we have
E{X4}=d4
dα4GX(α)α=0
=λZλ3
Ze3α+ 6λ2
Ze2α+ 7λZeα+ 1eα+λZeα−λZα=0
=λ4
Z+ 6λ3
Z+ 7λ2
Z+λZ
≤7λ4
Z+λ3
Z+λ2
Z+λZ.
Appendix C
Proof of Lemma 3
In the following, we provide the proof of Lemma 3. The method of proof is by contradiction, namely,
we assume that the condition given in (44) is violated and then we show that this leads to a contradiction
(sum of the type I and type II error probabilities converge to one).
24
Fix λ1, λ2>0. Let κ, δ > 0be arbitrarily small constants. Assume to the contrary that there exist two
messages i1and i2, where i16=i2, meeting the error constraints in (8), such that for all t∈[[n]], we have
1−vi2,t
vi1,t ≤ǫ′
n,(58)
where vik,t =ρuik,t +λ, k = 1,2. In order to show contradiction, we will bound the sum of the two error
probabilities, Pe,1(i1) + Pe,2(i2, i1), from below. To this end, define
Bi1=
y∈ Di1:1
n
n
X
t=1
yt≤ρP max +λ+δ
.(59)
Then, observe that
Pe,1(i1) + Pe,2(i2, i1) = 1 −X
y∈Di1
Wnyui1+X
y∈Di1
Wnyui2
≥1−X
y∈Di1
Wnyui1+X
y∈Di1∩Bi1
Wnyui2.(60)
Now, consider the sum over Di1in (60),
X
y∈Di1
Wnyui1=X
y∈Di1∩Bi1
Wnyui1+X
y∈Di1∩Bc
i1
Wnyui1
≤X
y∈Di1∩Bi1
Wnyui1+ Pr
1
n
n
X
t=1
Yt> ρP max +λ+δui1
.(61)
Next, we bound the probability on the right hand side of (61) as follows
Pr
1
n
n
X
t=1
Yt−1
n
n
X
t=1
E{Yt}> ρP max +δ−1
n
n
X
t=1
E{Yt}
(a)
≤
var n1
nPn
t=1 Ytui1o
ρP max +δ−1
nPn
t=1 E{Yt}2
(b)
=
1
n2Pn
t=1(ρui1,t +λ)
ρP max +δ−1
nPn
t=1 ρui1,t +λ2
(c)
≤ρPmax +λ
nδ2
≤κ , (62)
for sufficiently large n, where inequality (a)follows from Chebyshev’s inequality, for equality (b), we
exploited var{Yt|ui1,t}=E{Yt|ui1,t}=ρui1,t +λ, and for inequality (c), we used the fact that ui1,t ≤
Pmax ,∀t∈[[n]].
Returning to the sum of error probabilities in (60), exploiting the bound (62) leads to
Pe,1(i1) + Pe,2(i2, i1)≥1−X
y∈Di1∩Bi1Wnyui1−Wnyui2−κ . (63)
Now, let us focus on the summand in the square brackets in (63). By (3), we have
Wnyui1−Wnyui2=Wnyui11−Wnyui2/ W nyui1
25
=Wnyui1
1−
n
Y
t=1
e−(vi2,t−vi1,t ) vi2,t
vi1,t !yt
=Wnyui1
1−
n
Y
t=1
e−ǫ′
nvi1,t 1−ǫ′
nyt
,(64)
where for the last inequality, we employed
vi2,t −vi1,t ≤vi2,t −vi1,t≤ǫ′
nvi1,t ,(65)
and
1−vi2,t
vi1,t ≤
1−vi2,t
vi1,t ≤ǫ′
n,(66)
which follow from (58). Now, we bound the product term inside the bracket as follows:
n
Y
t=1
e−ǫ′
nvi1,t 1−ǫ′
nyt=e−ǫ′
nPn
t=1 vi1,t ·1−ǫ′
nPn
t=1 yt
(a)
≥e−nǫ′
n(ρP max+λ)·1−ǫ′
nn(ρP max+λ+δ)
=enǫ′
nδ·e−nǫ′
n(ρP max+λ+δ)·1−ǫ′
nn(ρP max+λ+δ)
(b)
≥enǫ′
nδ·e−nǫ′
n(ρP max+λ+δ)·1−nǫ′
nρP max+λ+δ
≥enǫ′
nδ·f(nǫ′
n)
(c)
> f(nǫ′
n)
(d)
≥1−3 (ρP max +λ+δ)nǫ′
n
= 1 −3 (ρP max +λ+δ)Pmax
nb
≥1−κ . (67)
for sufficiently large n. For inequality (a), we used
vi1,t ≤ρP max +λ , ∀t∈[[n]] ,(68)
and
n
X
t=1
yt≤n(ρPmax +λ+δ),(69)
where the latter inequality follows from y∈ Bi1, cf. (59). For (b), we used Bernoulli’s inequality
(1 −x)r≥1−rx , ∀x > −1,∀r > 0,(70)
[78, see Ch. 3]. For (c), we exploited enǫ′
nδ>1and the following definition:
f(x) = e−cx(1 −x)c,(71)
with c=λ+ρP max +δ. Finally, for (d), we used the Taylor expansion f(x) = 1 −2cx +O(x2)to obtain
the upper bound f(x)≥1−3cx for sufficiently small values of x.
26
Equation (64) can then be written as follows
Wnyui1−Wnyui2≤Wnyui1·h1−e−ǫ′
nPn
t=1 vi1,t ·1−ǫ′
nPn
t=1 yti
≤κ·Wnyui1.(72)
Combining, (63), (64), and (72) yields
Pe,1(i1) + Pe,2(i2, i1)
(a)
≥1−X
y∈Bi1Wnyui1−Wnyui2−κ
= 1 −X
y∈Bi1κ·Wnyui1−κ
(b)
≥1−2κ , (73)
where for (a), we replaced y∈ Bi1∩ Di1by y∈ Bi1to enlarge the domain and for (b), we used
Py∈Bi1Wnyui1≤1. Clearly, this is a contradiction since the error probabilities tend to zero as
n→ ∞. Thus, the assumption in (58) is false. This completes the proof of Lemma 3.
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