I Introduction
Unlike the classical wireless communication, diffusion based Molecular Communication (MC) utilizes molecules as the carriers of information between the communicating parties. Type, concentration, or the release time of molecules can be used for signaling by a molecular transmitter. As a result, a mechanism must be set in place for production of molecules at the transmitters [1]. For instance, this may be realized by chemical reactions inside the transmitter nodes. The impact of this molecule production process on the capacity of a molecular channel is the subject of this paper.
We assume that the transmitter includes a production unit as well as a storage unit. The production unit adds some amount of new molecules in each timeslot to the storage unit. The production rate of molecules may depend on the amount of molecules already exist in the storage unit, e.g., the chemical process responsible for molecule production might have a faster production rate if the storage unit is empty. The transmitter communicates its message by controlled opening of an outlet of the storage unit for a short period of time and thereby releasing a concentration of molecules into the environment at the beginning of each timeslot. This model is comparable with an Energy Harvesting (EH) system in the classical communications, in which the transmitter harvests energy and wishes to send its message such that its transmitted signal is amplitudeconstrained to the amount of harvested and stored energy at the transmitter [2, 3, 4, 5, 6, 7]. The transmitter (in the EH system) may have finite [6] or infinite [8] energy storage (battery) or it can be assumed with no battery [9].
There are several approaches and models of transmitters and receivers for diffusion based MC in the literature. We follow the common approach of choosing one of the models and adapting our analysis to it. In particular, we adopt the macroscale mode of MC and the molecular Poisson model for our study (see [10, 1]
and references therein for a review of different models and the related results). More specifically, we assume that the amount of released molecules is a deterministic concentration (in molar) and the Fick’s law of diffusion describes the medium. The reception noise is modeled by a Poisson random variable,
i.e.,the received signal has a Poisson distribution whose mean is proportional to the average concentration of molecules at the receiver.
In this paper, we consider two cases of the Poisson channel, with or without InterSymbol Interference (ISI). The ISI occurs when a nonnegligible portion of transmitted molecules from the previous timeslots remain in the medium and affect the communication in the current timeslot. We begin by providing a number of capacity results when the channel is without ISI. One should note that though the channel is memoryless in this case, the problem is still complicated due to the fact that the transmitter has memory; the number of released molecules in each transmission is limited by the level of the storage, which itself depends on the previous transmissions. A similar phenomenon occurs in the classical energy harvesting systems. Next, we consider a channel with ISI, i.e., a channel with memory, and provide a result on its capacity depending on whether the channel statistics are known at the transmitter or at the receiver.
Ia Related works
In practice, a MC transmitter suffers from constraints on its molecule production and storage processes [11, 12, 13]. The molecule production may be constrained by limitations on the chemical reactions or the availability of food and energy for molecule generation at the transmitter [13]. Moreover, in practical scenarios, the bionanomachines store the molecules that they produce internally or capture externally from the environment [13]. Thus, the limitation on molecule storage forces some constraints on the molecule transmitting process [14].
Most of the existing works in the literature of diffusion based MC networks assume availability of a constant number of molecules at the transmission times [15]. For instance in the on/off keying, the transmitter releases a fixed amount of molecules to send bit ‘1’ and stays silent to send bit ‘0’ [16, 17, 18, 15, 19]. However, there are a few works that consider the limitations on molecule production at the transmitter [11, 12, 13]: the random chemical reactions in the molecule production process is considered in [11]. This work studies the chemical description of the transmitters in terms of Langevin equation. In [12] the total molecule concentration at the transmitter is assumed to be a function of the number of random chemical reactions between different molecule types. More precisely, the inherent randomness in the availability of food and energy is assumed to affect the molecule generation process which limits the availability of molecules at the transmitter in the beginning of each timeslot [13]. Due to lack of sufficient molecules, the lengths of the symbol intervals may vary in practical MC systems and the constrained transmitter may not be able to emit molecules with a fixed release frequency [20]; the authors in [20] suggest using two types of molecules for communicating and symbol synchronization over the channel. In [21], it is assumed that the amount of available molecules in the present timeslot, which is referred to as the “state” of the transmitter, depends on the state and the inputs in the previous timeslots.
In another line of works, the transmitter is assumed to actively capture the needed molecules from the environment. This process is referred to as “molecule harvesting” [14]. Here the transmitter has the ability to harvest the arrived molecules, in addition to the background molecules that randomly hit the transmitter and the ones captured through chemical synthesis from other materials. In this model, the harvesting process is constrained by the size of the molecule storage at the transmitter. The state of the transmitter (i.e., the number of the molecules at the transmitter’s storage) is updated according to a weighted combination of the released, the received and the harvested molecules in the past timeslot [14].
In this paper, we study the transmitter molecule production constraint. Our work is novel since previous works do not consider capacity degradation due to the molecule production constraint at the transmitter. For our study, we adopt a molecular Poisson channel model, an important molecular channel adopted in many works in the literature [22, 19, 23, 24, 25, 26, 27],
The rest of the paper is organized as follows. In Section II, the MC channel and some preliminary definitions are presented. The main results of the paper on the MC networks with no ISI are presented in III, including the capacity in the case of infinite molecule storage, and inner and outer bounds on the capacity in the case of finite molecule storage. The main results of the paper on the MC networks with ISI are presented in IV. The paper is concluded in Section V.
Ii System Model
Iia Notation and Definitions
We use the notation to denote the sequence . Random variables (r.v.s) are denoted by uppercase letters, while their realizations are denoted by the lowercase letters. We say that random variables
form a Markov chain if
for all . We show this Markov chain relation by . A sequence of random variables for is said to be Asymptotically Mean Stationary (AMS) if(1) 
exists for all measurable . Under this condition,
is also a probability measure and is named the
stationary mean of the AMS sequence [33].IiB Channel Model
In our setting, we have a pointtopoint communication channel in which the information is conveyed by the molecule concentration released into the environment by the transmitter. A deterministic channel between the transmitter and the receiver, based on the Fick’s law of diffusion, is assumed. A reception noise is assumed at the receiver; the concentration of molecules is detected by a Poisson reception process at the receiver [28]. A common example of the Poisson reception process is the particle counting noise of a transparent receiver [29, 30]. A transparent receiver consists of a transparent sphere of a certain volume. It counts the number of molecules that fall into its sphere. A transparent receiver is passive in the sense that it does not affect the diffusion medium by imposing a boundary condition to the differential equation describing the diffusion process. The precise mathematical description of a Poisson reception process is as follows: if the concentration of molecules around the receiver is moles, the receiver’s measurement is distributed according to for some constant
. Since the variance of a Poisson distribution is proportional to its mean, the larger the
, the noisier the receiver’s reception will be.We assume a timeslotted transmission. The transmitter instantaneously releases moles of molecules into the environment at the beginning of each timeslot for . In other words, if the transmitter is located at and has a clock with frequency , the transmitter’s channel input is the impulse train
The Fick’s law of diffusion describes propagation of molecules in the environment. Here, we assume that the communication is invariant over time, and the boundary conditions are set to zero, i.e., there is no molecule production source besides the transmitter. As a result, the diffusion medium (described by the Fick’s law of diffusion) becomes a linear timeinvariant (LTI) system and can be characterized by its impulse response. Thus, the concentration of molecules at the receiver at the end of the th timeslot can be expressed as the convolution , where represents the channel impulse response coefficient. The receiver’s measurement follows a Poisson distribution as . Letting , we can write for . We refer to as the channel coefficients. We assume that the reception noise at the receiver in different time slots are mutually independent, i.e., ’s are conditionally independent given the transmission amounts . If the reception noise is the particle counting noise, to satisfy independence of the reception noises one should adopt the common assumption in the literature that the sampling period is not very small [31, Section IV], [10, Remark 1].
We say that the channel is without ISI if is negligible for . In this case, , and
(2) 
The channel is said to be with ISI if the output of the channel in th timeslot depends on the past inputs of the channel in the previous timeslots with weights (see Fig. 1).
IiC Transmitter Model
The transmitter opens the outlet of its molecule storage unit for a short period of time at the beginning of each timeslot. The channel input , at timeslot , represents the deterministic released molecule concentration (in molar) into the environment. We assume that the transmitter has moles of molecules in the beginning of th timeslot (in its storage unit). Moreover, the amount of molecules in the storage unit after recharging for the duration of the th timeslot is denoted by , where is a known function. We expect that , i.e., molecule production is nonnegative in the th time slot. Also, we expect to be a nondecreasing function, i.e., for . The intuition here is that if the transmitter starts off with mole of molecules, it will have more molecules after recharging than if it starts off with moles of molecules. In some practical scenarios, the production rate of new molecules () decreases in , the amount of molecules already existing in the storage unit. We do not need to make such a restrictive assumption on the function for our results to hold.
If is the amount of molecules in the transmitter at th timeslot, we assume that moles of molecules are released into the environment. Thus, we will have
(3) 
In other words, reduces to because of molecule release and then recharges to .
We assume that the transmitter starts off empty, i.e., . If we have no transmission () and just recharging, we will have for ; and thus, where
Since we assumed that for all , the equation implies that . Therefore, is a nondecreasing sequence of nonnegative reals. Let
(4) 
be the amount of molecules after recharging for infinite time (when there is no transmission); we set if the sequence tends to infinity (as discussed later). From (4), it follows that given any , it is possible to save molecules for a finite number of time instances and reach the molecular level or larger in the transmitter’s reservoir (when there is no transmission). That is .
We say that the transmitter has finite molecule capacity and saturates if , and has unbounded molecule capacity if . If , from (4) we have , i.e., is a fixed point of . If , from monotonicity of and (4) we have
which we can interpret as for by symbolically extending the domain of to include .
With the assumption that the transmitter starts off empty, we only need to define the function over the interval . The reason is that for any transmission sequence satisfying , we will have . This follows from induction
(5) 
where we use the monotonicity property of . Hence, we assume that the function is welldefined over .
IiD Channel Capacity
The transmitter wishes to send the message
which is uniformly distributed over the set
to the receiver in channel uses. The communication rate is bits per channel use. The rate is achievable if for any there exists a code (respecting the transmitter’s input constraints) with rate whose average error probability is less than . The channel capacity is the maximum achievable communication rate.Capacity calculation is complicated due to the memory introduced into the problem by the constraints on the transmitter. The transmitter should adapt its molecule release to the amount of produced molecules in the current timeslot and the consumed molecules in the previous timeslots. Thus, the transmitter has memory.
Iii MC Network Without ISI
In this section, we present our main results on the capacity of the MC network without ISI, i.e., for all .
For a given memoryless channel (here a Poisson channel) on input alphabet and a real , let us define:
(6) 
(7) 
The difference between and is that in we take supremum over all distributions on with an expected value be at most , while in we take supremum over all distribution whose support is .
Theorem 1
Let
(8) 
Then the channel capacity satisfies:
(9) 
Proof:
To see the upper bound, note that the number of molecules produced in each recharging period cannot exceed . Therefore, the average number of consumed molecules cannot be larger than :
(10) 
Observe that (6) is the channel capacity under the average input cost constraint given in (10) [32]. As a result, the upper bound then follows from the known result on the capacity of memoryless channels with an average input cost constraint [32]. Because we assume no ISI in this section, our channel is a memoryless Poisson channel and the result of [32] can be utilized.
It remains to prove the lower bound on the capacity: . Take an arbitrary and let
(11) 
We prove that is an achievable rate. This will complete the proof because has been taken arbitrary in , and furthermore
as . To show that is achievable, it suffices to show that for any arbitrary distribution on , the mutual information is an achievable rate.
Let . Then from (11) we have . We claim that . This is because where we used the monotonicity property of . Since , we can use the save strategy at the beginning and wait for a finite number of time instances to reach the molecule level or larger in the transmitter’s reservoir. If we wait for one more time step, we reach the molecule level . Hence, there is some finite number such that assuming that we do not transmit any molecules in the first time steps.
We would like to start transmitting in time instances . We claim that if we limit the transmission level to , then transmitter’s molecular reservoir will not drop below . In other words, if for we have for all . This follows from induction. Assuming that and using the fact that , we have
(12) 
The above argument shows that it is possible to transmit any sequence satisfying for all , as the molecular reservoir is always at least and hence never hits zero. By letting the blocklength tend to infinity, the initial finite time instances becomes negligible compared to transmission length and results in no rate loss. We obtain the achievability of via Shannon’s point to point achievability scheme, i.e., by constructing i.i.d. codewords from a defined on .
Theorem 2
Assume that . Let
(13) 
Then the channel capacity can be bounded from below as follows:
(14) 
Proof:
We prove that is an achievable rate for any . By the definition of , there exists some such that for any we have
(15) 
To prove that is achievable, we need to prove achievability of for an arbitrary distribution satisfying . The idea is to mimic the proof of [5] here and prove this by considering distributions on a finite support. Take some positive real number , and an arbitrary distribution of finite support on . Here may be larger than , but the distribution of is required to satisfy . It is shown in [5] that achievability of for such finite support distributions is enough to show the achievability of .
As in [5], we wait at the beginning for a finite amount of time and save molecules so that the amount of saved molecules exceeds a number (to be specified later). Because , we can take as large as we want.
As long as , we have and hence from (15) we have
(16) 
In other words, a production of moles of molecules is guaranteed as long as the molecular reservoir level stays above .
In [5], the update rule is considered. Here in the inequality (16), the sequence has a larger growth and dominates the process of [5]. This indicates that the same proof can be mimicked here since (16) shows that in each step, more molecules are produced than the update rule of [5]. The number should be chosen to ensure that with high probability, starting from molecules, remains above so that (16) remains valid. But this is possible with the same argument as in [5].
Remark 1
Theorems 1 and 2 give a tight characterization of capacity for the case of infinite battery capacity if . For example, this occurs if for some constant , i.e., when we have a constant production rate. Theorem 2 is comparable with an EH system with an infinite battery [5]. In [5], it is shown that the capacity of Gaussian EH channel with an infinite battery, with the average arrived energy , is equal to the capacity of the Gaussian pointtopoint channel with average power constraint .
In the following, we find a lower bound on the capacity when .
Theorem 3
Assume that . Let be a conditional distribution on
such that the Markov chain , with asymptotically converges to a stationary distribution , i.e., tends to in total variation distance as tends to infinity. Then the capacity is lower bounded as
(17) 
where where is the Poisson channel .
Proof:
Assume that we start with and choose from according to . Consider the update rule This yields a sequence for . Let be a memoryless channel. Then we have
(18) 
We claim that is an Asymptotically Mean Stationary (AMS) sequence. As argued in [33], the AMS condition allows us to conclude that
is a lower bound on the capacity of the channel. Next, we show that is an Asymptotically Mean Stationary (AMS) sequence. Observe that is a Markov chain (on a countable state space) starting from . Furthermore, from the definition of the in the statement of the theorem, is a limiting stationary distribution for this Markov chain and converges to in total variation distance. Because , the distribution of also converges to and will be proven to be an Asymptotically Mean Stationary (AMS) sequence.
To sum this up, we have that
is a lower bound on the capacity of the channel. Next, note that
where we used the fact that and form Markov chains, which are implied from (18). Because tends to as tends to infinity, we have
Iv MC Network With ISI
In this section, we present our main results on the capacity of the MC network with ISI. As we mentioned in Section IIB, in this channel and represent the channel input and the channel output, respectively (see Fig. 1). Let denote the capacity of a channel with coefficient sequence . Calculation of is complicated by the fact that the channel has memory.
Assume that the channel coefficient sequence belong to a class . Here, we consider two scenarios: the first scenario assumes that the transmitter is aware of the actual , but the receiver only knows the class and is unaware of the sequence . A rate is achievable if there is a sequence of coding strategies for the transmitter and receiver whose error probability vanishes regardless of the choice of . The second scenario is the other way around; it assumes that the receiver is aware of the actual , and the transmitter only knows the class . Thus, the two scenarios differ in terms of whether the “channel state information” is available at the transmitter or receiver.
To state our results, we begin by a definition:
Definition 1
Given two sequences and of channel coefficients, we say if
(19) 
for some nonnegative sequence satisfying .
Intuitively speaking, can be understood as a channel with a coefficient sequence having a more spread out channel coefficient profile than the channel with a coefficient sequence .
Example 1
As an example, given any we have where is defined as follows: for and is any number greater than or equal to .
Our main results are as follows:
Theorem 4
Assume that the molecule production function is a concave function (in addition to the constraints of Section IIC). Take some channel coefficient sequence such that for all . Then, is an achievable rate for any channel coefficient profile when is known only at the transmitter.
Theorem 5
Take some channel coefficient sequence such that for all . Then, for any molecule production function , is an achievable rate for any channel coefficient profile when is known only at the receiver.
Remark 2
Remark 3
From Example 1, we know that given any channel coefficient sequence , the sequence satisfies . Since corresponds to a memoryless channel, the above theorems confirms that ISI cannot be utilized to increase the channel capacity.
Proof:
In this case, the receiver is unaware of the . The receiver proceeds with the assumption that the true channel coefficient is . The idea is to show that any code for a channel with coefficient can be “simulated” by a code for channel at the transmitter. More specifically, choose an arbitrary code of rate designed for a channel with coefficients consisting of a set of codewords for . The codeword satisfies the transmitter molecule production constraint (i.e., transmission in each stage is less than the amount of molecules available in the transmitter reservoir). The receiver assumes that the actual channel coefficient is and uses the decoding algorithm for this channel code. The transmitter gets the true channel coefficient and uses the following strategy:

From , it computes a nonnegative sequence such that

To transmit message , it transmits
(20) on the channel, where is the codeword of the codebook for corresponding to message .
With this strategy, the receiver gets . Observe that
Therefore, , as if the codeword was transmitted over a channel with the coefficient sequence . Hence, (20) allows for the simulation of channel from channel .
The crucial and more difficult step is to show that the sequence of transmission defined in (20) satisfies the transmitter cost constraint. To prove this, we use the assumption that each codeword satisfies the transmitter cost constraint. That is,
where and is the number of molecules in transmitter’s reservoir at time if message is transmitted. Our claim follows from Lemma 1 in the Appendix.
Proof:
In this case, the transmitter is unaware of the . The transmitter proceeds with the assumption that the true channel coefficient sequence is . The idea is to convert the channel to the channel by an operation at the receiver. Choosing an arbitrary code of rate designed for a channel with a coefficient sequence consisting of a set of codewords for , the transmitter sends the codeword , assuming that the actual channel coefficient sequence is . The receiver gets the true channel coefficient sequence and uses the following strategy: from , the receiver computes a nonnegative sequence such that
Next, having received the sequence , the receiver produces the sequence as follows: for each , we produce random variables from a multinomial distribution with balls and probability sequence . We assume that the variables for different values of are produced independently. Then, we let
Note that are independent given . By the thinning property of Poisson distribution, and the fact that sum of independent Poisson random variables is again a Poisson random variable, we obtain that and furthermore, for different values of are mutually independent. We have
Furthermore, for different values of are mutually independent because they involve summation over disjoint sets of . As a result, is exactly what we obtain if were passed through the channel . This completes the proof of the receiver being able to apply a postprocessing to convert the channel to .
V Conclusions
In this paper, a molecular communication (MC) system is considered in which the information is encoded in the concentration of the molecules, and the molecule generation process causes some constraints on the transmitter. Moreover, it is assumed that the number of received molecules at the receiver follows a Poisson distribution of the channel input.
Two scenarios on the MC channel with no Intersymbol Interference (ISI) and the MC channel with ISI were studied. For the case of noISI scenario, lower and upper bounds on the channel capacity were derived. For the case of channels with ISI, we provided a capacity result for the cases where the channel coefficient sequence is known either at the transmitter or at the receiver.
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