1

ADMA: Amplitude-Division Multiple Access forBacterial Communication Networks

Bhuvana Krishnaswamy a, Yubing Jian a, Caitlin M. Austin b Jorge E. Perdomo b, Sagar C. Patel b

Brian K. Hammer c, Craig R. Forest b and Raghupathy Sivakumar a

a School of Electrical and Computer Engineering, b George W. Woodruff School of Mechanical Engineeringc School of Biology, Georgia Institute of Technology, Atlanta, GA, USA

Abstract—In a communication network with extremely highprocessing delays, an efficient addressing and multiple-accesscontrol mechanism to improve the throughput performance ofthe system is a necessity. This work focuses on source addressingin multiple-source, single-receiver bacterial communication net-works. We propose Amplitude-Division Multiple Access (ADMA),a method that assigns the amplitude of the transmitted signalas the address of the source. We demonstrate using geneticallyengineered Escherichia coli bacteria in a microfluidic device thatusing amplitude for addressing is feasible. We analyze the perfor-mance of the network with several addressing mechanisms andpropose an amplitude sequence and a low-complexity receiverdesign that minimizes error in resolving the source addresses inthe presence of collisions. Finally, we demonstrate that ADMAimplicitly solves the problem of multiple-access control.

I. INTRODUCTION

Nano-scale communication can be categorized into twobroad domains: electromagnetic communication (EM) at thenanoscale, which involves the extension of EM-based com-munication techniques for use in nonbiological applications[1], [2]; and molecular communication (MC), which involvesstrategies for use in biological applications [3]–[5]. Advance-ments in synthetic biology and nanotechnology now makeMC possible and practical for applications like toxicology,environmental monitoring [6], and drug delivery [7].

The ability to genetically engineer bacteria to introduce ordelete DNA for specific traits (e.g., bioluminescence, motility,adhesion), and their natural properties for communication [8],[9] have made bacteria emerge as candidates for nanoma-chines [10]. Bacterial nano-machines hold promise for use inbiological applications, including toxicology, biofueling, andbiosensing. For example, receiver bacteria are used as biosen-sors to detect the presence of metals [10], arsenic pollution[11], environmental monitoring [6], and drug delivery [7].

The context for this work is molecular communication be-tween bacterial populations. Specifically, we consider bacterialpopulations as transceivers connected through microfluidicpathways that act as conduits for molecular signals. Priorresearch in MC have focused on channel and system mod-eling [12], [13], capacity derivation [14]–[16], modulationtechniques [17]–[19] and analysis of channel and inter symbolinterference [16], [20], [21]. These studies focus on a singlelink and do not consider the challenges in implementing thealgorithms in a real-life environment. Our work focuses on astar topology with multiple sources and a single receiver, asshown in Figure 1. This topology is most commonly seen in

sensor networks, where multiple sensors communicate with asingle receiver/sink. The sensors broadcast information anddo not require a destination address. The receiver, on theother hand, receives a cumulative signal from multiple sources,making it necessary to have an efficient addressing mechanismthat uniquely identifies the sources.

In this paper, we make the following five contributions:

1) We propose an addressing mechanism, Amplitude-Division Multiple Access/Addressing (ADMA), for atopology with multiple sources and a single receiver.ADMA embeds the address of each source in the ampli-tude of its corresponding transmitted signal by assigning aunique amplitude for each source. We prove that a care-ful assignment of the source amplitudes can implicitlysolve the multiple-access control problem, and allow thereceiver to uniquely identify the sources.

2) We demonstrate the feasibility of ADMA using Es-cherichia coli (E. coli) bacteria genetically engineeredto exhibit fluorescence on the receipt of a specific signalmolecule (N-(3-Oxyhexanoyl)-L-homoserine lactone, orC6-HSL). We show that the response of a receiver bacte-rial population is significantly different for distinct inputamplitudes. We also demodulate the receiver response ob-tained from the experiments and evaluate the performanceof ADMA in a practical system.

3) We propose an optimum, low-complexity receiver designthat maximizes the network throughput in a multipleaccess channel. We then derive a theoretical upper boundfor the throughput performance of amplitude source ad-dressing under the proposed receiver designs.

4) We propose a heuristic algorithm that selects an addresssequence for minimizing the collision resolution error fora given number of sources in the network.

Receptor

Sampler

Decoder

Source1

Source2

Source3

Source4

Source5

Receiver

Fig. 1: Network Setup

2

5) We implement the proposed receiver design and am-plitude assignment algorithm in nanoNS3, a bacterialcommunication simulator built on top of NS3. nanoNS3simulates the response of receiver bacteria as modeledin [22] and introduces channel and receiver errors torepresent a practical system.

The remainder of the paper is organized as follows: Section IIpresents the advantages and disadvantages of different sourceaddressing mechanisms and provides experimental validationof the amplitude source addressing. Section III details ourproposed optimal source addressing mechanism. Section IVdescribes two receiver designs, and provides an upper boundon the maximum number of successful bits. Section V de-scribes our proposed address assignment algorithm that mini-mizes interference in a dense network. Section VI presents thesimulation results of the performance of amplitude sequencesproposed. Section VII concludes the paper by discussing someof the challenges and future work.

II. THE ADDRESSING PROBLEM

A. Application Scenario

We discuss the communication modules required for a prac-tical bacterial communication network with the help of a spe-cific application: pathogen-detection using bacterial sensors.Transmitter bacteria transmit information sensed by sensorbacteria, that are genetically engineered to detect specificpathogens to a receiver (sink). The transmitter needs a modu-lation scheme to communicate the information to the receiverthrough a channel or medium. Multiple sensors are deployed todetect different pathogens, and communicate with the receivervia point-to-point channels. Therefore, an addressing schemethat will uniquely identify each source and a multiple accesscontrol mechanism that will allow the sources to share thesingle receiver are also required.

Modulation techniques specifically targeting MC have beendeveloped by a number of researchers [16]–[18]. Concentra-tion shift keying [17] and molecule shift keying [16], [18]are two well-known methods that encode information in theconcentration (amplitude) of the signal and the molecule typerespectively. A majority of the existing research on MC hasfocused on channel and system modeling, and algorithms fora single link [14], [15], [19]. With the growth of networksizes, there is a need for algorithms that perform addressing,medium access control (MAC), routing and reliability. In[23], a distance-based addressing that estimates the distancebetween transmitter and receiver using beacons, propagationdelay, and path loss is proposed. [23] establishes a coordinatesystem from the distances measured and the molecules moveto the desired location. This addressing mechanism assumesthat the transmitter can identify its location in the channeland guide molecules to a particular direction. Developingsuch a transmitter using biological circuits is highly chal-lenging. Extending this scheme to more than one link willbe challenging as it requires accurate channel estimation.[21] proposes a MAC using molecule type, wherein eachtransmitter communicates with a distinct molecule, and thus donot interfere with other transmitters, analogous to frequency

division multiplexing. We explore this approach in detail inthe following section.

In this paper, we focus on source addressing, an addressingmechanism that can distinguish multiple sources communicat-ing with a single receiver in a star topology as shown in Figure1. We consider a system where sources transmit chemicalmolecules that propagate through a microfluidic channel andtrigger the receiver to generate Green Fluorescent Protein(GFP), which is then sampled, demodulated and decoded.Each source uses On-Off-Keying (OOK), a simple modulationtechnique that transmits a rectangular pulse m(t), as shown inEquation (1), with concentration A for T seconds (bit period)to transmit bit 1 and an absence of signal for T seconds totransmit bit 0.

m(t) =

{A 0 ≤ t ≤ T0 otherwise

(1)

B. Types of Addressing

Address Fields: In traditional communication systems, itis common to allocate a fixed number of bits in the packetheader for addressing, e.g. the IP address and MAC addressfields. In bacterial communication networks, on the other hand,very high processing delays at the transceiver nodes result inextremely low data rates (order of 10−5 bits per second [24]).Overheads in the form of address fields can result in additionalper-frame delays as well as a decreased per-user throughput.

With the use of modulation techniques such as concentrationshift keying [17], the number of symbols (signals) requiredto transmit an address can be reduced. However, a significantreduction in the network throughput cannot be avoided withoutfully eliminating the use of address fields. We propose Em-bedded Addressing, an addressing mechanism that eliminatesthe need for address fields by embedding the source addressin the transmitted signals. It uses the unique characteristics ofmolecular signals to identify the senders (sources).

This is a local addressing mechanism, i.e., the addressembedded is local to the network and not the sender’s uniqueglobal address. We assume that each source has already beenassigned a unique global address1. The receiver uses onlythe received signal and its knowledge of the characteristicsof the source signals to identify them locally. It is worthmentioning here that code division multiple access (CDMA) isalso a technique that uses embedded addressing by assigninga unique pseudo-random spreading code to each source. How-ever, it does not provide a solution for slow networks (that arecommon in MC) because of its use of spreading codes whichfurther decreases the data rate of the network.

In MC, we identify the following key characteristics of amolecular signal that allow us to embed the source addressesin the transmitted signals without compromising the networkthroughput: molecule type, signal duration, and signal ampli-tude. We next elaborate on each of these properties.

Molecule Type: The address is embedded in the signalingmolecule type with the same amplitude and duration for allsources. Each source is assigned a unique molecule, and

1Assignment of global addresses is out of the scope of this work.

3TABLE I: Source Addressing Mechanisms

Addressing Mechanism Strength WeaknessAddress Fields Global Address Increased Delays, Reduced ThroughputMolecule Type Increased Throughput, Fair Not Scalable, Complex Receiver CircuitsPulse Duration Simple Transceiver Circuits Reduced Throughput, Introduces Unfairness

Amplitude Simple Transceiver Circuits, Scalable, Fair Network Size Limited by Receiver

the receiver must be capable of receiving all the molecules,allowing all the sources equal access to the receiver withoutcontention. Thus, source addressing with molecule type alsosolves the MAC problem.

The other two characteristics, pulse amplitude and duration,allow all the sources to transmit the same molecule. Thereceiver accepts only one type of molecule, simplifying thereceiver and transmitter designs.

Pulse Duration: The amplitude of the signal and themolecule type are fixed across sources, with the addressembedded in the signal duration. Each source is assigned aunique pulse duration. When a source has bit 1 to transmit,it transmits a signal with a given amplitude and the durationassigned to it as the address. The sources are assigned distinctdurations, which leads to increased latency. The per-framedelay of each source is different from the others, leading tounfair throughput and increased network delays.

Signal Amplitude: In electromagnetic communication, am-plitudes have been used for modulating the signal. Here, weconsider using amplitudes as the address. Each source isassigned a unique amplitude, and transmits its signal withthe assigned amplitude for a duration fixed for the network.The receiver maps the received amplitude to the respectivesource. When multiple sources transmit at the same time,the receiver receives the sum of amplitudes. To identify eachsource, the receiver must determine the individual amplitudesfor all received sums.

Table I summarizes the strengths and weaknesses of theabove mechanisms, and we conclude that embedding theaddress in the amplitude of the signal is the most efficientamong the four source addressing mechanisms.

C. Experimental Validation of Amplitude Differentiation

Due to the challenges in the design of state-of-the-artmicrofluidic system [25], the proof-of-concept presented inthis paper extrapolates results from a system with one sourceand one receiver. We extrapolate experimental validation in thefollowing steps. Using the experimental setup, we verify thata receiver can distinguish between amplitudes from the GFPresponse of the receiver bacteria. [22] proved experimentallythat the response of the receiver bacteria is distinct for distinctinput amplitudes. We use the model developed in [22] that wasvalidated with experimental results to model the response ofreceiver bacteria to a molecular signal.

In this work, we implement the numerical inverse of theabove model as a demodulator. In the inverse model, thereceiver response is the input and an estimate of the molecularsignal that triggered this response is the output. We usea model that was experimentally validated to generate theresponse of receiver and an inverse module that demodulatesthe signal from the response of the receiver. We assume Nparallel point-to-point channels from N sources to 1 receiver

as shown in Figure 1. Therefore, the molecular signal reachingthe receiver is the sum of amplitudes of the signals transmittedby each source. The cumulative signal from the individualsources becomes the input to the receiver model. The responseof the receiver to the cumulative signal simulates a multiple-source-single-receiver topology. The modulation and channelerror of each channel is handled individually. In this work,we consider a system in which genetically-engineered bacte-rial populations are receivers connected through microfluidicpathways. Microfluidic pathways allow for dynamic changesin media composition. The constant stream of media keepsthe bacteria in ideal growth conditions, eliminating growthphase dependent variables from the experiments. The bacteriadensity does not change during the experiment. The bacteriaare first seeded into the trapping chambers and grow until thechamber is filled. The chamber is in direct fluidic contact withthe main channel, which has constant flow providing nutrientsto the bacteria and removing excess bacteria. This means thedensity of the chamber contents remains constant.

The bacteria used in all experiments were a genetically-engineered strain of DH5α E. coli. Methods and functionalityof the bacteria and the microfluidic device fabrication andspecifications can be seen in previous works [22], [24], butwill be briefly mentioned here. In this system, the chemicalstimulus is autoinducer N-acyl homoserine lactone (AHL), andthe bacterial response is the expression of GFP. To fabricate themicrofluidic devices, we utilize standard soft lithography [26]resulting in a polydimethylsiloxane (PDMS) device bondedto a glass coverslip. The design consists of the main channelwhich has direct fluidic contact with adjacent chambers thathouse the bacteria for the duration of the experiment, asseen in Figure 2. Figure 2 shows the receiver bacteria. The

Fig. 2: The experimental setup consists of microfluidic chan-nels in direct fluidic contact with trapping chambers housingbacteria. The main channel provides nutrients and AHL to thebacteria.

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“Main Channel” is the channel through which molecules carryinformation. After initial loading of the chip, the bacteria wereallowed to populate the chambers for 24 hours to reach capac-ity. During this time the bacteria were supplied with a constantflow of 2xYT lysogeny broth (LB) at 100 µl/hr. Once thebacteria had filled the chamber, flow rate was increased to 360µl/hr. One syringe was used for LB (at 350 µl/hr), while thesecond (10 µl/hr) was used to deliver varying concentrationsof AHL. An “AHL pulse” is the duration and concentrationover which this AHL was delivered to the bacteria chamber.Pulse durations of 50 minutes that we optimized [24] wereused for “on” or “bit 1” state; while 50 minutes of no AHLequated to “off” or “bit 0” state. Unless otherwise mentioned,we use the above pulse concentration and duration throughoutthe paper. Fluorescent images were captured once every 10minutes and post-processed using MATLAB. The intensityof the pixels within the bacteria chamber was averaged andthe background fluorescence was subtracted, yielding relativefluorescence (arbitrary units, or AU). Four dynamically pro-grammed signals of on and off states were delivered to thereceiver bacteria populations while their fluorescent outputswere recorded. The following input bit patterns were tested:1010101010, 1000100010, 1010000010, and 0000000000. Thebit patterns are chosen to represent different probabilities of bit1. Utilizing the receiver response model developed in [22], wecompared the experimental results to the predicted responseand found that the model captures the dynamics well. Figure3 shows the response of the receiver bacteria from experimentsand simulations to an input of 1000100010 in one trial.

To further test the model’s capabilities to demodulate the re-ceived signal, we used the experimental GFP results to decodethe original AHL input signal by using the inverse of the modelproposed in [22]. The decoding efficiencies of the model are,respectively: 90%, 100%, 80%, 100%. Each bit pattern wasrepeated four times, and the average decoding efficiency wepresent here is the average over the four repetitions. Theseexperiments are time intensive and challenging due to thenature of bacteria. There are numerous factors that affect theability of bacteria to process the AHL input and produce theGFP output. We have made great strides in controlling severalof these factors, such as temperature, population size, andnutrients, by utilizing a microfluidic device, but several remainout of our control. These uncontrollable factors can fluctuateand cause small variations in the bacterial response, whichcan lead to lower decoding efficiencies. This is most likelythe cause for the 80% decoding efficiency.

III. AMPLITUDE-DIVISION MULTIPLE ACCESS

A. Problem Definition

As summarized in Table I, relying on Address Fields,Pulse Duration, and Molecule Type as addressing mechanismscan result in increased network delays, decreased throughput,complex receiver circuits, and throughput unfairness. On theother hand, embedding address in the amplitude of the signalis both fair and throughput friendly, but the maximum numberof users is limited by the number of amplitude levels thereceiver can distinguish. In the arguments above, we showthat embedding the address in the amplitude is best suited

0

10

20

30

0 500 1000 1500

Relativeflu

oresen

ce(A

U)

Time(minute)

Experimentalvalidation

Simulation

Fig. 3: Bacterial Receiver Response

for source addressing in a super-slow network, such as abacterial communication network. Amplitude source address-ing requires simple source and receiver bacterial circuits,which will be described in Section V. [22] demonstrated thatthe response of receiver bacteria is distinct for each distinctamplitude, making the use of amplitude address practical.However, the receiver response model developed by [22] doesnot consider the distance between the source and the receiveras the microfluidic system in [22] uses flow channels, i.e.,the nutrients and signals are carried to the receiver at a givenflow rate. In this case, the distance between the source andthe receiver [24] affects only the propagation delay and hasno impact on the performance of ADMA.

To this end, we propose ADMA, a source addressing mech-anism that embeds the source address in the amplitude ofthe signal transmitted. In this section, we present the goalsand challenges of using ADMA in a practical system. Whenmultiple sources access the channel at the same time, theirsignals collide, and as a result, amplitudes are summed up inthe channel. In the following section, we present an optimalamplitude assignment that can achieve zero address resolutionerror. Goals of ADMA are,

• to resolve the bits transmitted by sources, given a receivedamplitude, with minimum error, and

• to maximize the number of users that can be addressed.

B. Optimal Amplitude Addressing

Amplitude source addressing can be 100% accurate if thereceiver can resolve every source given a received amplitude.Addressing and MAC has two main goals: 1) reliable packetdelivery 2) throughput fairness. We define the Collision Reso-lution Error (CoRE) as a metric to measure reliability. CoREis the ratio of the number of sources identified incorrectly tothe total number of sources. An optimal amplitude assignmentwill achieve zero CoRE. CoRE is determined by the choice ofamplitudes and the receiver design. We derive the conditions toachieve zero CoRE and propose an optimal amplitude assign-ment that achieves zero CoRE. Let bi be the bit transmittedby ith source with amplitude ai in a given time slot.

bi =

{1 if ith source transmits bit 10 if ith source transmits bit 0

5

When molecular signals collide in the channel, the receiver

obtains the sum of amplitudes transmitted y =N∑i=1

biai, where

N is the number of sources in the network. If the number ofpartitions of the received amplitude y is one, i.e., the numberof ways in which different ai can be added to reach a sum yis one, then CoRE will be zero. If the number of partitions ofy is greater than one, then CoRE is strictly greater than zero.Table II shows a sample network with three sources, eachtransmitting bi ∈ {0, 1}. The amplitudes assigned to sourcesare {1, 2, 3}, creating 23 possible combinations of bits. Werefer to each bit combination as a configuration, as shownin Column 4. The sum of amplitudes corresponding to eachconfiguration is defined as its magnitude, which depends onboth the configuration and the amplitudes assigned. Note thatthe configurations {0, 0, 1} and {1, 1, 0} have a magnitude of3, which implies that the number of partitions for 3 in thissetup, is two. On receiving an amplitude 3, the receiver mustchoose from the partitions of 3.

To achieve zero CoRE, the number of partitions of everyreceived amplitude must be less than or equal to one. In otherwords, the magnitude of each configuration must be unique.This problem is studied in Number Theory as “Distinct subsetsum (SSD)”. A set is defined an SSD if and only if the sum ofevery subset of the sequence is unique [27]. An example of anSSD is the binary set, the set of powers of 2, S = {1, 2, 4, 8}.Each subset has a unique sum as every configuration has aunique magnitude. The majority of research on SSD focuseson finding the limit of the maximum value in a subsetsum sequence [28]. In bacterial communication, the range ofamplitudes [Rmin, Rmax] that the receiver can distinguish isdetermined by the receiver circuit design. The receiver circuitthus determines the following parameters of the network.1) The minimum decodable amplitude at the receiver Rmin.2) The maximum receivable amplitude Rmax beyond which

the receiver saturates. If an amplitude greater than Rmax istransmitted, it is received as Rmax.

3) The step size of the levels of amplitudes that the receivercan distinguish. Rmax and step size δ of the amplitudesdetermines the number of amplitude levels that can be

distinguished, i.e., N =Rmax −Rmin

δlevels. By factoring

the step size out and subtracting Rmin, the amplitudes thatcan be assigned are integer values 1, 2, 3, . . . , N . Therefore,in ADMA, integer amplitudes are used to analyze theperformance of the network. The addresses proposed herecan be used in a network with a step size greater or lessthan “1” by multiplying the proposed addresses with δ.

Theorem 1 For a given maximum sum, the set of powers oftwo (binary set) is an optimum set of amplitudes that renderzero CoRE.

Proof: A set S with n elements has up to 2n−1 non-emptysubsets, hence 2n − 1 non-zero sums. To achieve zero CoRE,these 2n − 1 sums must be distinct. Each sum is differentfrom another by at least one, so, the sum of all elementsis at least 2n − 1. A binary set, consisting of powers of 2,satisfies the above condition. A binary set with all powers of2 is an optimum set of addresses that minimizes CoRE. The

maximum number of sources that can be supported by thebinary set is related to Rmax. �

A binary set can accommodate up to Nlog sources, whereNlog = blog2(Rmax + 1)c. By assigning addresses from thebinary set to sources, we also solve the multiple accessproblem. When signals from multiple sources with amplitudesfrom the binary set collide at the receiver, their sum ismapped to a unique configuration, allowing the receiver todecode with zero CoRE. As a corollary of Theorem 1, whenN > blog2(Rmax + 1)c, CoRE is strictly greater than zero.

Though the binary set achieves CoRE, it limits the numberof sources to Nlog. If N > Nlog, a set of addresses thatcan accommodate all the sources and minimizes CoRE isrequired. We propose a heuristic algorithm to select a setof amplitudes that approaches minimum CoRE, given N andRmax. The two major challenges in designing an ADMA systemthat minimizes CoRE are,• designing a sequence of amplitudes,• designing a scalable and low complexity decoder.

C. Components of the ADMA Architecture

In this section, we define and describe each component ofthe receiver architecture of ADMA. The three componentsof the receiver architecture are sampler, demodulator, anddecoder. The response of the receiver bacteria, shown inFigure 3, is input to the sampler, that samples it to discretereceived amplitudes. Sampling and demodulation utilize theinverse of the bacterial receiver response model derived in[22]. The output of the sampler (the inverse model) is the timesequence of the molecular signal samples received. Thus, theinverse model determines the sampler’s accuracy. The modeldeveloped in [22] is deterministic and does not account for thestochastic nature of receiver bacteria. In order to realize thestochastic nature of receiver response to input chemical signal,we introduce random noise to the “k parameters” of the inversemodel. The “k parameters” are the different rate constantsof the receiver bacteria (example, GFP expression rate) thatdefine the state of the receiver bacteria. By varying theseparameters randomly within a specified range, we implementsampling and demodulation errors in the receiver response.

The samples are then input to the decoder to resolve theaddresses and bits. We present decoder designs in detailin Section IV. The decoder outputs a vector of bits, theestimate of bits transmitted by all sources that sum to y[i], thereceived sample. Thus, the receiver design determines addressresolution on receiving a signal, which in turn, determinesCoRE. Here, we propose two receiver designs based on theprinciples of Maximum a posteriori detection: 1) a proba-bilistic receiver, and 2) a deterministic receiver. The receiverdecodes the samples, which are then used to decode the bits.The optimality of the receiver design and the time complexityof the decoder are analyzed at the sample level. The number ofsamples per bit is pre-determined based on the application andsystem constraints. Thus, there is no need for coordination ortime synchronization between sources, and the sources do notrequire additional processing to synchronize transmission, i.e.,each source can transmit data as and when it has informationto transmit. Assuming that all sources always have data to

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TABLE II: Configurations

bu1 bu2 bu3 Configuration(Ci) Mag(Ci) Pr(Ci) Received Amplitude0 0 0 0,0,0 0 (1− pt)

3 00 0 1 0,0,1 3 (1− pt)

2 · (pt) 30 1 0 0,1,0 2 (1− pt)

2 · (pt) 20 1 1 0,1,1 5 (1− pt) · (pt)2 31 0 0 1,0,0 1 (1− pt)

2 · (pt) 11 0 1 1,0,1 4 (1− pt) · (pt)2 31 1 0 1,1,0 3 (1− pt) · (pt)2 31 1 1 1,1,1 6 (pt)

3 3

transmit, the receiver is continuously receiving samples. Themost recurring value of samples is then used to determine bits.A receiver that maximizes the probability of success of eachsample, in turn, maximizes the probability of success of thebit decoded from these samples.

IV. ADMA RECEIVER DESIGNS

In the previous section, we presented an optimal ampli-tude addressing that achieves zero CoRE. But, the maximumnumber of sources N is limited by blog2(Rmax + 1)c. Asderived in Theorem 1, when N is > blog2(Rmax+1)c, averageCoRE will be strictly greater than zero. In a network withN > blog2(Rmax + 1)c, at least one amplitude has more thanone partition and therefore the receiver design also contributesto CoRE. In this section, we propose two receiver designs andderive an upper bound on the expected number of success inresolving address with each receiver design.

We make the following assumptions about the networkin the design and evaluation of the receiver designs. Rmaxis the maximum amplitude that the receiver can uniquelyidentify. Each source is assigned a distinct amplitude, andhence up to Rmax sources can be accommodated, i.e., N ≤Rmax. If N > Rmax, the network is divided into subnets.S = {a1, a2, . . . aN} is the set of amplitudes assigned tosources u1, u2, . . . uN respectively. The sources always havedata to transmit; at a given time, a source is either transmittingbit 1 or transmitting bit 0 and these N sources can betransmitting in one of the 2N configurations. Consider theexample in Table II. Columns 1 to 3 in Table II are the bitstransmitted by sources 1 to 3. Column 5 is the magnitude of theconfiguration, equal to the sum of amplitudes correspondingto each configuration and Column 7 is the amplitude receivedwhen the corresponding configuration is transmitted. Multipleconfigurations can add up to the same sum. All configurationswith the same magnitude are called the partitions of thatmagnitude. Since Rmax = 3, any amplitude ≥ 3 is receivedas 3. Here, {0, 0, 1}, {0, 1, 1}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1} arethe partitions of 3, as all these configurations are received as 3by the receiver. pt and 1− pt are the probabilities of a sourcetransmitting bit 1 and bit 0 respectively. The probability of aconfiguration occurring in the channel depends on pt and theaddresses. For example, the probability of C0 is (1 − pt)

N

as all N sources transmit bit 0, each with probability 1− pt.The probability of each configuration is shown in Column 6of Table II.

We propose Probabilistic Receiver to minimize the errorin decoding a configuration and Deterministic Receiver todecrease the bit error for individual sources. We also derivean upper bound on the expected number of successful addressresolutions for the above two receivers, which is then used tocalculate a bound on the throughput performance of ADMAwith each receiver.

A. Probabilistic Receiver (PR)

The Probabilistic Receiver is designed to minimize thenumber of errors in resolving the transmitted configuration.On receiving an amplitude, PR chooses the configuration thatminimizes the probability of error in decoding that amplitude,in turn, maximizing the probability of success in decodingthe received amplitude. PR follows the Maximum a Posteriori(MAP) detection rule [29] to minimize the bit error in decod-ing a configuration, given the received amplitude.

Observations on Optimality and Practicality of PR

The received amplitude is the sum of the magnitude ofthe transmitted configuration and the amplitude errors dueto channel noise. The receiver observes this noisy signal andestimates the transmitted configuration using a priori estimatesof the channel and the transmitter distributions. We use theprobability of error in decoding the configuration as the metricto evaluate the performance of the receiver. We follow the logicof MAP detection rule that minimizes the expected symboldecoding error to define an optimum receiver [29].

An optimum receiver is a receiver which minimizes theprobability of error in resolving the transmitted configurationgiven the received amplitude.

Minimizing the probability of error on receiving a configu-ration Ci in turn maximizes the probability of success.

Pr(Ci 6= Ci) = 1− Pr(Ci = Ci) (2)

where Ci is the configuration transmitted. The receiver esti-mates Ci on receiving y. The receiver decision is considered asuccess if the estimated configuration was the actual transmit-ted configuration. On receiving an amplitude y, the probabilityof success in decoding the transmitted configuration is,

Pr(Ci = Ci, y) = Pr(Ci = Ci | y) · Pr(y) (3)

Since Pr(y), the probability of receiving amplitude y, is aconstant for a known source distribution and channel model,the optimum receiver will choose Ci such that it maximizes the

7

conditional probability Pr(Ci = Ci | y). Thus, the probabilityof success in choosing a configuration Ci on receiving y is,

Pr(Ci | y) =Pr(y | Ci) · Pr(Ci)

Pr(y)(4)

where Pr(Ci), the probability of Ci being transmitted, isobtained from a priori estimates of the source distribution,Pr(y) is known for given amplitudes, and Pr(y | Ci) from thea priori estimates of the channel transition probabilities. Thus,for every received amplitude y, the optimum receiver choosesa configuration that maximizes the probability of success withaccurate a priori estimates of the source and the channel.

We design PR using the above MAP detection rule anditerate through all possible configurations and choose themost probable configuration which maximizes the overallprobability of success given by,

Rmax∑y=0

Pr(Ci, y) =

Rmax∑y=0

Pr(Ci | y) · Pr(y) (5)

PR is therefore an optimum receiver that maximizes the prob-ability of success in decoding the transmitted configuration fora received amplitude, with an accurate a priori estimate of thesource distribution and the channel transition probabilities.

Though PR maximizes the probability of success in estimat-ing the transmitted configuration, it is an idealized receiverthat assumes accurate a priori estimates of the source andchannel distributions. It is computationally complex to obtainan accurate estimate of these distributions. The computationalcomplexity of the receiver to iterate through all possible con-figurations for each received amplitude increases exponentiallywith the number sources as the number of configurationsincreases exponentially with the number of sources.

To overcome these challenges, we propose DeterministicReceiver (DR), a practical, low-complexity, heuristic receiverdesign later in this section. PR maximizes the probability ofsuccess in decoding the configuration while DR focuses onindividual bits, i.e., PR minimizes the symbol error rate whileDR focuses on reducing the bit error rate of individual sources.

We derive an upper bound on the expected number ofsuccesses using PR, which is then used to develop DR andthe amplitude assignment algorithm in Section V.

Probabilistic Receiver Architecture: Figure 4 is an illustra-tion of the PR architecture. The sampler in this architectureis the inverse model derived from [22]. The sampler takes asinput the response of bacteria and generates a time sequenceof amplitude samples as received by the receiver bacteria.The decoder of PR takes a sampled signal y[i] as inputand generates a table of partitions of y[i]. z1 to zj arethe partitions of y[i]. The decoder chooses a partition thatmaximizes the probability of success. The a priori channeltransition probabilities are considered in generating the tableof partitions for each received amplitude. If more than onepartition can maximize the probability of success, one ofthem is chosen randomly as it will not affect the performancestatistically.

Consider the example in Table II. Each configuration hasa magnitude and a received amplitude. For simplicity, we

Partitions of y[i]

z1 pz1

… ..zy pzy

Sampler y[i] 𝕣 = zi with prob. pzi

Decoder

-1 3 7 11 15

RxAmp(Y)

Time

Fig. 4: Probabilistic Receiver Illustration

do not present channel transition probabilities in Table II.In the presence of channel errors, the received amplitudefor each configuration will be a range of amplitudes witha probability associated to each amplitude, unlike the singlereceived amplitude shown here. Thus, in an ideal channel, onreceiving an amplitude 3, PR chooses configuration {1, 1, 0}with a probability (1−pt)(pt)

2

Pr(y=3) .It can be noted that the conditional probability is affected

by two types of errors,• channel error - channel induced errors alter the ampli-

tudes received,• collision error - when multiple sources collide in the

channel, receiver receives the sum of amplitudes.PR decodes Ci on receiving an amplitude y based on itsprobability of occurrence, i.e., Pr(Ci | y) is equal to Pr(Ci)

Pr(y) ,if the channel is noise free, as shown in Table III. With theknowledge of the transition probabilities of the channel, PRchooses a partition Ci as shown in Equation (4).

In the following section, we derive an upper bound onthe expected number of successful address resolutions usingPR. The number of successful address resolutions can bedetermined accurately for a known amplitude assignment.We derive the upper bound to compare different amplitudeassignment mechanisms. There are

(N

Rmax

)ways of assigning

Rmax amplitudes to N sources. The upper bound is then usedas a measure to evaluate the data-rate performance of differentsets of addresses. In the derivation of the upper bound, weconsider only the decoding error from collisions. The channelnoise is a function of the amplitudes assigned, and therefore itbecomes impossible to derive an upper bound in the presenceof channel noise, without the knowledge of the amplitudes. Inthe implementation and evaluation of ADMA in nanoNS3, weintroduce channel noise. As shown in Figure 4, PR maintainsthe list of partitions for each received amplitude. For a givenpt, the receiver determines the a priori probabilities of eachpartition. The decoder chooses one of the partitions with aprobability equal to its probability of occurrence. An exampleof PR decoder table for Rmax = 3, N = 3, pt = 0.5 is shownin Table III. Column 3 shows the conditional probabilities ofeach configuration given the received amplitude. On receivingamplitude 3, all five partitions have the equal probabilityof being chosen, and hence the conditional probability ofchoosing a partition on receiving amplitude 3 is 0.2. Theprobability of decoding a signal is proportional to that of itsoccurrence. A partition with a high probability of transmissionis received with high probability. Let Y be the random variablerepresenting the received amplitude and mk be the bit sampletransmitted by source uk at a given time and mk be the bitsample the receiver decodes. Decoding is successful for source

8

uk if mk = mk. The expected number of success, wheresuccess is the event of accurate address resolution, is given by

E(No. of succ) =Rmax∑y=1

E(No. of succ. | Y = y) · Pr(Y = y)

E(No. of succ. | Y = y) =

N∑k=1

Pr(mk = mk | Y = y) (6)

where Pr(mk = mk | Y = y) is the probability of successfor source uk on receiving y. We derive the probability ofsuccess for source uk using PR below. As the sources arealways backlogged, the message bit is either 1 or 0 and hencemk ∈ {0, 1}. We further condition on mk = 1 and mk = 0.Let py = Pr(Y = y) be the probability of receiving amplitudey. Using Bayes’s rule,

Pr(mk = mk,mk = 1 | Y = y)

=Pr(mk = mk,mk = 1, Y = y)

Pr(Y = y)

=Pr(mk = mk | mk = 1, Y = y) · pyk1

py

where pyk1 = Pr(mk = 1, Y = y) is the probability of sourceuk transmitting bit 1 and Y = y. A source always has a bitto transmit. Therefore, pyk1 + pyk0 = py

Pr(mk =mk,mk = 0 | Y = y)

=Pr(mk = mk | mk = 0, Y = y) · pyk0

py

=Pr(mk = mk | mk = 0, Y = y) · (py − pyk0)

py

Since PR chooses a partition with a probability equal to itsconditional probability, on receiving an amplitude y givenmk = 1, uk is successfully received, if the receiver choosesany one of the partitions such that mk = 1.

Pr(mk = mk | mk = 1, Y = y) =pyk1

py(7)

Pr(mk = mk | mk = 0, Y = y) =pyk0

py=py − pyk1

py

Pr(mk = mk,mk = 1 | Y = y) =pyk1 · pyk1

py · py(8)

Pr(mk = mk,mk = 0 | Y = y) =pyk0 · pyk0

py · pySubstituting above probabilities, we derive,

Pr(mk = mk | Y = y) =(pyk1

py

)2+(pyk0

py

)2(9)

The expected number of success depends on the probabilityof success of all sources and hence an upper bound on theprobability of success per source will provide an upper bound

TABLE III: Probabilistic Rx Example

Rx Amplitude Partitions Conditional Prob.0 {0,0,0} 11 {1,0,0} 12 {0,1,0} 13 {0,0,1},{0,1,1},{1,0,1},{1,1,0},{1,1,1} 0.2

on the expected number of success. Substituting Equation (9)in Equation (6),

E(No. of succ. | Y = y) =

N∑k=1

(pyk1

py

)2+(pyk0

py

)2(10)

E(No. of succ.) =Rmax∑y=1

N∑k=1

Pr(mk = mk | Y = y) · py

≤Rmax∑y=1

N ·( (pyk1)

py

2

+(py − pyk1)

py

2)(11)

From Equation (11), note that the expected number of successis maximized when pyk1 = py . This supports the theoremthat the number of successes is N only when the number ofpartitions is 1. Equation (11) can be further simplified as

E(No. of succ.) ≤ NRmax∑y=1

(py −

py − pyk1

py2pyk1

)(12)

pyk1 ≤ py . When the equality does not hold we can writepyk1 = r · py where r < 1

E(No. of succ.) ≤ NRmax∑y=1

{py · (1− 2r + 2r2)} (13)

Average network throughput = Expected number of successesPulse duration . For

each signal received, PR chooses one of the partitions withits probability of occurrence. For a set of integer amplitudes,the number of partitions increases exponentially [30] forlarge values of N . On receiving a signal, the receiver goesthrough the partitions of integers contributing O(eN ) to thetime complexity; where N is the number of sources. This isrepeated for each signal received and hence the overall timecomplexity is O(eN |Z|), where |Z| is the number of receivedsignals. As the number of partitions increases exponentially,the space required to store all the partitions is given by O(eN ).The exponential time and space complexity limit the practicalimplementation of PR. In the following section, we propose asimpler receiver with reduced time and space complexity andan improved network throughput.

Algorithm 1 Deterministic Rx Implementation

N ← Number of sources, R(y)← {}for y := 0 to Rmax do

for k := 1 to N dopyk1 ← Pr{Source k transmitting bit 1 | y}pyk0 ← Pr{source k transmitting bit 0}if pyk0 ≥ pyk1 then

R(y)← R(y), {0}else

R(y)← R(y), {1}end if

end forend for

9

B. Deterministic Receiver (DR)

PR maximizes the probability of success with accurateestimates of source distribution and channel noise model.Also, the decoder complexity is exponential to the numberof sources. To reduce the complexity of the receiver, we pro-pose Deterministic Receiver (DR), a heuristic receiver designwhich chooses a pre-determined configuration on receiving anamplitude to reduce the bit error of each source independently.

Deterministic Receiver Architecture: For each received am-plitude y, DR chooses a bit sample for individual sourcesindependently. DR goes through all the partitions of theamplitude y and chooses the most probable bit for each source.This is pre-determined during receiver setup. In this work,we focus on minimizing decoding errors due to collisions inDR analysis. Knowledge of channel transition probabilities isincorporated by including channel noise in the estimation ofthe most probable bit. DR thus differs from PR in buildingand updating the decoder table. With the help of Algorithm 1,we explain the operation of DR.

Line 2 loops over all receivable amplitudes. For eachamplitude received, DR compares pyk1 and pyk0 for eachsource, i.e., the probability of the source transmitting bit 1and bit 0 given that amplitude y was received, is compared. Inline 6, if pyk0 ≥ pyk1 for source uk, the receiver updates thereceived signal R(y) = 0 for source uk. R(y) is the vector ofbits decoded by the receiver on receiving y. R(y) is calculatedonce, and the receiver uses a constant-time lookup to decode.DR updates the vector of bits decoded for all amplitudes thatcan be received. DR maximizes the conditional probabilityof source uk transmitting a bit given the received signal y.The maximization is performed a priori, and hence the timecomplexity is O(1). The bit vector calculated independentlyfor each source may not be one of the partitions or configu-rations of the amplitude received. But this vector maximizesthe number of successes for each source independently. Westudy the average performance of DR for different states ofthe receiver. To study the performance of DR, we use the sameparameters as that of PR. The average number of successesusing DR can be described as,

E(No. of succ. | Y = y) ≤ N · max1≤k≤N

max(pyk1 , pyk0)

≤ N · max1≤k≤N

py ·max(r, 1− r)

It can be observed that the bound on the probability of successreaches its maximum value of 1 at r = 0 or r = 1. We alsoproved in Theorem 1 that a practical system can achieve aprobability of success of 1 only when N ≤ blog2(Rmax +1)c.We derive a practical upper bound on the expected number ofsuccesses with DR, analytically.

Case 1: Chosen configuration is a partition: If the chosenconfiguration is one of the partitions, then the expected numberof successes when it is transmitted is N . When the chosenconfiguration is transmitted, it will be decoded without error.All the partitions that differ from the chosen configuration byone bit will be received with N − 1 successes and so on.

TABLE IV: DR: Upper bound estimation

Max No. of successes Case 1 Case 2

N 1 0

N − 1 N − 1 1

N − 2 ≤(N2

)≤(N2

)N − 3 ≤

(N3

)≤(N3

).. .. ..

1 ≤(

NN−1

)≤(

NN−1

)0 0 1

Algorithm 2 Deterministic Receiver Upper bound

1: R← Rmax2: N ← Number of sources3: E← 0,Expected number of successes4: for i := 1 to R do5: E← E+ C[0 : i] ·N, C ← C[i : end]6: E← E+ C[0 : (R− i)] · (N − 1)7: C ← C[(R− i) : end]8: E← E+ C[0 :

(N1

)] · (N − 1)

9: C ← C[R+(N1

): end]

10: E← E+ C[0 :(N2

)] · (N − 2)

11: C ← C[R+(N1

)+(N2

): end]

12: E← E+ C[0 : (R− i) ·(N2

)] · (N − 2)

13: C ← C[R+(N1

)+(N2

)+ (R− i) ·

(N2

): end]

14: istart ← R+(N1

)+(N2

)+ (R− i) ∗

(N2

)15: iend ← istart +

(N3

)+(N2

)+ (R− i) ·

(N3

)16: for j = 3 to N do17: E← E+ C[istart : iend], istart ← iend18: iend ← istart + i ·

(Nj

)+ (R− i) ·

(Nj

)19: end for20: end for

All configurations that sum to y will be different from thechosen one by at least one bit. No two configurations thatdiffer by one bit can add to the same integer. For example, aconfiguration 1101 and 1100 differs by one bit. Let y1 be thesum of configuration 110. y1+a4 cannot be equal to y1. Thus,if the chosen configuration is one of the partitions, no partitionof that sum will have N−1 success. Configurations that map toRmax is an exception. Since all configurations that sum to Rmaxand greater are mapped to Rmax, two configurations that differby one bit can add to Rmax. For example, configurations 001and 011 are mapped to 3 in Table 4. Therefore, at most N −1configurations that differ by one bit from chosen configurationand mapped to Rmax can have N − 1 successes.

Following the same argument, no partition of a sum candiffer from the chosen one by two bits if the chosen config-uration is a partition. For example, 1101 and 1110. Let sumof amplitude 11 be y2. y2 + a4 cannot be equal to y2 + a3as all amplitudes are distinct. Therefore, if the chosen oneis one of the partitions, no configuration that sum to X willhave ≤ N − 3 successes. Configurations summing to Rmax

10

are an exception. Up to(N2

)configurations that differ from

the chosen configuration that add to Rmax can have N − 2successes. There can be up to

(N3

)partitions with a Hamming

distance of 3 between themselves and the chosen configurationand up to

(N4

)partitions with a Hamming distance of 4 from

the chosen partition and so on. Column 1 in Figure IV showsthe maximum number of successes and Column 2 defines themaximum number of configurations that can achieve thesesuccesses in Case 1.

Case 2: Chosen configuration is not a partition: If thechosen configuration is not one of the partitions, none of thepartitions of that integer can have N success. At most, onepartition can have N −1 success. At most, one partition has aHamming distance of one from the decoded configuration. Letus assume there are two partitions with a Hamming distanceof one between them and the decoded configuration. Then theHamming distance between these partitions is at most 2. Sincetwo partitions of an integer differ by at least 3 bits, this is notpossible. Thus, if the chosen configuration is not one of thepartitions, up to one configuration can have N − 1 success.There can be up to

(N2

)partitions that have a Hamming

distance of 2 from the chosen configuration. Similarly, therecan be up to

(N3

)partitions that are 3 Hamming distance away

from the chosen configuration. Columns 2 and 3 in FigureIV summarizes Case 1 and 2 respectively. There are Rmaxdistinct integers receiver can receive. For some amplitudes, thereceiver can choose one of its partitions; and for others, thereceiver chooses a configuration that is not a partition. We useAlgorithm 2 to find an upper bound using DR. C is the sortedarray of the probability of each configuration being transmittedin descending order and E is the expected number of successon receiving an amplitude sample. Given N and pt, there are(Nk

)configurations with k ones and N − k zeros, where k

varies from 0 to N . Lines 4 loops over i where i variesfrom 1 to R. i represents the number of received amplitudeswhose decoded configuration is one of the partitions. R − iamplitudes have decoded configurations that are not theirpartition. From the above table, we know that only thosedecoded configurations that are partitions of the amplitude canhave N success. Thus, in the for loop in Algorithm 2, up toi configurations can have N success. To calculate an upperbound, we assume that the highest probable i configurationscan have N success. Also, one configuration per receivedamplitude can have N−1 success, if the decoded configurationis not its partition. Thus, up to R− i configurations can haveN−1 success. All amplitudes greater than Rmax is received asRmax , and hence the highest receivable amplitude can have upto N partitions with N − 1 success and similarly up to

(N2

)configurations with N − 2 successes. Lines 16 to 19 loopthrough remaining configurations assigning N − j successesto(Nj

)· N configurations. Maximum E over all i gives the

upper bound on the expected number of success.

V. AMPLITUDE ASSIGNMENT ALGORITHM

As discussed in Section II, the two factors that impactCollision Resolution Error (CoRE) are the receiver design andthe source addresses. In this section, we present a heuristic

algorithm that chooses an amplitude sequence that minimizesCoRE based on the insights we gained from the derivation ofthe probability of success in Section IV. Following Theorem1, the maximum number of sources that can be accommodatedwith zero CoRE is blog2(Rmax + 1)c. Since the addressesare distinct, the maximum number of sources is limited byRmax. As the number of sources increases, the required Rmaxto achieve zero CoRE increases exponentially. In this section,we propose an algorithm to choose a set of amplitudes thatminimizes CoRE, for a fixed N and Rmax. From the probabilityof success derivation for receiver design in Section IV, weobtain the following insights, which is then used to proposefour amplitude sequences.

Insight 1 The higher the number of partitions, the lower isthe probability of success on receiving the sum.

Insight 2 At a low probability of transmitting a bit 1, thenumber of colliding signals is ≤ 2 with high probability.

When the number of collisions is less than or equal to two,an address sequence that can recover from two collisions willhave a high probability of success. We define shifted (natural)sequence as a sequence of integers with the first elementshifted by Rmax−1

2 , S ={

Rmax−12 , Rmax−1

2 + 1, . . . ,Rmax − 1}

.A maximum of Rmax

2 sources can be supported using thissequence. No element in the shifted sequence can be written asa sum of any two elements. ai+aj 6= ak where ai, aj , ak ∈ Sand i 6= j 6= k. Therefore, when one source transmits a bit 1, itwill be received without error. When two sources transmit bit1 and collide, the number of partitions of received amplitude isreduced, and therefore the error is smaller compared to otheraddress sequences since an error in one address will not affectothers, making it suitable for scenarios with a low probabilityof bit 1 collisions. A shifted natural sequence allows Rmax

2

sources. If N ≥ Rmax2 , we extrapolate the shifted sequence

as S = {Rmax −Nu − 1, Rmax −Nu − 2, ...Rmax − 1, Rmax}.The extrapolated sequence does not hold the property of ashifted sequence. The number of elements that can be writtenas sum of two other elements is small in this sequence sinceits elements are decreasing integers from Rmax.

Insight 3 At a high probability of transmitting a bit 1, thenumber of colliding signals is ≥ Nmax with high probability.

Let Nmax be the number of sources such that sum of am-plitudes up to Nmax is less than equal to Rmax. When morethan Nmax sources collide, a natural sequence, which is thesequence of integers beginning from 1 till N , will have ahigh probability of success. When the number of sourcestransmitting bit 1 at a given time increases, the probabilityof receiving configurations that sum to ≥ Rmax increases. Allsums greater than Rmax are received as Rmax. On receivinga sum Rmax, the receiver chooses one of the configurationswith sum ≥ Rmax. The probability of choosing a particularconfiguration is 1

N(x≥Rmax), where N(x ≥ Rmax) is the number

of configurations that sum up to Rmax and more. The fewerthe number of configurations with sum ≥ Rmax, the higher

11

the probability of success. Amplitudes that can minimizeN(x ≥ Rmax) will minimize CoRE. A natural sequence hasthe maximum number of configurations < Rmax, and henceminimum N(x ≥ Rmax).

Algorithm 3 Address Allocation

Rmax ← Maximum Receivable amplitudeNs ← Number of simultaneous sourcesNu ← Total number of sourcesS ← Set of addressesif Pr(Ns ≤ 2) > Pr(Ns > 2) and Nu ≤ Rmax

2 thenS ← {Rmax−1

2 , Rmax−12 + 1, . . . , Rmax − 1}

else if Pr(Ns ≤ 2) > Pr(Ns > 2) and Nu >Rmax2 then

S ← {Rmax −Nu, Rmax −Nu − 1, . . . , Rmax − 1}else if Pr(Ns ≤ Nlog) > Pr(Ns > Nlog) then

S ← {20, 21, . . . , 2Nlog , Rmax, Rmax − 1, . . .}else if Pr(Ns ≥ Nmax) > Pr(Ns < Nmax) then

S ← {1, 2, 3, . . . , Nu}else S ← {20, 21, . . . , 2Nlog , Rmax, Rmax − 1, . . .}end if

Insight 4 At an intermediate probability of transmitting abit 1, the number of colliding signals is ≤ Nlog with highprobability.

At intermediate values of pt, the per-source probability oftransmitting a bit 1, up to Nlog signals collide. A se-quence that has the maximum number of sums with uniqueconfigurations with up to N elements will have a highprobability of success. Extrapolated binary sequence S ={20, 21, 22, . . . Nlog, Rmax − N − Nlog, Rmax − N − Nlog −1, . . . , Rmax} combines a binary sequence and a shifted se-quence. Nlog = blog2(Rmax + 1c) is the number of sourcesa binary sequence can support. A binary sequence is anoptimum solution to achieve zero CoRE. An extrapolatedbinary sequence utilizes the maximum number of distinctsums when using a binary sequence. These distinct sums areobtained from the binary elements. The rest of the elementsare integers decreasing from Rmax, which reduces the numberof overlapping sums. As the number of collisions approachesNlog, the distinct sums contributed by the binary elements willimprove the probability of success. Algorithm 3 summarizesthe insights gained from the sequences observed and theprobability of success derivation.

Amplitude assignment : Practical Implementation

We presented an optimum receiver design and an algorithmto choose an amplitude sequence that minimizes CoRE. In thissection, we discuss network architecture and a practical bacte-rial communication system that considers practical amplitudeassignment. We describe how a receiver can assign amplitudesto the sources based on its Rmax. To the best of our knowledge,no existing work provides a practical transmitter design thatcan transmit different amplitudes using only components builtfrom bacterial populations. In Figure 5, we present a circuitdesign to implement a bacterial transmitter for ADMA. Asshown in Figure 5, each source consists of three major com-ponents viz., a transducer, an attenuator and a transmitter. Each

Transducer Attenuator Transmitter Receiver

Fig. 5: Practical Implementation: Illustration

source in Figure 1 is composed of the above three components.In a sensing network, the transducer is genetically engineered,based on the application, to sense a specific signal and convertto another signal, as it is now possible to design networks thatutilize multiple signal molecules with little cross-talk [31]. Thetransducer emits one quorum sensing signal (AIP for example,crosses) at a concentration proportional to the inducer (IPTG)concentration. The AIP thus generated is attenuated by theamplifier/attenuator, which produces a second signal (AI-2 forexample, circles). The amount of attenuation can be controlledexternally using feedback. This attenuated AI-2 signal isthe input to the transmitter which in turn emits a distinctsignal (C6-HSL for example, triangles) to be transmitted tothe receiver through a microfluidic channel. The attenuatormodifies the transmitted signal concentration, and therefore isthe address allocator in the architecture. Assuming a feedbackpath from the receiver to the attenuator, the receiver assigns theamplitude to each source of bacterial population by modifyingthe attenuator. The attenuator can be a bacterial populationthat emits AI-2 in a manner so that the concentration of AI-2 emitted is controlled by an external trigger (squares) tothe attenuator population or additional signal (such as AIPitself) from the receiver. Such a trigger is provided by thereceiver (or a centralized server) to each source based onthe amplitude assigned to the source. Thus, the receiver (orthe server) controls the amplitude/concentration of the signaltransmitted to the receiver which in turn is used to identifythe source on receiving a signal. It can be noted that theconcentration of molecules generated by all the sources is thesame. Thus, ADMA is also an energy fair mechanism.

VI. PERFORMANCE EVALUATION

We evaluate ADMA in progressive steps.

1) We evaluate the performance of amplitude assignmentalgorithm and the receiver designs, using BCS (BacterialCommunication Simulator), a custom-built Python-basedsimulator in idealized channel conditions, i.e.., assumingzero sampling and demodulation error at the receiver.

2) We introduce channel errors in BCS and analyze theperformance of ADMA with channel errors.

3) We implement the response of a receiver bacteria locatedin a microfluidic chip derived in [22] in nanoNS3 [32], thebacterial molecular communication simulator developed ontop of NS3 simulates source error, channel error, and sam-pling and demodulation error at the receiver. We implementamplitude assignment algorithm and deterministic receiver(DR) in nanoNS3 and evaluate the performance with errors.We also show using experimental results and nanoNS3,

12

0.5

0.7

0.9

1.1

0.1 0.3 0.5 0.7 0.9

Determ

inistic

Prob

abilistic

pt

Integer

ExtrapolatedBinary

Shifted

ExtrapolatedShifted

Fig. 6: Data-rate of Deterministic vs. Probabilistic Re-ceiver N = 5, Rmax = 15

50

60

70

80

90

100

110

0.1 0.3 0.5 0.7 0.9

PracticalM

aximum

Theo

reticalUpp

erBou

nd

pt

N=5 N=6 N=7

N=8 N=9 N=10

N=11

Fig. 7: Theoretical vs. Practical Upper Bound

that by using the inverse of the receiver model, receiverresponse can be demodulated.

A. Idealized network conditions

Unless otherwise mentioned, we make the following as-sumptions in the implementation and evaluation of ADMAin BCS. Each source transmits bit 1 with a probability ptand has an uninterrupted supply of data to transmit. Each datapoint in the results presented is averaged over 100 simulations.Before evaluating ADMA, we present the tightness of upperbound derived in Section IV.

Upper bound tightness: In Algorithm 2, we derive an upperbound on the expected number of success, assuming that thehighest probability configuration achieves the highest numberof success without considering the amplitudes assigned. Inpractice, configurations are not independent of the amplitudesassigned and the probability of success depends on the choiceof amplitudes and the receiver. We evaluate the theoreticalupper bound tightness by performing an exhaustive search onall possible address assignment and determine the practicalupper bound. Given Rmax, N , there are

(RmaxN

)possible address

sequences; N addresses chosen from [1, 2, 3, . . . , Rmax]. Theaverage expected number of success is calculated for eachaddress sequence possible and compared to the theoreticalupper bound. We limit our exhaustive search to a maximum

0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Prob

abilityofb

iterror

pt

NoError

ChannelError

NoError-TimeSync

ChannelError- TimeSync

Fig. 8: Bit error rate performance of ADMA, load awareDR,Rmax = 30, N = 14

0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Prob

abilityofb

iterror

pt

NoError

ChannelError

NoError-TimeSync

ChannelError- TimeSync

Fig. 9: Bit error rate performance of ADMA, load awareDR,Rmax = 70, N = 15

of Rmax = 15 as(RmaxN

)increases as the factorial of Rmax

increases and an exhaustive search is computationally notfeasible. Figure 7 shows the ratio of practical bound to that oftheoretical bound in y-axis as a function of pt, for Rmax = 15and number of sources N ranging from 5 to 11. At smallervalues of N , the probability of fewer “bit 1” collisions ishigher, i.e., the probability of collision of N

2 or fewer sourcesis higher than the collision of more than N

2 sources and theprobability of receiver saturation is small. As N increases, theprobability of collision of N

2 or more sources increases, furtherincreasing the probability of receiver saturation, which in turndecreases the effective throughput. For Rmax = 15, N = 5, thepractical bound is 99.3% of the theoretical bound at pt = 0.1and 91.2% on an average. Theoretical upper bound, on anaverage is 95.8% at pt = 0.1, 78.9% at pt = 0.5 and 91.3%at pt = 0.9 for Rmax = 15 and N varying from 5 to 11. Atvery low and very high pt, the theoretical bound is close to90% of the practical upper bound.

Load Aware Receiver

From Equation (1), we find that the parameters influencingCoRE are, 1) receiver design, 2) per-user probability oftransmitting bit 1, pt, 3) maximum receivable amplitude Rmax,and 4) number of sources N . We evaluate the performance

13

of ADMA by varying pt, N and Rmax. In Figure 6, thethroughput performance of PR is compared against that ofDR for N = 5, Rmax = 15 under idealized conditions. Thethroughput achieved using DR outperforms that of PR fordifferent sequences, which is attributed to the objective ofeach receiver design. The goal of DR is to reduce per-user biterror and increase the average per-user throughput, whereas PRmaximizes the joint throughput of the network. We evaluatethe performance of ADMA using DR in the rest of the sectiondue to its low time complexity in implementation. We use biterror rate as the metric to evaluate ADMA for different setsof Rmax, N and pt, which is then used to calculate per-userthroughput. We calculate the expected number of successfulbits received per user from bit error and average over totaltime taken to calculate the average throughput. Figures 8and 9 shows the bit error rate performance of ADMA forincreasing values of pt when receiver is aware of pt. Fourcurves in each graph plots the bit error rate performance ofADMA in the presence and absence of channel error, andpresence and absence of time synchronization. As discussedin Section IV, the receivers are designed to decode samplesand do not require any time synchronization. Channel error isintroduced by adding up to 20% amplitude error to 10% ofthe bits transmitted by each source. Figures 8 and 9 showsthat ADMA is robust to asynchronous transmissions evenin the presence of channel error. When integer amplitudesare assigned, DR chooses a configuration with the mostprobable bit for each source. The configuration chosen byDR for received amplitude y differs only by few sourcesfor amplitudes close to y, making ADMA robust to channelerror. For Rmax = 30, N = 14, the throughput performance ofADMA is 92.5% of the upper bound at pt = 0.1 and on anaverage 80.5% of the upper bound. For Rmax = 70, N = 15,the throughput performance of ADMA is 92.4% of the upperbound at pt = 0.1 and on an average 82.3% of the upperbound. As shown in Figure 7, the practical bound on anaverage is close to 90% of the upper bound. Extrapolatingthis result to the performance at Rmax = 70, N = 15 andRmax = 30, N = 14, the average performance of ADMA isclose to 90% of the absolute maximum.

Load Unaware Receiver

In the above results, we assume that the receiver is aware ofthe load distribution (pt). It may not be possible in a practicalsystem to estimate pt accurately. We propose a load unawarereceiver that updates the decoding table for pt = 0.5, whenpt is unknown. In a load unaware scenario, ADMA alwayschooses the integer address, i.e., {1, 2, 3, . . .}. Figures 10 and11 shows that the load unaware receiver does not affect the biterror rate performance of ADMA. For Rmax = 30, N = 14,the throughput performance of ADMA is 88.5% of the upperbound at pt = 0.1 and 76.1% of the upper bound on anaverage. For Rmax = 70, N = 15, the throughput performanceof ADMA is 91% of the upper bound at pt = 0.1 and 81.4%of the upper bound on an average. We observed that theinteger sequence has the best performance on an average whenthe receiver decodes using pt = 0.5, simplifying amplitudeassignment algorithm. On average, the performance of load

0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Prob

abilityofb

iterror

pt

NoError

ChannelError

NoError-TimeSync

ChannelError- TimeSync

Fig. 10: Bit error rate performance of ADMA, loadunaware DR, Rmax = 30, N = 14

0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Prob

abilityofb

iterror

pt

NoError

ChannelError

NoError-TimeSync

ChannelError- TimeSync

Fig. 11: Bit error rate performance of ADMA, loadunaware DR, Rmax = 70, N = 15

unaware DR using integer sequence remains within 95% ofthe load aware DR. Also, we note that the performance ofinteger sequence is within 99% of the algorithm performanceon average.

B. ADMA with Bacteria Receiver in Microfluidic Channel

So far, we evaluated the performance of ADMA assumingzero sampling and demodulation error. Here, we implementand evaluate ADMA in nanoNS3 to simulate a practicalbacterial communication system with source errors, channelerrors and sampling and demodulation errors. We also showusing experimental results and the receiver response model of[22] in nanoNS3 that the inverse of the receiver model can beused to demodulate the received signal with high accuracy. Weimplement the bacterial receiver response derived in [22] thatmodels the performance of receiver bacteria in a microfluidicchip. The receiver model outputs the GFP response of thebacteria in the chamber for a given input. nanoNS3 generatesa train of rectangular pulses for different sources and sums thesignals in the channel in time-domain. A channel attenuationmodel that attenuates the amplitude of the signal in thechannel introduces a 20% amplitude attenuation per source.The cumulative signal (from multiple sources) is input to thereceiver model, which generates the response of the receiver

14

0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Prob

abilityofb

iterror

pt

LoadUnaware

LoadAware

Fig. 12: Bit error rate performance of ADMA innanoNS3, Rmax = 30, N = 14

0

0.2

0.4

0.6

0.8

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Prob

abilityofb

iterror

pt

LoadUnaware

LoadAware

Fig. 13: Bit error rate performance of ADMA innanoNS3, Rmax = 70, N = 15

for the corresponding input. nanoNS3 builds an inverse of themodel numerically, taking GFP response as input and outputsan estimate of the molecular signal. The output of inversemodel is a time sequence of an estimated amplitude sample.Sampling and demodulation error is introduced at the receiverby varying “k parameters” of the inverse model which definedifferent rate constants in the receiver response and varyingthem introduces random errors at the receiver. Each source isassigned a bit duration of 50 min and sampled every 10 mingenerating five samples. The samples from the inverse modelare divided into blocks of 5 (5 samples per bit) and inputto the decoder that outputs bit 1 if the average amplitude ofa block of samples is greater than the average, and bit 0 ifbelow. For example, a time sequence output of DR block foruser u1 with amplitude a1, a1,a1,a1,0,a1, is decoded as bit 1and 0,0,0,a1,a1 is decoded as bit 0.

nanoNS3 thus uses forward response model to generateGFP response of receiver bacteria and the inverse modelto demodulate. The results from nanoNS3 thus represents apractical system by considering non-ideal conditions at thesource, channel, demodulator and the receiver. Figures 12 and13 show the bit error rate performance of ADMA in nanoNS3for Rmax = 30, N = 14 and Rmax = 70, N = 15. Theload unaware and load aware results are very close to each

other, as actual input distribution pt ≥ 0.5. This is due toreceiver saturation at Rmax. All amplitudes greater than Rmaxare received as Rmax. For pt ≥ 0.5, the probability of receivingRmax or higher is high, i.e., the receiver observes Rmax withhigh probability and the number of partitions mapped to Rmaxis much higher than other received amplitudes. Therefore, evenwith pt ≥ 0.5, the probability of bit 1 and bit 0 is close to 0.5on receiving Rmax. For Rmax = 30, N = 14, the throughputperformance of ADMA in nanoNS3 is 90.2% on average, and99.9% in best case, of that achieved using idealized simulatorBSC; at Rmax = 70, N = 15, throughput of ADMA is 89.1%on an average and a maximum of 99% of that of BSC.

ADMA also acts as a multiple access control (MAC)mechanism. One of the requirements of a MAC protocol isfairness. Using simulations, we also calculate the fairness ofADMA. We calculate the Jain’s fairness index [33] for differentsets of Rmax, N and pt for both load aware and load unawarereceivers. for N = 14, Rmax = 30 and N = 15, Rmax = 70respectively. Fairness achieved by ADMA with deterministicreceiver, on an average is 0.99 indicating high fairness.

VII. CONCLUSIONS AND FUTURE WORK

The following assumptions were made in deriving the bestaddress sequence for a network. We discuss in detail each ofthe assumptions below.pt is known: As shown in Figure 8, the performance of

an addressing sequence depends on the probability of sourcestransmitting bit 1. To select the best sequence, we must knowthe approximate range of pt. Figure 10 plots the performanceof an amplitude assignment algorithm when the receiver isunaware of input load; it assumes pt = 0.5. The integersequence under load unaware conditions performs close to thatof a load aware deterministic receiver (within 95%). Whilethe knowledge of pt can improve the performance of thesystem, when using an integer sequence with a load unawaredeterministic receiver, there is not a significant decrease inthe throughput performance. Thus, pt does not affect theperformance of ADMA.pt is same across sources: The insights developed in

ADMA assume same pt for all sources. Deriving the prob-ability of success and the best sequence for different pt acrosssources is a challenging problem. We showed that even whenthe receiver is not aware of pt, throughput performance is notaffected significantly. Following the same argument, if sourcestransmit bit 1 with a different probability, the performance ofthe system is not significantly affected.

Non empty data queue: We implement OOK modulation,where an absence of a signal indicates bit 0. In practice, itis necessary to differentiate between bit 0 and no data. Wepropose the use of start and end-of-frame sequences. A pre-assigned bit sequence can define the start and end of a frameand an absence of signal outside this start and end of framesis considered as no-data.

We propose ADMA and demonstrate that the amplitude ofthe transmitted signal can be used to address sources in a bac-terial communication network. We also verify experimentallyusing genetically engineered bacteria in the microfluidic chip

15

as a proof-of-concept that it is possible to implement ADMAin a practical system. The state-of-the-art microfluidic systemaccommodates only a single receiver. Due to constraints in thesystem design, the proof-of-concept presented in this paperextrapolates results from a system with one source and onereceiver. A microfluidic system design that can accommodatemultiple sources and multiple receivers is a work in progress.

We propose a heuristic algorithm to choose a sequence ofamplitudes, to minimize collision resolution error. We alsopropose an optimum receiver that can minimize collisionresolution error and maximizes the number of sources ac-commodated in the network. We derive a theoretical upperbound on the average number of successes using ADMA.Finally, we evaluate the proposed algorithm in nanoNS3, abacterial communication simulator module built on top ofNS3; and through simulations, determine that the optimumreceiver design is robust to channel errors and receiver errors.

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