# An $M$-ary Concentration Shift Keying With Common Detection Thresholds For
Multi-Transmitter Molecular Communication

Ethungshan Shitiri, and Ho-Shin Cho E. Shitiri is with the Department of
Computer Architecture, Universitat Politècnica de Catalunya, Barcelona, Spain,
and H.S. Cho is with the School of Electronic and Electrical Engineering,
Kyungpook National University, Daegu, South Korea. e-mail<EMAIL_ADDRESS>This study was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korean Government (MSIT) (2021R1A2C1003507).Manuscript
received XXX, 2022; revised XXX, 2022.

###### Abstract

Concentration shift keying (CSK) is a widely adopted modulation technique for
molecular communication-based nanonetworks, which is a key enabler for the
Internet of Bio-NanoThings (IoBNT). However, existing methods provide optimal
error performance at the cost of high operational complexity that scales
poorly as the number of transmitters, $K$, increases. This paper proposes a
novel $M$-ary CSK method termed CSK with Common detection Thresholds (CSK-CT).
CSK-CT uses common thresholds that are sufficiently low to ensure the reliable
detection of symbols transmitted by every transmitter, regardless of their
distance. We derive closed-form expressions to obtain the common thresholds
and release concentrations. To enhance the error performance, we optimize the
release concentration using a scaling exponent that further optimizes the
common thresholds. We evaluate the performance of CSK-CT in comparison to the
benchmark CSK for varying values of $K$ and $M$. In terms of the error
probability, CSK-CT offers between $10^{-7}$ and $10^{-4}$, which are a
substantial improvement from the $10^{-4}$ to $10^{-3}$ offered by the
benchmark. In terms of complexity, CSK-CT is $\mathcal{O}\big{(}n)$ and does
not scale with $K$ but $M$ ($M\ll K$), while the benchmark is
$\mathcal{O}\big{(}n^{2})$. Furthermore, CSK-CT showcased the ability to
mitigate inter-symbol interference, although this facet warrants further
investigation. Due to its low error probability, improved scalability, low
complexity, and potential ISI mitigation features, CSK-CT demonstrates
benefits in applications of IoBNT focused on data-gathering. Specifically, its
utility is well-noted in settings where a computationally strained receiver
collects sensitive health-related data from multiple transmitters.

###### Index Terms:

internet of bio-nano things, molecular communication, concentration shift
keying, modulation, inter-symbol interference, symbol detection, nanonetworks.

##  I Introduction

The most recent addition to the family of Internet of Things is the Internet
of Bio-Nano Things (IoBNT) that has been facilitated by the advances made in
the fields of synthetic biology and nanotechnology [1]. Architecturally, an
IoBNT comprises three networks at different length scales—the Internet at the
macro-scale, wireless body area networks at the micro-scale, and the emerging
nanonetwork at the nano-scale [2]. A nanonetwork is the interconnection of
bio-nanothings, which are based on living cells and their biochemical
functions, capable of executing simple sensing, communicating, and computing
tasks [3]. With their unique size advantage, nanonetworks are set to usher in
a myriad of unexplored wireless applications within extreme environments,
ranging from intra-body healthcare systems to hazardous chemical industries
[4, 5, 6, 7]. The small size, though advantageous, introduces unprecedented
challenges as it imposes rigid restrictions on energy consumption, computing
prowess, and processing capacity. This makes it extremely challenging to
directly employ the well-known, efficient, but complex wireless technologies,
driving the need for simple yet efficient communication technologies.

This brings us to one of the most promising communication paradigms for
nanonetworks known as molecular communication (MC). MC is a communication
paradigm that draws inspiration from nature and uses engineered molecules to
carry information symbols [8]. It is suited for nanonetworks due to its nano-
scale nature, energy efficiency, and compatibility with biological systems [9,
10, 11]. Though still a developing technology, MC is proving vital for IoBNT’s
potential healthcare innovation, such as smart drug delivery systems [9] and
early disease detection systems [12, 13]. This underscores the necessity for
focused research in this emergent area and is the focus of this paper.

Owing to the inherent stochastic nature of MC coupled with the rigid
restrictions mentioned above, conventional wireless techniques are not
straightforwardly applicable to MC-based nanonetworks [1, 8]. In this context,
one of the primary areas of focus in current research is the development of
novel modulation techniques. As a result, various innovative methods for
modulation are being explored, such as concentration [14, 15, 16, 17],
type[18, 19, 20, 21], timing [22, 23, 24], and direction [25, 26]. Among these
methods, concentration modulation, commonly referred to as concentration-shift
keying (CSK), is widely popular because of its simplicity.

CSK is analogous to the conventional amplitude shift keying method in that
information symbols modulate the number of molecules released [14, 15].
(Operation of CSK is described in Section II.) To achieve optimal error
performance, approaches using conventional estimation techniques have been
developed, including the robust maximum likelihood estimation (MLE) [16, 17].
These approaches assume that bio-nanothings possess adequate computing
resources and capabilities, which is supported by the recent advancements made
in nano-scale computing bio-hardware, such as DNA-based computing [27, 28].
However, the feasibility of running powerful but computationally intensive
techniques, such as MLE, on such bio-hardware is not fully understood yet.
Furthermore, integrating the computing and communication hardware blocks is an
open research challenge [29].

Regardless of advances in computing bio-hardware, it is always desirable to
maintain a low complexity profile alongside high energy efficiency, which can
be correlated to the level of complexity. The current methods require
$K\cdot(M-1)$ threshold computations, where $K$ and $M$ denote the number of
transmitters and the number of information symbols, respectively. This results
in a quadratic time complexity of $\mathcal{O}\big{(}n^{2})$, where $n$ is a
function of both $K$ and $M$. (Further details of how the complexity value is
obtained will be presented in Section V-A). This means that the complexity
scales poorly when $K$ goes up in the order of thousands, which is a practical
number in nanonetworks [3]. Therefore, to obtain the comprehensive
effectiveness of CSK methods, it is imperative to simultaneously address both
operational complexity and error performance, thereby reducing the gap between
theoretical CSK frameworks and their practical implementations.

To address the problems mentioned above, we propose a novel low-complexity
method termed CSK with Common detection Thresholds (CSK-CT). The term “common
detection thresholds” reflects the distinctive feature of CSK-CT. It sets the
detection thresholds for each information symbol sufficiently low to account
for the fact that not all transmitted symbols have high received energy levels
when they reach the receiver. Some, originating from distant transmitters,
will have lower energy levels at reception that may be insufficient to cross
over the given detection threshold. By employing that approach, CSK-CT aims to
ensure the reliable detection of symbols transmitted by every transmitter,
regardless of their distance. In contrast to current methods, where a unique
set of detection thresholds must be computed for each transmitter, CSK-CT only
needs to compute one common set of $(M-1)$ thresholds, valid for every
transmitter. Moreover, CSK-CT exhibits a linear time complexity, i.e.,
$\mathcal{O}\big{(}n)$, where $n$ is a function of $M$ and, therefore, does
not scale with $K$. Importantly, such complexity makes CSK-CT on par with
current methods for single transmitter systems and superior for multiple
transmitter systems.

The remainder of the paper is organized as follows: In Section II, we provide
a background on CSK. In Section III, we present the MC free-diffusion channel
model for describing the channel impulse response (CIR) along with the
formulation of the energy of the received signal and the average inter-symbol
interference (ISI) model. In Section IV, the benchmark CSK [16] is discussed
to highlight the limitations of current CSK methodologies before introducing
the proposed CSK-CT method. Specifically, this section derives the closed-form
expressions for calculating threshold values and release concentrations with a
scaling exponent. Time complexity and symbol error probability are analyzed in
Section V. Subsequently, in Section VI, we analyze the performance of CSK-CT
under varying values of $M$ (BCSK ($M=2$) and 4-CSK ($M=4$)), average
distance, scaling exponent, and symbol period duration. Additionally, we
demonstrate that CSK-CT can deliver superior performance compared to the
benchmark CSK based on the selection of suitable scaling exponent values.
Finally, Section VII offers a summary of the study and discusses potential
directions for future research.

##  II Background

In CSK, the number of molecules transmitted by a transmitter (a bio-nanothing)
is termed release concentration, denoted by $Q$. In general, $M$ release
concentration levels represent $M=2^{b}$ symbols, where $b$ is the number of
bits per symbol. The j-th information symbol is denoted by
$S_{\mathit{\\!j}}$, and its associated release concentration is denoted by
$Q_{\mathit{j}}$. At the receiver (also a bio-nanothing), the number of
molecules received is compared against the detection thresholds, denoted by
$\tau$, to decode the transmitted symbol. The total number of molecules
received is termed the energy of the received signal [30, 31], denoted by
$\mathcal{E}_{\text{tot}}$.

As depicted in Fig. 1, to identify $M$ symbols, there are $(M-1)$ detection
thresholds. The threshold between adjacent symbols $S_{\mathit{\\!j}}$ and
$S_{\mathit{\\!j}+1}$ is denoted by $\tau_{\mathit{j}},j\in[0,M-2]$.
Accordingly, the detection region of a symbol $S_{\mathit{\\!j}}$ is between
the threshold levels $\tau_{\mathit{j}-1}$ and $\tau_{\mathit{j}}$. For
example, suppose $S_{\\!1}$ is transmitted. If $\mathcal{E}_{\text{tot}}$
falls between $\tau_{0}$ and $\tau_{1}$, the symbol is decoded as $S_{\\!1}$.
If $\mathcal{E}_{\text{tot}}$ falls between $\tau_{1}$ and $\tau_{2}$, the
symbol is decoded as $S_{\\!2}$, and so on.

Figure 1: An illustration depicting the threshold levels and detection regions
of information symbols in CSK.

Given the infancy and intricacy of MC systems, it is a common practice to
analyze the performance of CSK methods using a simplified communication system
model comprising a transmitter and a receiver [14, 15, 16, 17, 32]. The
simplicity of this model has led to laying the foundation for fundamental
principles of MC. For instance, the channel impulse response (CIR) function
can be derived by representing the transmitter and receiver as point or
spherical objects [32]. This function describes the communication channel’s
response to an impulse input and characterizes its behavior, allowing for the
assessment of the channel’s quality. Most importantly, it allows the
derivation of $\mathcal{E}_{\text{tot}}$.

The characterization of $\mathcal{E}_{\text{tot}}$ enables the determination
of the standard system performance metric termed the error probability (or
rate), and its minimization is pivotal for determining the efficiency of a
modulation technique [31]. Furthermore, an MC channel has memory [32], and
molecules from a previous transmission may arrive during a current symbol
period. This leads to the ISI phenomenon [33]. The ISI molecules cannot be
distinguished from the signal molecules and are considered part of the current
symbol transmission, which can worsen the error probability.

With regards to the threshold computations required in current CSK methods, we
focus on the work of Singhal et. al. [16]. The MLE strategy is employed to
estimate the thresholds through a pairwise comparison of the Gaussian noise
probability density functions (PDFs) of adjacent release concentrations.
Consequently, a closed-form expression to determine the thresholds is
presented [16, eq. (13)]. However, recognizing the high computational
complexity of the derived expression, Singhal et. al. also developed a reduced
computation complexity method, where the thresholds are computed based on the
square root of the product of the received energy associated with adjacent
release concentrations.

While the original approach was developed for a single transmitter system, it
can be extended for multiple transmitters with relative ease. (We will
demonstrate this in Section IV.) However, as stated earlier, the receiver must
compute a total of $K\cdot(M-1)$ thresholds, which increase in proportion to
$K$. We reiterate that CSK’s high complexity could limit its practical
implementation, which is the primary reason we propose the CSK-CT.

Figure 2: Time-slotted channel model.

##  III System Model

We consider a star-network topology in a three-dimensional (3D) unbounded
medium, with $K$ point transmitters randomly distributed around a fully
absorbing spherical-shaped receiver with radius $r$. Accordingly, the random
distance between a transmitter and the receiver’s surface nearest to it,
denoted by $y$, is assumed to follow the uniform distribution, i.e.,
$Y\sim\mathcal{U}(y_{\text{min}},y_{\text{max}})$, where $y_{\text{min}}$ and
$y_{\text{max}}$ are the smallest and largest distances in the network,
respectively. $\mathcal{U}(a,b)$ represents the continuous uniform
distribution with $a$ and $b$ as the minimum and maximum values, respectively.
Accordingly, the probability density function of $y$ is
$f_{Y}(y)=1/y_{\text{max}}-y_{\text{min}}$.

The transmitters are assumed to be time-synchronized with the receiver and
access the channel in a time-slotted manner, similar to time-division multiple
access [6, 34, 35]. As shown in Fig. 2, time is divided into equal and
recurring intervals called symbol periods. During a symbol period, a symbol is
transmitted at the beginning of the period, and its reception occurs within
the period. Each transmitter is assigned one symbol period in each
transmission round. The duration of a symbol period, denoted as
$t_{\text{sym}}$, is further divided into sampling periods, each of length
$\Delta t$.

We assume that the channel has a memory of length $(K-1)$. This means that the
signal received during a period is comprised of both the intended signal from
the current transmission and the ISI from the previous $(K-1)$ transmissions.
Conversely, we say that a transmitted symbol is a potential ISI contributor to
the future $(K-1)$ symbols. As an example, in Fig. 3, transmitters TX1, TX2,
and TX3 transmit sequentially in the first, second, and third symbol periods.
In the first symbol period, the receiver receives only the signal of TX1 (red
color), and it has no ISI component since there are no previous transmissions.
In the second symbol period, the receiver receives the signal of TX2 (green
color) and the ISI component of TX1. In the third symbol period, the receiver
receives the signal of TX3 (blue color) and the ISI components of TX2 and TX1.
The exact details of the ISI formulation will be carried out in Section III-C.

Figure 3: Illustration of signal and ISI components of a received signal with
three transmitters and three symbol periods.

###  III-A Free-diffusion 3D Channel Model and CIR

In a diffusive channel, the molecules released by a transmitter move
independently, governed by the laws of Brownian motion [36]. When a molecule
hits the surface of the receiver, it gets absorbed. Given a fully absorbing
receiver, any point on the surface of the receiver acts as a point of
absorption [32]. Since the absorbed molecules are effectively removed from the
channel, they contribute to the received signal only once [37, 38]. If a
transmitter that is located at a distance $y$ from the receiver releases the
molecules at time $t=0$, then the hitting rate of the molecules at the
receiver at time $t>0$ is given by [39]

$f_{\text{hit}}(y,t)=\frac{r}{y+r}\frac{y}{\sqrt{4\pi
Dt^{3}}}\exp^{-{y}^{2}/{4Dt}},$ (1)

where $D$ is the diffusion coefficient of the molecule. The fraction of
molecules at the receiver till time $t$ can be obtained from (1) as [37]

$F_{\text{hit}}(y,t)=\frac{r}{y+r}\text{erfc}\left(\frac{y}{\sqrt{4Dt}}\right),$
(2)

where erfc$(\cdot)$ is the complementary error function. The CIR is the
probability that the molecules transmitted at time $t=0$ are absorbed by the
receiver at a distance $y$ during the i-th symbol period. This can be
expressed as [32]

$\displaystyle\begin{split}h(y,i)&=F_{\text{hit}}\left(y,it_{\text{sym}}\right)-F_{\text{hit}}\left(y,\left(i-1\right)t_{\text{sym}}\right)\end{split}$
(3)

According to the definition, $h(y,1)$ denotes the CIR of the signal, and
$h(y,i),\ i\geq 2$, denotes the CIR of the ISI. Using Fig. 3 as an example, we
consider the case where TX3 is currently transmitting. In this case,
$h(y_{\text{3}},1)$ denotes the CIR of the signal of TX3. Then,
$h(y_{\text{1}},3)$ and $h(y_{\text{2}},2)$ denotes the CIR of the ISI from
TX1 and TX2, respectively.

###  III-B Received Signal Energy Without ISI and Noise

As stated earlier, the received signal energy is the total number of molecules
received by the receiver per transmitted symbol. Let symbol
$S_{\mathit{\\!j}}\ (j=0,1,2,\ldots,M-1)$ be assigned the release
concentration $Q_{\mathit{j}}$ and equally transmitted with a probability of
$1/M$. Furthermore, $Q_{j+1}>Q_{j},\forall j$. Then, the energy of the
received signal when a transmitter with distance $y$ releases concentration
$Q$ can be expressed as

$\mathcal{E}_{\text{sig}|Q}=Q\,h(y,1).$ (4)

Since $Q$ and $y$ are independent random variables, the expected received
signal energy can be expressed as

$\displaystyle\mathcal{E}_{\text{sig}}$
$\displaystyle=\mathbb{E}\left[\mathcal{E}_{\text{sig}|Q}\right]$
$\displaystyle=\mathbb{E}\left[Q\,h(y,1)\right]$
$\displaystyle=\int_{y_{\text{min}}}^{y_{\text{max}}}\sum_{Q=Q_{0}}^{Q_{M-1}}Q\,h(y,1)\,\text{Pr}(Q)\,f_{Y}(y)\,dy$
$\displaystyle=\sum_{Q=Q_{0}}^{Q_{M-1}}Q\,\text{Pr}(Q)\,\int_{y_{\text{min}}}^{y_{\text{max}}}\,h(y,1)\,f_{Y}(y)\,dy$
$\displaystyle=\frac{Q_{0}+\ldots+Q_{M-1}}{M}\,\frac{1}{y_{\text{max}}-y_{\text{min}}}\int_{y_{\text{min}}}^{y_{\text{max}}}\,h(y,1)\,dy$
$\displaystyle=\bar{Q}\,\frac{1}{y_{\text{max}}-y_{\text{min}}}\int_{y_{\text{min}}}^{y_{\text{max}}}\,\left(F_{\text{hit}}(y,t_{\text{sym}})-F_{\text{hit}}\left(y,0\right)\right)\,dy$
$\displaystyle=\bar{Q}\,\frac{1}{y_{\text{max}}-y_{\text{min}}}\int_{y_{\text{min}}}^{y_{\text{max}}}\,\frac{r}{y+r}\text{erfc}\,\Bigg{(}\frac{y}{\sqrt{4Dt_{\text{sym}}}}\Bigg{)}\,dy$
$\displaystyle=\bar{Q}\,\mathcal{H}[1],$ (5)

where $\mathbb{E}[\cdot]$ is the expectation operator and
$\mathcal{H}[1]=\frac{1}{y_{\text{max}}-y_{\text{min}}}$
$\int_{y_{\text{min}}}^{y_{\text{max}}}\,\frac{r}{y+r}\text{erfc}\left(y/\sqrt{4Dt_{\text{sym}}}\right)\,dy$
is the average CIR of the signal. Note that the integral is intractable;
hence, we solve it numerically.

###  III-C ISI Model and Analysis

To perform the ISI analysis, we consider at most $k$ previous transmissions
$(1\leq k\leq K-1)$ as shown in Fig. 4. Then, the expected ISI is the sum
total of the interference from each interferer, expressed as

$\displaystyle\mathcal{E}_{\text{isi}}$
$\displaystyle=\sum_{i=2}^{k+1}\mathbb{E}\left[Q\ h(y,i)\right]$
$\displaystyle=\sum_{i=2}^{k+1}\int_{y_{\text{min}}}^{y_{\text{max}}}\sum_{Q=Q_{0}}^{Q_{M-1}}Q\,h(y,i)\,\text{Pr}(Q)\,f_{Y}(y)\,dy$
$\displaystyle\begin{split}&=\bar{Q}\,\sum_{i=2}^{k+1}\frac{1}{y_{\text{max}}-y_{\text{min}}}\int_{y_{\text{min}}}^{y_{\text{max}}}\,\Bigg{(}F_{\text{hit}}\left(y,it_{\text{sym}}\right)-\\\
&\qquad\qquad\qquad\qquad\qquad\qquad
F_{\text{hit}}\left(y,(i-1)t_{\text{sym}}\right)\Bigg{)}\,dy\end{split}$
$\displaystyle\begin{split}&=\bar{Q}\,\sum_{i=2}^{k+1}\frac{1}{y_{\text{max}}-y_{\text{min}}}\int_{y_{\text{min}}}^{y_{\text{max}}}\,\frac{r}{y+r}\Bigg{(}\text{erfc}\,\Bigg{(}\frac{y}{\sqrt{4D-it_{\text{sym}}}}\Bigg{)}-\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\text{erfc}\,\Bigg{(}\frac{y}{\sqrt{4D(i-1)t_{\text{sym}}}}\Bigg{)}\,\Bigg{)}\,dy\end{split}$
$\displaystyle=\bar{Q}\sum_{i=2}^{k+1}\mathcal{H}[i],\quad 1\leq k\leq K-1,$
(6)

where $\sum_{i=2}^{k+1}\mathcal{H}[i]$ is the sum of average CIR of the
interferes. As done previously, since the integral is intractable; hence, we
solve it numerically. Since we are considering interference from other
transmitters, (III-C) also describes the multi-user interference (MUI).
However, as stated previously, we consider one symbol per transmission per
transmitter; therefore, without loss of generality, we use the term ISI
instead of MUI.

###  III-D Received Signal Energy With ISI and Without Noise

Lastly, from (III-B) and (III-C), the total expected received signal energy
can be expressed as

$\displaystyle\mathcal{E}_{\text{tot}}$
$\displaystyle=\mathcal{E}_{\text{sig}}+\mathcal{E}_{\text{isi}}$
$\displaystyle=\bar{Q}\sum_{i=1}^{k+1}\mathcal{H}[i].$ (7)

Figure 4: Illustration of the CIRs of signal (green solid line) and ISI from
previous $k$ transmissions from other transmitters (red dashed-lines).

###  III-E Received Signal Energy With ISI and Noise

Since the released molecules move independently of each other, the number of
molecules absorbed during the i-th symbol period is random. It can be
described by the Binomial random variable with $Q$ trials and a success
probability $h(y,i)$ [40]. The success probability is sufficiently small if
$Q$ is sufficiently large and $\Delta t\rightarrow 0$. In this case, the
Normal distribution can approximate the Binomial random process. By
mathematical induction, the received signal energy is a random variable
described by the Normal distribution.

At this point, unlike the general $\bar{Q}$ that is considered in (III-B), we
consider the expectation of the total received signal energy conditioned on
any specific $Q_{\mathit{j}}$. This conditional energy can be expressed as

$\displaystyle\mathcal{E}_{\text{tot}|Q_{\mathit{j}}}$
$\displaystyle=\mathcal{E}_{\text{sig}|Q_{\mathit{j}}}+\mathcal{E}_{\text{isi}}.$
$\displaystyle=Q_{\mathit{j}}\mathcal{H}[1]+\bar{Q}\sum_{i=2}^{k+1}\mathcal{H}[i].$
(8)

Then, the received signal energy described by the Normal distribution can be
expressed as

$\hat{\mathcal{E}}_{\text{tot}|Q_{\mathit{j}}}\sim\mathscr{N}\left(\mathcal{E}_{\text{tot}|Q_{\mathit{j}}},\,\sigma^{2}_{\text{tot}|Q_{\mathit{j}}}\right),$
(9)

where $\mathscr{N}\left(a,b\right)$ represents the Normal distribution with
mean $a$ and variance $b$. The variance in (9) accounts for the noise inherent
to Brownian motion [41] and the interference. Since the received signal and
interference are independent of each other, the variance can be expressed as

$\sigma^{2}_{\text{tot}|Q_{\mathit{j}}}=\sigma^{2}_{\text{sig}|Q_{\mathit{j}}}+\sigma^{2}_{\text{isi}},$
(10)

where

$\displaystyle\sigma^{2}_{\text{sig}|Q_{\mathit{j}}}$
$\displaystyle=Q_{\mathit{j}}\,\mathcal{H}[1]\cdot\left(1-\mathcal{H}[1]\right)$
(11a) $\displaystyle\sigma^{2}_{\text{isi}}$
$\displaystyle=\bar{Q}\sum_{i=2}^{k+1}\mathcal{H}[i]\cdot\left(1-\mathcal{H}[i]\right),\
1\leq k\leq K-1.$ (11b)

##  IV Proposed CSK with Common Detection Thresholds

Before we introduce the proposed CSK-CT method, we will show the
generalization of Singhal et al.’s CSK method [16] to a multiple transmitter
setting. Moreover, we refer to it as the benchmark CSK, taking into account
the use of MLE.

###  IV-A Generalization of the benchmark CSK

For a given transmitter, denoted by $k$, located at a random distance $y_{k}$
from the receiver, the threshold value can be calculated as

$\tau_{k,j}=\sqrt{\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{k}}\,\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{k}}},\quad\forall
k,$ (12)

where

$\displaystyle\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{k}}$
$\displaystyle=Q_{\mathit{j}}\,h(y_{k},1)+\mathcal{E}_{\text{isi}}$ (13a)
$\displaystyle\text{and }\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{k}}$
$\displaystyle=Q_{\mathit{j+1}}\,h(y_{k},1)+\mathcal{E}_{\text{isi}},$ (13b)

and $\tau_{k,j}$ represents threshold $j$ between symbols $S_{\mathit{\\!j}}$
and $S_{\mathit{\\!j}+1}$ of transmitter $k$. It is evident from (12) that the
receiver must acquire $K$ CIRs information and determine $(M-1)$ thresholds
for each transmitter, resulting in a total of $K\cdot(M-1)$ thresholds.

###  IV-B Description of the proposed CSK-CT

CSK-CT is designed on the following principles. Since the received signal
energy decreases with increasing distance, the j-th threshold value must be
sufficiently low to ensure the received signal energy of symbol
$S_{\mathit{j}+1}$ from any transmitter can cross it. Formally, we state that
the j-th threshold is equal to or smaller than the weakest received signal
energy that is associated with $Q_{j+1}$. Similarly, the j-th threshold value
must also be sufficiently large to ensure the received signal energy of symbol
$S_{\mathit{j}}$ from any transmitter cannot cross it. As before, formally, it
also holds true that the j-th threshold is equal to or greater than the
strongest received signal energy that is associated with $Q_{j}$. Concretely,
these statements can be expressed as

$\underset{k}{\operatorname{argmax}}\,\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{k}}\leq\tau_{\mathit{j}}\leq\underset{k}{\operatorname{argmin}}\,\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{k}},\
\forall k\in K,$ (14)

and form the lower (second statement) and upper (first statement) bounds of
$\tau_{\mathit{j}}$. These bounds can be further simplified into an elegant
form. Let $y_{\text{min}}$ and $y_{\text{max}}$ denote the minimum and maximum
$y$. If $K$ is sufficiently large, the following approximations hold:

$\displaystyle\underset{k}{\operatorname{argmax}}\,\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{k}}$
$\displaystyle\approx\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}$
(15a)
$\displaystyle\underset{k}{\operatorname{argmin}}\,\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{k}}$
$\displaystyle\approx\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}},$
(15b)

where $\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}$ and
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$ denote the
strongest and weakest received signal energies that are associated with
$Q_{j}$ and $Q_{j+1}$ at $y_{\text{min}}$ and $y_{\text{max}}$, respectively.
For ease of reference, we name
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}$ and
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$ as the limits of
received signal energies because they are associated with the distance limits,
i.e., $y_{\text{min}}$ and $y_{\text{max}}$, respectively. Then, (14) is
reduced to

$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}\leq\tau_{\mathit{j}}\leq\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}},$
(16)

where

$\displaystyle\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}$
$\displaystyle=Q_{\mathit{j}}\,h(y_{\text{min}},1)+\mathcal{E}_{\text{isi}}$
(17a) $\displaystyle\text{and
}\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$
$\displaystyle=Q_{\mathit{j}+1}\,h(y_{\text{max}},1)+\mathcal{E}_{\text{isi}}.$
(17b)

Accordingly, the release concentrations strictly maintain

$\displaystyle Q_{0}<Q_{1}<\dots<Q_{M-1},$ (18a) s.t.
$\displaystyle\mathcal{E}_{\text{tot}|Q_{0},y_{\text{min}}}\leq\mathcal{E}_{\text{tot}|Q_{1},y_{\text{max}}},\
\mathcal{E}_{\text{tot}|Q_{1},y_{\text{min}}}\leq\mathcal{E}_{\text{tot}|Q_{2},y_{\text{max}}},\ldots$
$\displaystyle\qquad\qquad\qquad\mathcal{E}_{\text{tot}|Q_{M-2},y_{\text{min}}}\leq\mathcal{E}_{\text{tot}|Q_{M-1},y_{\text{max}}}.$
(18b)

From (16) to (18), it is evident that to derive the thresholds, we first need
to derive the release concentrations. In this context, we note that studies
widely use values of $Q$ that are independent of the distance since they
consider a single transmitter [14, 15, 16, 17]. However, when multiple
transmitters are present, using values of $Q$ independent of the distance will
not necessarily guarantee (18b) even if they satisfy (18a). In that regard, we
derive a closed-form expression to determine the values of $Q$ that will
satisfy (18b), as discussed below.

####  IV-B1 Determining the release concentrations

For simplicity, first, we relax the inequality conditions in (16). Then, it
can be shown that

$\displaystyle\tau_{\mathit{j}}=\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}=\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}.$
(19)

Note that doing so does not invalidate (16) because it defines
$\tau_{\mathit{j}}$ to be at most equal to
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$ or at least equal
to $\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}$. This simplifies
the math by removing the intractability posed by the inequality condition.
However, in the later part of this section, we show an elegant way (cf. (23))
to realize the inequality condition.

From (19), we can observe that the received signal energies of symbols
$S_{\mathit{j}}$ and $S_{\mathit{j}+1}$ given their respective distance limits
are equal. This indicates that their associated release concentrations,
$Q_{\mathit{j}}$ and $Q_{\mathit{j}+1}$, are tightly correlated with each
other. More generally, $Q_{\mathit{j}+1}$ has to be a function of $Q_{j}$ for
the equality condition to be maintained. Then, it can be shown that the
solution to determine the values of $Q$ lies in the ratio between the adjacent
release concentrations. First, we substitute (17) in (19) to obtain

$\displaystyle Q_{\mathit{j}+1}\,h(y_{\text{max}},1)+\mathcal{E}_{\text{isi}}$
$\displaystyle=Q_{\mathit{j}}\,h(y_{\text{min}},1)+\mathcal{E}_{\text{isi}}.$
(20)

Then, rearranging (20), it can be shown that

$\displaystyle\frac{Q_{\mathit{j}+1}}{Q_{\mathit{j}}}=\frac{h(y_{\text{min}},1)}{h(y_{\text{max}},1)}\triangleq\varGamma.$
(21)

Thereafter, solving (21) for each $Q_{j}$, we obtain the release
concentrations as

$Q_{\mathit{j}}=Q_{0}\,\varGamma^{\,j},$ (22)

where $Q_{0}$ is a system parameter. For the cases when $Q_{\mathit{j}}$ is
not an integer, we may choose the integer that is closest to $Q_{\mathit{j}}$.

Figure 5: $\varGamma$ versus $t_{\text{sym}}$ for
$y_{\text{min}}=$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, $D=$
$79.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}$,
$r=$ $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and $\Delta t=$
$0.32\text{\,}\mathrm{s}$.

######  Remark 1.

From (21), we can infer that $\varGamma$ is a function of the time
$t_{\text{sym}}$ (cf. (3)) and the distances $y_{\text{min}}$ and
$y_{\text{max}}$. To illustrate these relationships, we present Fig. 5, in
which we demonstrate the behavior of $\varGamma$ versus $t_{\text{sym}}$ for
varying values of
$y_{\text{max}}=\\{17,18,19,20,21\\}\,$\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ while fixing $y_{\text{min}}$ at $6\text{\,}\mathrm{\SIUnitSymbolMicro
m}$. We can observe that $\varGamma$ decreases with $t_{\text{sym}}$ even
though $h(y_{x},1)$, $x\in\\{\text{min},\text{max}\\}$, is increasing and
converging to $\frac{r}{y_{x}+r}$ for large $t_{\text{sym}}$ [42]. However,
$h(y_{\text{min}},1)$ has a relatively faster rate of increase because
$y_{\text{min}}<y_{\text{max}}$. This difference indicates that
$h(y_{\text{max}},1)$ is still increasing when $h(y_{\text{min}},1)$ is
already converging. Hence, $\varGamma$ decreases with $t_{\text{sym}}$.
Furthermore, we can observe that $\varGamma$ increases with $y_{\text{max}}$.
This is because of the following reasons. 1) $h(y_{\text{max}},1)$ decreases
with $y_{\text{max}}$. 2) $\varGamma$ is inversely proportional to
$h(y_{\text{max}},1)$ as defined by (21). Hence, $\varGamma$ increases with
$y_{\text{max}}$. Thus, we note that $\varGamma$ is inversely proportional to
$t_{\text{sym}}$ and proportional to $y_{\text{max}}$ when $y_{\text{min}}$ is
fixed (or inversely proportional to $y_{\text{min}}$ when $y_{\text{max}}$ is
fixed). $\hfill\blacksquare$

Fig. 6 illustrates the PDFs of the received energies using (13) and (17). The
blue colored lines denote the PDFs of
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{k}},\forall k$, the red colored
lines denote the PDFs of
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{k}},\forall k$. The solid lines
denote the PDFs of the limits of received signal energies, and the dashed
lines denote the PDFs of the received energies at distances other than
$y_{\text{min}}$ and $y_{\text{max}}$. We can observe that the PDFs of the
limits of received signal energies overlap with each other and are centered
around the threshold because of relaxing the inequality. This overlapping will
result in a high error probability near the threshold values, and as a result,
the overall system’s performance can be reduced.

Figure 6: Illustration of the PDFs of the received energies when
$\tau_{\mathit{j}}=\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}=\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}$.
Figure 7: Illustration of the PDFs of the received energies and the limits
spacing when
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}<\tau_{\mathit{j}}<\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$.

(a) $\rho=1$

(b) $\rho=1.12$

Figure 8: PDFs of the limits of received signal energies highlighting the
effects of $\rho$ on the limits spacing in CSK-CT for $M=4$,
$y_{\text{min}}=$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$y_{\text{max}}=$19\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, $D=$
$79.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}$,
$K=$ $14\text{\,}$, $r=$ $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $\Delta
t=$ $0.32\text{\,}\mathrm{s}$, and $t_{\text{sym}}=$
$21.12\text{\,}\mathrm{s}$.

To reduce the errors near the threshold values, the PDFs of the limits of
received signal energies must have minimal or no overlapping at the
thresholds. This implies that
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}<\tau_{\mathit{j}}<\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$
must hold, which is the inequality condition of (16). In other words, the
space between the limits of received signal energies must be wider. We name
this spacing the limits spacing as illustrated in Fig. 7.

Simply put, the limits spacing must be greater than $0$ to reduce the errors.
This implies that the release concentrations must be adjusted. To achieve
that, we note from (22) that the release concentrations are proportional to
$\varGamma$. Then, we could increase $\varGamma$ to increase the release
concentrations. However, for a given $y_{\text{min}}$ and $y_{\text{max}}$,
$\varGamma$ is fixed and cannot be modified. Hence, we add an auxiliary
parameter on $\varGamma$ that allows us to obtain the n-th power of
$\varGamma$, namely, the scaling exponent $\rho$, a positive real number.
Thus, we re-formulate the release concentrations as

$Q_{\mathit{j}}=Q_{0}\cdot\left(\varGamma^{\,j}\right)^{\,\rho},\
\rho\in\mathbb{R}_{\geq 1}.$ (23)

Equation (23) is the final expression to calculate the release concentration
in CSK-CT.

####  IV-B2 Determining the thresholds

For the $\rho=1$ case, $\tau_{\mathit{j}}$ can be calculated using (19), but
not for the $\rho>1$ cases. In the latter case, we can apply the method of
pairwise comparison of the PDFs as done in [16] to determine
$\tau_{\mathit{j}}$ since
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}<\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}},\
\forall j\in M$, must hold when $\rho>1$. Therefore, we obtain
$\tau_{\mathit{j}}$ for $\rho>1$ cases as

$\displaystyle\tau_{\mathit{j}}=\sqrt{\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}\,\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}},$
(24a) s.t.
$\displaystyle\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}<\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}.$
(24b)

For the sake of brevity, we refer readers to [16] for the detailed methodology
of deriving (24a). Notice that when
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}=\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$
is applied to (24a), it reduces to (19). Taking into account the various cases
of $\rho$ described by (19) and (24a), we can obtain the final expression of
$\tau_{\mathit{j}}$ for CSK-CT as

$\tau_{\mathit{j}}=\begin{cases}\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}},&\text{if
}\rho=1\smash{\raisebox{-4.30554pt}{$\ ,\ \forall j.$}}\\\
\sqrt{\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}\,\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}},&\text{if}\
\rho>1\end{cases}$ (25)

Equation (25) is the final expression to calculate the thresholds in CSK-CT.

######  Remark 2.

Fig. 8 depicts the effects of $\rho$ on the limits spacing of CSK-CT with
$M=4$. Dashed lines represent the PDFs of
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}$ and solid lines
represent the PDFs of
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$. When $\rho=1$, we
can observe the relationship
$\tau_{\mathit{j}}=\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}=\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}$,
indicated by the overlapping between the thresholds and the limits of received
signal energies. Similarly, when $\rho=1.12$, the relationship
$\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{\text{min}}}<\tau_{\mathit{j}}<\mathcal{E}_{\text{tot}|Q_{\mathit{j}+1},y_{\text{max}}}$
can be observed, as indicated by the non-overlapping between the thresholds
and the limits of received signal energies. This demonstrates that increasing
$\rho$ can widen the limits spacing. Lastly, from Fig. 8(b), we can infer that
the limits spacing widens with $\tau_{j}$, which implies that the errors will
be highest around $\tau_{0}$ and lowest around $\tau_{M-1}$.
$\hfill\blacksquare$

##  V Performance Analysis

###  V-A Time Complexity Analysis

To assess the level of operational complexity that a CSK method can impose on
a receiver, we seek to quantify the rate at which the detection threshold
computations grow with $M$ and $K$. For tractability purposes, we take a
holistic view where we perceive the operations involved in computing a
threshold as one elementary operation rather than tallying the number of
distinct mathematical operations per threshold computation. Hence, the
operational complexity can be described by the number of thresholds computed
by a receiver.

To determine the asymptotic limits of operational complexity, we employ the
widely-used indicator, namely the time complexity. It estimates the run-time
of an algorithm by counting the number of elementary operations it performs
and is expressed using the big $\mathcal{O}$ notation [43, 44]. The time
complexity applies here because computing a threshold is an elementary
operation.

We can infer from (25) that with the proposed CSK-CT, the number of thresholds
the receiver must compute in CSK-CT equals $(M-1)$ the requires two CIRs
pertaining to $y_{\text{min}}$ and $y_{\text{max}}$. This results in a time
complexity of $\mathcal{O}\big{(}n\big{)}\approx\mathcal{O}(M-1)$. In
contrast, with the benchmark CSK, as previously mentioned, the receiver must
compute $K\cdot(M-1)$ thresholds that requires $K$ CIRs pertaining to
$y_{1},y_{2},\ldots,y_{K}$. This results in a time complexity of
$\mathcal{O}\big{(}n^{2}\big{)}\approx\mathcal{O}(K\cdot(M-1))$.

Fig. 9 depicts the time complexity in terms of the number of threshold
computations for varying $K$. As expected, CSK-CT yields a very low number of
threshold computations independent of $K$. Therefore, we get $1$ and $3$ for
BCSK and 4-CSK, respectively. Similarly, as expected, the benchmark CSK’s
threshold computations grow with $K$.

Figure 9: Time complexity in terms of the number of threshold computations
versus $K$.

###  V-B Error Probability Analysis

Given that symbol $S_{\mathit{\\!j}}$ was transmitted, the receiver can decode
the j-th transmitted symbol of the current transmitter according to the
following detection rule.

$\hat{S_{\mathit{\\!j}}}=\begin{cases}S_{\\!0},&\text{if
}{\mathcal{E}}_{\text{tot}}<\tau_{0},\ \ j=0,\\\ S_{\mathit{\\!j}},&\text{if
}\tau_{\mathit{\\!j}-1}\leq{\mathcal{E}}_{\text{tot}}<\tau_{\mathit{j}},\ \
j=1,2,\ldots,M-2,\\\ S_{\mathit{\\!M}-1},&\text{if
}{\mathcal{E}}_{\text{tot}}\geq\tau_{M-2},\ \ j=M-1.\end{cases}$ (26)

Then, the probability of error when $S_{\mathit{\\!j}}$ is transmitted by
transmitter $k$ with distance $y_{k}$ can be computed as

$\displaystyle
P_{\mathit{\\!e,k}}(S_{\mathit{\\!j}})=\begin{cases}\Pr\left({\mathcal{E}}_{\text{tot}}\geq\tau_{0}\right)=0.5\,\text{erfc}\left(\frac{\tau_{0}-\mathcal{E}_{\text{tot}|Q_{0},y_{k}}}{\sqrt{2}\,\sigma_{\text{tot}|Q_{0},y_{k}}}\right),\
j=0,\\\
\Pr\left({\mathcal{E}}_{\text{tot}}<\tau_{\mathit{\\!j}-1}\right)+\Pr\left({\mathcal{E}}_{\text{tot}}\geq\tau_{\mathit{j}}\right)\\\
\ \ \
=0.5\,\text{erfc}\left(\frac{\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{k}}-\tau_{\mathit{\\!j}-1}}{\sqrt{2}\,\sigma_{\text{tot}|Q_{\mathit{j}},y_{k}}}\right)+0.5\,\text{erfc}\left(\frac{\tau_{\mathit{j}}-\mathcal{E}_{\text{tot}|Q_{\mathit{j}},y_{k}}}{\sqrt{2}\,\sigma_{\text{tot}|Q_{\mathit{j}},y_{k}}}\right),\\\
\qquad\qquad j=1,2,\ldots,M-2,\\\
\Pr\left({\mathcal{E}}_{\text{tot}}<\tau_{M-2}\right)=0.5\,\text{erfc}\left(\frac{\mathcal{E}_{\text{tot}|Q_{M-1},y_{k}}-\tau_{M-2}}{\sqrt{2}\,\sigma_{\text{tot}|Q_{M-1},y_{k}}}\right),\\\
\qquad\qquad j=M-1.\end{cases}$ (27)

From (27), the probability of error for transmitter $k$ when each
$S_{\mathit{\\!j}}$ has an equal transmission probability of $1/M$ can be
calculated as

$\displaystyle
P_{\mathit{\\!e,k}}=\frac{1}{M}\,\sum_{j=0}^{M-1}P_{\mathit{\\!e,k}}(S_{\mathit{\\!j}}).$
(28)

Then, the total error probability can be expressed as

$\displaystyle
P_{\mathit{\\!e}}=\frac{1}{K}\sum_{k=1}^{K}P_{\mathit{\\!e,k}}.$ (29)

TABLE I: Simulation parameters Parameter | Values
---|---
Diffusion coefficient, $D$ |  $79.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}$ [37]
Radius of receiver, $r$ |  $5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [37]
Sampling period duration, $\Delta t$ |  $0.32\text{\,}\mathrm{s}$ [45]
Symbol period duration, $t_{\text{sym}}$ | $1.28\text{\,}\mathrm{s}32\text{\,}\mathrm{s}$
Min. distance between transmitters and receiver, $y_{\text{min}}$ | $6\text{\,}\mathrm{\SIUnitSymbolMicro m}$
Max. distance between transmitters and receiver, $y_{\text{max}}$ | $17\text{\,}\mathrm{\SIUnitSymbolMicro m}21\text{\,}\mathrm{\SIUnitSymbolMicro m}$
No. of transmitters, $K$ | $12\text{\,}16\text{\,}$
Scaling exponent, $\rho$ | $1\text{\,}4\text{\,}$
$Q_{0}$ |  $1000$ molecules
TABLE II: Values of parameters of CSK-CT and benchmark CSK. $t_{\text{sym}}$ is set to $21.12\text{\,}\mathrm{s}$. Scheme | $\bar{d}$ | $\varGamma$ | $\rho$ | BCSK ($M=2$) | 4-CSK ($M=4$)
---|---|---|---|---|---
$Q_{0}$ | $Q_{1}$ | $\tau_{0}$ | $Q_{0}$ | $Q_{1}$ | $Q_{2}$ | $Q_{3}$ | $\tau_{0}$ | $\tau_{1}$ | $\tau_{2}$
CSK-CT | 11.5 | 2.386 | 1 | 1000 | 2386 | 478 | 1000 | 2386 | 5693 | 13581 | 606 | 1184 | 2563
12 | 2.538 | 1000 | 2538 | 483 | 1000 | 2538 | 6441 | 16344 | 642 | 1283 | 2911
12.5 | 2.695 | 1000 | 2695 | 487 | 1000 | 2695 | 7262 | 19568 | 684 | 1391 | 3295
13 | 2.857 | 1000 | 2858 | 492 | 1000 | 2858 | 8163 | 23323 | 733 | 1507 | 3720
13.5 | 3.025 | 1000 | 3025 | 496 | 1000 | 3025 | 9151 | 27681 | 789 | 1633 | 4188
11.5 | 2.386 | 1.24 | 1000 | 2940 | 534 | 1000 | 2940 | 8641 | 25400 | 780 | 1678 | 4318
12 | 2.538 | 1000 | 3174 | 544 | 1000 | 3174 | 10070 | 31955 | 862 | 1876 | 5093
12.5 | 2.695 | 1000 | 3419 | 554 | 1000 | 3419 | 11687 | 39951 | 961 | 2098 | 5982
13 | 2.857 | 1000 | 3676 | 564 | 1000 | 3676 | 13511 | 49663 | 1081 | 2348 | 7001
13.5 | 3.025 | 1000 | 3946 | 574 | 1000 | 3946 | 15567 | 61419 | 1224 | 2629 | 8165
Benchmark CSK | n.a. | n.a. | n.a. | 1000 | 1500 | n.a. | 1000 | 1500 | 2000 | 3000 | n.a. | n.a. | n.a.

##  VI Numerical Analyses and Results

The transmitters are located at distances between
$\left[y_{\text{min}},y_{\text{max}}\right]$, where
$y_{\text{min}}=$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$y_{\text{max}}\in\\{17,18,19,20,21\\}$ $\text{\,}\mathrm{\SIUnitSymbolMicro
m}$. Without loss of generality, the distance between adjacent distances
$y_{k}$ and $y_{k+1}$ is $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. For
example, when $y_{\text{max}}=$17\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the
distances are $\\{y_{1}=$6\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,\,y_{2}=$7\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,\ldots,\,y_{K}=$17\text{\,}\mathrm{\SIUnitSymbolMicro m}$\\}$.
Accordingly, the total number of transmitters is
$K=y_{\text{max}}-y_{\text{min}}+1$, and the k-th transmitter receives ISI
from $(K-1)$ transmitters (or equivalently from the previous $(K-1)$ symbol
periods). We define the average distance as
$\bar{d}=(y_{\text{min}}+y_{\text{max}})/2\,$\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$, which yields
$\bar{d}=\\{11.5,12,12.5,13,13.5\\}\,$\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$. Unless specified otherwise, Table I lists the values of the remaining
simulation parameters.

(a) $\bar{d}=$11.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$

(b) $\bar{d}=$13\text{\,}\mathrm{\SIUnitSymbolMicro m}$$

Figure 10: PDFs of the limits of received signal energies highlighting the
effects of $\bar{d}$ on the limits spacing in CSK-CT for $M=4$, $\rho=1.12$,
and $t_{\text{sym}}=$ $21.12\text{\,}\mathrm{s}$.

(a) BCSK

(b) 4-CSK

Figure 11: $P_{\mathit{\\!e}}$ versus $\bar{d}$ and varying $\rho$ for
$t_{\text{sym}}=$21.12\text{\,}\mathrm{s}$$.

(a) BCSK

(b) 4-CSK

Figure 12: $P_{\mathit{\\!e}}$ versus $\rho$ and varying $\rho$ for
$t_{\text{sym}}=$21.12\text{\,}\mathrm{s}$$ and
$\bar{d}=$11.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The $x$\- and $y$-axis
for BCSK is cropped for better visualization, and the black cross in 4-CSK
indicates the optimal value of $\rho$.

In Table II, we present the $Q$ values of CSK-CT and benchmark CSK alongside
the values of $\varGamma$ and $\tau$ of CSK-CT for varying values of $\bar{d}$
and $\rho$. Note that the values of $\varGamma$, $Q$, and $\tau$ of CSK-CT are
obtained using (21), (23), and (25), respectively. On the one hand, we can
observe that CSK-CT exhibits higher $Q$ values than the benchmark CSK. This
can be attributed to the combination of two underlying factors. First, the $Q$
values of CSK-CT are determined to ensure that the expected weakest received
signal energy crosses the threshold as defined by (14) and, thus, increases
with $\bar{d}$. Second, the $Q$ values of symbols other than $Q_{0}$ are an
increasing function of the j-th power of $Q_{0}$ (refer (22)). On the other
hand, the $Q$ values of benchmark CSK, which do not depend on $\bar{d}$ [14,
15, 16, 17], are comparable to those used in [16]. We leave the optimization
of the $Q$ values of CSK-CT aside for future work.

Additionally, we can observe from Table II that $\varGamma$ is independent of
$M$, but depends on $\bar{d}$ (as indicated in Remark 1). Furthermore,
$\tau_{0}$ is larger in 4-CSK than in BCSK despite both schemes sharing the
same $Q_{0}$ and $Q_{1}$ values. This is because the average release
concentration in 4-CSK is larger owing to the significantly high $Q_{2}$ and
$Q_{3}$ values. Hence, the larger release concentration increases the expected
ISI, resulting in larger thresholds. Lastly, as $\rho$ increases, we can
observe an increasing trend in the values of $Q_{\mathit{j}},j\neq 0$, which
is expected because $Q_{\mathit{j}},j\neq 0$, is directly proportional to
$\rho$ (refer (23)). This implies that increasing $\rho$ also increases the
ISI and, subsequently, the thresholds.

Fig. 10 illustrates the effects of $\bar{d}$ on the limits spacing of CSK-CT
for $\rho=1.12$. We can observe that the limits spacing widens with $\bar{d}$.
This is because of the aforementioned fact that $\varGamma$ increases with
$\bar{d}$, which further causes the release concentrations to increase (refer
(23)). Subsequently, the limits of received signal energies increase
proportionally, resulting in the observation of a wider limits spacing, which
contributes positively to reducing $P_{\mathit{\\!e}}$.

Based on our current findings, we can anticipate the following outcomes:
Firstly, the increase in release concentration yields both advantageous (wider
limit spacing) and detrimental (increased ISI) effects on $P_{\mathit{\\!e}}$.
Secondly, the values of $\bar{d}$ and $\rho$ are responsible for determining
the release concentration in a proportional manner.

Fig. 11 depicts the behavior of the $P_{\mathit{\\!e}}$ for varying $\bar{d}$.
We can observe that $P_{\mathit{\\!e}}$ improves in both BCSK and 4-CSK as
$\bar{d}$ increases for the given $\rho$ (The impact of $\rho$ on
$P_{\mathit{\\!e}}$ will be separately analyzed in Fig. 12.). While both
positive and negative effects on $P_{\mathit{\\!e}}$ occur simultaneously with
$\bar{d}$, the primary reason $P_{\mathit{\\!e}}$ generally improves as
$\bar{d}$ increases in most scenarios is due to the dominance of the positive
effect of wider limits spacing over the negative effects of increasing ISI on
$P_{\mathit{\\!e}}$. This highlights that CSK-CT can be intentionally designed
to mitigate the effects of ISI. In the case of the benchmark CSK, the
performance degrades with $\bar{d}$ because the release concentration and the
limits spacing are not determined considering $\bar{d}$.

Lastly, it is shown that CSK-CT outperforms the benchmark CSK for larger
$\rho$. For example, for $\rho=1.36$, CSK-CT achieves $P_{\mathit{\\!e}}$ up
to $10^{-7}$ to $10^{-4}$ for BCSK and 4-CSK, respectively. This is at least
an order of magnitude lower than the benchmark CSK’s, which is between
$10^{-4}$ to $10^{-3}$ for BCSK and 4-CSK, respectively. Even greater
performance enhancement can be attained by increasing $\rho$ (Refer to the
analysis in Fig. 12). This demonstrates the superiority of CSK-CT and, most
importantly, underscores the simplicity with which CSK-CT can be designed to
attain a desired $P_{\mathit{\\!e}}$ by selecting a suitable $\rho$ value. In
contrast, achieving a target $P_{\mathit{\\!e}}$ with the benchmark CSK might
require an exhaustive search to determine the combinations of release
concentrations.

Fig. 12 shows the behavior of $P_{\mathit{\\!e}}$ in CSK-CT for varying
$\rho$. In the absence of ISI, both BCSK and 4-CSK exhibit $P_{\mathit{\\!e}}$
improvement with increasing $\rho$. This is due to the positive effect of
increased release concentration, which solely contributes to
$P_{\mathit{\\!e}}$ reduction without ISI. However, when ISI is present, the
behavior can change. While BCSK still demonstrates $P_{\mathit{\\!e}}$
improvement with increasing $\rho$, 4-CSK reaches an optimal point of $\rho$
($2.56$ in this case), beyond which $P_{\mathit{\\!e}}$ starts to deteriorate.
This behavior suggests that the positive effect of wider limits spacing on
$P_{\mathit{\\!e}}$ remains dominant over the negative effect of increasing
ISI until a certain point, after which the negative effect dominates the
positive effect.

Figure 13: $P_{\mathit{\\!e}}$ versus $t_{\text{sym}}$ for $\rho=1.24$ and
$\bar{d}=$11.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

Fig. 13 shows the behavior of $P_{\mathit{\\!e}}$ for varying
$t_{\text{sym}}$. In the absence of ISI, $P_{\mathit{\\!e}}$ for both BCSK and
4-CSK tends to increase with $t_{\text{sym}}$. This is because $\varGamma$ is
inversely proportional to $t_{\text{sym}}$, as stated in Remark 1. A smaller
$\varGamma$ indicates that the $Q$ values are close to each other, as shown in
Table II. This proximity causes the limits spacing to reduce, which
consequently causes $P_{\mathit{\\!e}}$ to increase. However, as
$t_{\text{sym}}$ continues to increase, the increments in $P_{\mathit{\\!e}}$
become minimal. This can be attributed to the fact that the received signal
energy converges to a steady state for larger $t_{\text{sym}}$ [42]. As a
result, the limits spacing reaches a point where further reduction is
negligible. We also observe that 4-CSK performs better than BCSK. This is
because $P_{\mathit{\\!e}}$ for 4-CSK includes the errors around $\tau_{1}$
and $\tau_{2}$, where the limits spacing is considerably larger than around
$\tau_{0}$. In contrast, for BCSK, the determination of $P_{\mathit{\\!e}}$
relies solely on errors around $\tau_{0}$. In the presence of ISI, as
expected, the $P_{\mathit{\\!e}}$ of 4-CSK is worse compared to that of BCSK.
Furthermore, we observe that for smaller $t_{\text{sym}}$ values, 4-CSK
exhibits poor performance. This is because, for shorter symbol periods, there
is no assurance that the energy of the received signal is much larger than the
energy of the interference, which is ideal. As $t_{\text{sym}}$ increases and
consequently the received signal energy grows, the $P_{\mathit{\\!e}}$ of
4-CSK gradually improves and stabilizes, reflecting the convergence of
received signal energy. Conversely, BCSK displays a slight deterioration in
$P_{\mathit{\\!e}}$ with $t_{\text{sym}}$. This behavior can be attributed to
the effects of the decreasing limits spacing.

##  VII Conclusion and Discussions

We proposed the CSK with received energy-aware threshold (CSK-CT) to address
the high time complexity of the existing CSK methods for MC-based
nanonetworks, one of the key components of IoBNT. By utilizing the fact that
the weakest received signal energy is from the farthest transmitter, CSK-CT
offers a time complexity of $\mathcal{O}\big{(}n\big{)}$, which is superior to
the $\mathcal{O}\big{(}n^{2}\big{)}$ time complexity of the benchmark CSK. We
demonstrated through numerical experiments that CSK-CT yields lower errors
compared to the benchmark CSK when the scaling exponent value is appropriately
selected. Other notable findings include an optimum scaling exponent value for
4-CSK ($M=4$) and an increase in errors for BCSK ($M=2$) for longer symbol
period durations. Additional advantages of CSK-CT over the benchmark CSK
include requiring only two CIRs and ISI combating ability. Owing to lower time
complexity and errors, we believe CSK-CT can be advantageous for MC-based
nano-sensor networks targeted toward data-gathering applications, such as in-
body medical sensor networks, toxic chemical industrial sensor networks, etc.
These applications require deploying many nano-sensors to sense data and
report those readings to a nano-sink, which typically has computational and
power constraints.

Lastly, it is important to note that CSK-CT requires larger release
concentration values than the benchmark CSK. However, large release
concentrations result in a corresponding increase in transmitting energy costs
and ISI. Although some studies have used values of release concentrations
similar to those of CSK-CT [15, 17], we acknowledge the trade-off between
energy cost and error probability. Hence, future research directions include
optimizing the release concentrations to reduce the energy cost, making CSK-CT
more robust to ISI so that it can be implemented in MC-based nanonetworks that
have higher throughput and stricter reliable data transfer requirements, and
evaluating CSK-CT under mobile transmitter scenarios and flow-assisted
diffusion channels.

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|  Ethungshan Shitiri (Member, IEEE) received the B.E. degree in electronics
and communication engineering (2006), the M.Tech. degree in communication
systems (2013), and the Ph.D. degree in information and communication
engineering (2018) from Thanthai Periyar Government Institute of Technology
(India), Christ University (India), and Kyungpook National University (KNU)
(South Korea), respectively. He was a Postdoctoral Research Fellow with the
IDEC Center, KNU (2018-2021) and the Electronics and Electrical Engineering
Advancement Institute, KNU (2021-2023). Since 2023, he has been with N3Cat at
Universitat Politècnica de Catalunya, working on intra-body nanoscale
communication systems. His research interests include resource allocation,
handshaking protocols, synchronization, and multiple access techniques for
wireless communication networks—namely cellular networks, M2M networks,
underwater acoustic sensor networks, and molecular communication networks. He
was a recipient of the KNU Honors Scholarship (2014), Qualcomm Innovation
Award (2016), Best Thesis Award of School of Electronics (2018), and Best
Paper Award, KICS (2021).
---|---
|  Ho-Shin Cho (Senior Member, IEEE) received the B.S., M.S., and Ph.D.
degrees in electrical engineering from the Korea Advanced Institute of Science
and Technology (KAIST), in 1992, 1994, and 1999, respectively. From 1999 to
2000, he was a Senior Member of the Research Staff with the Electronics and
Telecommunications Research Institute (ETRI), where he was involved in
developing a base station system for IMT-2000. From 2001 to 2002, he was a
Faculty Member with the School of Electronics, Telecommunications, and
Computer Engineering, Hankuk Aviation University. In 2003, he joined the
School of Electronics Engineering, Kyungpook National University, as a Faculty
Member, where he is currently a Professor. His research interests include
traffic engineering, radio resource management, and medium access control
protocol for wireless communication networks, cellular networks, underwater
communication, and molecular communication. Prof. Cho is a member of IEICE,
IEEK, KICS, and ASK. He received the Rising Researcher Fellowship of NRF, in
1998.
---|---