I. INTRODUCTION AND MOTIVATION
Nanotechnology has the potential to have a significant impact on a number of applications
areas including health care [1], [2], [3], bio-hybrid implants [4], [5], food and water quality
control [6], defense systems against biological and chemical attacks [7], air pollution control [8],
and biodegradation [9]. Possibility of building components at the nano scale revolutionized the
way we think about systems by enabling myriad possibilities, that were simply impossible
otherwise. At the same time, countless challenges were raised in system design. One such
challenge is to build components that act together to handle complex tasks that require physically
separate components to work in unison. To achieve coordination, these components have to be
capable of communicating reliably, either with a central controller or amongst themselves. In
this research, we propose to build analytical foundations to analyze and design nanonetworks,
consisting of individual stations communicating over a wireless medium using nanotransceivers
with nanotube antennas. We give a simple nanoreceiver design and analyze its basic limitations.
Based on the insights drawn, we propose a communication theoretic framework to design
reliable and robust nanoreceivers. With the basic limitations of the nanocommunications via
nanoantennas in mind, it is possible to develop mathematical tools to help construct nanonetworks
that execute basic sequential tasks in a reliable manner with minimal amount of communication
and computation required.
The physical operation of the nanotube antenna is significantly different from that of a classical
antenna for wireless communication. Nanotube absorbs electromagnetic waves and converts them
C. Emre Koksal and Eylem Ekici are with the Department of Electrical and Computer Engineering, The Ohio State University,
Columbus, OH (email: {koksal, ekici}@ece.osu.edu).
2
to mechanical vibrations. Since the nanotube is charged, these vibrations cause changes in the
electric field at the cathode right accross the nanotube and the induced current is processed to
recover the incident signal. Since the mechanism of the nanotube antenna is electromechanical,
the nanoreceiver has to overcome the acoustic noise, which is effective directly on the nanotube,
as well as the thermal noise, effective at the rear end of the antenna. The unconventional physics
of the nanotube antennas and the various noise sources one has to consider, makes the design
of nanoreceivers and the analysis of their performance a highly challenging task.
Another challenge in the design of nanoreceivers stems from the limitations of components at
the nano scale. Nanoelectronic manufacturing is at its infancy stage and the nanoreceiver cannot
involve hardware components apart from some very basic ones. For example, a detection scheme
cannot involve any computations beyond the most trivial ones, such as the mere comparison.
Consequently, the associated team tasks have to be simple to implement.
The analysis presented in this paper lays the groundwork to design nanonetworks, consisting
of nodes individually responsible for a certain number of basic tasks. To achieve these tasks,
the nodes must be activated, either externally by a centralized controller, or internally in a
sequential manner as the activation signal propagates in the network in a multihop fashion.
The activation signal is an electromagnetic wave of a certain duration, tuned according to the
resonance frequency of the nanotube antennas of the nodes to be activated. Note that, this
construct can be useful in a number of applications. For instance, in a drug delivery system,
once activated, each node can be responsible for releasing a certain drug in some desired order.
The node can send the activation signal for the desired next hop in the sequence, once releasing
its chemicals after being activated. Coupled with sensing devices, automated dosage control
can be integrated into the drug delivery system, as well. Also, one can consider a nano-RFID
application, in which each node can be viewed as an RFID, emitting a certain signal once active.
Furthermore, multihop activation of nodes can mimic the effect of nervous system through the
propagation of activation signals in a bio-hybrid implant such as artificial muscle tissues to
coordinate the activation of different parts of the tissue.
In designing such systems, reliability is the major issue and one has to be careful in the
possibility of the following events:
(1) A node can go active, even in the absence of an activation signal. This can be catastrophic
in a number of applications including drug delivery. One has to make sure that the probability
3
of the false activation event is very low even in the time scale of months/years of continual
operation.
(2) A node can remain inactive, even after the activation signal is sent to it. This event can be
highly undesirable, especially in applications that require timely response. One has to guarantee
that the probability of the unsuccessful activation event is below a certain threshold each time
the activation signal is sent to a node. There is clearly a tradeoff between the probability of false
activation and the probability of unsuccessful activation.
(3) There are a number of imperfections associated with the system components. Examples of
such imperfections include the frequency and timing mismatch between the activator and the
receiver, and model imperfections and parameter uncertainties at the nano scale. Nanoreceiver
design should be robust with respect to such imperfections.
(4) If the activation signals are chosen to be “close” to each other, the probabilities of unsuccessful
and false activation events increase, due to crosstalk among different nodes. This imposes a
fundamental limit on the number of activation signals, and hence the tasks the network can
handle, for a given limited amount of usable bandwidth. The network must be designed with
this limitation taken into account.
(5) The delay in the execution of certain tasks can be too high to be tolerable. This can be caused
by a high activation time of individual events in a sequence or through failures in sensing physical
phenomena. Designed systems must ensure not only the eventual completion of a sequence of
tasks, but also their successful completion within tolerable delay limits.
In this paper, we present a communication-theoretic analysis of networks of nano-scale nodes
equipped with carbon-nanotube-based receivers and transmitters. Our objective is to analyze
performance characteristics of nano-scale nodes and expose their fundamental capabilities and
limitations. The presented analysis is intended to serve as the basis of nanonetwork design
enabling various applications.
The remainder of the paper is organized as follows: In Section II, a brief summary of related
work in the literature is presented. The system architecture and the network model are presented
in Section III. Analysis of single node activation is discussed in Section IV, followed by multinode
activation analysis in Section V. The paper is concluded in Section VII.
4
II. RELATED WORK
Recent advances in nanotechnology has resulted in development of nano-scale machinery such
as molecular elevators [10], nano switches [11], chemical sensors [12], [13], nano actuators [14],
[15], carbon nanotube receivers [16] and transmitters [17]. Main mechanisms to manufacture
nano-machinery involve three approaches [9]: The top-down approach aims to manufacture
nano-machinery through the miniaturization of micro-scale devices to nanoscale. Electron beam
lithography [18], [19] and micro-contact printing [20] are two main tools used in this approach.
Great strides are being made in the manufacturing of individual components [21], and simple
structures such as nano-gears are fabricated via this approach [22]. The bottom-up approach
aims to form nano-machines using molecules as the basic building blocks. Also called molecular
manufacturing [4], this approach has theoretically been shown to be very useful to fabricate a
variety of components including molecular pumps [23]. Finally, the third approach to nanomachinery
manufacturing is the bio-hybrid approach, which models the nano-machinery after
biological structures such as cells. Using bacteria to propel micro-scale objects follows the biohybrid
approach to nano-scale manufacturing [24].
These nano-machines cannot accomplish complex tasks outlined in Section I by themselves
and should be combined into nano-nodes. Complex tasks and applications require interaction
among nano-nodes, exchange of information, and execution of tasks conditioned on environmental
inputs. Communication between nano-nodes can occur via nano-mechanical interactions,
diffusion and exchange of chemicals, through pressure waves, or electromagnetic signals [25].
Among these options, molecular diffusion techniques [26], [27] and nano-mechanical interactions
[28] have been envisioned as means of long and short/medium range communication for
nanonetworks, respectively. Long considered as infeasible due to projected size and complexity
resulting from miniaturization-based transceiver design [25], RF-based communication has
captured the limelight of nanocommunication research through the development of nano-scale
receivers [16] and transmitters [17]. Tests performed on prototypes of these devices have proved
not only the feasibility of nano-scale transceivers, but also manufacturing possibilities using
today’s technology. Therefore, carbon nanotube based transmitters and receivers are considered
as most promising enablers of RF-based communication at nano-scale. However, fundamental
properties of communication via carbon nanotube-based transmitters and receivers have not been
5
investigated in detail, which is the focus of this paper.
In the literature, there are a limited number of network architecture proposals using nanomachinery
and nano-nodes. In [9], high-level network architectures involving communication
via molecular motors, diffusion based calcium signaling, and pheromones have been introduced
and discussed. A bio-hybrid architecture involving bacteria and nanomotors have been introduced
in [28]. The common thread in these proposals is the simplicity in interaction and specificity of
tasks to be performed, which parallels design principles of our vision for nanonetworks. As the
first step of system design, we provide a rigorous characterization of RF-based communication
in nanonetworks.
III. SYSTEM ARCHITECTURE AND MODELING
A. Physics of Nanotube Antennas
A nanotube antenna is composed of a carbon nanotube attached to a cathode. The carbon
nanotube vibrates according to the incident electric fields’ intensity and frequency, which causes
variations in the electron emission from the tip of the carbon nanotube. The induced current
formed at the cathode is detected at the output of the rear end of the nanotube antenna, which
we will refer to as the nanoantenna. In the rest of this section, we briefly describe the physical
behavior of the nanoantenna (for the detailed description, see [16]) and our abstract model for
the nanoantenna as well as our nanoreceiver will be given in the following section.
In Figure 1(a), the schematic of the nanoantenna and actual images of the nanotube when
excited at its resonance frequency and off its resonance frequency are shown. The resonance
frequency f0 of the carbon nanotube is given as f0 = 0.56
L2 qY I
A, where L is the length of the
nanotube, Y is the Young’s modulus, I is the areal moment of inertia with I = ( /4)(r4
o − r4
i )
for a cylinder with outer and inner radii of ro and ri, is the density, and A is the cross
sectional area. Typical values for these parameters, L ≈ 250nm and r ≈ 5nm, results in
resonance frequency range of 10-100 MHz [16]. The amplitude of the vibrations Yo is given by
Yo = qErad/meff
4 2√(f2−f2
0 )2+(ff0/Q)2 , where q is the charge of the tip of the nanotube with q ≈ 3×10−17C,
Erad is the amplitude of the electric field of the incoming transmission, meff = 0.24m is the
effective mass of the nanotube with m being the actual weight, f0 is the resonance frequency,
f is the frequency of the incoming transmission, and Q is the quality factor with typical values
around 500.
6
(a) Nanotube radio receiver (b) Nanotube radio transmitter
Fig. 1. (a) Nanotube radio receiver. Incident EM waves cause oscillations of the tip of the nanotube, changing the electron
emission rate. (b) Nanotube radio transmitter. Field emission from the tip of the nanotube can induce self-oscillations in the
nanotube. In combination with the excess charge in the tip of the nanotube, these mechanical oscillations effectively transmit a
radio signal [17].
The field emission current from the tip of the carbon nanotube is described as Ir = c1A(Eext)2e
−
c2
Eext ,
where A is the area from which the nanotube emits electrons, Eext is the external applied electric
field, c1 and c2 are constants specific to nanotube behavior, and is the local field enhancement
factor. The field enhancement factor can be approximated as = 3.5+h
r , where h is the height of
the tip of the nanotube above the cathode (see Figure 1(a)) and r is the radius of the nanotube. As
the nanotube vibrates, the height of its tip oscillates resulting in a time-varying field enhancement
factor (t) = 0+ (t). Expanding to second order in powers of (t)/0 and filtering out the
zeroth and first powers of (t)/0, which correspond to dc and radio frequency terms, yields
Ir(t) = I0(1 + + 2/2)( (t)/0)2; = c2
0Eext
.
The above described setup accounts for the reception of an RF signal using a carbon nanotube.
Similarly, in [17], the basic setup for a nanomechanical transmitter is depicted. The main idea
is to have the nanotube carry a charge at its tip and have it mechanically oscillate, effectively
producing electromagnetic waves. To this end, the property of the nanotube that mechanical
self-oscillations can be induced in a single-clamped nanoscale resonator by applying only a dc
voltage is leveraged. Self-oscillations are dependent on field emission from the nanotube to a
counter electrode. This concept can be applied to the nanotube transmitter by adjusting Vbias,
in Figure 1(b), to a dc voltage that will cause both field emission and self-oscillations in the
7
nanotube. The nanotube will oscillate at the mechanical resonance frequency. By changing the
tension on the nanotube with a voltage on the Vtension electrode in Figure 1(b), the nanotube would
bend and therefore change the resonant frequency. The information signal could be applied to
this electrode to modulate the frequency of the self-oscillations.
B. Nanoreceiver Model
The abstract model for our nanoreceiver based on the physical properties of the described
nanoantenna is given in Fig. 2. The basic components of the front end include the nanoantenna
and the square-law detector. Here, hr(t) is the impulse response of the linear filter that captures
the input-output behavior of the nanoantenna, where the input Yi(t) is the incoming electromagnetic
field and the output Yo(t) is the amplitude of the associated vibrations. The frequency
response of the nanoantenna was given in the previous section to be
Hr(f) =
q/meff
4 2p(f2 − f2
0 )2 + (ff0/Q)2
. (1)
Since the antenna response is symmetric with respect to the resonance frequency f0, the 3-dB
bandwidth, B, of Hr(f) can be found by solving Hr(f0)/√2 = Hr(f0+B). With the assumption
that f0 ≫ B, one can find B ≈ f0
2Q. Note that the assumption is highly accurate for the typical
values of Q (i.e., between 500-1000). We have the square-law device, since the observed current,
Ir(t) is proportional to the square1 of the amplitude of the vibrations of the nanotube.
We assume that the signal is corrupted by additive white Gaussian noise (AWGN) at two
levels: The acoustic noise, Wa(t), is the mechanical component that affects the amplitude of the
vibrations Yo(t), whereas the thermal noise, WT (t) is added to the detected current. We denote
the two sided power spectral densities of the acoustic noise and the thermal noise with Na/2
and NT /2 respectively.
For a nanoreceiver to be feasible, low complexity is one of the main constraints. At the
nanoscale, even slightly complex components become infeasible. Hence, to achieve node activation,
we use the simple energy detector as shown in Fig. 2. Since the signal Ir(t) is the current
at the output of the front-end of the receiver, the integrator can be realized by a mere capacitor.
The integrator is followed by the sampler, sampling the output of the integrator once every T
1Note that the detected current is proportional to the square of (t), which in turn is proportional to the amplitude Yo(t).
8
( . ) 2 Ir (t)
hr(t)
Yo(t)
+
Ys [k ]
+ ( . )
(t)
1,2 >do not
activate
front end energy detector
t=kT
dt
t−T
Yi t
activate
Wa(t) WT (t)
Fig. 2. System model of our nanoreceiver.
seconds, which we refer to as the activation period. We assume that the activation period is
much longer than the reciprocal of the 3-dB bandwidth of the antenna response. Hence, we have:
1
T ≪ B ≪ f0. (2)
Finally, each sample Ys[k] is compared with a pair of predetermined thresholds 1 and 2 (as will
be explained in the next section) and the node becomes active depending on these comparisons.
The combination of the square-law device and the integrator acts as a demodulator for the
waveform Yo(t). However, note that, Yo(t) also includes the noise component Wa(t), filtered
by the antenna response hr(t). Since we do not have any front-end filter to remove the outof-
band noise components, the performance of the system degrades. This is the price paid to
avoid realizing the filter. Notice that, in our energy detector, the only components we used are a
capacitor, a sampler and a series of comparators. To activate multiple events at a node, we do not
rule out the possibility that a node has multiple energy detectors, with multiple nanoantennas
of different-sizes (and hence different resonance frequencies). This will enable us to activate
multiple functions in the same node, without a need for post-processing of the received signal.
C. Network Model
The network model is based on a low complexity node architecture and their predefined
interactions. The basic structure of a nano-node under consideration consists of power, sensing,
actuation, and transceiver units. Although these components are also found in traditional wireless
sensor nodes, our proposed architecture is significantly different than sensor nodes since a
major component, i.e., processor, is not present. The reason for this exclusion is that a nanoscale
processor with significant computational power is not likely to materialize in the near
future. We acknowledge recent efforts and research attempts in realizing processor components
at nano-scale. As an example, nano-scale circuits and components have been proposed in recent
9
years [11], [29], [30], [31]. However, a computational component as we understand today requires
significant research and development efforts. On the other hand, research on other components
of the node architecture, i.e., energy storage [32], chemical sensors [12], [13], molecular-level
actuators [14], [15] and nanotransceivers [16], [17] as well as simple methods to connect them
into system have matured far more significantly [33]. Each node is assumed to capable of sensing
(with a possible binary outcome), receiving electromagnetic signals (to activate actions in a nanonode),
possibly relay another activation signal, and to act based on the RF-based activation and
local stimuli.
Our hypothesis is that one can achieve significant advancements in development of nano-scale
A
A
ACTIVATOR
f0
carrier
at
inactive
activated
IA IA
(a) Nodes activated centrally
ACTIVATOR
inactive
f0
f0’’
f0’
carrier
at
task 2
task 3
task 1
A
A2
A3
IA
1
(b) Nodes activated centrally and by other nodes for
sequential tasks
Fig. 3. Various nanonetwork architectures for activation of sequential tasks. Multiple antennas enable multiple potential tasks
actiavated for a certain node.
machinery even without using any significant processing power2. The logical operations to
combine received “activation” signals and locally sensed information can be managed through
physical pathways that control the actuators and the transmitter blocks. We consider two main
architectures of nano-networks: The first one is the centralized (cellular) architecture where an
external activation signal is transmitted to a set of nano-nodes as illustrated in Fig. 3(a). Nodes
are activated upon reception of the RF signal and they do not directly interact among themselves.
Hence, the control of the system is given to the centralized external controller that emits the
activation signals. These signals can be transmitted to sequentially activate nano-nodes one by
one or in groups. The selection of the activation sets is determined by the resonance frequency
2If and when more sophisticated processing capabilities are available, the effectiveness of such systems would improve, without
diminishing the contributions of the proposed work.
10
of the antennas, and consequently, the transmission frequencies. The second architecture is a
multi-hop architecture where nodes are not only activated from an external source but can also
activate each other as shown in Fig. 3(b). A multi-hop architecture is crucial to accomplish nontrivial
sequences of tasks. Combined with the local sensing and actuation capabilities, a network
of nodes can be designed to perform tasks in an event-driven manner, accounting for local
conditions as well as activation signals external and internal to the nanonetwork. Furthermore,
the interactions between nodes can be designed to implement condition-based branching and
loops to achieve target stimulus levels without the involvement of an outside controller in the
loop.
IV. SINGLE NODE ACTIVATION
In this section, we present a communication theoretic analysis of our receiver, given in Fig. 2.
We will consider a single link and provide fundamental limitations involved in node activation.
In particular, we will analyze the necessary signal duration T to meet a desired probability of
successful activation. This gives us an idea of how much energy is required to achieve a certain
task. We use this analysis as an initial point for further system design, including the tradeoffs
involved in the number of tasks achievable in a given nanonetwork, which we consider in the
next section.
To activate a node, we assume that the activator uses pure sinusoids of duration T. Consequently,
at the input of the the receiver, we have Yi(t) = a cos(2 f0t + -), where - is the
random phase. The most energy-efficient way of activating a node is to choose the frequency of
the sinusoid, identical to the resonance frequency f0 of the nanoantenna.The signal at the output
of the antenna is thus,
Yo(t) =

a cos(2 f0t + -)Hr(fo) + ˜Wa(t), activation attempt
˜Wa(t), otherwise
, (3)
where Hr(f) is the frequency response of the nanoantenna and the acoustic white Gaussian
noise, filtered by the antenna response is denoted by ˜Wa(t), which is also a Gaussian process.
Hence, the pre-thermal noise portion of the current at the output of the front-end of the receiver
11
can be written as
Ir(t) =



a2Hr(f0)2 cos2(2 f0t + -) + 2aHr(f0) cos(2 f0t + -) ˜Wa(t)
+ ˜W 2
a (t), activation attempt
˜W
2
a (t), otherwise
.
(4)
The energy detector integrates Ir(t) + WT (t) over the past T seconds and a sampler samples
the output of the integrator every T seconds. We initially disregard the issues of timing and
frequency mismatch between the activator and the nanoreceiver in the following analysis. We
will analyze the impacts of these imperfections later on.
One can realize that there are three components of Ir(t) under the activation attempt, as
given in Eq. 4. The first one is the signal component, the second one is the signal-noise cross
component, which is a Gaussian process, and the last one is the noise-noise cross component,
which has Chi-squared samples. Next we analyze the contribution of each component as well
as the thermal noise to the detected sample Ys[1] under an activation attempt in the scheduling
period k = 1:
(1) Signal component: Given an activation attempt in the scheduling period k = 1, the signal
component of Ys[1] can be written as:
Y (s)
s [1] = Z T
0
a2Hr(f0)2 cos2(2 f0t + -) dt =
1
2
Ta2Hr(f0)2. (5)
Note that phase recovery comes for free due to the square-law device and hence our nanoreceiver
avoids the associated complex circuitry for that task.
(2) Signal-noise cross component: Given an activation attempt in scheduling period k = 1, the
signal-noise cross component can be written as:
Y (s-n)
s [1] = Z T
0
2aHr(f0) cos(2 f0t + -) ˜Wa(t) dt. (6)
Since ˜Wa(t) is a Gaussian process, Y (s-n)
s [1] is a Gaussian random variable with mean E Y (s-n)
s [1] =
0. To find the variance, we calculate
var 􀀀Y (s-n)
s [1] = E Z T
0 Z T
0
4a2Hr(f0)2 cos(2 f0t + -) cos(2 f0 + -) ˜Wa(t) ˜Wa( ) dt d
= 4a2Hr(f0)2 Z T
0 Z T
0
1
2
[cos(2 f0(t + ) + 2-)
+cos(2 f0(t − ))]K˜Wa(t − ) dt d , (7)
12
where K ˜Wa(·) is the autocovariance function of the filtered noise process ˜Wa(t). The associated
power spectral density can be written as S ˜Wa(f) = Na
2 Hr(f)2. Here, let us define ˆHr(f) such
that
Hr(f)2 = Hr(f0)2 ˆHr f − f0
B + ˆHr f + f0
B , (8)
where B = f0/2Q is the 3-dB bandwidth of Hr(f). Thus, ˆHr(f) is the the baseband representation
of a sidelobe (sidelobes are symmetric) of Hr(f)2, normalized to have a unit gain
at DC frequency and a unit 3-dB bandwidth. Hence, the time-domain response of Hr(f)2 is
2 cos(2 f0t)Bˆhr(Bt)Hr(f0)2, where ˆhr(t) is the inverse Fourier transform of ˆHr(f). Consequently,
the autocovariance function of the ˜Wa can be written as
K˜Wa(t) = NaBHr(f0)2 cos(2 f0t)ˆhr(Bt). (9)
One can realize that the variance of white noise, filtered by the antenna response is 2
˜W
a
=
NaBHr(f0)2, where BHr(f0)2 can be viewed as the “energy” of the filter response. With
this, we can evaluate the variance of the signal-noise cross component as:
var 􀀀Y (s-n)
s [1] = 2a2NaHr(f0)4 Z T
0 Z T
0 cos2(2 f0(t − ))
+cos(2 f0(t − )) cos(2 f0(t + ))]Bˆhr(B(t − )) dt d
= 2a2NaHr(f0)4 Z T
0 Z T
0
1
2
[1 + cos(4 f0(t − ))]Bˆhr(B(t − )) dt d
+ Z T
0 Z T
0
1
2
[cos(4 f0t) + cos(4 f0 )]Bˆhr(B(t − )) dt d (10)
≈ a2NaHr(f0)4 Z T
0 Z T
0
Bˆhr(B(t − )) dt d (11)
≈ Ta2NaHr(f0)4, (12)
where (11) follows since the integral of the cosines are inversely proportional to f0 and they
become negligible with respect to the integral of the constant term. Also, (12) follows since
1/B ≪ T and thus Bˆhr(B(t− )) is identical to 0 for almost all pairs of (t, ) except for those
that are very close to each other. Since the are under ˆhr(t) is 1, Bˆhr(B(t − )) acts as a unit
impulse function (t − ).
As a result, Y (s-n)
s [1] ∼ N(0, Ta2NaHr(f0)4). Note that, the strategy of increasing the transmit
signal power, a2/2, in order to reduce the time T to activate a node generally fails due to the
13
signal-noise cross component, since the noise level is also amplified by the the signal amplitude
a.
(3) Noise-noise cross component: Regardless of whether there is an activation attempt in
scheduling period k = 1, the noise-noise cross component will be observed at the output of
the antenna. The contribution of the noise-noise cross component on the sample Ys[1] can be
found as:
Y (n-n)
s [1] = Z T
0
˜W
2
a (t) dt. (13)
The noise-noise cross component has a non-zero mean: E Y (n-n)
s [1] = T 2
˜W
a
= TNaBHr(f0)2.
To find the variance, we note that ˜W 2
a has a power spectral density with a 3-dB bandwidth
identical to 2B. Since 1/2B ≪ T, for any pair (t, ) ∈ (0, T)2, K ˜W 2
a
(t − ) is very close to 0,
unless t ≈ . Due to the large bandwidth, we write K ˜W 2
a
(t − ) ≈ 2
˜W
2
a
(t − ). Consequently,
var 􀀀Y (n-n)
s [1] = Z T
0 Z T
0
K ˜W 2
a
(t − ) dt d
≈ Z T
0 Z T
0
2
˜W
2
a
(t − ) dt d
= Z T
0 Z T
0
2 2
˜W
a
(t − ) dt d (14)
= 2T 􀀀BNaHr(f0)2 2
. (15)
One cannot disregard the noise-noise cross component, since it can potentially be large due to
the lack of a front-end filter3.
(4) Thermal noise: The contribution of thermal noise on the sample Ys[1] can be found as:
Y (T)
s [1] = Z T
0
WT (t) dt. (16)
Clearly, E Y (T)
s [1] = 0 and var 􀀀Y (T)
s [1] = T NT
2 .
With the above observations, we can write the following conditional distributions for Ys[1],
using a Gaussian approximation for the noise-noise cross component:
3Note that the nanoantenna acts as a front-end filter to some extent. However, the bandwidth of the antenna response is fairly
wide (depends on the Q factor) and it lets a significant amount of noise through.
14
Ys[1]
pdf of
t2 t1
no activation
attempt
activation
attempt
Fig. 4. The pdf of the detected signal under activation attempt and no activation attempt.
Given an activation attempt,
Ys[1] ∼ N

THr(f0)2(a2/2 + NaB)
{z } μp
, T NaHr(f0)4(a2 + 2B2Na) +
NT
2
{z } 2
p
;
Given no activation attempt,
Ys[1] ∼ N

THr(f0)2NaB
{z } μn
, T 2 􀀀BNaHr(f0)2 2 +
NT
2
{z } 2n

.
In the subsequent analysis, we deal with the probability of two events: unsuccessful activation
attempt and false activation. In the former, the activator attempts to activate a node, but the
node remains inactive, whereas in the latter, the node goes active without an activation signal. We
define the optimal detector [34], [35] as the one that minimizes the probability of unsuccessful
activation attempt, pua, subject to a given probability of false activation, pfa. Since the
signal-noise cross component is 0 without the activation signal, the total noise variance differs
with and without the activation attempt as shown in Fig. 4. Thus, the optimal detector involves
comparisons with multiple thresholds. Let the us define SNRa , a2
2Na
as the signal to acoustic
noise ratio and SNRT , a4
4NT
as the (power of the) observed current to thermal noise ratio. We
also define pa as the prior probability for an activation attempt in any given activation period
and probability of error as the total probability of an undesirable event: pe = papua+(1−pa)pfa.
Next, we present the detector performance in what follows.
Fig. 5 illustrates the performance of the nanoreceiver with our energy detector. To obtain these
15
10−1 100 101 102 10−7
10−6
10−5
10−4
10−3
10−2
10−1
activation time (secs)
probability of error (p
e
)
SNR
a
=10 dB
SNR
a
=15 dB
SNR
a
=20 dB
(a) pe vs. T
10 15 20 25 30
10−10
10−5
SNR
a
(dB)
probability of error (p
e
)
SNR
T
=0 dB
SNR
T
=5 dB
SNR
T
=8 dB
acoustic
noise limited
thermal
noise limited
(b) pe vs. SNRa
10−10 10−5 100 10−10
10−8
10−6
10−4
10−2
100
probability of false activation (p
fa
)
probability of uncuccessful activation (p
ua
)
T=2.5 sec
T=1.5 sec
T=0.5 sec
(c) pua vs. pfa
Fig. 5. System performance and various tradeoffs and are illustrated for a single nanoreceiver.
curves, we chose the parameters of the system as f0 = 15 MHz, Q = 1000 and pa = 10−3.
We also assumed that 1
T ≪ B ≪ f0, where B = f0/2Q is the 3-dB bandwidth of the antenna
response Hr(f).
In Fig. 5(a), we plot the probability of error as a function of the activation time T for various
values of SNRa. Here we assume that the dominant source of noise is the acoustic noise, i.e.,
SNRa ≪ SNRT . One can find that the maximum likelihood decision rule activates the node if
Ys[1] > (ML)
1 or Ys[1] < (ML)
2 , where the thresholds (ML)
1 and (ML)
2 satisfy
(ML)
1 − μn
n
= rT
2
B
−1 +vuut 1 +
SNRa
B 1 +
2
TSNRa
log " 1
pa − 1 2 1 +
SNRa
B #!
 ,
(17)
(ML)
2 − μn
n
= rT
2
B
−1 −vuut 1 +
SNRa
B 1 +
2
TSNRa
log " 1
pa − 1 2 1 +
SNRa
B #!
 .
(18)
Note that, here we neglect TNT /2 in 2n
, since acoustic noise is assumed to be the dominant noise
source. The minimum error probability (pe associated with the maximum likelihood detector)
p(ML)
e = pap(ML)
ua + (1 − pa)p(ML)
fa , where
p(ML)
ua = (ML)
1 − μp
p − (ML)
2 − μp
p , (19)
p(ML)
fa = (ML)
2 − μn
n + 1 − (ML)
1 − μn
n . (20)
16
One can find evaluate ( (ML)
i − μp)/ p for i = 1, 2 using ( (ML)
i − μn)/ n as given in Eq. (17,18)
as follows:
(ML)
i − μp
p
= (ML)
i − μn
n −rT
2
SNRa
B ! r1 +
SNRa
B2 !−1
. (21)
One can observe from Fig. 5(a) that, roughly with every 5 dB increment of the SNR, the
activation time necessary to achieve a certain probability of error decreases by an order of
magnitude. It is notable that, with SNRa = 20 dB, only 2 secs is sufficient to achieve a pe < 10−6.
In Fig. 5(b), we illustrate the probability of error as a function of SNRa for various values
of SNRT and activation time T = 1 sec. For low values of SNRa, the acoustic noise is the
determining factor for pe, while the thermal noise starts to dominate as SNRa is increased beyond
a certain point. One can realize that, the impact of thermal noise is relatively less detrimental,
compared to that of the acoustic noise. Indeed, even an SNRT as low as 5 dB allows for
pe = 10−7, if SNRa is sufficiently high. The reason for the impact of acoustic noise being more
severe is that, due to the lack of a front-end filter, a big portion of the cross components cannot
be filtered out by the integrator.
In Fig. 5(c), we illustrate the receiver operating characteristics (ROC); i.e., minimum pua
achievable by our nanoreceiver for a given pfa a la Neyman-Pearson criterion [34]) for various
values of the activation time T. Here, we assume that the acoustic noise is the dominant noise
source and we take SNRa = 20 dB. To plot these curves, we used the pair of thresholds (NP)
1 and
(NP)
2 , and calculated the associated values of pua and pfa, based on the decision rule, “activate
node if Ys[1] > 1 or Ys[1] < 2” and remain inactive otherwise. The Neyman-Pearson thresholds
(NP)
1 and (NP)
2 turn out to be identical to (ML)
1 as in (17) and (ML)
2 as in (18) respectively, evaluated
as the value of pa is varied in [0, 1]. Note that, this does not imply that the values of pua and
pfa depend on the probability of activation at all. Here, pa is merely a parameter that gives us
the Neyman-Pearson thresholds to find the minimum pua subject to a given pfa. The ROC is
sketched by plotting (19) vs. (20) as the thresolds are varied using pa as described above.
Fig. 5(c) illustrates the reliability of our nanoreceiver. If T = 2.5 secs is chosen as the
activation period, one can achieve a pua = 10−4 at a pfa = 10−8. Note that, at this pfa and
T pair, even over a month of continuous operation, the probability of false activation of our
node remains below 10−2. Even with such a conservative selection, it is possible to achieve a
probability of unsuccessful activation 10−4.
17
We just illustrated that, even with our simple nanoreceiver, the activation of a node with very
10 15 20 25 30
10−10
10−8
10−6
10−4
10−2
SNR
a
p
e
Q=100
Q=500
Q=1000
(a) pe vs. Q
0 1 2 3 4
x 10−3
10−15
10−10
10−5
100
Dw0
/w0
p
e
SNR
a
=20 dB
SNR
a
=23 dB
SNR
a
=25 dB
(b) pe vs. f0/f0
Fig. 6. Sensitivity of system performance with respect to the Q parameter and frequency mismatch.
low probability of error is possible with reasonably short activation signals and at values of
overall signal to noise ratio as low as 15 dB. Moreover, our nanoreceiver is reliable: it can be
designed to operate over months without a false activation event and at the same time to achieve
fairly low probabilities of unsuccessful activation, even with activation periods of as low as a few
seconds. Lastly, the receiver performance is independent of the random phase, -, of the carrier
signal (due to the square-law device). Hence, our nanoreceiver avoids the complex circuitry for
phase recovery.
On the other hand, the performance of our nanoreceiver is highly sensitive to the Q parameter.
To illustrate this, in Fig. 6(a), we plot the probability of error as a function SNR for various values
of the Q parameter of the nanoantenna for T = 1 sec. Recall that the Q parameter is related
to the 3-dB bandwidth, B of the nanoantenna response by equation B = f0/2Q. Therefore, the
higher the bandwidth, the worse the performance since the noise power allowed is proportional
to the bandwidth.
Also, frequency mismatch is an important factor that affects the system performance. In
Fig. 6(b), we plot the probability of error as a function of the amount of frequency mismatch,
f0, normalized with respect to the resonance frequency, f0, of the nanoantenna for various
values of SNRa at T = 2.5 sec. Here, we take Q = 500, f0 = 15 MHz, hence B = 15 kHz.
One can observe that, even with a 0.1% of frequency mismatch with respect to the resonance
frequency, the probability of error increases by multiple degrees of freedom. This degradation
will be even more severe for nanoantennas with higher Q factors (even though the improvement
18
is also significant). The associated power penalty for a 0.1% increase in the frequency mismatch
can be as high as 2 dB at values of SNRa around 20 dB.
These results illustrate the feasibility of individual node activation in real environments using
simple and realizable, and yet reliable nanoreceivers. However, the performance is highly sensitive
with respect to the nanoantenna parameters and possible imperfections due to frequency
and timing mismatch.
We would like to finalize this section noting that one can achieve multiple tasks per node by
waiting for multiple activation periods and interpreting sequences of pulses. Taking this idea one
step further, we can actually use the nanoreceiver for digital data communication. Indeed, the
reciprocal, 1/T, of the activation period of the system can be viewed as the data rate of the
communication system at the associated probability, pe, of error. For instance, a rate of 1 bit/sec
is achievable at a pe = 10−6 at a signal to noise ratio of 20 dB. Data communication enables the
possibility of more complex tasks for each node. We propose to explore nanoreceiver designs
for communication of data. Important fundamental questions include whether there exists more
efficient detectors than the mere energy detector and whether the required computational power
to achieve the task is feasible at the nano level.
V. MULTI-NODE ACTIVATION
In our nanonetwork, various tasks are activated by exciting nanoreceivers at appropriate frequencies.
Our basic system vision allows for a single task to be activated per nanoreceiver. Unless
nodes have sufficient computational capabilities to interpret sequences of bits, the number of
distinct antennas constitute a fundamental limit on the number of different tasks the nanonetwork
can accommodate. In the following, we extend our analysis to a single-hop multi-receiver system.
Each antenna has a different length, and hence a different center frequency. As the number
of tasks we would like to activate increases, the necessary number of antennas with different
center frequencies increases. To accommodate this requirement, the channels need to be stacked
closer and closer to each other. This raises the issue of cross-channel crosstalk: Due to the
non-atomic response of the nanoantennas, antennas with center frequencies close to each other
start to interfere; hence the tradeoff between the communication performance and the number
of different tasks in the network. We first quantify this tradeoff.
At any given time t, consider an activation attempt at a neighboring channel, with an activation
19
signal Y ′
i (t) = a′ cos(2 (f0 + c)t+-′), where c is the frequency spacing between neighboring
channels and -′ is the random phase associated with the carrier of the neighboring channel.
The contribution of this signal at the output of the antenna response is Y ′
o (t) = a′Hr(f0 +
c) cos(2 (f0+ c)t+-′). In what follows, we treat the crosstalk as additive Gaussian noise and
assume Y ′
o (t) ∼ N(0, a′2Hr(f0 + c)2/2). Thus, the observed signal to crosstalk plus acoustic
noise ratio can be written as:
SCNRa =
a2Hr(f0)2
a′2Hr(f0 + c)2 + NaHr(f0)2 .
To plot the following curves, we take identical carrier amplitudes, a′ = a and assume that
all crosstalk, beyond the adjacent channel is negligible. Note that we will also observe in the
following analysis that this assumption is highly reasonable, since the impact of crosstalk from
another channel vanishes fairly quickly beyond a certain frequency spacing.
In Fig. 7(a), we illustrate pua achievable by our nanoreceiver with Q = 1000, for a given pfa
10−15 10−10 10−5 100 10−10
10−8
10−6
10−4
10−2
100
p
fa
p
ua
dc
=10 MHz
dc
=3 MHz
dc
=1 MHz
(a) pua vs. pfa
104 105 106 107 108
10−15
10−10
10−5
100
dc
p
e
SNR
a
=20 dB
SNR
a
=23 dB
SNR
a
=25 dB
(b) pe vs. c
104 105 106 107 108 10−7
10−6
10−5
10−4
10−3
dc
p
e
Q=300
Q=500
Q=800
(c) pe vs. Q
Fig. 7. System performance and various tradeoffs and are illustrated for a multiuser nanonetwork.
(a la Neyman Pearson) for various values of c, at SNRa = 20 dB and T = 2.5 secs. One can
observe that, while the performance remains almost unchanged for values of c higher than 3
MHz, the performance degrades abruptly as c decreases from 3 MHz to 1 MHz. This phase
transition phenomenon somewhat simplifies system design. For this above set of parameters for
instance, a value frequency spacing above 3 MHz is sufficient to achieve a high performance,
and above this value of c, the performance is somewhat insensitive to the variations of c. Recall
that the feasible values for the lengths of nanoantennas enable us to utilize a usable bandwidth
between 10 − 100 MHz. Thus, the number of tasks that the network can handle, utilizing this
roughly 90 MHz of bandwidth is ∼ 30 for the above set of parameters.
20
In Fig. 7(b), we illustrate pe as a function of frequency spacing c at T = 1, for various
values of SNRa and with the assumption that the acoustic noise is the dominant noise source,
Q = 1000 and pa = 10−3. Similar to the ROC plots given in Fig. 7(a), here we can also observe
the phase transition phenomenon that occurs in pe as the frequency spacing decreases. Beyond
a certain point, increasing c does not increase the performance significantly. That cutoff point
(3-5 MHz here) is an appropriate choice for determining the number of different tasks to be
activated in the nanonetwork.
Finally, in Fig. 7(c), we plot pe as a function the frequency spacing, c, of neighboring channels,
for SNRa = 23 dB at T = 1 and pa = 10−3, for various values of Q and with the assumption
that the acoustic noise is the dominant noise source. Similar to the single receiver case, we
observe the sensitivity of the error probability with respect to the Q parameter. This observation
emphasizes the importance of antenna design to achieve a high performance.
To summarize, we observed that here is a fundamental tradeoff between the number of different
tasks that a nanonetwork can execute and reliable communication over the network. Thus, one
has to be careful in planning the tasks and divide the available bandwidth carefully between
each node according to the task they are supposed to execute. Notably, the error performance is
highly insensitive with respect to the frequency spacing, except for a phase transition at a certain
point. One should be careful in order not to remain on the unfavorable side of the error curve
when designing the system. However, the system performance is highly tied to the parameters
of the nanoantenna, hence nanoantenna design is a critical component of network design.
VI. EXAMPLE APPLICATION: CHEMICAL CONCENTRATION CONTROL
In this section, we introduce an application of the basic communication in a system designed
to control the the concentration of a chemical in an environment. This example is a simple
nanonetwork that can be enabled by the proposed communication paradigm. The envisioned
system is composed of a nanonode that can release a predetermined amount of a chemical
when activated via a nanoreceiver (Node A2 in Figure 8), and another nanonode that measures
the concentration of the chemical and emits an activation signal via its nanotransmitter if the
measured concentration is below a particular threshold (Node A1 in Figure 8). The information
flow from A2 to A1 occurs via release and measurement of chemicals in the medium. We refer
to this mode of communication as communication over the process channel. In the following,
21
we will analyze this control system with the assumption that the only source of uncertainty is
the communication channel. The process channel is assumed to work perfectly, and so do the
nanonodes in their measurement and chemical release functions.
Fig. 8. Example nanonetwork architecture controlling a chemical’s concentration.
At any integer time value t, the concentration of the chemical in interest is measured to be
Ct. Between each fixed activation period of length D, let the concentration decay to (D) of
its previous value, where 0 < (D) ≤ 1. The system depicted in Figure 8 aims to maintain
the concentration at a desired level Cdes. The system accomplishes this by releasing a fixed
amount of chemicals that raises the concentration level by . Consequently, the concentration
Ct measured at time t is given by
Ct = (D) Ct−1 + 􀀀a(t − 1)1Pca(D) + (1 − a(t − 1))1Pfa(D) , (22)
where (D) is the rate of decay, a(t) is 1 if A2 is activated to release its chemical and 0
otherwise, Pca(D) is the probability of correct activation, defined as Pca(D) , 1 − p(ML)
ua (see
Equation 19) and Pfa(D) is the probability of false activation, defined as Pfa(D) , p(ML)
fa (see
Equation 20), both for an activation period of D. A large D improves the probability of correct
reception and reduces probability of false activation4. At the same time, a large D value decreases
, which affects the control accuracy negatively. Hence, it is expected that there exist an optimal
value of D. Similarly, is another design parameter that affects the expected error term. The
4Note that p(ML)
ua and p(ML)
fa values are also functions of (ML)
i , i = 1, 2. Hence, for the same D value, multiple (p(ML)
ua , p(ML)
fa )
values can be computed for different values of (ML)
i , i = 1, 2.
22
problem of minimizing the expected error over the operation time can be formulated as follows:
min
,D
E min
a(t−1)
E (Ct − Cdes)2 Ct−1 . (23)
Deferring the discussion on the minimization of the error term over D and to a later point,
we first concentrate on the minimization of expected error for a given system (i.e., fixed D and
) given Ct−1. In other words, we would like to minimize the expected error term over the
activation actions, i.e., mina(t−1) E [(Ct − Cdes)2 Ct−1]. Since D does not change at operation
time, we drop this notation in the following discussion. The decision problem can be written as
E (Ct − Cdes)2 Ct−1, a(t − 1) = 1
a=0
R
a=1
E (Ct − Cdes)2 Ct−1, a(t − 1) = 0 . (24)
Substituting Equation 22 above, we obtain
E ((Ct−1 + 1Pca) − Cdes)2 Ct−1
a=0
R
a=1
E ((Ct−1 + 1Pfa) − Cdes)2 Ct−1
2Ct−1(Pca − Pfa) + 2(Pca − Pfa)
a=0
R
a=1
2Cdes(Pca − Pfa)
(2 Ct−1 + − 2Cdes) (Pca − Pfa)
{z } >0
a=0
R
a=1
0. (25)
In Equation 25, any reasonably designed system that has a greater correct activation probability
than false activation probability would satisfy the inequality independent of the (Pca−Pfa) term.
Therefore, given a measured concentration Ct−1, the action that minimizes the expected error is
computed as follows:
Ct−1
a=0
R
a=1
Cdes
−
2
, c, (26)
where c is the threshold below which the node A2 should be activated. It is worth noting that
the decision threshold is independent of the correct and false activation probabilities for any
reasonably well designed system, though dependence on the delay associated with activation is
implicitly present through the parameter . When Ct−1 > c, the expected error term can be
computed as
E (Ct − Cdes)2 Ct−1 > c = (Ct−1 − Cdes)2 + Pfa[(2Ct−1 + ) − 2Cdes]. (27)
Similarly, when Ct−1 < c, we obtain
E (Ct − Cdes)2 Ct−1 < c = (Ct−1 − Cdes)2 + Pca[(2Ct−1 + ) − 2Cdes]. (28)
23
The minimization problem of Equation 23 requires the computation of the expectation of
the conditional error term E mina(t−1) E [(Ct − Cdes)2 Ct−1] , which in turn depends on the
distribution of the Ct. If this expectation is known as a function of and D, then the problem of
Equation 23 can be solved implicitly or using numerical methods and optimal design values of
and D can be determined. Here, instead of analyzing the distribution of Ct, we provide numerical
results obtained through simulations. Results depicted in Figure 9 depict the mean square error
as a function of the parameter for four different values of D. The target concentration is
selected as Cdes = 2 and the concentration evolution rate as (D) = 0.995D. Other system
related parameters include Q = 1000, center frequency 90MHz, and acoustic SNR 15dB. For a
given D value, we scan the (ML)
i , i = 1, 2, threshold values and select the pair that minimizes
the MSE.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
10−4
10−3
10−2
a
MSE
D = 0.25 sec
D = 0.50 sec
D = 1.00 sec
D = 2.00 sec
D = 4.00 sec
Fig. 9. Mean Square Error for a sample system.
The results in Figure 9 suggest that smaller values of require smaller reaction times, i.e.,
smaller D to minimize the error. Although a small D value also reduces the correct activation
probability, it is compensated by a larger rate and rapid release of additional chemicals when
needed. On the other hand, if the system is capable of delivering large quantities of chemicals
in a single shot, i.e., is large, then D must be increased, as well. A large D value means that
the decay will be faster, and when the system needs to release additional chemicals, this must be
done with high probability of correct activation. Also, we observe diminishing minimum MSE
values as D increases from 0.25 to 2 with the appropriate selection of the parameter. However,
the minimum MSE value for D = 4 is greater than that for D = 2, which suggests that there is
an optimal combination of D and that minimizes MSE for a given system.
24
VII. CONCLUSIONS AND FUTURE WORK
In this work, we have presented an analysis of a nano-scale communication systems based
on carbon nanotube antennas. The system operation is based on the mechanical vibrations of
the carbon nanotube when subjected to electromagnetic radiation. The mechanical response is
primarily a function of the frequency of the EM waves and the length of the carbon nanotube,
among other factors. The resulting communication system consists of a front end composed of
the carbon nanotube antenna and an energy detector.
We explored the detection theoretic capabilities of this simple and practical receiver, used to
enable basic binary tasks that involve activation of nanodevices using pure carrier signals tuned
to the resonance frequencies of the nanoantennas. We showed that, at reasonable values of SNR,
our receiver is capable of being activated successfully with very high probability within a matter
of few seconds and at the same time it avoids false activation even for periods of operation
as long as several months. We also showed that a network of nanoreceivers can handle a large
number of distinct tasks, activated simultaneously over a shared medium without a significant
detriment in reliability. These analyses gave us a conceptual justification and provided design
guidelines to accomplish more sophisticated tasks via interactions among multiple nanonodes
equipped with carbon nanotube-based communication devices. To further illustrate this point, we
presented an example application scenario where two nodes interact to control the concentration
of a chemical in a given environment.
This work is the first step to develop and analyze more sophisticated nano scale networks.
Based on the properties explored in this paper, it is possible to construct multi-hop networks consisting
of nanonodes that can accomplish more complex tasks without increasing the complexity
of individual nodes. To that end, the interaction between nodes and the supporting communication
subsystem must be placed under the loupe. The example presented in this work is only a simple
example in that direction. Most importantly, a general framework that identifies key constructs
of such networked systems and establishes their interactions is most desired. Furthermore,
system-wide analysis of latency, application-level accuracy and reliability are essential pieces
of information to foster the design of nanonetworks. In our future work, we will explore these
aspects with an emphasis on effects deployment of media and scenarios