# Interference Alignment Using Reaction in Molecular Interference Channels

Maryam Farahnak-Ghazani, Mahtab Mirmohseni, and Masoumeh Nasiri-Kenari
Sharif University of Technology

###### Abstract

Co-channel interference (CCI) is a performance limiting factor in molecular
communication (MC) systems with shared medium. Interference alignment (IA) is
a promising scheme to mitigate CCI in traditional communication systems. Due
to the signal-dependent noise in MC systems, the traditional IA schemes are
less useful in MC systems. In this paper, we propose a novel IA scheme in
molecular interference channels (IFCs), based on the choice of
releasing/sampling times. To cancel the aligned interference signals and
reduce the signal dependent noise, we use molecular reaction in the proposed
IA scheme. We obtain the feasible region for the releasing/sampling times in
the proposed scheme. Further, we investigate the error performance of the
proposed scheme. Our results show that the proposed IA scheme using reaction
improves the performance significantly.††This work was supported in part by
the Iran National Science Foundation (INSF) Research Grant on Nano-Network
Communications and in part by the Research Center of Sharif University of
Technology. ††The authors are with the Department of Electrical Engineering,
Sharif University of Technology, Tehran, Iran (email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>mnasiri@sharif.edu).

##  I Introduction

In MC networks, where molecules are used as information carriers [1], multiple
transmitter-receiver pairs may operate in a shared medium [2]. In this case,
if the transmitters use the same molecule type for communication and operate
at the same time, there is a co-channel interference (CCI) and we have a
molecular interference channel (IFC). Inter-symbol interference (ISI) and CCI
are two performance limiting factors in MC systems [3]. ISI mitigating
techniques in MC systems are presented in [4, 5, 6].

The effect of CCI is considered in [3, 7]. To avoid CCI in MC systems, the
common approach is to use different molecule types for each transmitter-
receiver pair, which is analogous to using different frequencies in wireless
communication systems. As shown in [3], the effect of CCI is negligible after
a certain distance called molecular re-use distance and the same molecule type
can be used after this distance, which is similar to the concept of frequency
re-use distance in wireless communication systems. However, when the distance
is less than the molecular re-use distance, CCI is not negligible, and we
should use different molecule types to avoid CCI. Using different molecule
types, we achieve the degrees of freedom (DoF) of $\frac{1}{2}$. However, as
shown in [8], a $K$-user IFC with single antenna at the transmitters and the
receivers has the DoF of $\frac{K}{2}$, i.e., each user can reach half of the
capacity it can reach in the absence of interference. Interference alignment
(IA) is a technique to increase the DoF of a network when the distances of the
transmitter-receiver pairs are less than the re-use distance, and the
interference signal is comparable to the desired signal. The idea of IA is to
align the signal vectors at the receivers such that the interference signals
occupy the same space, while the desired signal lies in a separate space [9].
With the help of IA techniques, we can use the resources in the network more
efficiently.

In molecular communication (MC) systems, due to the low data rate and the
limitation in the number of molecule types, the usage of IA techniques in the
networks can be very beneficial. In this paper, we propose an IA scheme to
mitigate the effect of CCI when the distance is less than the re-use distance.
In classic communication systems, for $K$-user IFC with time/frequency varying
channel coefficients, an asymptotic IA scheme is proposed to achieve the DoF
of $\frac{K}{2}$ as the number of time slots or frequencies used for a super-
symbol increases [8]. This scheme can be applied to MC systems using multiple
time slots or multiple molecule types to form a super-symbol. However, this
scheme is not practical since to achieve the DoF of $\frac{K}{2}$, the number
of time slots or molecule types should be very large. Another scheme in
classic IFCs is to align interference signals by changing the channel
coefficients, or changing the propagation delays [8]. To adopt this idea to MC
systems, in this paper, we make use of changing the channel coefficients by
choosing different releasing times at the transmitters and/or sampling times
at the receivers. Based on this idea, we propose an IA scheme by the choice of
releasing/sampling times in molecular IFCs. The transmitters and receivers
should be synchronized for the proposed IA scheme. The synchronization methods
in MC systems are investigated in [10, 11, 12]. In this paper, to avoid
complexity, we assume perfect synchronization among the transmitters and
receivers.

We also face another challenge in applying IA schemes to MC systems. Most IA
schemes in classic communications, e.g., the asymptotic IA scheme, require
high signal-to-noise ratio (SNR) [9]. However, in MC systems, there is usually
a signal dependent receiver noise [1], which results in high noise levels for
high signal levels, and hence, the classic IA schemes may not be very useful
in MC systems. In this paper, we propose using molecular reaction [5] to
cancel the aligned interference signals in the medium and reduce the signal
dependent noise. We investigate the error performance and show that the
proposed IA scheme using reaction improves the performance significantly.

Our main contributions in this paper are as follows:
$\mathbin{\vbox{\hbox{\scalebox{0.75}{$\bullet$}}}}$ We apply the asymptotic
IA scheme in the classic communications to a $K$-user molecular IFC.
$\mathbin{\vbox{\hbox{\scalebox{0.75}{$\bullet$}}}}$ We propose an IA scheme
based on the choice of releasing/sampling times in molecular systems for a
3-user IFC using two molecule types, and use the reaction in the proposed IA
scheme to cancel the aligned interference signals in the medium and to reduce
the signal dependent noise.
$\mathbin{\vbox{\hbox{\scalebox{0.75}{$\bullet$}}}}$ We obtain the feasible
region for the releasing/sampling times in the proposed IA scheme with and
without reaction. For a special case of the releasing/sampling times, we
simplify the equations.
$\mathbin{\vbox{\hbox{\scalebox{0.75}{$\bullet$}}}}$ We investigate the error
performance of the interference channel using the proposed IA scheme with and
without reaction. It is seen that the IA scheme with reaction improves the
performance of the system significantly.

The structure of the paper is as follows: In Section II, we describe the
system model. In Section III, we describe the asymptotic IA and the proposed
IA schemes. In Section IV, we investigate the optimum and sub-optimum sampling
times and in Section V, we obtain the error probabilities of the IA schemes.
The numerical results are given in Section VI. Finally, in Section VII we
conclude the paper.

Notation: Throughout the paper, vectors and matrices are shown with bold
letters, $\bm{X}^{\textrm{T}}$ shows the transpose of vector or matrix
$\bm{X}$, $\textrm{diag}(\bm{A})$ is a diagonal matrix whose entries are the
elements in the vector $\bm{A}$, and $\bm{A}\equiv\bm{B}$ is equivalent to
$\textrm{column-span}(\bm{A})=\textrm{column-span}(\bm{B})$.

##  II System Model

We consider an IFC with $K$ molecular transmitter-receiver pairs (Fig. 1). The
location of the $j$-th transmitter ($\textrm{Tx}_{j}$) and the $i$-th receiver
($\textrm{Rx}_{i}$), for $i,j\in\\{1,...,K\\}$, are noted by
$\bm{r}_{j}^{\textrm{Tx}}$ and $\bm{r}_{i}^{\textrm{Rx}}$, respectively, and
the message of $\textrm{Tx}_{j}$ is noted by $M_{j}\in\\{0,...,M-1\\}$. The
time is slotted with duration $T_{\textrm{s}}$.

Figure 1: $K$-user molecular IFC.

Transmitter: Each $\textrm{Tx}_{j},j=1,...,K$ uses the same $L$ molecule
types, and releases $X_{j}^{[l]}=N_{j}V_{j}^{[l]},l=1,...,L$ molecules of type
$l$ in time $\tilde{t}_{j}^{[l]}$, where
$0<\tilde{t}_{j}^{[l]}<T_{\textrm{s}}$ and
$N_{j}\in\\{\zeta_{0},...,\zeta_{M-1}\\}$ is the total number of molecules of
all types released from $\textrm{Tx}_{j}$, i.e.,
$\sum_{l=1}^{L}X_{j}^{[l]}=N_{j}$, which requires
$\sum_{l=1}^{L}V_{j}^{[l]}=1$. We use concentration shift keying (CSK)
modulation at the transmitters, i.e., for the message $M_{j}=m$,
$m=0,...,M-1$, the total released molecules at $\textrm{Tx}_{j}$ is
$N_{j}=\zeta_{m}$. We denote the number of released molecules of all types
from $\textrm{Tx}_{j}$ with vector
$\bm{X}_{j}=[X_{j}^{[1]},...,X_{j}^{[L]}]^{\textrm{T}}=N_{j}\bm{V}_{j}$, where
$\bm{V}_{j}=[V_{j}^{[1]},...,V_{j}^{[L]}]^{\textrm{T}}$ is called the
beamforming vector.

Channel: The channel impulse response from $\textrm{Tx}_{j}$ to
$\textrm{Rx}_{i}$ for the $l$-th molecule type, noted by $h_{ij}^{[l]}(t)$, is
defined as the concentration of the $l$-th molecule type at $\textrm{Rx}_{i}$
and in time $t$ when one molecule is released from $\textrm{Tx}_{j}$ in time
$\tilde{t}_{j}^{[l]}$. If different molecule types do not interfere with each
other, $h_{ij}^{[l]}(t)$ can be obtained from advection-diffusion equation for
3-D space as[1]

$\displaystyle h_{ij}^{[l]}(t)=\frac{1[t>\tilde{t}_{j}^{[l]}]}{(4\pi
D_{l}(t-\tilde{t}_{j}^{[l]}))^{\frac{3}{2}}}e^{-\frac{||\bm{r}_{i}^{\textrm{Rx}}-\bm{r}_{j}^{\textrm{Tx}}-\bm{\nu}(t-\tilde{t}_{j}^{[l]})||^{2}}{4D_{l}(t-\tilde{t}_{j}^{[l]})}},$
(1)

for $i,j\in\\{1,...,K\\}$ and $l\in\\{1,...,L\\}$, where $D_{l}$ is the
diffusion coefficient of the $l$-th molecule type and $\bm{\nu}$ is the flow
velocity of the medium. For simplicity of exposition, for the analytical
derivations, we assume that the ISI is negligible in the channel, which can be
satisfied by using enzymes that remove the remaining molecules from the
previous transmissions in the medium [6]. In Section VI, we investigate the
error probability of the proposed IA scheme in the presence of ISI using
simulation. In the absence of ISI, the concentration of the $l$-th molecule
type at $\textrm{Rx}_{i}$ from $\textrm{Tx}_{j}$ can be obtained as
$C_{ij}^{[l]}(t)=X_{i}^{[l]}h_{ij}^{[l]}(t)$.

Receiver: Each receiver $\textrm{Rx}_{i},i=1,....,K$ is assumed to be
transparent with volume $V_{\textrm{R}}$ (radius $r_{\textrm{R}}$), and counts
the number of molecules that enters its volume. We assume that
$\textrm{Rx}_{i}$ uses one-sample decoder with sampling time $t_{i}^{[l]}$ for
the $l$-th molecule type, where
${\tilde{t}_{j}^{[l]}<t_{i}^{[l]}<T_{\textrm{s}}}$,
$i,j\in\\{1,...,K\\},l\in\\{1,...,L\\}$.111The one sample decoder at the
receiver is commonly used in MC systems, as in [6, 13]. The releasing and
sampling times at each transmitter and receiver are shown in Fig. 1 for two
molecule types, i.e., $L=2$. Assuming uniform concentration in the receiver
volume, which is valid if the distance of the transmitter and the receiver is
sufficiently large compared to the receiver radius [14], the mean number of
counted molecules at $\textrm{Rx}_{i}$ from $\textrm{Tx}_{j}$ is obtained as
$\mu_{ij}^{[l]}=C_{ij}^{[l]}(t_{i}^{[l]})V_{\textrm{R}}=X_{i}^{[l]}h_{ij}^{[l]}(t_{i}^{[l]})V_{\textrm{R}}$.
We denote $H_{ij}^{[l]}=h_{ij}^{[l]}(t_{i}^{[l]})V_{\textrm{R}}$. Hence, the
mean number of counted molecules of type $l$ at $\textrm{Rx}_{i}$ transmitted
from all transmitters is
$\mu_{i}^{[l]}=\sum_{j=1}^{K}\mu_{ij}^{[l]}=\sum_{j=1}^{K}X_{j}^{[l]}H_{ij}^{[l]}$.
We denote the vector of mean number of counted molecules of all types at
$\textrm{Rx}_{i}$ with vector
$\bm{\mu}_{i}=[\mu_{i}^{[1]},...,\mu_{i}^{[L]}]^{\textrm{T}}$. Hence,
$\bm{\mu}_{i}=\sum_{j=1}^{K}\bm{H}_{ij}\bm{X}_{j}$, where

$\displaystyle\bm{H}_{ij}=\textrm{diag}(H_{ij}^{[1]},H_{ij}^{[2]},...,H_{ij}^{[L]}),\qquad
i,j\in\\{1,...,K\\}.$ (2)

We assume that the channel state information (CSI) is known at the
transmitters and receivers. Further, we assume that the transmitters and
receivers are synchronized.

Noise: Let $Y_{i}^{[l]}$ be the number of counted molecules of type $l$ at
$\textrm{Rx}_{i}$ transmitted from all transmitters. Assuming the counting
noise at the receivers and an environment noise with mean
$\mu_{\textrm{n}}^{[l]}$ for the $l$-th molecule type, $Y_{i}^{[l]}$ has a
Poisson distribution conditioned on the transmitter messages with mean
$\mu_{i}^{[l]}+\mu_{\textrm{n}}^{[l]}$ [1]. So, we have a signal dependent
noise at the receivers. For simplicity of the analysis, we assume
$\mu_{\textrm{n}}^{[l]}=0$. However, in Section VI, we investigate the effect
of the environment noise.

##  III Interference Alignment

In this section, we first apply the asymptotic IA scheme of [8] to $K$-user
molecular IFC, where we use different molecule types or time slots for sending
a super-symbol and increase their number to reach the DoF of $\frac{K}{2}$,
asymptotically. However, we should note that increasing the number of molecule
types or the number of time slots is not practical. A practical solution is to
partition the $K$-user pairs to clusters with $3$-user pairs and use this
scheme with $3$ molecule types in each cluster, as in [15], where we reach the
DoF of $\frac{4}{3}$. Further, this scheme is only useful in systems with high
SNRs. Since in MC systems, we face a signal dependent receiver noise, which
makes the SNR low, the traditional IA schemes are less useful. To overcome
this issue, in the second part, we propose a non-asymptotic IA scheme by the
choice of releasing/sampling times for 3-user IFC using 2 molecule types,
which has a DoF of $\frac{3}{2}$. We use reaction in the proposed IA scheme to
reduce the signal dependent noise. We should note that using the asymptotic IA
scheme for 3 user pairs, we can’t reach the DoF of $\frac{3}{2}$ with limited
number of molecule types (for 3 molecule types, we can reach the DoF of
$\frac{4}{3}$) and we should increase the number of molecules types to reach
the DoF of $\frac{3}{2}$, asymptotically.

Asymptotic IA scheme: We consider a super-symbol at each transmitter
$\textrm{Tx}_{i}$ consisting of $n_{\textrm{s},i}$ symbols and send it using
different variations of the channel to reach the DoF of $\frac{K}{2}$,
asymptotically. To have the varying channel gains, we can use either different
molecule types in one time slot, or one molecule type in multiple time slots
with different releasing/sampling times in each time slot. Here, we assume $L$
different molecule types in one time slot. Hence, according to the system
model, the number of released molecules of all types from $\textrm{Tx}_{j}$ is
obtained from $\bm{X}_{j}=N_{j}\bm{V}_{j}$. Using the method in [8], we assume
$n_{\textrm{s},1}=(n+1)^{N}$, $n_{\textrm{s},i}=n^{N},i=2,...,K$, and
$L=(n+1)^{N}+n^{N}$, in which $N=(K-1)(K-2)-1$ and $n$ is an auxiliary
variable, which is going to be increased in order to achieve the DoF of
$\frac{K}{2}$. The beamforming vectors at the transmitters are chosen as

$\displaystyle\bm{V}_{1}$
$\displaystyle=\bigg{\\{}\prod_{\begin{subarray}{c}g,q\in\\{2,...,K\\},\\\
m\neq
g,(g,q)\neq(2,3)\end{subarray}}(\bm{T}_{gq})^{\alpha_{gq}}\bm{w}:\forall\alpha_{gq}\in\\{0,...,n\\}\bigg{\\}},$
$\displaystyle\bm{V}_{j}$ $\displaystyle=\bm{S}_{j}\bm{B},\qquad j=2,...,K,$
(3)

where $\bm{w}$ is a $L\times 1$ vector as $\bm{w}=[1,...,1]^{\textrm{T}}$, and

$\displaystyle\bm{S}_{j}=(\bm{H}_{1j})^{-1}\bm{H}_{13}(\bm{H}_{23})^{-1}\bm{H}_{21},\qquad
j=2,...,K,$
$\displaystyle\bm{T}_{ij}=(\bm{H}_{i1})^{-1}\bm{H}_{ij}\bm{S}_{j},\qquad
i,j=2,...,K,\quad i\neq j,$
$\displaystyle\bm{B}=\bigg{\\{}\prod_{\begin{subarray}{c}g,q\in\\{2,..K\\},\\\
g\neq
q,(g,q)\neq(2,3)\end{subarray}}(\bm{T}_{gq})^{\alpha_{gq}}\bm{w}:\forall\alpha_{gq}\in\\{0,...,n-1\\}\bigg{\\}}.$

The DoF is equal to $\frac{(n+1)^{N}+(K-1)n^{N}}{(n+1)^{N}+n^{N}}$ and by
increasing $n$, we can reach the DoF of $\frac{K}{2}$.

Proposed non-asymptotic IA scheme by choice of releasing/sampling times: We
assume $K=3$ and $L=2$. For the 3-user IFC, the vector of mean number of
counted molecules at $\textrm{Rx}_{i}$, $i\in\\{1,2,3\\}$, is

$\displaystyle\bm{\mu}_{i}$
$\displaystyle=\sum_{j=1}^{3}\bm{H}_{ij}X_{j}=\bm{H}_{i1}\bm{X}_{1}+\bm{H}_{i2}\bm{X}_{2}+\bm{H}_{i3}\bm{X}_{3}$
$\displaystyle=\bm{H}_{i1}\bm{V}_{1}N_{1}+\bm{H}_{i2}\bm{V}_{2}N_{2}+\bm{H}_{i3}\bm{V}_{3}N_{3}.$
(4)

For IA, we should have

$\displaystyle\textrm{Rx}_{1}:\quad\bm{H}_{12}\bm{V}_{2}\equiv\bm{H}_{13}\bm{V}_{3}\not\equiv\bm{H}_{11}\bm{V}_{1},$
(5)
$\displaystyle\textrm{Rx}_{2}:\quad\bm{H}_{21}\bm{V}_{1}\equiv\bm{H}_{23}\bm{V}_{3}\not\equiv\bm{H}_{22}\bm{V}_{2},$
$\displaystyle\textrm{Rx}_{3}:\quad\bm{H}_{31}\bm{V}_{1}\equiv\bm{H}_{32}\bm{V}_{2}\not\equiv\bm{H}_{33}\bm{V}_{3}.$

From (5), if the channel matrices are such that
$T=\bm{H}_{13}^{-1}\bm{H}_{12}\bm{H}_{21}^{-1}\bm{H}_{23}\bm{H}_{32}^{-1}\bm{H}_{31}\equiv
I$, where $I$ is an identity matrix of order 2, or equivalently,

$\displaystyle\frac{H_{13}^{[2]}}{H_{13}^{[1]}}.\frac{H_{12}^{[1]}}{H_{12}^{[2]}}.\frac{H_{21}^{[2]}}{H_{21}^{[1]}}.\frac{H_{23}^{[1]}}{H_{23}^{[2]}}.\frac{H_{32}^{[2]}}{H_{32}^{[1]}}.\frac{H_{31}^{[1]}}{H_{31}^{[2]}}=1,$
(6)

we can choose the following beamforming vectors that satisfy (5):

$\displaystyle\bm{V}_{1}$
$\displaystyle\equiv[1,1]^{\textrm{T}},\quad\bm{V}_{2}\equiv\bm{H}_{32}^{-1}\bm{H}_{31}\bm{V}_{1}=\left[\frac{H_{31}^{[1]}}{H_{32}^{[1]}},\frac{H_{31}^{[2]}}{H_{32}^{[2]}}\right]^{\textrm{T}},$
$\displaystyle\bm{V}_{3}$
$\displaystyle\equiv\bm{H}_{23}^{-1}\bm{H}_{21}\bm{V}_{1}=\left[\frac{H_{21}^{[1]}}{H_{23}^{[1]}},\frac{H_{21}^{[2]}}{H_{23}^{[2]}}\right]^{\textrm{T}}.$
(7)

We should normalize the beamforming vectors such that we have a unit norm,
i.e., $|\bm{V}_{i}|_{1}=1$. Further, (5) requires the following conditions:

$\displaystyle\bm{H}_{12}\bm{V}_{2}\not\equiv\bm{H}_{11}\bm{V}_{1},~{}\bm{H}_{21}\bm{V}_{1}\not\equiv\bm{H}_{22}\bm{V}_{2},~{}\bm{H}_{31}\bm{V}_{1}\not\equiv\bm{H}_{33}\bm{V}_{3}.$
(8)

We call the condition in (6) as the IA condition and the conditions in (8) as
the independency conditions. Our goal is to change the channel coefficients
such that the IA and independency conditions hold, thus we can align the
interference signals at the receivers by choosing the beamforming vectors at
the transmitters as (III).
Using the beamforming vectors in (III), the mean signal vectors at the
receivers are obtained from (III) as

$\displaystyle\textrm{Rx}_{1}:\bm{\mu}_{1}$
$\displaystyle=\frac{N_{1}}{2}\left[\begin{matrix}H_{11}^{[1]}\\\
H_{11}^{[2]}\end{matrix}\right]+\frac{N_{2}}{A}\left[\begin{matrix}H_{12}^{[1]}\frac{H_{31}^{[1]}}{H_{32}^{[1]}}\\\
H_{12}^{[2]}\frac{H_{31}^{[2]}}{H_{32}^{[2]}}\end{matrix}\right]+\frac{N_{3}}{B}\left[\begin{matrix}H_{13}^{[1]}\frac{H_{21}^{[1]}}{H_{23}^{[1]}}\\\
H_{13}^{[2]}\frac{H_{21}^{[2]}}{H_{23}^{[2]}}\end{matrix}\right],$ (9)
$\displaystyle\textrm{Rx}_{2}:\bm{\mu}_{2}$
$\displaystyle=\frac{N_{1}}{2}\left[\begin{matrix}H_{21}^{[1]}\\\
H_{21}^{[2]}\end{matrix}\right]+\frac{N_{2}}{A}\left[\begin{matrix}H_{22}^{[1]}\frac{H_{31}^{[1]}}{H_{32}^{[1]}}\\\
H_{22}^{[2]}\frac{H_{31}^{[2]}}{H_{32}^{[2]}}\end{matrix}\right]+\frac{N_{3}}{B}\left[\begin{matrix}H_{21}^{[1]}\\\
H_{21}^{[2]}\end{matrix}\right],$ $\displaystyle\textrm{Rx}_{3}:\bm{\mu}_{3}$
$\displaystyle=\frac{N_{1}}{2}\left[\begin{matrix}H_{31}^{[1]}\\\
H_{31}^{[2]}\end{matrix}\right]+\frac{N_{2}}{A}\left[\begin{matrix}H_{31}^{[1]}\\\
H_{31}^{[2]}\end{matrix}\right]+\frac{N_{3}}{B}\left[\begin{matrix}H_{33}^{[1]}\frac{H_{21}^{[1]}}{H_{23}^{[1]}}\\\
H_{33}^{[2]}\frac{H_{21}^{[2]}}{H_{23}^{[2]}}\end{matrix}\right],$

where $A=\frac{H_{31}^{[1]}}{H_{32}^{[1]}}+\frac{H_{31}^{[2]}}{H_{32}^{[2]}}$
and $B=\frac{H_{21}^{[1]}}{H_{23}^{[1]}}+\frac{H_{21}^{[2]}}{H_{23}^{[2]}}$.
From (9), it can be seen that the interference signals at Rx2 and Rx3 are
aligned without resorting to (6). However, to align the interference signals
at Rx1, (6) is needed to hold. For this purpose, we change the channel
coefficients by the choice of the releasing times at the transmitters,
$\tilde{t}_{j}^{[l]}$, and/or the sampling times at the receivers,
$t_{i}^{[l]}$, to meet the conditions in (6) and (8).222In MC systems,
releasing time is usually chosen as zero, i.e., the molecules are released at
the beginning of the time slot, and the sampling time is usually chosen such
that the concentration of the molecules is maximized at the time [6]. However,
it can be shown that if we use these times for each transmitter-receiver pair,
the desired signal and the interference signals will be all aligned and the
independency condition cannot be met.  In the following, we simplify the
alignment and independency equations in (6) and (8) to obtain a feasible
region for $t_{i}^{[l]}$s and $\tilde{t}_{j}^{[l]}$s.

$\displaystyle r_{13}^{2}.f(\Delta t_{13}^{[1]},\Delta
t_{13}^{[2]})-r_{12}^{2}.f(\Delta t_{12}^{[1]},\Delta
t_{12}^{[2]})+r_{21}^{2}.f(\Delta t_{21}^{[1]},\Delta
t_{21}^{[2]})-r_{23}^{2}.f(\Delta t_{23}^{[1]},\Delta t_{23}^{[2]})$ (10)
$\displaystyle\quad+r_{32}^{2}.f(\Delta t_{32}^{[1]},\Delta
t_{32}^{[2]})-r_{31}^{2}.f(\Delta t_{31}^{[1]},\Delta
t_{31}^{[2]})+6\ln(\frac{\Delta t_{13}^{[1]}}{\Delta
t_{13}^{[2]}}.\frac{\Delta t_{12}^{[2]}}{\Delta t_{12}^{[1]}}.\frac{\Delta
t_{21}^{[1]}}{\Delta t_{21}^{[2]}}.\frac{\Delta t_{23}^{[2]}}{\Delta
t_{23}^{[1]}}.\frac{\Delta t_{32}^{[1]}}{\Delta t_{32}^{[2]}}.\frac{\Delta
t_{31}^{[2]}}{\Delta t_{31}^{[1]}})=0.$ $\displaystyle\textrm{Rx}_{1}:$
$\displaystyle r_{11}^{2}.f(\Delta t_{11}^{[1]},\Delta
t_{11}^{[2]})-r_{12}^{2}.f(\Delta t_{12}^{[1]},\Delta
t_{12}^{[2]})+r_{32}^{2}.f(\Delta t_{32}^{[1]},\Delta
t_{32}^{[2]})-r_{31}^{2}.f(\Delta t_{31}^{[1]},\Delta
t_{31}^{[2]})+6\ln(\frac{\Delta t_{11}^{[1]}}{\Delta
t_{11}^{[2]}}.\frac{\Delta t_{12}^{[2]}}{\Delta t_{12}^{[1]}}.\frac{\Delta
t_{32}^{[1]}}{\Delta t_{32}^{[2]}}.\frac{\Delta t_{31}^{[2]}}{\Delta
t_{31}^{[1]}})\neq 0,$ $\displaystyle\textrm{Rx}_{2}:$ $\displaystyle
r_{22}^{2}.f(\Delta t_{22}^{[1]},\Delta t_{22}^{[2]})-r_{21}^{2}.f(\Delta
t_{21}^{[1]},\Delta t_{21}^{[2]})+r_{31}^{2}.f(\Delta t_{31}^{[1]},\Delta
t_{31}^{[2]})-r_{32}^{2}.f(\Delta t_{32}^{[1]},\Delta
t_{32}^{[2]})+6\ln(\frac{\Delta t_{22}^{[1]}}{\Delta
t_{22}^{[2]}}.\frac{\Delta t_{21}^{[2]}}{\Delta t_{21}^{[1]}}.\frac{\Delta
t_{31}^{[1]}}{\Delta t_{31}^{[2]}}.\frac{\Delta t_{32}^{[2]}}{\Delta
t_{32}^{[1]}})\neq 0,$ $\displaystyle\textrm{Rx}_{3}:$ $\displaystyle
r_{33}^{2}.f(\Delta t_{33}^{[1]},\Delta t_{33}^{[2]})-r_{31}^{2}.f(\Delta
t_{31}^{[1]},\Delta t_{31}^{[2]})+r_{21}^{2}.f(\Delta t_{21}^{[1]},\Delta
t_{21}^{[2]})-r_{23}^{2}.f(\Delta t_{23}^{[1]},\Delta
t_{23}^{[2]})+6\ln(\frac{\Delta t_{33}^{[1]}}{\Delta
t_{33}^{[2]}}.\frac{\Delta t_{31}^{[2]}}{\Delta t_{31}^{[1]}}.\frac{\Delta
t_{21}^{[1]}}{\Delta t_{21}^{[2]}}.\frac{\Delta t_{23}^{[2]}}{\Delta
t_{23}^{[1]}})\neq 0.$ (11)

Using (1), the IA condition in (6) simplifies to (10) and the independency
conditions in (8) simplify to (III), for
$\tilde{t}_{j}^{[l]}<t_{i}^{[l]}<T_{\textrm{s}},i,j\in\\{1,2,3\\}$,
$l\in\\{1,2\\}$, where $f(\Delta t_{ij}^{[1]},\Delta
t_{ij}^{[2]})=\frac{1}{D_{1}\Delta t_{ij}^{[1]}}-\frac{1}{D_{2}\Delta
t_{ij}^{[2]}}$, $\Delta t_{ij}^{[l]}=t_{i}^{[l]}-\tilde{t}_{j}^{[l]}$, and
$r_{ij}^{2}=||\bm{r}_{i}^{\textrm{Rx}}-\bm{r}_{j}^{\textrm{Tx}}||^{2},i,j\in\\{1,2,3\\}$,
$l\in\\{1,2\\}$. The conditions in (10) and (III) form a feasible region for
$t_{i}^{[l]}$s and $\tilde{t}_{j}^{[l]}$s noted by the set
$\mathcal{T}_{\textrm{nr}}$, where “nr” stands for “no reaction”. In Lemma 1,
we simplify these conditions for a special case of the releasing and sampling
times.

######  Remark 1.

It can be seen that the conditions do not depend on the flow velocity of the
medium.

######  Lemma 1.

For
$t_{i}^{[1]}=t_{i}^{[2]}=t_{i},\tilde{t}_{j}^{[1]}=\tilde{t}_{j}^{[2]}=\tilde{t}_{j}$,
$i,j\in\\{1,2,3\\}$, the IA condition reduces to:

$\displaystyle\frac{r_{13}^{2}}{t_{1}-\tilde{t}_{3}}-\frac{r_{12}^{2}}{t_{1}-\tilde{t}_{2}}+\frac{r_{21}^{2}}{t_{2}-\tilde{t}_{1}}$
$\displaystyle\quad-\frac{r_{23}^{2}}{t_{2}-\tilde{t}_{3}}+\frac{r_{32}^{2}}{t_{3}-\tilde{t}_{2}}-\frac{r_{31}^{2}}{t_{3}-\tilde{t}_{1}}=0,$
(12)

and the independency conditions reduce to

$\displaystyle\textrm{Rx}_{1}:\frac{r_{11}^{2}}{t_{1}-\tilde{t}_{1}}-\frac{r_{12}^{2}}{t_{1}-\tilde{t}_{2}}+\frac{r_{32}^{2}}{t_{3}-\tilde{t}_{2}}-\frac{r_{31}^{2}}{t_{3}-\tilde{t}_{1}}\neq
0,$ (13)
$\displaystyle\textrm{Rx}_{2}:\frac{r_{22}^{2}}{t_{2}-\tilde{t}_{2}}-\frac{r_{21}^{2}}{t_{2}-\tilde{t}_{1}}+\frac{r_{31}^{2}}{t_{3}-\tilde{t}_{1}}-\frac{r_{32}^{2}}{t_{3}-\tilde{t}_{2}}\neq
0,$
$\displaystyle\textrm{Rx}_{3}:\frac{r_{33}^{2}}{t_{3}-\tilde{t}_{3}}-\frac{r_{31}^{2}}{t_{3}-\tilde{t}_{1}}+\frac{r_{21}^{2}}{t_{2}-\tilde{t}_{1}}-\frac{r_{23}^{2}}{t_{2}-\tilde{t}_{3}}\neq
0.$

The conditions in (1) and (13) form a feasible region for $t_{i}$s and
$\tilde{t}_{j}$s, noted by the set $\mathcal{T}_{\textrm{nr,s}}$, where “s”
stands for “special case”.

###### Proof.

The proof is straightforward from (10) and (III). ∎

Using Reaction in the IA scheme: To cancel the aligned interference in the
proposed IA scheme and reduce the signal dependent noise, we exploit molecular
reaction. Consider the following reactions among the molecules of type 1 and 2
($\textrm{M}_{1}$ and $\textrm{M}_{2}$) at $\textrm{Rx}_{i}$:

$\displaystyle\textrm{Rx}_{i}:c_{i}\textrm{M}_{1}+\textrm{M}_{2}+\textrm{E}_{i}\rightarrow\textrm{products},$
(14)

for $i\in\\{1,2,3\\}$, where $\textrm{E}_{i}$ is an enzyme released from
$\textrm{Rx}_{i}$, which enables $\textrm{M}_{1}$ and $\textrm{M}_{2}$ to
react with coefficient $c_{i}$ and with high reaction rate. The products are
not detectable by the receivers. Further, we assume that the sampling times of
different molecule types at the each receiver are the same, i.e.,
$t_{i}^{[1]}=t_{i}^{[2]}=t_{i}$, and enzymes are released at the sampling time
$t_{i}$ from each receiver. According to (14), to cancel the interference
signals at the receivers, we require that the ratio of the interference signal
of $\textrm{M}_{1}$ to the interference signal of $\textrm{M}_{2}$ at
$\textrm{Rx}_{i}$ be equals to $c_{i}$. Hence, using the signal vectors in
(9), we should have

$\displaystyle\textrm{Rx}_{1}:\frac{H_{12}^{[1]}\frac{H_{31}^{[1]}}{H_{32}^{[1]}}}{H_{12}^{[2]}\frac{H_{31}^{[2]}}{H_{32}^{[2]}}}=c_{1},~{}\textrm{Rx}_{2}:\frac{H_{21}^{[1]}}{H_{21}^{[2]}}=c_{2},~{}\textrm{Rx}_{3}:\frac{H_{31}^{[1]}}{H_{31}^{[2]}}=c_{3}.$
(15)

We call the conditions in (15) as the reaction conditions. Hence, if it is
feasible to choose the releasing/sampling times such that along with the IA
and independency conditions in (6) and (8) the reaction conditions also hold,
the aligned interference signals at the receivers will be canceled using
reaction. Using (1), the reaction conditions in (15) simplifies to (III),
where $f(.,.)$ is defined in (10). Note that in the IA, independency, and
reaction conditions in (10), (III), and (III), we should further have
$t_{i}^{[1]}=t_{i}^{[2]}=t_{i}$.

$\displaystyle\textrm{Rx}_{1}:$ $\displaystyle r_{12}^{2}.f(\Delta
t_{12}^{[1]},\Delta t_{12}^{[2]})+r_{31}^{2}.f(\Delta t_{31}^{[1]},\Delta
t_{31}^{[2]})-r_{32}^{2}.f(\Delta t_{32}^{[1]},\Delta
t_{32}^{[2]})+6\ln(\frac{D_{1}}{D_{2}}\frac{\Delta t_{12}^{[1]}}{\Delta
t_{12}^{[2]}}\frac{\Delta t_{31}^{[1]}}{\Delta t_{31}^{[2]}}\frac{\Delta
t_{32}^{[2]}}{\Delta t_{32}^{[1]}})$
$\displaystyle+||\bm{\nu}||^{2}(\frac{\Delta t_{12}^{[1]}+\Delta
t_{31}^{[1]}-\Delta t_{32}^{[1]}}{D_{1}}-\frac{\Delta t_{12}^{[2]}+\Delta
t_{31}^{[2]}-\Delta
t_{32}^{[2]}}{D_{2}})-2(\bm{r}_{1}^{\textrm{Rx}}-\bm{r}_{1}^{\textrm{Tx}}).\bm{\nu}(\frac{1}{D_{1}}-\frac{1}{D_{2}})=-4\ln{c_{1}},$
$\displaystyle\textrm{Rx}_{2}:$ $\displaystyle r_{21}^{2}.f(\Delta
t_{21}^{[1]},\Delta t_{21}^{[2]})+6\ln(\frac{D_{1}}{D_{2}}\frac{\Delta
t_{21}^{[1]}}{\Delta t_{21}^{[2]}})+||\bm{\nu}||^{2}(\frac{\Delta
t_{21}^{[1]}}{D_{1}}-\frac{\Delta
t_{21}^{[2]}}{D_{2}})-2(\bm{r}_{2}^{\textrm{Rx}}-\bm{r}_{1}^{\textrm{Tx}}).\bm{\nu}(\frac{1}{D_{1}}-\frac{1}{D_{2}})=-4\ln{c_{2}},$
$\displaystyle\textrm{Rx}_{3}:$ $\displaystyle r_{31}^{2}.f(\Delta
t_{31}^{[1]},\Delta t_{31}^{[2]})+6\ln(\frac{D_{1}}{D_{2}}\frac{\Delta
t_{31}^{[1]}}{\Delta t_{31}^{[2]}})+||\bm{\nu}||^{2}(\frac{\Delta
t_{31}^{[1]}}{D_{1}}-\frac{\Delta
t_{31}^{[2]}}{D_{2}})-2(\bm{r}_{3}^{\textrm{Rx}}-\bm{r}_{1}^{\textrm{Tx}}).\bm{\nu}(\frac{1}{D_{1}}-\frac{1}{D_{2}})=-4\ln{c_{3}}.$
(16)

Two approaches are possible: (i) We may choose the releasing and/or sampling
times such that they satisfy the IA and independency conditions in (10) and
(III), and then obtain $c_{1},c_{2},$ and $c_{3}$ from (III). However, we may
not find reactions among $\textrm{M}_{1}$ and $\textrm{M}_{2}$ with these
coefficients. (ii) We may choose $c_{1},c_{2}$, and $c_{3}$ arbitrarily and
then choose the releasing and/or sampling times such that they satisfy the IA,
independency, and reaction conditions in (10), (III), and (III). These
conditions form a feasible region for $t_{i}^{[l]}$s and
$\tilde{t}_{j}^{[l]}$s, noted by the set $\mathcal{T}_{\textrm{r}}$, where “r”
stands for “reaction”. We take the second approach and for a special case,
simplify the IA, independency, and reaction conditions in the following Lemma.

######  Lemma 2.

For
$t_{i}^{[1]}=t_{i}^{[2]}=t_{i},\tilde{t}_{j}^{[1]}=\tilde{t}_{j}^{[2]}=\tilde{t}_{j}$,
$i,j\in\\{1,2,3\\}$, $\bm{\nu}=0$, and $c_{1}=c_{2}=c_{3}=c$, the IA and
reaction conditions reduce to

$\displaystyle\Delta\tilde{t}_{12}(1-\frac{r_{12}^{2}}{r_{32}^{2}})-\Delta\tilde{t}_{13}(1-\frac{r_{13}^{2}}{r_{23}^{2}})=s.(\frac{r_{12}^{2}}{r_{32}^{2}}.r_{31}^{2}-\frac{r_{13}^{2}}{r_{23}^{2}}.r_{21}^{2}),$
(17)

and the independency conditions reduce to

$\displaystyle\Delta\tilde{t}_{12}\neq
s.\frac{r_{22}^{2}.r_{31}^{2}-r_{21}^{2}.r_{32}^{2}}{r_{32}^{2}-r_{22}^{2}},$
$\displaystyle\Delta\tilde{t}_{13}\notin\big{\\{}s.\frac{r_{13}^{2}.r_{21}^{2}-r_{11}^{2}.r_{23}^{2}}{r_{23}^{2}-r_{13}^{2}},s.\frac{r_{33}^{2}.r_{21}^{2}-r_{31}^{2}.r_{23}^{2}}{r_{23}^{2}-r_{33}^{2}}\big{\\}},$
(18)

where $\Delta\tilde{t}_{1j}=\tilde{t}_{1}-\tilde{t}_{j},j=2,3$,
$s=-\frac{\frac{1}{D_{1}}-\frac{1}{D_{2}}}{4\ln{c}-6\ln{\frac{D_{2}}{D_{1}}}}$.
Further, it is needed that $\Delta t_{ij}>0,i,j\in\\{1,2,3\\}$, which results
in $s>0$ and

$\displaystyle\begin{cases}\Delta\tilde{t}_{13}>-s\min\\{r_{21}^{2},r_{31}^{2}\\},\qquad&\textrm{if}~{}r_{13}>r_{23}\\\
-s\min\\{r_{21}^{2},r_{31}^{2}\\}<\Delta\tilde{t}_{13}<-\frac{s.r_{21}^{2}}{1-\frac{r_{23}^{2}}{r_{13}^{2}}},~{}&\textrm{if}~{}r_{13}<r_{23}\end{cases}.$
$\displaystyle\Delta\tilde{t}_{12}>-s\min\\{r_{21}^{2},r_{31}^{2}\\}.$ (19)

The conditions in (17)-(2) form a feasible region for $\Delta\tilde{t}_{12}$
and $\Delta\tilde{t}_{13}$, noted by $\mathcal{T}_{\textrm{r,s}}$. After
choosing $\Delta\tilde{t}_{12}$ and $\Delta\tilde{t}_{13}$ in the feasible
region, we choose one of the releasing times arbitrarily and obtain the other
two releasing times from the values of $\Delta\tilde{t}_{12}$ and
$\Delta\tilde{t}_{13}$. Then, $t_{1}$, $t_{2}$, and $t_{3}$ can be obtained
from $\tilde{t}_{1}$ and $\tilde{t}_{3}$ as follows:

$\displaystyle
t_{1}=\frac{r_{13}^{2}}{r_{23}^{2}}\tilde{t}_{1}+(1-\frac{r_{13}^{2}}{r_{23}^{2}})\tilde{t}_{3}+s.\frac{r_{13}^{2}}{r_{23}^{2}}.r_{21}^{2},$
$\displaystyle t_{2}=\tilde{t}_{1}+s.r_{21}^{2},\quad
t_{3}=\tilde{t}_{1}+s.r_{31}^{2}.$ (20)

###### Proof.

The proof is provided in Appendix A. ∎

######  Remark 2.

Without loss of generality we assume $D_{1}>D_{2}$. Hence, for the spacial
case of the times in Lemma 2, we should choose molecules to have reaction with
$c>(\frac{D_{2}}{D_{1}})^{\frac{3}{2}}$ to garantee $s>0$.

##  IV Optimum Sampling and Releasing Times

To obtain the optimum sampling and releasing times, we minimize the total
error probability of the system over the feasible region of the times. Using
the error probability at $\textrm{Rx}_{i}$, noted by $P_{\textrm{e},i}$,
$i\in\\{1,2,3\\}$, the total error probability of the system is
$P_{\textrm{e}}=\frac{1}{3}(P_{\textrm{e},1}+P_{\textrm{e},2}+P_{\textrm{e},3}).$

IA without reaction: For the general case of the sampling and releasing times,
the times are obtained from the following optimization problem:

$\displaystyle\bm{t}_{\textrm{nr}}=\arg\min_{\bm{t}\in\mathcal{T}_{\textrm{nr}}}P_{\textrm{e,nr}},$
(21)

where
$\bm{t}=[\tilde{t}_{1}^{[1]},\tilde{t}_{1}^{[2]},\tilde{t}_{2}^{[1]},\tilde{t}_{2}^{[2]},\tilde{t}_{3,}^{[1]},\tilde{t}_{3}^{[2]},t_{1}^{[1]},t_{1}^{[2]},t_{2}^{[1]},t_{2}^{[2]},t_{3}^{[1]}$,
$t_{3}^{[2]}]$, $P_{\textrm{e,nr}}$ is the total error probability in the IA
scheme without reaction, and as defined in Section III,
$\mathcal{T}_{\textrm{nr}}$ is the feasible region of the times in the IA with
no reaction. For the special case of the times when $t_{i}^{[l]}=t_{i}$ and
$\tilde{t}_{j}^{[l]}=\tilde{t}_{j}$, $i,j\in\\{1,2,3\\}$, we should solve

$\displaystyle\bm{t}_{\textrm{nr,s}}=\arg\min_{\bm{t}_{\textrm{s}}\in\mathcal{T}_{\textrm{nr,s}}}P_{\textrm{e,nr}},$
(22)

where $\bm{t}_{\textrm{s}}=[\tilde{t}_{1},\tilde{t}_{2},\tilde{t}_{3},t_{1}$,
$t_{2},t_{3}]$ and $\mathcal{T}_{\textrm{nr,s}}$ is characterized in Lemma 1.

IA with reaction: Similar to the no reaction case, for the general case of the
times, we have

$\displaystyle\bm{t}_{\textrm{r}}=\arg\min_{\bm{t}\in\mathcal{T}_{\textrm{r}}}P_{\textrm{e,r}},$
(23)

where $P_{\textrm{e,r}}$ is the total error probability in the IA scheme with
reaction, and as defined in Section III, $\mathcal{T}_{\textrm{r}}$ is the
feasible region of the times in the IA with reaction. For the special case of
the times, according to Lemma 2, the feasible region is obtained for
$\Delta\tilde{t}_{12}$ and $\Delta\tilde{t}_{13}$. Hence, we should solve the
following optimization problem

$\displaystyle[\Delta\tilde{t}_{12,\textrm{r,s}},\Delta\tilde{t}_{13,\textrm{r,s}}]=\arg\min_{[\Delta\tilde{t}_{12},\Delta\tilde{t}_{13}]\in\mathcal{T}_{\textrm{r,s}}}P_{\textrm{e,r}}.$
(24)

where $\mathcal{T}_{\textrm{r, s}}$ is characterized in Lemma 2. The values of
$t_{i}$s and $\tilde{t}_{j}$s can be obtained from
$\Delta\tilde{t}_{12,\textrm{r,s}},\Delta\tilde{t}_{13,\textrm{r,s}}$, which
are noted by $\tilde{t}_{1,\textrm{r,s}}$, $\tilde{t}_{2,\textrm{r,s}}$,
$\tilde{t}_{3,\textrm{r,s}}$, $t_{1,\textrm{r,s}}$, $t_{2,\textrm{r,s}}$, and
$t_{3,\textrm{r,s}}$.

The optimum sampling times in each case are obtained numerically in Section
VI. We note that the values of the channel coefficients and hence the error
probability depend on $\Delta t_{ij}^{[l]}$s. Hence, if we shift the releasing
and sampling times by a same value, the error probability does not change and
thus the optimum times are not unique. We choose the times that are all
positive and the smallest one is zero.

##  V Error Performance Analysis of the Proposed IA Scheme

In the proposed IA scheme, according to (9), the mean signal vector at
$\textrm{Rx}_{i}$, assuming no reaction is

$\displaystyle\bm{\mu}_{i}=\tilde{\bm{H}}_{i}N_{i}+\tilde{\bm{H}}_{\textrm{I},i}N_{\textrm{I},i},\qquad
i\in\\{1,2,3\\},$ (25)

where $N_{\textrm{I},1}=\frac{N_{2}}{A}+\frac{N_{3}}{B}$,
$N_{\textrm{I},2}=\frac{N_{1}}{2}+\frac{N_{3}}{B}$, and
$N_{\textrm{I},3}=\frac{N_{1}}{2}+\frac{N_{2}}{A}$; $A$ and $B$ are defined in
(9). Further,
$\tilde{\bm{H}}_{i}=[\tilde{H}_{i}^{[1]},\tilde{H}_{i}^{[2]}]^{\textrm{T}}$
and
$\tilde{\bm{H}}_{\textrm{I},i}=[\tilde{H}_{\textrm{I},i}^{[1]},\tilde{H}_{\textrm{I},i}^{[2]}]^{\textrm{T}}$
can be obtained from the channel coefficients using (9).

IA without reaction: The message of $\textrm{Tx}_{i}$ ($M_{i}$) can be
obtained from the number of released molecules, $N_{i}$. Using (25), $N_{i}$
can be obtained at $\textrm{Rx}_{i}$ from the mean number of received
molecules by zero-forcing as
$N_{i}=\frac{\tilde{H}_{\textrm{I},i}^{[2]}\mu_{i}^{[1]}-\tilde{H}_{\textrm{I},i}^{[1]}\mu_{i}^{[2]}}{\tilde{H}_{i}^{[1]}\tilde{H}_{\textrm{I},i}^{[2]}-\tilde{H}_{i}^{[2]}\tilde{H}_{\textrm{I},i}^{[1]}}$.
Due to the counting noise, the number of received molecules of type $l$,
$Y_{i}^{[l]}$, has Poisson distribution with mean $\mu_{i}^{[l]}$. Using the
noisy observations, we obtain a random variable
$\tilde{N}_{i}=\frac{\tilde{H}_{\textrm{I},i}^{[2]}Y_{i}^{[1]}-\tilde{H}_{i}^{[1]}Y_{i}^{[2]}}{\tilde{H}_{i}^{[1]}\tilde{H}_{\textrm{I},i}^{[2]}-\tilde{H}_{i}^{[2]}\tilde{H}_{\textrm{I},i}^{[1]}}$,
with mean $N_{i}$, whose distribution can be obtained from the distribution of
$Y_{i}^{[1]}$ and $Y_{i}^{[2]}$. We use maximum a-posteriori (MAP) decision
rule on $\tilde{N}_{i}$ to obtain $M_{i}$:

$\displaystyle\mathrm{P}(\tilde{N}_{i}=\tilde{n}_{i}|M_{i}=1)\overset{M_{i}=1}{\underset{M_{i}=0}{\gtrless}}\mathrm{P}(\tilde{N}_{i}=\tilde{n}_{i}|M_{i}=0).$
(26)

For $\textrm{Rx}_{1}$, the above equation results in

$\displaystyle\sum_{m_{2},m_{3}\in\\{0,1\\}}\mathrm{P}(\tilde{n}_{1}|M_{1}=1,M_{2}=m_{2},M_{3}=m_{3})\overset{M_{1}=1}{\underset{M_{1}=0}{\gtrless}}$
$\displaystyle\sum_{m_{2},m_{3}\in\\{0,1\\}}\mathrm{P}(\tilde{n}_{1}|M_{1}=0,M_{2}=m_{2},M_{3}=m_{3}).$
(27)

To obtain the MAP decision rule, we need to obtain the distribution of
$\tilde{N}_{i}$ conditioned on the messages of the transmitters.
$\tilde{N}_{i}$ is a linear combination of two Poisson random variables
$Y_{i}^{[1]}$ and $Y_{i}^{[2]}$, i.e.,
$\tilde{N}_{i}=a_{i}Y_{i}^{[1]}+b_{i}Y_{i}^{[2]}$, where
$a_{i}=\frac{\tilde{H}_{\textrm{I},i}^{[2]}}{\tilde{H}_{i}^{[1]}\tilde{H}_{\textrm{I},i}^{[2]}-\tilde{H}_{i}^{[2]}\tilde{H}_{\textrm{I},i}^{[1]}}$
and
$b_{i}=\frac{-\tilde{H}_{\textrm{I},i}^{[1]}}{\tilde{H}_{i}^{[1]}\tilde{H}_{\textrm{I},i}^{[2]}-\tilde{H}_{i}^{[2]}\tilde{H}_{\textrm{I},i}^{[1]}}$.
This yields to a very complex problem. For simplicity, we assume Gaussian
approximation to the Poisson distribution of $Y_{i}^{[l]}$, i.e.,
${Y_{i}^{[l]}}_{|m_{1},m_{2},m_{3}}\sim\mathcal{N}(\mu_{i}^{[l]},\sqrt{\mu_{i}^{[l]}})$,
which is a good approximation if the number of released molecules is large.
Since different molecule types do not interfere with each other, $Y_{i}^{[1]}$
and $Y_{i}^{[2]}$ are independent conditioned on the messages of the
transmitters. Hence, $\tilde{N}_{i}$ has a Gaussian distribution conditioned
on $M_{1}=m_{1}$, $M_{2}=m_{2}$, and $M_{3}=m_{3}$ with mean
$\tilde{\mu}_{i}(m_{1},m_{2},m_{3})=a_{i}\mu_{i}^{[1]}+b_{i}\mu_{i}^{[2]}$ and
variance
$\tilde{\sigma}_{i}^{2}(m_{1},m_{2},m_{3})=a_{i}^{2}\mu_{i}^{[1]}+b_{i}^{2}\mu_{i}^{[2]}$.
Therefore, the MAP decision rule in (V) becomes

$\displaystyle\sum_{m_{2},m_{3}\in\\{0,1\\}}\big{[}\frac{1}{\sqrt{2\pi\tilde{\sigma}_{i}^{2}(1,m_{2},m_{3})}}e^{-\frac{(\tilde{n}_{1}-\tilde{\mu}_{i}(1,m_{2},m_{3}))^{2}}{2\tilde{\sigma}_{i}^{2}(1,m_{2},m_{3})}}$
(28)
$\displaystyle\qquad\qquad-\frac{1}{\sqrt{2\pi\tilde{\sigma}_{i}^{2}(0,m_{2},m_{3})}}e^{-\frac{(\tilde{n}_{1}-\tilde{\mu}_{i}(0,m_{2},m_{3}))^{2}}{2\tilde{\sigma}_{i}^{2}(0,m_{2},m_{3})}}\big{]}\overset{M_{1}=1}{\underset{M_{1}=0}{\gtrless}}0.$

However, obtaining a closed-form of this decision rule is not possible and we
derive it numerically in Section VI. Since the MAP decision rule does not
yield to a simple form, we cannot obtain a closed-form equation for the error
probability of this scheme and we obtain the error probability using
simulation in Section VI.

IA with reaction: For the proposed IA scheme with perfect reaction, the
average signal vector at $\textrm{Rx}_{i}$ after reaction is, [5],

$\displaystyle\bm{\mu}_{\textrm{r},i}=[\mu_{\textrm{r},i}^{[1]},\mu_{\textrm{r},i}^{[2]}]^{\textrm{T}}=\begin{cases}\big{[}(\tilde{H}_{i}^{[1]}-c_{i}\tilde{H}_{i}^{[2]})N_{i},0\big{]}^{\textrm{T}},&\textrm{if}~{}\frac{\tilde{H}_{i}^{[1]}}{\tilde{H}_{i}^{[2]}}>c_{i}\\\
\big{[}0,(\tilde{H}_{i}^{[2]}-\frac{1}{c_{i}}\tilde{H}_{i}^{[1]})N_{i}\big{]}^{\textrm{T}},&\textrm{if}~{}\frac{\tilde{H}_{i}^{[1]}}{\tilde{H}_{i}^{[2]}}<c_{i}\end{cases}.$

Again, the number of received molecules of type $l$ after reaction,
$Y_{\textrm{r},i}^{[l]}$, has Poisson distribution with mean
$\mu_{\textrm{r},i}^{[l]}$ [5]. If
$\frac{\tilde{H}_{i}^{[1]}}{\tilde{H}_{i}^{[2]}}>c_{i}$, we have
$y_{\textrm{r},i}^{[2]}=0$, and hence, we use $y_{\textrm{r},i}^{[1]}$ to
detect $M_{i}$ using MAP decision rule,
$\mathrm{P}(y_{\textrm{r},i}^{[1]}|M_{i}=1)\overset{M_{i}=1}{\underset{M_{i}=0}{\gtrless}}\mathrm{P}(y_{\textrm{r},i}^{[1]}|M_{i}=0),$
which results in a simple threshold rule as
$y_{\textrm{r},i}^{[1]}\overset{M_{i}=1}{\underset{M_{i}=0}{\gtrless}}\gamma_{i}^{[1]}$,
where
$\gamma_{i}^{[1]}=\frac{(\tilde{H}_{i}^{[1]}-c_{i}\tilde{H}_{i}^{[2]})(\zeta_{1}-\zeta_{0})}{\ln(\frac{\zeta_{1}}{\zeta_{0}})}$.
Hence, for $\frac{\tilde{H}_{i}^{[1]}}{\tilde{H}_{i}^{[2]}}>c_{i}$, the error
probability at $\textrm{Rx}_{i}$, $i\in\\{1,2,3\\}$, is obtained as

$\displaystyle P_{\textrm{e},i}$
$\displaystyle=\frac{1}{2}(P\\{Y_{\textrm{r},i}^{[1]}>\gamma_{i}^{[1]}|M_{i}=0\\}+P\\{Y_{\textrm{r},i}^{[1]}<\gamma_{i}^{[1]}|M_{i}=1\\})$
$\displaystyle=\frac{1}{2}\big{[}1-\sum_{y_{\textrm{r},i}^{[1]}=0}^{\lfloor\gamma_{i}^{[1]}\rfloor}\big{(}\frac{((\tilde{H}_{i}^{[1]}-c_{i}\tilde{H}_{i}^{[2]})\zeta_{0})^{y_{\textrm{r},i}^{[1]}}e^{-(\tilde{H}_{i}^{[1]}-c_{i}\tilde{H}_{i}^{[2]})\zeta_{0}}}{y_{\textrm{r},i}^{[1]}!}$
$\displaystyle\quad-\frac{((\tilde{H}_{i}^{[1]}-c_{i}\tilde{H}_{i}^{[2]})\zeta_{1})^{y_{\textrm{r},i}^{[1]}}e^{-(\tilde{H}_{i}^{[1]}-c_{i}\tilde{H}_{i}^{[2]})\zeta_{1}}}{y_{\textrm{r},i}^{[1]}!}\big{)}\big{]}.$
(29)

The error probability for the case of
$\frac{\tilde{H}_{i}^{[1]}}{\tilde{H}_{i}^{[2]}}<c_{i}$ can be obtained
similarly.

##  VI Simulation and Numerical Results

We evaluate the performance of the proposed IA scheme in a 3-user IFC. We
consider the parameters in Table VI.

TABLE I: Simulation and numerical analysis parameters
Parameter | Symbol | Value
---|---|---
Location of Tx1 | $r_{1}^{\textrm{Tx}}$ | $(0,0,0)~{}\mu\textrm{m}$
Location of Tx2 | $r_{2}^{\textrm{Tx}}$ | $(0,20,10)~{}\mu\textrm{m}$
Location of Tx3 | $r_{3}^{\textrm{Tx}}$ | $(0,0,30)~{}\mu\textrm{m}$
Location of Rx1 | $r_{1}^{\textrm{Rx}}$ | $(0,150,0)~{}\mu\textrm{m}$
Location of Rx2 | $r_{2}^{\textrm{Rx}}$ | $(0,200,10)~{}\mu\textrm{m}$
Location of Rx3 | $r_{3}^{\textrm{Rx}}$ | $(0,300,20)~{}\mu\textrm{m}$
Diffusion coefficient of M1 | $D_{1}$ | $10^{-8}~{}\textrm{m}^{2}/\textrm{s}$
Diffusion coefficient of M2 | $D_{2}$ | $5\times 10^{-8}~{}\textrm{m}^{2}/\textrm{s}$
Reaction coefficients at Rx1, Rx2, and Rx3 |  $c_{1}$, $c_{2}$, $c_{3}$ | $2$
Receiver radius | $r_{\textrm{R}}$ | $15~{}\mu\textrm{m}$
Medium flow velocity | $\bm{\nu}$ | $\bm{0}$

Figure 2: Feasible region of $\Delta\tilde{t}_{12}$ and $\Delta\tilde{t}_{13}$
in the proposed IA scheme with reaction.

Figure 3: The error probability versus $\Delta t_{12}$ for the proposed IA
scheme with reaction.

Figure 4: The total error probability versus $\zeta_{0}$ for the proposed IA
scheme with and without reaction.

The feasible region in the proposed IA scheme with reaction for
$\Delta\tilde{t}_{12}$ and $\Delta\tilde{t}_{13}$ for the special case of the
releasing and sampling times
($t_{i}^{[1]}=t_{i}^{[2]}=t_{i},\tilde{t}_{j}^{[1]}=\tilde{t}_{j}^{[2]}=\tilde{t}_{j}$,
$i,j\in\\{1,2,3\\}$), i.e., $\mathcal{T}_{\textrm{r, s}}$, based on Lemma 2,
is shown in Fig. 4, which is a line segment between the points
$(-0.303,-0.466)$ and $(0.303,0.6414)$, where three points $(-0.0538,0.0005)$,
$(-0.0492,0.0092)$, and $(-0.0434,0.0201)$ are excluded from it. The error
probability at the receivers and the total error probability of the system
versus $\Delta\tilde{t}_{12}$ in the feasible region is shown in Fig.
4.333Note that since the sampling times are not necessarily chosen in the peak
times of the received concentrations, $P_{\textrm{e}}$ does not go below
$0.0012$, while $P_{\textrm{e},1}$ is near $10^{-14}$ for some sampling and
releasing times. However, if we do not use IA, for 3 users with 2 molecules
types, two transmitters should use one molecule type and the error probability
of them becomes near $0.5$ because of the high co-channel interference, and
hence the total error probability of system will be $P_{\textrm{e}}\approx
0.5$, which is not acceptable. Here, we assume
$\zeta_{0}=\frac{\zeta_{1}}{2}=2\times 10^{6}$. It can be seen that the
optimum value of $\Delta\tilde{t}_{12}$ which minimizes the total error
probability is $\Delta\tilde{t}_{12,\textrm{r,s}}=-0.2323$, and using (17), we
have $\Delta\tilde{t}_{13,\textrm{r,s}}=-0.3317$. Now, assuming
$\tilde{t}_{1,\textrm{r,s}}=0$, we obtain the other releasing times
($\tilde{t}_{2,\textrm{r,s}}$ and $\tilde{t}_{3,\textrm{r,s}}$), and then from
(2), we obtain $t_{i,\textrm{r,s}}$s, which are provided in Table VI.444Since
$\Delta\tilde{t}_{12,\textrm{r,s}}$ and $\Delta\tilde{t}_{13,\textrm{r,s}}$
are both negative, by assuming $\tilde{t}_{1,\textrm{r,s}}=0$, we can assure
to have positive times with smallest one equal to zero. For the general case
of the times in the IA scheme with reaction, the optimum times are obtained
numerically from (23) in Table VI. The optimum sampling and releasing times in
the IA scheme without reaction are also provided in Table VI. The optimum
times are obtained numerically from (22) for the special case, and from (21)
for the general case,

TABLE II: Optimum releasing and sampling times
|  $\tilde{t}_{1}^{[1]}$ |  $\tilde{t}_{1}^{[2]}$ |  $\tilde{t}_{2}^{[1]}$ |  $\tilde{t}_{2}^{[2]}$ |  $\tilde{t}_{3}^{[1]}$ |  $\tilde{t}_{3}^{[2]}$ |  $t_{1}^{[1]}$ |  $t_{1}^{[2]}$ |  $t_{2}^{[1]}$ |  $t_{2}^{[2]}$ |  $t_{3}^{[1]}$ |  $t_{3}^{[2]}$
---|---|---|---|---|---|---|---|---|---|---|---|---
IA with reaction (Special case) |  $0$ |  $0$ |  $0.232$ |  $0.232$ |  $0.332$ |  $0.332$ |  $0.410$ |  $0.410$ |  $0.466$ |  $0.466$ |  $1.051$ |  $1.051$
IA with reaction (General case) |  $0.012$ |  $0$ |  $0.335$ |  $0.396$ |  $0.329$ |  $0.323$ |  $0.411$ |  $0.411$ |  $0.471$ |  $0.471$ |  $1.055$ |  $1.055$
IA without reaction (Special case) |  $0.592$ |  $0.592$ |  $0$ |  $0$ |  $0.623$ |  $0.623$ |  $0.674$ |  $0.674$ |  $0.666$ |  $0.666$ |  $1.758$ |  $1.758$
IA without reaction (General case) |  $1.262$ |  $0.004$ |  $0$ |  $1.079$ |  $1.102$ |  $1.555$ |  $1.636$ |  $1.575$ |  $1.392$ |  $2.385$ |  $1.877$ |  $1.814$

The total error probabilities of the system using the IA scheme with and
without reaction for the optimum times in the general and special cases are
shown in Fig. 4 versus the number of released molecules for bit 0 (i.e.,
$\zeta_{0}$). Here, we assume $\zeta_{1}=2\zeta_{0}$. For the IA with
reaction, we obtain the error probability using the analytical result in (V)
and simulation, and for the IA without reaction, we obtain the decision rule
numerically from (28) and obtain the error probability using simulation. It is
seen that using reaction in the IA scheme improves the total error probability
significantly. Further, for the IA scheme with reaction, the simulations and
the analysis provide the same result.

Figure 5: The total error probability versus $\zeta_{0}$ for the proposed IA
scheme with reaction in the presence of ISI. Figure 6: The total error
probability versus $\zeta_{0}$ for the proposed IA scheme with reaction in the
presence of the environment noise.

Fig. 5 investigates the ISI effect in the system. In this figure, the total
error probability of the system using IA with reaction assuming one slot
memory in the channels along with the error probability in the no ISI case are
shown versus $\zeta_{0}$. We obtain the error probabilities for the optimum
decision rule, the sub-optimum decision rule assuming ISI as noise, and the
adaptive decision rule using the previously detected bits introduced in [16].
Here, we assume $T_{\textrm{s}}=10~{}\textrm{s}$. It can be seen that the
error performance degrades in the presence of ISI. However, we have an
acceptable performance using adaptive and optimum decision rules. The effect
of the environment noise on the proposed IA scheme with reaction is shown in
Fig. 6 for $\mu_{n}^{[1]}=14,17,21$ and $\mu_{n}^{[2]}=5$, which shows that
the proposed scheme performs well in the presence of noise. It should be noted
that, due to the perfect reaction between the molecules of type 1 and 2 with
coefficient $c$ at the receivers, the average noise concentration for molecule
type 1 is $\textrm{max}\\{0,\mu_{\textrm{n},1}-c\mu_{\textrm{n},2}\\}$, and
the average noise concentration for molecule type 2 is
$\textrm{max}\\{0,\mu_{\textrm{n},2}-\mu_{\textrm{n},1}/c\\}$. Therefore, the
error probability depends on the value of
$\mu_{\textrm{n},1}-c\mu_{\textrm{n},2}$ and increases as
$\mu_{\textrm{n},1}-c\mu_{\textrm{n},2}$ goes away from zero.

##  VII Conclusion

In this paper, we proposed an interference alignment (IA) scheme for a 3-user
molecular interference channel (IFC) based on the choice of the releasing
times of the molecules at the transmitters and the sampling times at the
receivers using two molecule types. We proposed using reaction in the IA
scheme to cancel the aligned interference signal in the medium and reduce the
signal dependent noise. Further, we applied the asymptotic IA scheme in
classic communications to a $K$-user molecular IFC. For the proposed IA
scheme, we obtained the feasible region for the releasing/sampling times and
simplified it for a special case. We investigated the error performance of the
IFC using the proposed IA scheme with and without reaction and showed that the
proposed IA scheme using reaction improves the performance significantly.

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##  Appendix A Proof of Lemma 2

For
$t_{i}^{[1]}=t_{i}^{[2]}=t_{i},\tilde{t}_{j}^{[1]}=\tilde{t}_{j}^{[2]}=\tilde{t}_{j}$,
$i,j\in\\{1,2,3\\}$, $\bm{\nu}=0$, and $c_{1}=c_{2}=c_{3}=c$, the reaction
conditions in (III) simplify to

$\displaystyle\textrm{Rx}_{1}:$
$\displaystyle\frac{r_{12}^{2}}{t_{1}-\tilde{t}_{2}}+\frac{r_{31}^{2}}{t_{3}-\tilde{t}_{1}}-\frac{r_{32}^{2}}{t_{3}-\tilde{t}_{2}}=\frac{1}{s},$
(30) $\displaystyle\textrm{Rx}_{2}:$
$\displaystyle\frac{r_{21}^{2}}{t_{2}-\tilde{t}_{1}}=\frac{1}{s},\quad\textrm{Rx}_{3}:\frac{r_{31}^{2}}{t_{3}-\tilde{t}_{1}}=\frac{1}{s},$

where
$s=-\frac{\frac{1}{D_{1}}-\frac{1}{D_{2}}}{4\ln{c}+6\ln(\frac{D_{1}}{D_{2}})}$.
Combining the IA condition in (1) and the reaction conditions in (30) results
in

$\displaystyle
r_{12}^{2}.(t_{3}-\tilde{t}_{2})=r_{32}^{2}.(t_{1}-\tilde{t}_{2}),$ (31a)
$\displaystyle
r_{13}^{2}.(t_{2}-\tilde{t}_{3})=r_{23}^{2}.(t_{1}-\tilde{t}_{3}),$ (31b)
$\displaystyle t_{2}-\tilde{t}_{1}=s.r_{21}^{2},\qquad
t_{3}-\tilde{t}_{1}=s.r_{31}^{2}.$ (31c)

From (31b)-(31c), $t_{1}$, $t_{2}$, and $t_{3}$ can be obtained from
$\tilde{t}_{1}$ and $\tilde{t}_{3}$ as in (2) and if we substitute them in
(31a), we obtain (17). Combining the independency conditions in (13) and the
reaction conditions in (30) results in

$\displaystyle\textrm{Rx}_{1}:t_{1}-\tilde{t}_{1}\neq
s.r_{11}^{2},\textrm{Rx}_{2}:r_{32}^{2}.(t_{2}-\tilde{t}_{2})\neq
r_{22}^{2}.(t_{3}-\tilde{t}_{2}),$
$\displaystyle\textrm{Rx}_{3}:r_{33}^{2}.(t_{2}-\tilde{t}_{3})\neq
r_{23}^{2}.(t_{3}-\tilde{t}_{3}).$ (32)

Now, substituting $t_{i}$s from (2), we obtain (2).
Further, we should have $\Delta t_{ij}=t_{i}-\tilde{t}_{j}>0$,
$i,j\in\\{1,2,3\\}$. From (2), the values of $\Delta t_{ij}$ are related to
$\Delta\tilde{t}_{12}$ and $\Delta\tilde{t}_{13}$ as follows:

$\displaystyle\Delta
t_{11}=t_{1}-\tilde{t}_{1}=(\frac{r_{13}^{2}}{r_{23}^{2}}-1)\Delta\tilde{t}_{13}+s.\frac{r_{13}^{2}}{r_{23}^{2}}.r_{21}^{2},$
$\displaystyle\Delta
t_{12}=t_{1}-\tilde{t}_{2}=\Delta\tilde{t}_{12}+(\frac{r_{13}^{2}}{r_{23}^{2}}-1)\Delta\tilde{t}_{13}+s.\frac{r_{13}^{2}}{r_{23}^{2}}.r_{21}^{2},$
$\displaystyle\Delta
t_{13}=t_{1}-\tilde{t}_{3}=\frac{r_{13}^{2}}{r_{23}^{2}}\Delta\tilde{t}_{13}+s.\frac{r_{13}^{2}}{r_{23}^{2}}.r_{21}^{2},$
$\displaystyle\Delta t_{21}=t_{2}-\tilde{t}_{1}=s.r_{21}^{2},\Delta
t_{22}=t_{2}-\tilde{t}_{2}=\Delta\tilde{t}_{12}+s.r_{21}^{2},$
$\displaystyle\Delta
t_{23}=t_{2}-\tilde{t}_{3}=\Delta\tilde{t}_{13}+s.r_{21}^{2},$
$\displaystyle\Delta t_{31}=t_{3}-\tilde{t}_{1}=s.r_{31}^{2},\Delta
t_{32}=t_{3}-\tilde{t}_{2}=\Delta\tilde{t}_{12}+s.r_{31}^{2},$
$\displaystyle\Delta
t_{33}=t_{3}-\tilde{t}_{3}=\Delta\tilde{t}_{12}+s.r_{31}^{2}.$ (33)

Hence, $\Delta t_{ij}>0$ and (A) result in (2).