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arXiv:submit/4607498 [cs.IT] 20 Nov 2022
ARXIV PREPRINT
Molecular Communication
Deterministic Identification For MC ISI-Poisson Channel
Mohammad Javad Salariseddigh1| Vahid Jamali2| Uzi Pereg3|
Holger Boche4| Christian Deppe5| Roebrt Schober6
1Institute For Communication Engineering,
Technical University of Munich, Munich, 80333,
Germany, E-mail: mjss@tum.de
2Group of Resilient Communication Systems,
Technical University of Darmstadt, E-mail:
vahid.jamal@tu-darmstadt.de
3Department of Electrical and Computer Eng.,
Technion – Israel Institute of Technology, Haifa,
80333, Israel, E-mail: uzi.pereg@tum.de
4Chair of Theoretical Information Technology,
Technical University of Munich, Munich, 80333,
Germany, E-mail: boche@tum.de
5Institute For Communication Engineering,
Technical University of Munich, Munich, 80333,
Germany, E-mail: christian.deppe@tum.de
6Institute for Digital Communications,
Friedrich-Alexander-UniversityErlangen-Nürnberg,
Erlangen, Germany, E-mail: robert.schober@fau.de
Funding Informations
1. 6G-Life Project, GNT: 16KISK002
2. DFG, GNT: JA 3104/1-1
3. Israel CHE Prog. QD. Sci. Tech., GNT: 86636903
4. BMBF, GNT:16KIS1003K; MAMOKO, GNT:
16KIS0914
5. BMBF, GNT:16KIS1005, 16KISQ028; 6G-Life
Project, GNT: 16KISK002
6. MAMOKO, GNT: 16KIS0913
Several applications of molecular communications (MC) feature an alarm-
prompt behavior for which the prevalent Shannon capacity may not be the
appropriate performance metric. The identification capacity as an alterna-
tive measure for such systems has been motivated and established in the
literature. In this paper, we study deterministic identification (DI) for the
discrete-time Poisson channel (DTPC) with inter-symbol interference (ISI)
where the transmitter is restricted to an average and a pea k molecule release
rate constraint. Such a channel serves as a model for diffusive MC systems
featuring long channel impulse responses and employingmolecule counting
receivers. We derive lower and upper bounds on the DI capacity of the DTPC
with ISI when the number of ISI channel taps Kmay gr ow with the codeword
length n(e.g., due to increasing symbol rate). As a key finding, we establish
that for deterministic encoding, the codebook size scales as 2(nlog n)Rassum-
ing that the number of ISI channel taps scales as K= 2κlog n, where Ris the
coding rate and κis the ISI rate. Moreover, we show that optimizing κleads
to an effective identification rate [bits/s] that scales linearly with n, which is
in contrast to the typical transmission r ate [bits/s] that is independent of n.
KEYWORDS
Channel capacity, deterministic identification, molecular communication,
Poisson channel, and memory
1|INTRODUCTION
Molecular communications (MC) is a bio-inspired promising paradigm for communication between nanoma-
chines or different biological entities, such as cells and organs [1] and realizes the exchange of information
via the transmission, propagation, and reception of signaling molecules [2, 3]. In the past decade, different
aspects of synthetic MC have been explored in the literature from distinctive viewpoints, including channel
modeling [4, 5], modulation and detection design [6], biological building blocks for transceiver design [7], and
information-theoretical and relevant mathematical foundations [8–10]. Moreover, several proof-of-concept im-
plementations of synthetic MC systems have been reported in the literature, see, e.g., [11–13]. Furthermore,
the ongoing progress in synthetic biology [7, 14] is expected to enable sophisticated MC systems in the future,
capable of performing the complex computation and communication tasks required for realizing the Internet
of Bio-nano Things [15–18]. Also, the authentication problem [19] which exhibit affinity to the identification
1
2 M. J. S.
problem is considered in [20].
In particular, one of the basic and widely-accepted abstract models for MC systems with molecule counting
receivers is the discrete-time Poisson channel (DTPC) model with inter-symbol interference (ISI) model [8, 21,
22]. The DTPC model with memory has been used to study the performance limits of MC systems. Despite
the recent theoretical and technological advancements in the field of MCs, the transmission capacity of most
MC systems with DTPC with memory model are still unknown [8]. However, a number of approaches to
examining the behavior of Poisson channel are being explored. For instance, an analytic expression for the
transmission capacity of a DTPC with memory under an average power constraint alone, is still open [8, 23, 24].
However, several bounds and asymptotic behaviors for the DTPC with memory in different setups have been
established. For instance, analytical lower and upper bounds on the transmission capacity of the DTPC with
input constraints and memory are provided in [25]. Bounds on the transmission capacity of the DTPC with
memory are developed in [26, 27]. The design of optimal code for DTPC with memory under a peak and
average power constraint is studied in [28]. In [29], the impact of memory on the performance for a diffusive
MC channel is characterized. Performance analysis of modulation schemes for diffusive MC with memory is
considered in [30] and impact of degree of memory on the performance is shown. Design of the filter and
detector parts in a receiver for Poisson channel with time-varying mean when transmitted symbols are exposed
to the ISI is studied in [31]. The Code design problem for diffusive MC channel under ISI is considered in [32]
where influence of the ISI is incorporated into the code design. The authors in [33, 34] studied the DTPC in
absence of ISI, i.e., K=1, and established lower and upper bounds on the DI capacity where the codebook size
scales as ∼2(nlog n)R.
Numerous applications of MC within the scope of future generation wireless networks (XG) [35, 36] are
linked with event-triggered communication systems. In such systems, Shannon’s message transmission capacity,
as studied in [8–10, 25–27, 37–42], may not be the appropriate performance metric, instead, the identification
capacity is deemed to be an essential quantitative measure. In particular, in object-finding or event-detection
scenarios, where the receiver aims to determine the presence of an object or determine the occurrence of an
specific event in terms of a reliable Yes / No answer, the so-called identification capacity is the key applicable
performance measure [43]. Concrete examples of the identification problem within the MC context include
health monitoring [44, 45] where, e.g., one may desire in whether or not the pH value of the cerebrospinal fluid
of brain exceeds a crisis level; targeted drug delivery [1, 46] and cancer treatment [47–49], where, e.g., a nano-
device’s purpose is to identify whether or not an specific cancer biomarker exist in the vicinity of the target
tissue. Moreover, identification problems can also be found in various natural MC systems. For instance, in
bionic nose setting [50], or in natural pheromone communications [51, 52] where, e.g., animals involved in mating
seeks sexual pheromones to realize the presence of an opposite sex. In fact, the olfactory systems of animals
have the capability of recognizing the presence of extremely large numbers of different molecule mixtures
(e.g., pheromones, odors, etc.) [53, 54], which has inspired researchers to regard them as role models for the
design of bio-inspired synthetic MC systems [55]. Motivated by this discussion, in this paper, we investigate the
fundamental performance limits of identification problem in MC systems, which can be modelled by the DTPC
with ISI.
1.1 |Related Work on Identification Capacity
In Shannon’s communication paradigm [56], message encoding is conducted by sender, Alice, in a certain
way that guarantee a reliable recovering of the original message by the receiver, Bob. In contrast, for the
identification setting, the coding scheme is designed to accomplish a different objective [43]. Specifically, the
M.J.S . 3
decoder’s task is to verify whether a particular message was sent or not. Ahlswede and Dueck [43] introduced
a randomized-encoder identification (RI) scheme, in which the codewords are tailored according to their cor-
responding distributions. Although employing such distributions does not bring advantage in terms of gain in
the Shannon’s message transmission capacity [57] or codebook size, Ahlswede and Dueck [43] established that
providing local randomness at the encoder, reliable identification is yielded a remarkable attribute regarding
the codebook size, namely, the codebook size exhibit a double-exponentially growth in the codeword length
n, i.e., ∼22nR [43], where Ris the coding rate. This observation is extremely different from the conventional
message transmission problem, which has an exponential codebook size in the codeword length, i.e., ∼2nR.
The construction of randomized identification (RI) codes is considered in [58–60]. For example, a binary code
using three-layer concatenated constant-weight code is established in [61]. Nevertheless, the realization of such
codes entails extra complexity [33, see Sec. 1].
In the deterministic encoding setting of identification, also referred to as deterministic identification (DI)
[62] or identification without randomization [63], the codewords are selected by a deterministic function. In our
recent works [34, 62, 64, 65], we target DI for channels with power constraint, including discrete memoryless
channels (DMCs), Gaussian channels with fast and slow fading, and the memoryless discrete-time Poisson
channel (DTPC), respectively. The codebook size of DI for DMCs, similar to the transmission problem grows
exponentially in the codeword length [43, 63, 65–67], however, the achievable identification rates are significantly
higher compared to the transmission rates [62, 65]. For the Gaussian channel [64, 68] and DTPC [34, 69], the
codebook size scales as ∼2(nlog n)R. Deterministic codes often have the advantage of simpler implementation,
simulation [70, 71], and explicit construction [72]. DI problem for Gaussian channels is also studied in [64, 73–
75]. Further, DI may be preferred over RI in complexity-constrained applications of MC systems, where the
generation of random codewords is challenging1. Motivated by this discussion, in this paper, we investigate the
fundamental performance limits of the DI problem in MC systems, which can be modelled by the DTPC with
ISI.
1.2 |Contributions
In this paper, we consider MC systems employing molecule counting receivers with a large number of released
molecules at the transmitter, see [4, Sec. IV]. Further, we assume that the received signal experiences ISI and
follows the Poisson distribution. We formulate the problem of DI over the DTPC with memory under average
and peak molecule release rate constraints to account for the limited molecule production /release rates of the
transmitter. As our main objective, we investigate the fundamental performance limits of DI over the DTPC
with ISI. In particular, this paper makes the following contributions:
Generalized ISI Model: In MC systems, often the number of channel taps Kcan be large, particularly
for non-degrading signalling molecules in bounded environments, which leads to a long channel impulse
response (CIR). In addition, the value of Kincreases not only with the dispersiveness of the channel but
also with the symbol rate. Therefore, it is of interest to investigate the asymptotic limits of the system for
large symbol rates (leading to large K) and large codeword lengths n. To do so, we consider a generalized
ISI model that captures the ISI-free channel (i.e., K=1), ISI channels with constant K > 1, and ISI channels
for which Kincreases with the codeword length n(e.g., due to increasing symbol rate). To the best of the
1On the other hand, we note that the biological hardware of MC systems (e.g., reaction networks) features an inherent stochastic
nature [76] which can potentially be exploited for realizing RI.
4M.J.S.
authors’ knowledge, such a generalized ISI model has not been studied in the literature, yet.
Codebook Scale: We establish that the codebook size of the DTPC with ISI for deterministic encoding
scales in nsimilar to the memoryless DTPC [33], namely super-exponentially in the codeword length
(∼2(nlog n)R), even when the number of ISI taps scale as K= 2κlog nfor any κ∈[0,1), which we refer to
as the ISI rate. This observation suggests that memory does not change the scale of the codebook derived
for memoryless DTPC [33] and Gaussian channels [64].
Capacity Bounds: We derive DI capacity bounds for the DTPC with constant K≥1and growing ISI
K=2κlog n, respectively. We show that for constant K, the proposed lower and upper bounds on Rare
independent of K, whereas for growing ISI, they are functions of the ISI rate κ. Moreover, we show that
optimizing κleads to an effective identification rate [bits/s] that scales linearly with n, which is in contrast
to the typical transmission rate [bits/s] that is independent of n.
Technical Novelty in the Capacity Proof: To obtain the proposed lower bound, the existence of an
appropriate sphere packing within the input space, for which the distance between the centers of the
spheres does not fall below a certain value, is guaranteed. This packing incorporates the effect of ISI as a
function of κ. In particular, we consider the packing of hyper spheres inside a larger hyper cube, whose
radius grows in both the codeword length nand the ISI rate κ, i.e., ∼n1+κ
4. For derivation of the upper
bound, we assume that for given sequences of codes with vanishing error probabilities, a certain minimum
distance between the codewords is asserted, where this distance depends on the ISI rate and decreases as
Kgrows.
1.3 |Organization
The remainder of this paper is structured as follows. In Section 2, system model is explained and the required
preliminaries regarding DI codes are established. Section 3 provides the main contributions and results on the
message identification capacity of the DTPC with ISI. Finally, Section 4 of the paper concludes with a summary
and directions for future research.
1.4 |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 let-
ters X, Y , Z, . . . stand for random variables. Lower case bold symbol xand ystand for row vectors. Bold
symbol 1¯nindicates the all-one row vector of size ¯n. 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 set of non-negative real numbers is denoted by R+. 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
of vector xare 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
M.J.S . 5
Messages TX
Enc Release Diffus./Advec./Reac. Process.
ISIChannel RX
Reception Dec
j
Yes/ No
iuiY
FIGURE 1 End-to-end transmission chain for DI communication in a generic MC system modelled as a DTPC. Relevant processes
in the molecular channel include diffusion, advection, and chemical reactions. The transmitter maps message ionto a codeword ci.
The receiver is provided with an arbitrary message j, and given the channel output vector Y, it asks whether jis identical to ior not.
by Q0(n, A) = {x∈Rn
+:0≤xt≤A, ∀t∈[[n]]}. We denote the DTPC with KISI channel taps by P.
2|SYSTEM MODEL AND PRELIMINARIES
In this section, we present the adopted system model and establish some preliminaries regarding DI coding.
2.1 |System Model
We consider an identification-focused communication setup, where the decoder seeks to accomplish the fol-
lowing task: Determining whether or not an specific message was sent by the transmitter2; see Figure 1. To
attain this objective, a coded communication between the transmitter and the receiver over nchannel uses of
an MC channel3is established. We consider the Poisson channel Pwhich arises as a channel model in the
context of MC for molecular counting receivers [8]. Let X ∈ R≥0and Y ∈ N0denote random variables (RVs)
modeling the rate of molecule release by the transmitter and the number of molecules observed at the receiver,
respectively. We consider a stochastic release model, where for the t-th channel use, the transmitter releases
molecules with rate xt(molecules/second) over a time slot of Tsseconds into the channel [8]. These molecules
propagate through the channel via diffusion and/or advection, and may even be degraded in the channel via
enzymatic reactions [4]. The receive is assumed to be equipped with a counting-type mechanism which is able
to enumerate the number of received molecules observed in a determined volume.
The channel memory is modelled by a length Ksequence of probability values, i.e., p= [p0, p1, . . . , pK−1].
The value pkin specifies the probability that a given molecule released by the transmitter at the beginning time
slot t, is observed at the receiver during time slot t+kand depends on the propagation environment (e.g., diffu-
sion, 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 p
for various MC setups. Let ρk
def
=pkTswhere the value pk∈(0,1] denotes the probability that a given molecule
released by the transmitter at the beginning time slot t, is observed at the receiver during time slot t+k.
When the number of released molecules is large but only a small fraction of them arrives at the receiver,
2We assume that the transmitter does not know which message the decoder is interested in. This assumption is justified by the fact
that otherwise, entire communication setting is specialized to transmission of only one indicator bit between Alice and Bob.
3The proposed performance bounds works regardless of whether or not an specific code is used for communication, although proper
codes may be required to approach such performance limits.
6M.J.S.
the relation of channel output Yand input Xis characterized as follows [4, 8]:
Yt= Pois Xρ
t+λ,(1)
where
Xρ
t
def
=
K−1
X
k=0
ρkXt−k,(2)
is the mean number of observed molecules due to the release of the transmitter and the constant 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. Let x∗
t
def
= (xt−K+1, . . . , xt) be the vector of the Kmost
recently released symbols. Then, the letter-wise transition probability law is given by
V(yt|x∗
t) =
e−xρ
t+λxρ
t+λyt
yt!.(3)
We assume that different channel uses given any Kprevious input symbols are statistically independent, which
is a valid assumption for, e.g., fully absorbing receivers [4]. Therefore, for nchannel uses, the transition
probability law is given by
V¯n(y|x) =
¯n
Y
t=1
V(yt|x∗
t) =
¯n
Y
t=1
e−xρ
t+λxρ
t+λyt
yt!,(4)
where x= (x1, . . . , xn) and y= (y1, . . . , y¯n) denote the transmitted codeword and the received signal, respectively,
with ¯n=n+K−1. We assume that xt= 0 when t > n or t < 0. The peak and average molecule release rate
constraints on the codewords are
0≤xt≤Pmax and 1
n
n
X
t=1
xt≤Pavg ,(5)
respectively, ∀t∈[[n]], where Pmax >0 and Pavg >0 constrain the rate of molecule release per channel use
and over the entire nchannel uses in each codeword, respectively.
Remark (Input Constraint Interpretation) 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) [8].
2.2 |DI Coding for the DTPC
The definition of a DI code for the DTPC Pis given below.
ISI-Poisson DI Code An (n, M(n, R), K(n , κ), e1, e2) DI code for a DTPC Punder average and peak molecule
release rate constraints of Pave and Pmax, respectively, and for integers M(n, R ) and K(n, κ), respectively, where
M.J.S . 7
ui,1ui,2ui,3ui,4ui ,5p0p1
0.5 0.5
uiY
DTPC with 2-ISI
y1y2y3y4y5y6
FIGURE 2 A DTPC with 2-ISI channel with p= (0.5,0.5). Channel takes an input sequence of non-negative real numbers and
outputs a sequence with length n+K−1= 5+2−1= 6of integer numbers where each integer is a Poisson distributed random
variable whose mean is sum of previous marked as accumulation of different colors. The constant interference λis depicted in black.
nand Rare the codeword length and coding rate, respectively, is defined as a system (C,T), which consists of
a codebook C=cii∈[[ M]] ⊂Rn
+, such that
0≤ci,t ≤Pmax and 1
n
n
X
t=1
ci,t ≤Pavg ,(6)
∀i∈[[M]], ∀t∈[[n]], and a collection of decoding regions T={Ti}i∈[[M]] with
M(n,R)
[
i=1 Ti⊂N¯n
0.(7)
Given a message i∈[[M]], the encoder transmits ci, and 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 (type I) or acceptance of a false message (type II). The corresponding error probabilities of the DI
code (C,T) are given by
Pe,1(i) = 1 −X
y∈Ti
V¯n(y|ci) and Pe,2(i, j ) = X
y∈Tj
V¯n(y|ci),(8)
and satisfy the following bounds Pe,1(i)≤e1and Pe,2(i, j )≤e2,∀i, j ∈
i6=j[[M]] and every e1, e2>0. A rate R > 0
is called achievable if for every e1, e2>0 and sufficiently large n, there exists an (n, M (n, R), K(n, κ ), e1, e2) DI
code. The DI capacity of the DTPC Pis defined as the supremum of all achievable rates, and is denoted by
CDI (P, M, K ).
8M.J.S.
3|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 with ISI. Subsequently, we provide the detailed proofs of these bounds.
3.1 |Main Results
The DI capacity theorem for DTPC with ISI Pis stated below.
Theorem 1 Consider the DTPC with ISI Pand assume that the number of ISI channel taps scales sub-
linearly with codeword length n, i.e., K(n, κ) = 2κlog n, where κ∈[0,1). Then the DI capacity of Psubject to
average and peak molecule release rate constraints of the form n−1Pn
t=1 ci,t ≤Pave and 0≤ci,t ≤Pmax,
respectively, and a codebook of super-exponential scale, i.e., M(n, R) = 2(nlog n)R, is bounded by
1−κ
4≤CDI (P, M, K )≤3
2+κ . (9)
Proof The proof of Theorem 1 consists of two parts, namely the achievability and the converse proofs, which
are provided in Sections 3.2 and 3.3, respectively.
Remark The result in Theorem 1 comprises the following three special cases in terms of K:
K= 1:This cases accounts for an ISI-free setup (κ= 0), which is valid when the symbol duration is large
(Ts≥Tcir), and implies K= 1 and κ= 0. Thereby, ¯
Reff scales logarithmically with the codeword length
n. This is in contrast to the transmission setting in which ¯
Reff is independent of n(e.g., the well-known
Shannon formula for the Gaussian channel). This result is known in the identification literature [33, 43].
Constant K > 1:When Tsis constant and Ts< Tcir , we have constant K > 1 which implies κ→0 as
n→ ∞. Surprisingly, our capacity result in Theorem 1 reveals that the bounds for the DTPC with memory
are in fact identical to those for the memoryless DTPC given in [33].
Growing K:Our capacity results reveal that reliable identification is possible even when Kscales with the
codeword length as ∼2κlog n. Moreover, the impact of ISI rate κis reflected in the capacity lower and
upper bounds in (9), where the bounds respectively decrease and increase in κ. While the upper bound on
Reff increases in κ, too, the lower bound in (11) suggests a trade-off in terms of κ, which is investigated in
the Corollary 2.
Corollary 1 (Effective Identification Rate) Assuming Ts=Tcir/K =Tcir 2−κlog n,κ∈[0,1), the effective
identification rate, defined as
¯
Reff
def
=log M(n, R)
nTs
(10)
(in bits/s), under average and peak molecule release rate constraints is bounded by
(1−κ)nκlog n
4Tcir ≤¯
Reff ≤(3 + 2κ)nκlog n
2Tcir
.(11)
Proof The proof follows directly by substituting the capacity results in Theorem 1 into the definition of the
effective rate and making further mathematical simplifications.
M.J.S . 9
Corollary 2 (Optimum ISI Rate) The lower bound given in Corollary 1 is maximized for the following ISI
rate κmax(n), n ∈N, with
κmax(n) = 1 −1
ln n.(12)
The above κmax gives the following lower bound on ¯
Reff(n):
¯
Reff(n)≥log e
4eTcir ·n . (13)
Thereby,
lim inf
n→∞
¯
Reff(n)
n≥log e
4eTcir
.(14)
Proof The proof follows from differentiating the lower bound in Corollary 1 with respect to κand equating it
to zero.
The effective identification rate ¯
Reff [bits/s] in (10) consists of two terms, namely the identification rate per
symbol logM(n, R)
n[bits/symbol] (which decreases with κfor the lower bound in (9)) and the symbol rate 1
Ts
[symbol/s] (which increases with κ). The above corollary reveals that in order to maximize ¯
Reff, it is optimal to
set the trade-off for κsuch that the identification rate, i.e.,
log M(n, R)
n=(1 −κmax) log n
4
=log e
4(15)
becomes independent of nbut the symbol rate, i.e.,
1
Ts
=n
eTcir
(16)
linearly scales with n. As a result, in contrast to the typical transmission settings where the effective rate is
independent of n, here, the effective identification rate ¯
Reff for the optimal κlinearly grows in n.
3.2 |Achievability
The achievability proof consists of the following two main steps. Step 1: First, we propose a codebook construc-
tion and derive an analytical lower bound on the corresponding codebook size using inequalities for sphere
packing density. Step 2: Then, to prove that this codebook leads to an achievable rate, we propose a decoder
and show that the corresponding type I and type II error rates vanished as n→ ∞.
|Codebook construction
Let
A= min (Pave, P max).(17)
10 M.J.S.
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 (5). 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=√nθnin a hyper cube
with edge length A, where
θn=a√K
n1
2(1−b)=a
n1
2(1−(b+κ)) ,(18)
and a > 0 is a non-vanishing fixed constant and 0 < b < 1 is an arbitrarily small constant4.
Let Sdenote a sphere packing, i.e., an arrangement of Mnon-overlapping spheres Sci(n, r0), i∈[[M]], that
are packed inside the larger cube Q0(n, A) with an edge length A. As opposed to standard sphere packing coding
techniques [77], the spheres are not necessarily entirely contained within the cube. That is, we only require
A/2
√nεn
A√n
FIGURE 3 Illustration of a saturated sphere packing
inside a cube, where small spheres of radius r0=
√nεncover 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.
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
ratio of the saturated packing volume to the cube volume
Vol Q0(n, A), i.e.,
∆n(S),Vol SM
i=1 Sci(n, r0)
Vol Q0(n, A).(19)
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 simi-
lar flavor as that observed in the Minkowski–Hlawka theorem
for saturated packing [77]. Specifically, consider a saturated
packing arrangement of
M(n,R)
[
i=1 Sci(n, pnθn) (20)
spheres with radius r0=√nθnembedded within cube Q0(n, A). Then, for such an arrangement, we have the
following lower [78, Lem. 2.1] and upper bounds [77, Eq. 45] on the packing density
2−n≤∆n(S)≤2−0.599n.(21)
In our subsequent analysis, we use the above lower bound which can be proved as follows: For the saturated
4we recall that our achievability proof works for any b∈(0,1); however, arbitrarily small values of bare of interest since they result
in the tightest lower bound.
M.J.S . 11
packing arrangement given in (20), there cannot be a point in the larger cube Q0(n, A) with a distance of more
than 2r0from all sphere centers. Otherwise, a new sphere could be added which contradicts the assumption
that the union of M(n, R) spheres with radius √nθnis saturated. Now, if we double the radius of each sphere,
the spheres with radius 2r0cover thoroughly the entire volume of Q0(n, A), that is, each point inside the hyper
cube Q0(n, A) belongs to at least one of the small spheres. In general, the volume of a hyper sphere of radius
ris given by [77, Eq. (16)]
Vol Sx(n, r)=πn
2
Γ(n
2+ 1) ·rn.(22)
Hence, if the radius of the small spheres is doubled, the volume of SM(n,R)
i=1 Sci(n, √nθn) is increased by 2n.
Since the spheres with radius 2r0cover Q0(n, A), it follows that the original r0-radius packing has a density of
at least 2−n5. We assign a codeword to the center ciof each small sphere. The codewords satisfy the input
constraint as 0 ≤ci,t ≤A,∀t∈[[n]], ∀i∈[[M]], which is equivalent to
kcik∞≤A . (23)
Since the volume of each sphere is equal to Vol(Sc1(n, r0)) and the centers of all spheres lie inside the cube,
the total number of spheres is bounded from below by
M=
Vol SM
i=1 Sci(n, r0)
Vol(Sc1(n, r0))
=∆n(S)·Vol Q0(n, A)
Vol(Sc1(n, r0))
≥2−n·An
Vol(Sc1(n, r0)) ,(24)
where the first inequality holds by (19) and the second inequality holds by (21). The above bound can be further
simplified as follows
log M≥log An
Vol Sc1(n, r0)!−n
(a)
=nlog A
√πr0+ log n
2!−n
(b)
=nlog A−nlog r0+1
2nlog n−nlog e+o(n),(25)
where (a) exploits (22) and (b) holds by Stirling’s approximation6. Now, for
r0=pnθn=√an 1+b+κ
4,(26)
5We note that the proposed proof of the lower bound in (21) is non-constructive in the sense that, while the existence of the respective
saturated packing is proved, no systematic construction method is provided.
6we recall that the packing of hyper spheres with with growing radius ∼n1+2κ
4in the codeword length inside a hyper cube with finite
edge length is indeed counter-intuitive since the volume of such hyper sphere diverges to infinity as n→ ∞; see Appendix B for
more details and explanations.
12 M.J.S.
we obtain
log M≥nlog A
√a−1
4(1 + b+κ)nlog n+1
2nlog n−nlog e+o(n)
=1−(b+κ)
4nlog n+nlog A
e√a+o(n),(27)
where the dominant term is of order nlog n. Hence, for obtaining a finite value for the lower bound of the
rate, R, (27) induces the scaling law of Mto be 2(nlog n)R. Therefore, we obtain
R≥1
nlog n"1−(b+κ)
4nlog n+nlog A
e√a+o(n)#,(28)
which tends to 1−κ
4when n→ ∞ and bÏ0.
|Encoding
Given message i∈[[M]], transmit x=ci.
|Decoding
Let
τn=cρ2
0θn=acρ2
0n1
2((κ+b)−1) ,(29)
where 0 < b < 1 is an arbitrarily small constant and 0 < c < 2 is a constant.Before we proceed, for the sake of
brevity of analysis, we introduce the following conventions:
•Let Yt(i)∼Pois(cρ
i,t +λ) denote the channel output at time tgiven that x=ci.
•Let Y(i) = (Y1(i), . . . , Y¯n(i))
•Let Ix
t
def
=λ+PK−1
k=1 ρkxt−k.
•Let yt(i)def
=yt(i)−(ρ0ci,t +λ) where yt(i) is a realization of Yt(i).
Remark (Convoluted Symbol) Observe that cρ
i,t =PK−1
k=0 ρkci,t−kis only one symbol but is constructed from
a linear combination of Kmost recent symbols weighted by coefficients ρk.
To identify whether message j∈ M was sent, the decoder checks whether the channel output ybelongs to the
following decoding set:
Tj=ny∈ Y¯n:T(y;cj)≤τno,(30)
where
T(y;cj) = 1
¯n
¯n
X
t=1 yt−(ρ0cj,t +λ)2−ρ0cj,t +Icj
t−Icj
t−λ2(31)
M.J.S . 13
is referred to as the decoding metric evaluated for observation vector yand codeword cj.
|Error Analysis
Fix e1, e2>0 and let ζ0, ζ1>0 be arbitrarily small constants. Consider the type I errors, i.e., the transmitter
sends ci, yet Y/∈ Ti. For every i∈[[M]], the type I error probability is bounded by
Pe,1(i) = Pr T(Y(i); ci)> τn,(32)
where the condition means that x=ciwas sent. In order to bound Pe,1(i), we apply Chebyshev’s inequality,
namely
Pr T(Y(i); ci)−ET(Y(i); ci)> τn≤VarT(Y(i) ; ci)
τ2
n
.(33)
Let us derive the expectation of the decoding metric as follows:
E[T(Y(i); ci)] (a)
=1
¯n
¯n
X
t=1
E[Y2
t]−(ρ0ci,t +Ici
t)−(Ici
t−λ)2
(b)
=1
¯n
¯n
X
t=1
Var[Yt] + (E[Yt])2−(ρ0ci,t +Ici
t)−(Ici
t−λ)2
(c)
=1
¯n
¯n
X
t=1
Var[Yt(i)] + (E[Yt(i)])2−(ρ0ci,t +Ici
t)−(Ici
t−λ)2
= 0 .(34)
where (a) follows from the linearity of expectation, (b) holds since Var[Yt] = E[Y2
t]−(E[Yt])2, and (c) follows
since Var[Yt(i)] = Var[Yt(i)] = ρ0ci,t +Ici
tand E[Yt] = (ρ0ci,t +Ici
t)−(ρ0ci,t +λ) = Ici
t−λ.
Second, in order to compute the upper bound in (33) we proceed to compute the variance of the decoding
metric. Let us define
ψVar
def
=
¯n
X
t=1
Var[(Yt(i))2].(35)
we obtain
Var T(Y(i); ci)=ψVar
¯n2,(36)
since, conditioned on ci, the channel outputs conditioned on the Kmost recent input symbols are independent.
Remark (Output Correlation) The pair (Yt, Y t′) is correlated as long as t−t′≤K; cf (1). That is, output
symbols with a distance of no more than Kare dependent on the same input symbols, and hence are correlated;
cf (1). However, in our error analysis, we always assume that the argument of the variance is conditioned on
the given codeword x=ci, hence, (Yt(i), Y t′(i)) are safely uncorrelated.
14 M.J.S.
Now, based on Appendix A, we provide an upper bound for the summand in (35) as follows
Var h(Yt(i))2i≤6(A+λ)41 + e8
λ1 + (A+λ) + (A+λ)2+ (A+λ)3,(37)
Thereby,
ψVar
def
=
¯n
X
t=1
Var h(Yt(i))2i
≤6¯n(A+λ)41 + e8
λ1 + (A+λ) + (A+λ)2+ (A+λ)3
def
=ψUB
Var .(38)
Therefore, exploiting (33), (34) and (38) we can bound the type I error probability in (32) as follows
Pe,1(i) = Pr T(Y(i); ci)> τn
≤ψUB
Var
¯n2τ2
n
=
6(A+λ)41 + e8
λ1 + (A+λ) + (A+λ)2+ (A+λ)3
c2ρ4
0a2¯nn(κ+b)−1
=
6(A+λ)41 + e8
λ1 + (A+λ) + (A+λ)2+ (A+λ)3
c2ρ4
0a2nκ+b
≤e1,(39)
where the last equality follows from (38) and n < ¯n. Hence, Pe,1(i)≤e1holds for sufficiently large nand
arbitrarily small e1>0.
Next, we address type II errors, i.e., when Y∈ Tjwhile the transmitter sent ci. Then, for every i, j ∈[[M]],
where i6=j, the type II error probability is given by
Pe,2(i, j ) = Pr T(Y(i); cj)≤τn.(40)
where T(Y(i); cj) = β−αwith
β=1
¯n
¯n
X
t=1 Yt(i)−(ρ0ci,t +λ) + ρ0ci,t −cj,t 2,(41)
α=1
¯n
¯n
X
t=1 ρ0cj,t +Icj
t+Icj
t−λ2.(42)
Observe that term βitself can be expressed by β=β1+β2where
β1=1
¯n
Y(i)−(ρ0ci+λ1¯n)
2+
ρ0ci−cj
2,(43)
M.J.S . 15
β2=2ρ0
¯n
¯n
X
t=1 ci,t −cj,t Yt(i)−ρ0ci,t +λ.(44)
Then, define the following events
Hj
i=|β−α| ≤ τn,(45)
E0=|β2|> τn,
E1={β1−α≤2τn}.(46)
Exploiting the reverse triangle inequality, i.e., |β| − |α| ≤ |β−α|, we obtain the following upper bound on the
type II error probability
Pe,2(i, j ) = Pr Hj
i
= Pr |β−α| ≤ τn
≤Pr |β| − |α| ≤ τn
(a)
= Pr (β−α≤τn),(47)
where (a) follows since α≥0 and β≥0. Now, applying the law of total probability to event B=β−α≤τn
over E0and its complement Ec
0, we obtain
Pe,2(i, j )≤Pr (B ∩ E0)+ Pr B ∩ Ec
0
(a)
≤Pr (E0)+ Pr B ∩ Ec
0
(b)
≤Pr (E0)+ Pr (E1),(48)
where inequality (a) follows from B ∩ E0⊂ E0and inequality (b) follows from Pr B ∩ Ec
0≤Pr (E1), which is
proved in the following. Observe,
Pr B ∩ Ec
0= Pr β−α≤τn∩|β2| ≤ τn
= Pr β1−α≤τn−β2∩|β2| ≤ τn
(a)
≤Pr β1−α≤2τn
= Pr (E1),(49)
where inequality (a) holds since τn−β2≤2τnconditioned on |β2| ≤ τn.
We now proceed with bounding Pr (E0). By Chebyshev’s inequality, the probability of this event can be
bounded as follows
Pr(E0)≤
VarnP¯n
t=1 ci,t −cj,t Yt−ρ0ci,t +λo
¯n2τ2
n/(4ρ2
0)
16 M.J.S.
=4ρ2
0P¯n
t=1(ci,t −cj ,t )2·Var[Yt(i)]
¯n2τ2
n
=4ρ2
0P¯n
t=1(ci,t −cj ,t )2·(ρ0ci,t +Ici
t)
¯n2τ2
n
≤4ρ2
0(ρ0A+Ici
t)P¯n
t=1(ci,t −cj ,t )2
¯n2τ2
n
=4ρ2
0(ρ0A+Ici
t)
ci−cj
2
¯n2τ2
n
.(50)
Observe that
ci−cj
2(a)
≤kcik+
cj
2
(b)
≤√nkcik∞+√n
cj
∞2
(c)
≤√nA +√nA2
= 4nA2,(51)
where (a) holds by the triangle inequality, (b) follows since k·k ≤ √nk·k∞, and (c) is valid by (23). Hence, we
obtain
Pr(E0)≤16nρ2
0(ρ0A+Ici
t)A2
¯n2τ2
n
≤16ρ2
0(ρ0A+Ici
t)A2
nτ2
n
=16(ρ0A+ (K−1)A)A2
c2ρ2
0a2nκ+b
≤16(ρ0+nκ)A3
c2ρ2
0a2nκ+b
≤16(ρ0+ 1)A3
c2ρ2
0a2nb
≤ζ0,(52)
for sufficiently large n, where ζ0>0 is 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 √nθn, that is
ci−cj
2≥4nθn.(53)
Thus, we can establish the following upper bound for event E1:
Pr(E1) = Pr
1
¯n
Y(i)−(ρ0ci+λ1n)
2+
ρ0ci−cj
2−
¯n
X
t=1 ρ0cj,t +Icj
t+Icj
t−λ2
≤2τn
M.J.S . 17
(a)
≤Pr
1
¯n
Y(i)−(ρ0ci+λ1n)
2−
¯n
X
t=1 ρ0cj,t +Icj
t+Icj
t−λ2
≤2(c−2)ρ2
0θn
= Pr
1
¯n
¯n
X
t=1 Yt(i)−ρ0ci,t +λ2−ρ0cj,t +Icj
t+Icj
t−λ2≤2(c−2)ρ2
0θn
(b)
≤
Var 1
¯nP¯n
t=1 Yt(i)−ρ0ci,t +λ2
2(c−2)ρ2
0θn2
(c)
≤ψUB
Var
2¯n(c−2)ρ2
0θn2
≤
6(A+λ)41 + e8
λ1 + (A+λ) + (A+λ)2+ (A+λ)3
4(c−2)2ρ4
0a2¯nn(2κ+b)−1
≤
6(A+λ)41 + e8
λ1 + (A+λ) + (A+λ)2+ (A+λ)3
4(c−2)2ρ4
0a2n2κ+b
≤ζ1,(54)
for sufficiently large n, where ζ1>0 is an arbitrarily small constant. Here, (a) follows from (53) and (29), (b)
holds by Chebyshev’s inequality as given in (33), and (c) follows by Appendix A. Therefore,
Pe,2(i, j )≤Pr(E0) + Pr(E1)≤ζ0+ζ1≤e2,
hence, Pe,2(i, j )≤e2holds for sufficiently large nand arbitrarily small e2>0. We have thus shown that for
every e1, e2>0 and sufficiently large n, there exists an (n, M (n, R), K(n, κ), e1, e2) code.
3.3 |Converse Proof
The proof of the converse is based on the following two steps. Step 1: First, we show in Lemma 1 that for
any achievable rate (for which the type I and type II error rates vanish as n→ ∞), the distance between any
selected entry of one codeword with any entry of another codeword should be at least larger than a threshold.
Step 2: Then, using Lemma 1, we derive an upper bound on the codebook size of achievable identification
codes.
We start with the following lemma regarding the ratio of a function of the letters for every pair of codewords
where such a function is defined as di ,t =ρ0ci,t +Ici
t,∀t∈[[n]].
Lemma 1 Suppose that Ris an achievable rate for the DTPC P. Consider a sequence of (n, M (n, R), K(n, κ),
e(n)
1, e(n)
2)codes (C(n),T(n))such that e(n)
1and e(n)
2tend to zero as nÏ ∞. Then, given a sufficiently large n,
the codebook C(n)satisfies the following property. For every pair of codewords, ci1and ci2, there exists at
least one letter t∈[[n]] such that
1−ρ0ci2,t +Ici2
t
ρ0ci1,t +Ici1
t
> θ′
n,(55)
18 M.J.S.
for all i1, i2∈[[M]], such that i16=i2, with
θ′
n=Pmax
Kn1+b=Pmax
n1+b+κ,(56)
where b > 0is an arbitrarily small constant and Iciz
t
def
=λ+PK−1
k=1 ρkciz,t−k, z ∈ {1,2}.
Proof The proof is given in Appendix C.
Next, we use Lemma 1 to prove the upper bound on the DI capacity. Observe that since
di,t =ρ0ci,t +Ici
t> λ , (57)
Lemma 1 implies
ρ0ci1,t −ci2,t=di1,t −di2,t
(a)
> θ′
ndi1,t
(b)
> λθ′
n,(58)
where (a) follows by (55) and (b) holds by (57). Now, since
ci1−ci2
≥ci1,t −ci2,t, we deduce that the distance
between every pair of codewords satisfies
ci1−ci2
>λθ′
n
ρ0
.(59)
Thus, we can define an arrangement of non-overlapping spheres Sci(n, λθ ′
n
ρ0), i.e., spheres of radius λθ′
nthat are
centered at the codewords ci. 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 M, is bounded by
M=
Vol SM
i=1 Sci(n, r0)
Vol(Sc1(n, r0))
=∆n(S)·Vol Q0(n, P max)
Vol(Sc1(n, r0))
≤2−0.599n·Pn
max
Vol(Sc1(n, r0)) ,(60)
where the last inequality follows from inequality (21). Thereby,
log M≤log Pn
max
Vol Sc1(n, r0)!−0.599n
=nlog Pmax −nlog r0−nlog √π+1
2nlog n
2−n
2log e+o(n)−0.599n , (61)
where the dominant term is again of order nlog n. Hence, for obtaining a finite value for the upper bound of
M.J.S . 19
the rate, R, (61) induces the scaling law of Mto be 2(nlog n)R. Hence, by setting
M(n, R) = 2(nlog n)R,(62)
and
r0=λθ′
n
2ρ0
=λP max
2ρ0n1+b+κ,(63)
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+κ)nlog n−n log λ√πe
2ρ0
+ 1.0599!+o(n)
,(64)
which tends to 3
2+κas n→ ∞ and b→0. This completes the proof of Theorem 1.
4|SUMMARY AND FUTURE DIRECTIONS
In this work, we studied the DI problem over the DTPC with Knumber of ISI channel taps. We assumed that
K=K(n, κ) = 2κlog n=nκwhere κ∈[0,1) scales sub-linearly with the codeword length n. In practice, the
DTPC exhibits memory [8], therefore, our results in this paper may serve as a model for event-triggered based
tasks in the context of many practical MC applications. Especially, we obtained lower and upper bounds on
the DI capacity of the DTPC with memory subject to average and peak power constraints with the codebook
size of M(n, R) = 2(nlog n)R=nnR . Our results for the DI capacity of the DTPC with memory revealed that
the super-exponential scale of nnR is the appropriate scale for codebook size. This scale coincides the scale
for codebook of memoryless DTPC and Gaussian channels [62, 64] and stands considerably different from the
traditional scales as in transmission and RI setups where corresponding codebooks size grows exponentially
and double exponentially, respectively.
We show the achievability proof using a packing of hyper spheres and a distance decoder. In particular,
we pack hyper spheres with radius ∼n1+2κ
4where κ:= lognK∈[0,1) is the ISI rate, inside a larger hyper
cube. While the radius of the spheres in a similar proof for Gaussian channels vanishes, as nincreases [64],
the radius here similar to the case for memoryless DTPC [33] diverges to infinity. Yet, likewise as in [33] we
can obtain a positive rate while packing a super-exponential number of spheres fulfilling the molecule release
rates and error constraints.
For the converse proof, we follow a similar approach as in our previous work for the memory-less DTPC
[33]. In [33], we established a minimum distance between each pair of shifted codewords when the amount of
shift was the constant interference signal λ > 0. Here, we let the value of shift vary according to the related
codeword where it is lower bounded by λ > 0. In general, the derivation here is more involved than the
derivation in the Gaussian case [64]. In our previous work on Gaussian channels with fading [64], the converse
proof was based on establishing a minimum distance between each pair of codewords (with no shift). Here,
on the other hand, we use the stricter requirement that the ratio of the letters of every two different shifted
codewords is different from 1 for at least one index.
20 M.J.S.
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:
•Continuous Alphabet Conjecture: Our observations for the codebook size of the memoryless DTPC and
Gaussian channels [64] lead us to conjecture that the codebook size for any continuous alphabet channel
with/out memory is a super-exponential function, i.e., 2(nlog n)R. However, a formal proof of this conjecture
remains unknown.
•Multi User: The extension of this study (point-to-point system) to multi-user scenarios (e.g., broadcast and
multiple access channels) or multiple-input multiple-output channels may seems more relevant in complex
MC nano-networks.
•Fekete’s Lemma: Investigation of the behavior of the DI capacity in the sense of Fekete’s Lemma [79]:
To verify whether the pessimistic (C= lim inf n→∞ logM(n,R )
nlog n) and optimistic (C= lim supn→∞ logM(n ,R)
nlog n)
capacities [80] coincide or not; see [79] for more details.
•Channel Reliability Function: A complete characterization of the asymptotic behavior of the decoding
errors as a function of the codeword length for 0 < R < C requires knowledge of the corresponding
channel reliability function (CRF) [81]. To the best of the authors’ knowledge, the CRF for DI has not been
studied in the literature so far, neither for the Gaussian channel [62] nor the Poisson channel [33, 34, 69].
•Explicit Code Construction: Explicit construction of DI codes with incorporating the ISI effect and the
development of low-complexity encoding/decoding schemes for practical a designs where the associated
efficiency of such codes can be evaluated with regard to to the our derived performance bounds in Section 3.
•ISI Gain: We have not exploited the ISI knowledge in the decoding procedure. We observed that for
a DTPC with constant degree of ISI, capacity bounds coincide the bounds as of the memoryless DTPC.
This observation suggest that testing a different decoding method which takes effect of ISI into account by
conducting a symbol by symbol detection and exploits the previous Kinput symbols might probably yields
different and more accurate capacity bounds.
A|UPPER BOUND FOR VARIANCE
Let Yt(i)∼Pois ρ0ci,t +Ici
tdenote the channel output at time tgiven that x=ci. Recall that yt(i)def
=yt(i)−
(ρ0ci,t +λ), then we have
Var h(Yt(i))2i=Var Yt−ρ0ci,t +λ2
(a)
≤EYt(i)−ρ0ci,t +λ4
(b)
=EhY4
t(i)−4Y3
t(i)ρ0ci,t +λ+ 6Y2
t(i)ρ0ci,t +λ2−4Yt(i)ρ0ci,t +λ3+ρ0ci,t +λ4i
(c)
≤EhY4
t(i)i−4λEhY3
t(i)i+ 6(ρ0ci,t +λ)2EhY2
t(i)i−4(ρ0ci,t +λ)3EYt(i)+ (ρ0ci,t +λ)4
(d)
≤EhY4
t(i)i−4λEhY3
t(i)i+ 6(A+λ)2EhY4
t(i)i−4(ρ0ci,t +λ)3EYt(i)+ (A+λ)4
(e)
≤EhY4
t(i)i+ 4λEhY4
t(i)i+ 6(A+λ)2EhY4
t(i)i+ 4(A+λ)3EhY4
t(i)i+ (A+λ)4
≤EhY4
t(i)i1 + 4λ+ 6(A+λ)2+ 4(A+λ)3+ (A+λ)4
M.J.S . 21
(f)
≤(A+λ)4e8
λ1 + 4λ+ 6(A+λ)2+ 4(A+λ)3+ (A+λ)4
= (A+λ)41 + e8
λ1 + 4λ+ 6(A+λ)2+ 4(A+λ)3
= 6(A+λ)41 + e8
λ1 + (A+λ) + (A+λ)2+ (A+λ)3,(65)
where (a) follows from Var{Z} ≤ E[Z2
t] with Zt=Yt−ρ0ci,t +λ2, (b) holds by the 8-th order binomial
expansion, (c) follows by the linearity of the expectation operator, (d) and (e) follows from ci,t ≤A , ∀t∈[[n]],
and since expectation is an increasing function, that is, for integers p , q we have
Yp
t(i)< Yq
t(i)ÑEhYp
t(i)i<EhYp
t(i)i,(66)
(f) holds by employing an upper bound on the non-central moment of a Poisson random variable with mean
λZas follows (see [82, Coroll. 1])
EhZki≤λk
Zexp (k2
2λZ).(67)
B|VOLUME OF A HYPER SPHERE WITH GROWING RADIUS
To solidify the idea of packing spheres within a hyper cube, we explain about the packing of hyper spheres
with growing radius in the codeword length n. Despite the fact that radius of the hyper sphere’s diverges to
infinity as n→ ∞ as ∼n1+κ
4, still the associated volume converges to zero super-exponentially inverse as of
order ∼n−(1+κ)n
4. This makes an accommodation of super-exponential number of such hyper spheres inside
the hyper cube possible. The ratio of the spheres in our construction grows with n, as ∼n1+κ
4. Volume of
an n-dimensional unit-hyper sphere, i.e., with a radius of r0= 1, tends to zero, as n→ ∞ [77, 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 Sc1(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
,(68)
where the last equality follows by Stirling’s approximation [8 3, P. 52], that is, log n! = nlog n−nlog e+o(n). The
last expression in (68) tends to zero for all r0=ncwith c∈(0,1
2). Observe that when n→ ∞, the volume of a
hyper cube Q0(n, A) with edge length Awhen A < 1 tends to zeros, that is, limn→∞ Vol Q0(n, A)= limn→∞ An=
0.
Now, to count the number of spheres that can be packed inside the hyper cube Q0(n, A), we derive the
22 M.J.S.
log-ratio of the volumes as follows
log Vol Q0(n, A)
Vol Sc1(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+n log A
√πe −3
2!+o(n),(69)
where the last equality follows from r0=nc. Now, since the dominant term in (69) involves nlog n, we deduce
that codebook size should be M(n, R) = 2(nlog n)R, thereby by (24) we obtain
R≥1
nlog n
log Vol Q0(n, A)
Vol Sc1(n, r0)!−n
=1
nlog n
1
2−cnlog n+n log A
√πe −3
2!+o(n)
,(70)
which tends to 1
2−cwhen n→ ∞. As a result, (70) induces that condition c < 1
2with cnot being arbitrary
approaching 1
2to derive a meaningful (non-zero) lower bound. Since c=1+κ
4we obtain
1 + κ
4<1
2Ñκ < 1.(71)
C|PROOF OF LEMMA 1
In the following, we provide the proof of Lemma 1. The method of proof is by contradiction, namely, we
assume that the condition given in (55) is violated and then we show that this leads to a contradiction, namely,
sum of the type I and type II error probabilities converge to one, i.e., limn→∞ Pe,1(i1) + Pe,2(i1, i2)= 1.
Recall that Yt(i)∼Pois(cρ
i,t +λ) denote the channel output at time tgiven that x=ci. Fix e1, e2>0. Let
η, δ > 0 be 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−di2,t
di1,t ≤θ′
n,(72)
where diz,t =ρ0cik,t +Iciz
t, z = 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
Ri1=
y∈ Ti1:1
¯n
¯n
X
t=1
yt≤ρ0Pmax +Ici1
t+δ
.(73)
M.J.S . 23
Then, observe that
Pe,1(i1) + Pe,2(i2, i1) = 1 −X
y∈Ti1
V¯nyci1+X
y∈Ti1
V¯nyci2
≥1−X
y∈Ti1
V¯nyci1+X
y∈Ti1∩Ri1
V¯nyci2.(74)
Now, consider the sum over Ti1in (74),
X
y∈Ti1
V¯nyci1=X
y∈Ti1∩Ri1
V¯nyci1+X
y∈Ti1∩Rc
i1
V¯nyci1
≤X
y∈Ti1∩Ri1
V¯nyci1+ Pr
1
¯n
¯n
X
t=1
Yt(i1)> ρ0Pmax +Ici1
t+δ
.(75)
Next, we bound the probability on the right hand side of (75) as follows
Pr
1
¯n
¯n
X
t=1
Yt(i1)−1
¯n
¯n
X
t=1
E[Yt(i1)] > ρ0Pmax +δ−1
¯n
¯n
X
t=1
E[Yt(i1)]
(a)
≤
Var h1
¯nP¯n
t=1 Yt(i1)i
ρ0Pmax +δ−1
¯nP¯n
t=1 E[Yt(i1)]2
(b)
=
1
n2P¯n
t=1(ρ0ci1,t +Ici1
t)
ρ0Pmax +δ−1
¯nP¯n
t=1 ρ0ci1,t +Ici1
t2
(c)
≤ρ0Pmax +Ici1
t
nδ2
≤Pmax +λ+ (K−1)A
nδ2
≤Pmax +λ+A
n1−κδ2
≤η , (76)
for sufficiently large n, where inequality (a) follows from Chebyshev’s inequality, for equality (b), we exploited
Var[Yt(i1)] = E[Yt(i1)] = ρ0ci1,t +Ici1
t, and for inequality (c), we used the fact that ci1,t ≤Pmax ,∀t∈[[n]].
Returning to the sum of error probabilities in (74), exploiting the bound (76) leads to
Pe,1(i1) + Pe,2(i2, i1)≥1−X
y∈Ti1∩Ri1V¯nyci1−V¯nyci2−η . (77)
Now, let us focus on the summand in the square brackets in (77). By (4), we have
V¯nyci1−V¯nyci2=V¯nyci1·1−V¯nyci2/ V ¯nyci1
24 M.J.S.
=V¯nyci1·
1−
¯n
Y
t=1
e−(di2,t −di1,t ) di2,t
di1,t !yt
=V¯nyci1·
1−
¯n
Y
t=1
e−θ′
ndi1,t 1−θ′
nyt
,(78)
where for the last inequality, we employed
di2,t −di1,t ≤di2,t −di1,t≤θ′
ndi1,t and 1 −di2,t
di1,t ≤
1−di2,t
di1,t ≤θ′
n,(79)
which follow from (72). Now, we bound the product term inside the bracket as follows:
¯n
Y
t=1
e−θ′
ndi1,t 1−θ′
nyt=e−θ′
n¯n
t=1 di1,t ·1−θ′
n¯n
t=1 yt
(a)
≥e−nθ′
nρ0Pmax+Ici1
t·1−θ′
nnρ0Pmax+Ici1
t+δ
=enθ′
nδ·e−nθ′
nρ0Pmax+Ici1
t+δ·1−θ′
nnρ0Pmax+Ici1
t+δ
(b)
≥enθ′
nδ·e−nθ′
nρ0Pmax+Ici1
t+δ·1−nθ′
nρ0Pmax+Ici1
t+δ
≥enθ′
nδ·f(nθ′
n)
(c)
> f(nθ′
n)
(d)
≥1−3ρ0Pmax +Ici1
t+δnθ′
n
= 1 −3(ρ0Pmax +λ+KA +δ)Pmax
nb+κ
= 1 −3(ρ0Pmax +λ+Anκ+δ)Pmax
nb+κ
= 1 −3(ρ0Pmax +λ+A+δ)Pmax
nb
≥1−η . (80)
for sufficiently large nwhere (a) follows since
di1,t ≤ρ0Pmax +Ici1
t,∀t∈[[n]],and
¯n
X
t=1
yt≤nρ0Pmax +Ici1
t+δ,(81)
where the latter inequality follows from y∈ Ri1, cf. (73). For (b), we used Bernoulli’s inequality
(1 −x)r≥1−rx , ∀x > −1,∀r > 0,(82)
[84, see Ch. 3]. For (c), we exploited enθ ′
nδ>1 and the following definition: f(x) = e−cx (1 −x)cwith c=
Ici1
t+ρ0Pmax +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.
M.J.S . 25
Equation (78) can then be written as follows
V¯nyci1−V¯nyci2≤V¯nyci1·1−e−θ′
n¯n
t=1 di1,t ·1−θ′
n¯n
t=1 yt
≤η·V¯nyci1.(83)
Combining, (77), (78), and (83) yields
Pe,1(i1) + Pe,2(i2, i1)(a)
≥1−X
y∈Ri1V¯nyci1−V¯nyci2−η
= 1 −X
y∈Ri1η·V¯nyci1−η
(b)
≥1−2η , (84)
where for (a), we replaced y∈ Ri1∩Ti1by y∈ Ri1to enlarge the domain and for (b), we used Py∈Ri1V¯nyci1≤
1. Clearly, this is a contradiction since the error probabilities tend to zero as nÏ ∞. Thus, the assumption in
(72) is false. This completes the proof of Lemma 1.
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