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\begin{document}
\title{  Modeling and Analysis of Interference in Listen-Before-Talk Spectrum Access Schemes \\
%        {\normalsize  date: 12/05/05}
} \vspace{2cm}
\begin{small}
\author{
 \begin{center}
  \begin{tabular}{lll}
    \multicolumn{3}{l}{Alexe E. Leu$^1$, Mark McHenry$^1$, and Brian L. Mark$^2$,} \\
          & & \\
         $^1$Shared Spectrum Company & \hspace{10mm} & ECE Dept., MS 1G5   \\
         1595 Spring Hill Road   & \hspace{10mm} & George Mason University \\
         Vienna, VA 22182   & \hspace{10mm} & Fairfax, VA 22030Suite 300
   \end{tabular}
  \end{center}
}
\end{small}


\maketitle

\setcounter{page}{1}

\setcounter{totalnumber}{8}


\begin{abstract}
Spectrum measurement studies have shown that substantial portions of the
allocated wireless spectrum are highly underutilized.  Frequency agile radios (FARs) have
the potential to make opportunistic
use of such spectrum holes without causing harmful interference
to the users of the allocated spectrum.  Toward this goal, we develop
a framework for modeling the interference caused by FARs employing spectrum
access mechanisms based on the simple Listen-Before-Talk (LBT) scheme.
Two variations of LBT are considered:  individual LBT, whereby the
FARs act independently of each other, and collaborative LBT, whereby the FARs
communicate with each other in order to more accurately identify the spectrum
holes.  Our analysis of the
LBT scheme reveals the fundamental interdependencies among key system
design metrics and provides a basis for analyzing more complex
spectrum access methods.  In particular, the analysis of LBT
provides a lower bound on the capacity gain achievable by FARs employing spectrum sharing schemes.
Our numerical results show that even the individual LBT scheme
can provide capacity gains, while even more gain can be achieved
using the collaborative LBT schemes.  Our analysis suggests
that much greater gains should be achievable via
spectrum access schemes that incorporate location information
and/or more sophisticated group behaviors.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

In conventional wireless systems, the wireless spectrum is statically allocated
among various transmitters located over a geographic coverage area.  Recent studies
of wireless spectrum usage~\cite{McHenry03,Staple04} suggest that significant portions of
the wireless spectrum are highly underutilized.  An open research
question is whether such ``spectrum holes'' can be exploited
without causing harmful interference to the primary users, i.e.,
transmitters and receivers, of the allocated wireless spectrum.  In this paper,
we model the interference caused to the primary users by frequency agile radios (FARs),
which attempt to identify and make use of the spectrum holes by using an access
mechanism called Listen-Before-Talk (LBT).

Frequency agile radios~\cite{McHenry03}, also known as spectrum agile radios, have the
ability to transmit and receive signals on dynamically tunable frequency
ranges.  Spectrum agility is a key property of the next
generation of cognitive radios~\cite{Haykin05,Mitola99}.
If a group of FARs could detect the presence of
a spectrum hole in a given frequency range,
the group would be able to communicate on frequency
channels lying within the hole.  In this scenario, the FARs
must transmit with sufficient power to communicate with each other, but
must not cause harmful interference to the primary users.  Identification
of spectrum holes is made possible by highly sensitive detectors~\cite{Leu05}

The FARs may collaborate with each other to identify the spectrum holes.
In general, the FAR nodes do not communicate with the primary users.
For this reason, the primary nodes are sometimes referred to as
{\it non-cooperative} nodes.
The non-cooperative nodes may be divided into transmitters and receivers.
We refer to the transmitters as primary transmitters and
the receivers as {\it victim} nodes, to emphasize that the FARs
may cause some interference to these receivers.  A spectrum sharing
access scheme must ensure that such interference is maintained
below a certain tolerable threshold.

The main objective of this paper is to model the interference to the
victim nodes caused by a group of FAR nodes employing two variations of
the LBT scheme:  individual LBT and collaborative LBT.  In the individual
LBT scheme, each FAR node listens to a given frequency channel for
a short time interval called the {\it listen period} before
attempting to transmit on the channel.  If the received signal
measurement at the FAR node exceeds a certain threshold, called
the {\it detection threshold}, the FAR node abandons
the current frequency channel and moves on to a different channel.
Otherwise, the FAR node transmits on the channel for a short time
interval called the {\it transmit period}.  In the collaborative
LBT scheme, if at least one FAR node in a group detects the
presence of a signal in a given frequency channel
during its listen period, all FAR nodes
in the group are alerted to refrain from transmitting on the channel.
Collaborative LBT requires the exchange of special alert messages
among the FAR nodes.  Other variations of LBT are possible, but
modeling the interference caused by
these simple LBT schemes provides a basis
for studying more sophisticated spectrum access schemes (cf. \cite{McHenry05}).

Most of the previous research on resource management for wireless networks
has focused on systems in which all of the nodes may
cooperate with one another, typically in an infrastructured
environment (cf.\ \cite{Fong98}).  By contrast, the scenario
considered in this paper divides consists of two subsystems:
a non-cooperative system consisting of the set of primary
users and a cooperative system consisting of the set of FAR nodes.
The goal of the FAR nodes is to make use of the wireless
spectrum that is unused by the primary users without causing
harmful interference.  Thus, an analysis of the interference to the victim
nodes caused by the FAR nodes employing a spectrum access scheme such as LBT is
an important step in developing effective spectrum sharing
technologies.

The remainder of the paper is organized as follows.
Section~\ref{s:LBT} describes the LBT schemes in further
detail.  Section~\ref{s:int_model} describes
the basic model of interference between
the FAR nodes employing LBT and the primary users.  Analytical
expressions for the probability of interference for individual LBT
are derived in section~\ref{s:individual_LBT}.  Section~\ref{s:collaborative_LBT}
extends this analysis to the collaborative LBT scheme.
Section~\ref{s:numerical_results} presents numerical
results showing the performance of the two types of
LBT schemes under a wide range of parameter settings.
Finally, the paper is concluded in section~\ref{s:conclusion}.


\section{Listen-Before-Talk Spectrum Access}
\label{s:LBT}

\subsection{Overview of Spectrum Access Methods}

Spectrum methods can loosely be classified as cooperative
or noncooperative~\cite{McHenry05}.  \nocite{Raychaudhuri03,Popescu03}

\subsection{Individual LBT}

\begin{figure}
  \centering
  \includegraphics[scale=0.35,clip=true]{\FIGDIR/lbt.eps}
  \caption{State transition diagram of individual LBT.}
\label{f:lbt}
\end{figure}

The Listen-Before-Talk (LBT) algorithm is a simple scheme for an
FAR~node to access a radio frequency channel opportunistically.
Consider a radio frequency channel~$c$ centered at the carrier
frequency $f_c$ and spanning the range $[f_c - \Delta_f, f_c +
\Delta_f]$. Thus, the bandwidth of the channel is given by $B_c =
2\Delta_f$.  The LBT scheme consists of two states:  1) {\it listen}
or {\it off} state and 2) the {\it talk} or {\it on} state. During
the listen state, FAR node does not transmit a signal and estimates
the received signal power $R$ in the radio channel~$c$, given by
\begin{align}
R = \int_{f_c - \Delta}^{f_c+ \Delta} G_R(f) df ,
\end{align}
where $G_R(f)$ is an estimate of the (one-sided)
power spectral density of the received
signal.

In the {\it off} state, the FAR~node also estimates a transmit power
level, $s^*$, which we refer to as the {\it maximum
interference-free transmit power} (MIFTP).  The MIFTP is defined as
the maximum power at which the FAR~node can transmit without causing
harmful interference to any of the victim nodes.  If the received
signal power $R$ from the primary transmitter falls below a value
$\eta$ called the {\it detection threshold}, the FAR~node transitions
to the {\it on} state. Otherwise, the FAR~node remains in the off
state either for the same channel~$c$ or a different frequency
channel~$c'$. For simplicity, our analysis shall only consider the
case where the FAR~node stays in the same channel~$c$. During the on
state, the FAR~node transmits at the MIFTP power level $s^*$ for a
maximum duration $\tau_{max}$, and then returns to the off state.
During the off state, the FAR~node continues to listen to the channel
and returns to the off state if $R < \eta$.  Fig.~\ref{f:lbt}
illustrates the LBT algorithm by means of a state transition
diagram.

In this paper, we shall ignore the channel contention among FAR~nodes
accessing the same channel.  Such contention can be resolved using a
suitable medium access control (MAC) protocol. Thus, the maximum
transmission duty cycle of an FAR~node on a given channel is given by
\begin{align}
d_{max} = \frac{\tau_{on}}{\tau_{on} + \tau_{off}} , \label{e:duty_cycle}
\end{align}
where $\tau_{off}$ is the average duration in the off state.
Clearly, the amount of interference caused to the victim nodes
by an FAR~node performing LBT depends on the values
of the key parameters $\eta$, $\tau_{on}$,
and the MIFTP $s^*$.  The interpendencies among these parameters
are analyzed in section~\ref{s:individual_LBT}.

\subsection{Collaborative LBT}

Collaborative LBT extends the individual LBT scheme
by allowing a group of FAR~nodes to share information gathered
during the listen period with each other via alert messages.
In collaborative LBT, each FAR~node in the group executes
the individual LBT algorithm for a given frequency
channel~$c$ as discussed earlier with the following modification.
If at least one FAR~node in the group detects the presence
of a signal from a primary transmitter, then all of the
FAR~nodes in the group turn off and revert to the listen state.
Otherwise, the channel~$c$ is then accessed by the group
of FAR~nodes using a medium access control (MAC) protocol.
Again, we shall not concern ourselves
with the details of the MAC protocol used among the FAR~nodes.
We shall simply choose a particular FAR~node in the group and assume that
it can transmit at MIFTP during its talk period without
causing interference to the other FAR~nodes.


\section{Interference Model}
\label{s:int_model}

In this section, we develop a simple model to characterize
the interference caused by an FAR~node to a victim receiver.

\subsection{Interference scenario}

\begin{figure}
  \centering
  \includegraphics[scale=0.35,clip=true]{\FIGDIR/int_scenario.eps}
  \caption{Interference scenario.}
\label{f:int_scenario}
\end{figure}


Consider a scenario consisting of a three nodes:  a primary
transmitter~$p$, a victim receiver~$v$, and a frequency agile node
denoted by $a$, as shown in Fig.~\ref{f:int_scenario}. The
transmitter~$p$ transmits on frequency channel~$c$, while node~$a$
performs LBT on the channel.  The receiver~$v$, expects to receive
the transmissions of node~$p$.  However, node~$a$ may cause
interference to node~$v$ during the talk state of LBT.

Denote the geographic locations of the nodes by $(x_i, y_i)$,
where $i \in \{a, p, v \}$. Let $d_{ij}$ and $L_{ij}$ denote,
respectively the distance and propagation loss
from node~$i$ to node~$j$, where $i, j \in \{ a, p, v\}$.
Clearly, the propagation loss $L_{ij}$
depends on the propagation distance $d_{ij}$.
We model the primary transmitter~$p$ and
the FAR node~$a$ as on-off sources.  In the off-states, both
sources do not transmit signals.  In the on-states, node~$p$
transmits with power $s_p$, while node~$a$ transmits with power $s_a$
in the off-state.  The received signal power from the primary
transmitter~$p$ during its on-state
at nodes~$a$ and $v$ can be expressed as follows:
\begin{align}
R_a &= s_p - L_{p,a}, \label{e:R_a}\\
R_v &= s_p - L_{p,v} . \label{e:R_v}
\end{align}
Similarly, we represent the interference
signal power from node~$a$ received at node~$v$
is given by
\begin{align}
I_v &= s_a - L_{a,v} \label{e:I_v}
\end{align}
Mathematically, all of the quantities in (\ref{e:R_a})-(\ref{e:I_v})
are stochastic processes, although the time parameter~$t$
has been suppressed in the above equations
to emphasize the steady-state condition.

For successful reception of the primary signals from node~$p$ at
node~$v$, certain conditions must be satisfied.
Let $r_{min}$ denote the minimum received signal level
required by the primary receiver~$v$ from the primary transmitter~$p$.
Denote by $i_{max}$ the maximum allowable interference that node~$v$
can tolerate.
Let $E^{(on)}_a$ and $E^{(on)}_p$ denote, respectively, the events that
nodes~$a$ and node~$p$ are in the on state.
Then the event that node~$a$ causes harmful
interference to node~$v$, i.e., the {\it interference event}
can be expressed as follows:
\begin{align}
 E_{int} =  \{ R_v \geq r_{min}, E^{(on)}_p \} \cap  \{ I_v \geq i_{max} , E^{(on)}_a\} .
 \label{e:E_int}
\end{align}
Thus, the interference event occurs if and only if
the primary transmitter~$p$ is on and the received signal
at node~$v$ from $p$ exceeds $r_{min}$ {\it and}
node~$a$ is on and the received interference signal
at node~$v$ from $a$ exceeds $i_{max}$.



\subsection{Propagation model}

The signal propagation loss is modeled as a sum of
three components:  path loss, lognormal shadowing noise, and
Rayleigh fast fading.
We shall assume that the fast fading can be eliminated by
averaging the received signal measurements over time.
Thus, the propagation loss between nodes $i$ and $j$ can be
expressed as
\begin{align}
L_{ij} = D_{ij} + W_{ij} , \label{e:propagation_loss}
\end{align}
where $D_{ij}$ and $W_{ij}$ represent the path loss
and shadowing noise, respectively.  In general, the path
loss is a function several variables:
\begin{itemize}
 \item the locations of nodes~$i$ and~$j$ and the nature of the geographic terrain between them;

 \item the transmitter antenna height $h^{(t)}_i$;

 \item the receiver antenna height $h^{(r)}_j$;


 \item the carrier frequency $f_c$;

 \item the antenna polarization $o_i$ (horizontal or vertical).

\end{itemize}
The shadowing component $W_{ij}$ is assumed to be zero-mean white Gaussian
noise process with variance $\sigma_{ij}^2$ and independent of
the path loss $D_{ij}$.

The path loss $D_{ij}$ component can be modeled using the
empirical propagation model (EPM-73)~\cite{Lustgarten77}
or more complicated propagation prediction models such
as the Longley-Rice model~\cite{Longley68,Longley71} or the TIREM (Terrain Integrated
Rough Earth Model)~\cite{Weiss82}.  For simplicity, we shall assume
the EPM-73 model in this paper.  According to the
EPM-73 model, the mean path loss for a given
terrain type (hilly, swamp, etc.) depends only
on the distance between the two nodes.  Hence, we can write
\begin{align}
D_{ij} = g(d_{ij}, h^{(t)}_{i} , h^{(r)}_j, f_c, o_i) , \label{e:D_ij}
\end{align}
where the function $g(\cdot)$ is given in~\cite{Lustgarten77}.
For notational convenience, we shall simply write $D_{ij} = g(d_{ij})$
and suppress the dependence of the path loss
on the remaining four parameters.
Since the function $g(\cdot)$ is invertible, the distance
can be expressed in terms of the path loss by
\begin{align}
d_{ij} = g^{-1} (D_{ij} ).
\end{align}


\section{Analysis of Individual LBT}
\label{s:individual_LBT}

In this section, we derive expressions for the
interference probability for the individual LBT
schemes based on the interference model discussed
in section~\ref{s:int_model}.  We also derive several
other key system parameters and discuss their performance impacts.

\subsection{Outage probability and coverage distance}


The {\it outage probability} (of node~$v$) is defined by
\begin{align}
P_{out} \triangleq P\{ R_v < r_{min} | E^{(on)}_p \} .
\end{align}
Hence,
\begin{align}
P\{ R_v \geq r_{min} | E^{(on)}_p \} = ( 1 -  P_{out}) . \label{e:Prob_R_v}
\end{align}
Assuming the EPM-73 propagation path loss model,
from (\ref{e:R_v}), (\ref{e:propagation_loss}), and (\ref{e:D_ij}), we can write
\begin{align}
R_v = s_p - g(d_{p,v}) + W_{p,v} . \label{e:R_v2}
\end{align}
Therefore,
\begin{align}
P_{out} =  1 - Q \left ( \frac{r_{min} + g(d_{p,v}) - s_p}{\sigma_{p,v}} \right ) .
\label{e:P_out}
\end{align}
Closely related to the outage probability is the {\it coverage distance}, $d_{cov}$,
defined as the maximum distance between the primary transmitter and the
victim node such that the outage probability does not
exceed a value $\epsilon_{out}$:
\begin{align}
d_{cov}(\epsilon_{out}) \triangleq \max \{ d_{p,v} : P_{out} \leq \epsilon_{out} \} .
\label{e:d_cov_def}
\end{align}
From  (\ref{e:P_out}) and (\ref{e:R_v2}), we can write
\begin{align}
d_{cov} = g^{-1} ( s_p - r_{min} + \sigma_{p,v} Q^{-1} ( 1 - \epsilon_{out} ) ) .
\label{e:d_cov}
\end{align}


We define the {\it detection probability}, $P_{det}$, at the FAR node as
the probability that the received signal from the primary transmitter is
greater than or equal to
a given threshold $\eta$.  That is, $P_{det}$ is given by
\begin{align}
P_{det} = P \{ R_{a} \geq \eta \} =
1 - Q \left ( \frac{s_p - g(d_{p,a}) - \eta}{\sigma_{p,v}} \right ) .
\label{e:alpha}
\end{align}
The {\it detection distance}, $d_{det}$ at the FAR node is defined as the
maximum distance between the FAR node and the primary transmitter such
that the probability of detection at the FAR node exceeds
a value $\epsilon_{det}$:
\begin{align}
d_{det}(\epsilon_{det}) \triangleq \max \{ d_{a,p} : P_{det} \geq \epsilon_{det} \}
\label{e:d_det_def}
\end{align}
Assuming that the function $g^{-1}(\cdot)$ is smooth,
from (\ref{e:alpha}) and (\ref{e:d_det_def}), we have
\begin{align}
d_{det}(\epsilon_{det}) = g^{-1}
  ( s_p - \eta - \sigma_{p,v} Q^{-1} (1- \epsilon_{det}) ) , \label{e:d_det}
\end{align}
The detection distance decreases monotonically with the bound
on the detection probability, $\epsilon_{out}$, which we will show numerically in
section~\ref{s:numerical_results}.


\subsection{Interference probability and distance}

The signal power received by node~$v$ from node~$p$ in the on-state is
independent of the interference power it receives from node~$a$ in
the on-state.
Therefore, from (\ref{e:E_int}) we have
\begin{align}
P_{int} = P\{ R_v \geq r_{min}, E^{(on)}_p \} \cdot
P\{ I_v \leq i_{max} , E^{(on)}_a \} .
\label{e:P_int_prod}
\end{align}
From (\ref{e:I_v}), (\ref{e:propagation_loss}),
and (\ref{e:D_ij}), we can write
\begin{align}
I_v = s_a - g(d_{a,v}) + W_{a,v} .
\end{align}
Therefore,
\begin{align}
P\{ I_v \leq i_{max} | E^{(on)}_a \}
 = Q \left ( \frac{i_{max} + g(d_{a,v}) - s_a}{\sigma_{a,v}} \right ) ,
 \label{e:Prob_I_v}
\end{align}
where $Q(x) \triangleq \frac{1}{\sqrt{2 \pi}} \int_x^\infty e^{-t^2/2} ~dt$
is the standard Q-function.
Using (\ref{e:P_int_prod}), (\ref{e:Prob_R_v}), and (\ref{e:Prob_I_v}),
we obtain the following expression for the interference probability:
\begin{align}
P_{int} = ( 1 -  P_{out}) \cdot P(E^{(on)}_p) P(E^{(on)}_a) \nonumber \\
\cdot Q \left ( \frac{i_{max} + g(d_{a,v}) - s_a}{\sigma_{a,v}} \right ) .
\label{e:P_int_final}
\end{align}
From (\ref{e:P_int_final}), we can obtain an expression for the transmit
signal power of the FAR node in terms of the interference and outage probabilities
as follows:
\begin{align}
s_a &=
\sigma_{a,v} Q^{-1} \left (
1 - \frac{P_{int}}{(1 - P_{out}) P(E^{(on)}_p) P(E^{(on)}_a)}  \right ) \nonumber \\
&+ i_{max} + g(d_{a,v}) .
\end{align}

To proceed further with the interference analysis of LBT,
we make the following worst-case assumptions about the on-off source
behavior of the primary transmitter and the FAR node:
\begin{itemize}
  \item The primary transmitter is always in the on state,
i.e., $P(E^{(on)}_p) = 1$.

  \item The FAR node is in the on state if and only if the received signal from the
primary transmitter falls below the threshold $\eta$.
\end{itemize}
With reference
the LBT state diagram of Fig.~\ref{f:lbt}, the second assumption
is tantamount to the assumption that $\tau_{max} = \infty$.  In this case,
we have that
\[
P(E^{(on)}_a) = 1 - P_{det}.
\]
Under the above two assumptions, the duty cycle of the FAR node executing
LBT is given by (cf.~(\ref{e:duty_cycle}))
\[
d_a = P(E^{(on)}_a) = 1 - P_{det} .
\]
Let $\epsilon_{int}$ be the maximum interference probability
that can be tolerated by the victim node~$v$.
Under these assumptions, we define the {\it interference distance},
$d_{int}(\epsilon_{int} )$, as the minimum permissible distance
between the FAR node~$a$ and the victim node~$v$
such that the interference probability does not exceed $\epsilon_{int}$.
\begin{align}
d_{int}(\epsilon_{int}) \triangleq
 \min \{ d_{a,v} : P_{int} \leq \epsilon_{int} \} .
\end{align}
From  (\ref{e:P_int_final}), we
obtain the following expression for the interference distance:
\begin{align}
d_{int} \! &= \! g^{-1} \!
 \left ( s_a \! - \! i_{max} \! - \!
  \sigma_{a,v}
    Q^{-1} \left ( 1 \! - \! \frac{\epsilon_{int}}{(1 - P_{out}) (1 - P_{det})} \right )
    \right ) . \label{e:d_int}
\end{align}


\subsection{Avoidance of harmful interference}

The victim node is said to suffer {\it harmful interference}
from the FAR node under the following conditions:
\begin{align}
P_{out} \leq \epsilon_{out},~ P_{det} \leq \epsilon_{det} , ~P_{int} > \epsilon_{int} .
\label{e:harmful_cond}
\end{align}
The first condition states that the outage probability of the victim
node should not exceed the threshold $\epsilon_{out}$.  If the
outage probability exceeds $\epsilon_{out}$, the victim node is
oblivious to interference from the FAR node.  The second condition
requires the detection probability not to exceed the detection
probability threshold
$\epsilon_{det}$, which implies that the duty cycle of the FAR node would be
at least $1-\epsilon_{det}$.  Finally, the third condition requires the
probability of interference to be greater than a threshold
$\epsilon_{int}$.

From (\ref{e:P_int_final}), we can derive an expression
for the maximum power at which the FAR node can transmit
while avoiding harmful interference to node~$v$ for a given
inter-nodal distance $d_{a,v}$:
\begin{align}
s^*_a &=
i_{max} + g(d_{a,v}) +
\sigma_{a,v} Q^{-1}\left (
 1 - \frac{\epsilon_{int}}{(1 - P_{out}) ( 1 - P_{det} ) } \right ) . \label{e:MIFTP}
\end{align}
We call $s^*_a$ the {\it maximum interference-free transmit power} (MIFTP).
The MIFTP plays a key role in LBT-based spectrum access.

Assume that the transmit powers of the FAR node and the primary transmitter
are fixed at $s_a$ and $s_p$, respectively.  The following proposition
gives a simple sufficient condition for the the victim node {\it not} to
suffer harmful interference in terms of an inequality relating the
coverage distance, the detection distance, and the interference distance.
\begin{proposition}
Suppose that the first two conditions in (\ref{e:harmful_cond})
hold, i.e.,
\begin{align}
P_{out} \leq \epsilon_{out} ~\mbox{and}~ P_{det} \leq \epsilon_{det} .
\end{align}
Then the victim node does not suffer harmful interference if
\begin{align}
d_{int} \leq d_{det} - d_{cov}
\label{e:noninterfere_ineq}
\end{align}
\label{prop:noninterfere_cond}
\end{proposition}

\begin{proof}
By definition, the conditions $P_{out} \leq \epsilon_{out}$ and
$P_{det} \leq \alpha$ imply
\begin{align}
d_{p,v} \leq d_{cov} ~\mbox{and}~ d_{a,p} \geq d_{det}
,
\label{e:cov}
\end{align}
respectively.   Using the triangle inequality, we have
\begin{align}
d_{a,v}  \geq d_{a,p} - d_{p,v} . \label{e:det}
\end{align}
From (\ref{e:cov}) and (\ref{e:det}), we have
\begin{align}
 d_{a,v} \geq d_{det} - d_{cov} \geq d_{int} ,
 \label{e:d_det_cond}
\end{align}
which implies that
\begin{align}
P_{int} \leq \epsilon_{int} .
\end{align}
Hence, the harmful interference condition  (\ref{e:harmful_cond}) does not hold.
\end{proof}

\begin{figure}
  \centering
  \includegraphics[scale=0.35,clip=true]{\FIGDIR/int_condition.eps}
  \caption{Interference condition.}
\label{f:int_condition}
\end{figure}

Clearly, the converse of Proposition~(\ref{prop:noninterfere_cond})
does not hold in general.  The FAR node can always be positioned
sufficiently far from the victim and primary nodes in order to avoid
harmful interference.  A graphical illustration of
Proposition~\ref{prop:noninterfere_cond} is given in
Fig.~\ref{f:int_condition}.  In Fig.~\ref{f:int_condition}, we have
that both victim nodes $v_1$ and $v_2$ are within a distance
$d_{cov}$ from the primary transmitter.  Hence, the outage
probabilities of the victim nodes do not exceed $\epsilon_{out}$.
The FAR node $a$ lies just outside the distance $d_{det}$ from the
primary transmitter.  Hence, the detection probability does not
exceed $\epsilon_{det}$.

Observe that although node $v_1$ lies beyond the
detection distance, $d_{det}$, from the agile node~$a$, the node
$v_2$ does not.  Hence, node~$v_2$ suffers from harmful
interference, i.e., the interference probability experienced by
node~$v_2$ exceeds $\epsilon_{int}$.  Moreover, any victim node
lying within the shaded region would suffer harmful interference.
The shaded region can be reduced by lowering the transmit power,
$s_a$.  The absence of the shaded region corresponds to the
noninterference condition (\ref{e:noninterfere_ineq}).  If this
condition holds, no victim node can suffer harmful interference from
the FAR node.


\section{Analysis of Collaborative LBT}
\label{s:collaborative_LBT}

In collaborative LBT, we consider a group of $n$ FAR nodes
$\cG = \{ a_1, \cdots, a_n \}$.  Each of these nodes executes the individual
LBT algorithm with the proviso that if at least one node
in the group detects a signal from the primary transmitter~$p$,
all nodes in the group turn to the off state.  Otherwise,
each node is in the on state transmitting at a certain power level.
Let $R_{a_i}$ denote the signal power from the primary transmitter
that is received at FAR node~$a_i$ (cf.\ (\ref{e:R_a})):
\begin{align}
R_{a_i} = s_p + L_{p,a_i} . \label{e:R_ai}
\end{align}
Then the probability that the FAR nodes are all in
the on state is given by
\begin{align}
P(E^{(on)}_a) = P \left [
 \bigcap_{i=1}^n \{ R_{a_i} < \eta \} \right ] ,
\end{align}
where $\eta$ is the detection threshold.  Assuming
that the received signal strengths are independent,
\begin{align}
P(E^{(on)}_a) &= \prod_{i=1}^n P \{ R_{a_i} < \eta \} \nonumber \\
    &= \prod_{i=1}^n
    Q \left [ \frac{s_p - \eta - g(d_{p,a_i})}{\sigma_{p,a_i}}
    \right ] .
\end{align}

Let $\alpha$ be the probability that all the FAR nodes
are in the off state, i.e.,
\begin{align}
\alpha &= 1 - P(E^{(on)}_a) \nonumber \\
&= 1 -
\prod_{i=1}^n
    Q \left [ \frac{s_p - \eta - g(d_{p,a_i})}{\sigma_{p,a_i}}
    \right ] .
\end{align}
If all nodes in $\cG$ are equidistant from node~$p$ and the
propagation characteristics from $p$ to the FAR nodes
are {\it homogeneous}, then
\begin{align}
\alpha =
 1 - Q^n \left ( \frac{s_p - \eta - g(d_{p,a})}{\sigma_{p,a}}
    \right ) , \label{e:alpha_homo}
\end{align}
where $d_{p,a}$ is the common distance from node~$p$ to
each FAR node, and $\sigma_{p,a}$ is the common shadowing
standard deviation.  Under the assumption of homogeneity
of the FAR nodes with respect to node~$p$,
we can define a detection distance
analogous to (\ref{e:d_det}), with $\alpha$ given by (\ref{e:alpha_homo}).
Similarly, we can define an interference distance analogous to
(\ref{e:d_int}) and an MIFTP analogous to (\ref{e:MIFTP}).


\section{Numerical Results}
\label{s:numerical_results}

In this section, we present numerical results showing performance metrics
for individual and collaborative LBT over a range of
parameter values using the
expressions derived in sections~\ref{s:individual_LBT}
and~\ref{s:collaborative_LBT}.  These results
provide insight into the performance impacts of the various
key system parameters in LBT spectrum access.

\subsection{Interference, coverage, and detection distances}

\begin{figure*}
  \centering
  \includegraphics[scale=0.7,clip=true]{\FIGDIR/IDD_200MHz.eps}
  \caption{Interference, coverage, and detection distances for $f_c = 200$~MHz.}
\label{f:IDD_200MHz}
\end{figure*}

\begin{figure*}
  \centering
  \includegraphics[scale=0.7,clip=true]{\FIGDIR/IDD_2400MHz.eps}
  \caption{Interference, coverage, and detection distances for $f_c = 2.4$~GHz.}
\label{f:IDD_2400MHz}
\end{figure*}

Figs.~\ref{f:IDD_200MHz} and ~\ref{f:IDD_2400MHz} show plots of the
detection distance $d_{det}$ as a function of the maximum detection
probability $\epsilon_{det}$ for different values of the detection
threshold $\eta$ in individual LBT.  Fig.~\ref{f:IDD_200MHz} shows
curves obtained for a carrier frequency of $f_c = 200$~MHz, while
Fig.~\ref{f:IDD_2400MHz} corresponds to $f_c = 2.4$~GHz. The
frequency $f_c = 200$~MHz lies within the transmission band of
television towers, while $f_c = 2.4$~GHz lies within the
transmission band of MMDS (Multipoint Microwave Distribution System)
systems. The detection distance curves were obtained using
(\ref{e:d_det}). The graphs also show the sum of the interference
and coverage distances, i.e., $d_{int} + d_{cov}$, as a function of
the detection probability $\epsilon_{det}$. The values of $d_{int}$
and $d_{cov}$ were calculated using (\ref{e:d_int}) and
(\ref{e:d_cov}), respectively. The other system parameters of
interest are set as follows:
\begin{itemize}
 \item FAR transmitter power $s_a = 40$~dBm,
 \item FAR and victim node receiver antenna heights
    \[ h^{(r)}_a = h^{(r)}_v = 3~\mbox{m} , \]
 \item FAR and primary transmitter antenna heights
    \[ h^{(t)}_a = h^{(t)}_p = 10~\mbox{m} , \]
 \item Primary transmitter power $s_p = 80$~dBm,
 \item Interference probability $P_{int} = 0.01$,
 \item Maximum outage probability $P_{out} = 0.1$.
\end{itemize}
The detection threshold ranges from $\eta = -136$~dBm to $\eta =
-106$~dBm.

Observe that both the detection distance and the interference
distance are monotonically decreasing functions of the detection
probability. To avoid harmful interference, the detection
threshold~$\eta$ should be chosen such that the non-interference
condition (\ref{e:noninterfere_ineq}) holds over the entire range of
values of $\epsilon_{int}$, since we have no prior knowledge of the
detection probability. With respect to Figs.~\ref{f:IDD_200MHz} and
~\ref{f:IDD_2400MHz}, the non-interference condition requires the
$d_{det}$~curve to lie above the $(d_{int} + d_{cov})$~curve for all
values of $\alpha$.

We observe that the non-interference condition holds when the
detection threshold satisfies $\eta = -136$~dB for both carrier
frequencies. This implies a lower bound on the sensitivity of the
detector of the FAR node to ensure that it does not cause harmful
interference to the victim node. Note that at the higher carrier
frequency $f_c = 2.4$~GHz, the interference, detection, and coverage
curves are all significantly lower than at $f_c = 200$~MHz. When
$f_c = 2.4~$~GHz, the peak value of $d_{int} + d_{cov}$ is about
$175$~km, whereas the corresponding value at $f_c = 200$~MHz is
about $460$~km.

\subsection{MIFTP vs.\ Detection Threshold}

\begin{figure*}
  \centering
  \includegraphics[scale=0.7,clip=true]{\FIGDIR/MIFTP_vs_eta.eps}
  \caption{MIFTP vs.\ LBT detection threshold $\eta$.}
\label{f:MIFTP_vs_eta}
\end{figure*}

Fig.~\ref{f:MIFTP_vs_eta} show curves of the MIFTP as a function of
the detection threshold $\eta$ for the carrier frequencies
$f_c=200$~MHz and $f_c=2.4$~GHz, respectively. Observe that the
MIFTP curve for $f_c = 2$~GHz lies below the curve for $f_c =
200$~MHz. Thus, at lower carrier frequencies the FAR node can
transmit with higher signal power without causing harmful
interference to the victim receiver.

\begin{figure*}
  \centering
  \includegraphics[scale=0.8,clip=true]{\FIGDIR/MIFTP_Pint_200MHz.eps}
  \caption{MIFTP vs.\ detection threshold and interference probability for $f_c=200$~MHz.}
\label{f:MIFTP_Pint_200MHz}
\end{figure*}

\begin{figure*}
  \centering
  \includegraphics[scale=0.8,clip=true]{\FIGDIR/MIFTP_Pint_2400MHz.eps}
  \caption{MIFTP vs.\ detection threshold and interference probability for $f_c=2.4$~GHz.}
\label{f:MIFTP_Pint_2400MHz}
\end{figure*}




\begin{figure*}
  \centering
  \includegraphics[scale=0.7,clip=true]{\FIGDIR/MIFTP_Antenna_200MHz.eps}
  \caption{MIFTP vs.\ detection threshold and FAR transmitter antenna height for $f_c=200$~MHz.}
\label{f:MIFTP_Antenna_200MHz}
\end{figure*}

\begin{figure*}
  \centering
  \includegraphics[scale=0.7,clip=true]{\FIGDIR/MIFTP_Antenna_2400MHz.eps}
  \caption{MIFTP vs.\ detection threshold and FAR transmitter antenna height for $f_c=2.4$~GHz.}
\label{f:MIFTP_Antenna_2400MHz}
\end{figure*}




\begin{figure*}
  \centering
  \includegraphics[scale=0.7,clip=true]{\FIGDIR/MIFTP_vs_f.eps}
  \caption{MIFTP vs.\ carrier frequency and detection threshold.}
\label{f:MIFTP_vs_f}
\end{figure*}

\subsection{MIFTP vs.\ interference probability}

Figs.~\ref{f:MIFTP_Pint_200MHz} and ~\ref{f:MIFTP_Pint_2400MHz} show
surfaces of the MIFTP as a function of the detection threshold
$\eta$ and the interference probability $P_{int}$ for the carrier
frequencies $f_c=200$~MHz and $f_c=2.4$~GHz, respectively. We
observe that the surface corresponding to $f_c=2$~GHz lies below the
surface for $f_c = 200$~MHz.  The decreasing trend of the MIFTP with
respect to the detection threshold is similar at both carrier
frequencies.

\subsection{MIFTP vs.\ FAR transmitter antenna height}

Figs.~\ref{f:MIFTP_Antenna_200MHz} and
~\ref{f:MIFTP_Antenna_2400MHz} show surfaces of the MIFTP as a
functions of the detection threshold $\eta$ and the FAR transmitter
antenna height $h^{(t)}_a$ for $f_c = 200$~MHz and $f_c = 2$~GHz,
respectively.  At both carrier frequencies, the MIFTP increases
linearly with increasing antenna height.  Note that the MIFTP
surface at the higher carrier frequency has a larger slope with
respect to the FAR transmitter antenna height compared to the MIFTP
surface at the lower frequency. In fact, the two surfaces, cross
each other at an antenna height of approximately $150$~m.

\subsection{MIFTP vs.\ carrier frequency}

Fig.~\ref{f:MIFTP_vs_f} shows an MIFTP surface as a function of the detection
threshold $\eta$ and the carrier frequency $f_c$.
The MIFTP surface is a decreasing function of $\eta$, but is not a monotonic
function of $f_c$.  As can be observed in Fig.~\ref{f:MIFTP_vs_f}, the MIFTP
decreases roughly linearly as $f_c$ is increased from $0$~Hz to about $500$~MHz.
Then the MIFTP increases linearly as $f_c$ increases from $500$~MHz to a maximum value
of $65$~MHz when $f_c$ is approximately $1200$~Hz.  Finally, the MIFTP decreases
linearly for $f_c$ larger than $1200$~Hz.

\subsection{Collaborative LBT}

\begin{figure*}
  \centering
  \includegraphics[scale=0.7]{\FIGDIR/MIFTP_coll_200MHz.eps}
  \caption{MIFTP vs.\ detection threshold and FAR group size for
   collaborative sensing at $f_c=200$~MHz.}
\label{f:MIFTP_coll_200MHz}
\end{figure*}

\begin{figure*}
  \centering
  \includegraphics[scale=0.7]{\FIGDIR/MIFTP_coll_200MHz.eps}
  \caption{MIFTP vs.\ detection threshold and FAR group size for
    collaborative sensing at $f_c = 2$~GHz}
\label{f:MIFTP_coll_2000MHz}
\end{figure*}

Figs.~\ref{f:MIFTP_coll_200MHz} and \ref{f:MIFTP_coll_2000MHz}
show MIFTP surfaces for collaborative LBT under homogeneous
scenarios for carrier frequencies $f_c=200$~MHz and $f_c=2$~GHz,
respectively.  We see that the MIFTP increases
with the number of FAR nodes in the group.

\section{Conclusion}
\label{s:conclusion}

We developed a framework to evaluate the performance of the basic
Listen-Before-Talk (LBT) scheme and an extension of LBT called collaborative
LBT that allows a group of FAR radios to collaborate.
We introduced several important performance metrics including
detection distance, interference distance, and maximum
interference-free transmit power.  These metrics can provide the basis for
evaluating any spectrum access method.  We derived expressions
to evaluate these metrics with respect to the individual and collaborative
LBT schemes and presented numerical results
illustrating the relationships among them.

The numerical results show that the simple LBT
can be used to harvest unused spectrum without causing harmful interference
to existing users.  The collaborative LBT scheme provides significantly
higher spectrum capacity gains than individual LBT.  The poorer performance
of individual LBT is due to the fact that the nodes act without any knowledge
about the location of the primary transmitter and the victim nodes.  The
collaborative LBT scheme provides additional information about the location
of the primary implicitly through the use of alert messages.
Spectrum sharing schemes employing more sophisticated collaborative
behaviors should be able to harvest substantially more spectrum.  In ongoing work,
we are applying the analysis techniques developed in this paper to
evaluate the performance of spectrum access with more complex group behaviors.
The formulas derived in this paper for the performance metrics were
based on the EPM-73 propagation model, which has a relatively simple closed
form.  In future work, we plan to evaluate the performance of spectrum access
using more sophisticated models such as the TIREM and Longley-Rice models,
together with spectrum measurement data obtained through drive test experiments.


\bibliographystyle{ieeetr}
\bibliography{lbt}

\end{document}
