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\begin{document}

% paper title
\title{Outage Performance of a Hybrid DS/FH Spread Spectrum Signal in an ISM band}
\author{Hanyu~Li,~Mubashir~Syed,~and~Yu-Dong~Yao$^{\dag}$
%        } % <-this % stops a space
%\thanks{Manuscript received January 20, 2002; revised November 18, 2002.
%        This work was supported by the IEEE.}% <-this % stops a space
\thanks{Hanyu Li and Mubashir Syed are Ph.D. students in the Wireless Information Systems Engineering Laboratory (WISELAB),
Department of Electrical \& Computer Engineering, Stevens
Institute of Technology. Email: \{hli5, msyed\}@stevens.edu}%
\thanks{Yu-Dong Yao is the Director of WISELAB and an Associate Professor
in the Department of Electrical \& Computer Engineering, Stevens
Institute of Technology.}%
\thanks{$^{\dag}$Correspondence to: Yu-Dong Yao, Dept. of Electrical and
Computer Engineering, Stevens Institute of Technology, Castle
Point on Hudson, Hoboken, NJ 07030. Tel: +1 201 216-5264. Fax: +1
201 216-8246.
Email: \{yyao\}@stevens.edu } }%

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\begin{abstract}
This paper develops an analytical model for characterizing the
coexistence interference in the unlicensed industrial, scientific
and medical (ISM) bands, based on radio frequency measurement
results in the 2.4 GHz ISM band. The modeling approach presented
here offers a simple methodology for profiling the interference
from devices occupying a common spectrum. Outage performance using
the interference model is exemplified through the analysis for a
hybrid direct-sequence frequency hopping spread spectrum signal.
The work presented in this paper looks to provide insight by means
of quantitative evaluation of signal outage in a coexistence
environment that can be used for engineering design of
applications for unlicensed bands as well as function as a tool in
the development process for future spectrum management policy
models.
\end{abstract}

\begin{keywords}
Outage probability, ISM band, frequency hopping, direct-sequence
spread spectrum.
\end{keywords}
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\section{Introduction}\label{sec:intro}
\PARstart{L}{icensing} radio frequencies for commercial use has
long been the mechanism adopted by regulatory bodies for managing
the radio frequency (RF) spectrum. An exclusive license was
granted with the intention to protect the licensee's service from
interference, but it also allowed the licensee to exclude shared
use even when this may result in higher utilization of the
allocated spectrum. Justification, in part, for this approach to
spectrum management was predicated on technological limitations of
old. Although technology has evolved over time and in the process
annulled most of these limitations, regulatory spectrum management
methodology has failed to keep pace. On the other hand, the
industrial, scientific, and medical (ISM) radio bands, originally
allocated for non-commercial uses, were later modified to allow
for more services \cite{CFR47-15}, prompting an influx of wireless
communications applications (such as wireless local area networks
or WLANs, bluetooth etc.) that exploit these bands for
license-free operation. This can be seen as an indication of the
role that unlicensed bands are set to play in the evolution of
wireless services. The traditional philosophy of interference
management through licensing and regulatory processes is to
protect the licensee from potential sources of interference in the
licensed spectrum bands. The unlicensed bands, in contrast, offer
no such protection from interference caused by coexisting systems,
and resolving interference related problems is a challenge that
researchers and system designers have to address.

The dramatic increase in applications associated with the
unlicensed bands has brought to the fore two issues that demand
synergistic solutions. Motivation for the work presented in this
paper is derived from this inference. The first issue is
coexistence in unlicensed bands. Successful resolution of this
challenge is incumbent on technological innovation. Over time,
technological advancement has enabled more efficient and
(dynamically) adaptive wireless systems that can allow efficacious
simultaneous frequency reuse. Technologies like smart antennas,
\cite{Win98}$^,$\cite{Jos99}$^,$\cite{Ale04} %\cite{Win98,Jos99,Ale04}
spread spectrum modulation, software defined radio (SDR)
\cite{Jef02} and cognitive radio \cite{HAY05} make it feasible for
transceivers to dynamically change their frequency, modulation, or
power levels to enable more efficient and intelligent spectrum
sharing. For example, advances in signal processing technology
(e.g., multi-user detection theory \cite{MOS96}) and cooperative
networking (e.g., ad hoc networks \cite{CHA01,BAS04} or grid
computing \cite{DAR05}) can exploit the fact that there are
multiple signal sources sharing the same frequency bands to
improve reception gain. The notion of coexistence would in turn
drive technological evolution. In light of the progress of
technology and evolution of wireless systems, the traditional
exclusive-use licensing approach to tackling the interference
problem is increasingly being seen as inefficient and restrictive
(as also corroborated by the findings of the FCC's Spectrum Policy
Task Force \cite{SPTF-REP02}). This is the second issue that needs
to be addressed. It is critical that a resolution to this problem
abets the evolution of the current `command and control' approach
to spectrum policy into one that promotes greater reliance on
market forces for managing how spectrum is used. To this end,
characterization of the spectral emissions in the ISM band due to
the current range of uncoordinated wireless services that occupy
this spectrum, will provide valuable insight.

The sanction for unlicensed use of the ISM band predicates that
efficient coexistence is essential for successful operation of
systems in this band. This calls for a multifaceted approach to
system design, encompassing spectrum occupancy measurements,
modeling of coexistence interference, performance evaluation and
development of optimum waveforms. A number of studies of spectrum
occupancy measurements in the ISM band have already been reported
in the literature.%\cite{KEN97,HUA92,GIA00,BIG04}
\cite{KEN97}$^,$\cite{HUA92}$^,$\cite{GIA00}$^,$\cite{BIG04} At
the Stevens Institute of Technology, an investigative study is
being carried out for distributed spectrum occupancy measurements
in the 2.4 GHz ISM band \cite{THE05}. The focus of this paper is
on the modeling of interference in the 2.4 GHz ISM band (based on
measurement data), and outage performance analysis using the
developed interference model. Based on measurement data in typical
environments (indoor, outdoor etc.) that provide insight into the
spectral emission characteristics and the corresponding sources,
we develop an analytical model of the coexistence interference in
the 2.4 GHz ISM band. The model exemplifies a simple approach to
interference modeling due to uncoordinated sources/technologies
sharing a common band of frequencies, given an understanding of
their emission characteristics. It is evident that different
statistics are needed to characterize the interfering signals in
the ISM band. A simple yet general mathematical approach for
interference characterization would serve many a purpose.

Generally, in wireless system engineering, assessment of the
effects of interference on system operation is done through the
probability of outage, which lends itself well as a statistical
measure of radio link performance. In simple terms, outage
probability can be defined as the probability that the desired
signal-to-interference-plus-noise ratio (SINR) falls below an
appropriate threshold that is determined based on system design
requirements. The development of mathematical tools for outage
analysis has received a lot of attention since the deployment of
commercial cellular phone networks \cite{ANN01}. In this paper, we
derive the outage probability based on the developed interference
model. An analysis based on numerical evaluation of outage and
error probabilities is also presented.

The rest of the paper is structured as follows. In the next
section, we discuss the 2.4 GHz ISM band occupancy scenarios and
describe our approach to characterizing interference for a typical
application environment. The signal model that is considered for
the analysis in this paper is explained in the subsequent section.
Section \ref{sec:capture} presents mathematical derivations of
expressions for capture probabilities; outage and error
performance analysis is discussed in Section \ref{sec:outage}.
Results of numerical evaluation of the derived performance
measures are presented in Section \ref{sec:num}. Finally, the work
presented in this paper is summarized and conclusions are drawn in
Section \ref{sec:concl}.

\section{Interference Scenarios and Modeling} \label{sec:int model}
It is clear that an increasing number of devices occupy the ISM
band frequencies between 2.4 and 2.4835 GHz. As the services that
these devices provide are complementary rather than competing,
coexistence is seen as the operational norm. Also evident is that
these devices, which run the gamut from communications equipment
to kitchen aids, are based on different technologies and,
therefore, have different emission characteristics. Fig.
\ref{fig:1} shows some sample measurements \cite{THE05} in the 2.4
GHz ISM band that highlight spectrum occupancy by typical devices.
From the temporal and spectral emission characteristics, various
applications have been identified, including Bluetooth, IEEE
802.11b WLAN and microwave oven emissions. Additional tone and
narrowband emissions have also been observed.
\begin{figure*}[p]
\centerline { \subfigure[Observations of Bluetooth packets]
{\includegraphics[width = 0.33 \textwidth]{fig-1-a-mod.pdf}
\label{fig:1a}} \hfil \subfigure[Observations of Bluetooth packets
and microwave oven emissions]{\includegraphics[width = 0.33
\textwidth]{fig-1-b-mod.pdf} \label{fig:1b}} \hfil
\subfigure[Observations of IEEE 802.11b
packets]{\includegraphics[width = 0.33
\textwidth]{fig-1-c-mod.pdf} \label{fig:1c}} } \caption{ISM band
spectral emission measurement \cite{THE05}.} \label{fig:1}
\end{figure*}
%% When using eps figures instead of pdf
%\begin{figure*}[p]
%\centerline { \subfigure[Observations of Bluetooth packets]
%{\includegraphics[width = 0.33 \textwidth]{fig-1-a-mod.eps}
%\label{fig:1a}} \hfil \subfigure[Observations of Bluetooth packets
%and microwave oven emissions]{\includegraphics[width = 0.33
%\textwidth]{fig-1-b-mod.eps} \label{fig:1b}} \hfil
%\subfigure[Observations of IEEE 802.11b
%packets]{\includegraphics[width = 0.33
%\textwidth]{fig-1-c-mod.eps} \label{fig:1c}} } \caption{ISM band
%spectral emission measurement \cite{THE05}.} \label{fig:1}
%\end{figure*}
%%

In view of the variations in bandwidth occupancy patterns of
coexisting wireless devices and several other factors (such as
proximity constraints and application environment etc.), design
considerations for effective introduction of additional signal
transmissions in the ISM band necessitate that a three-fold
approach be employed. The first requirement is to minimize the
interference to coexisting services. The second is to quantify the
interference from coexisting users. To accurately characterize
every interfering signal in the ISM would require quite a few
different statistics and would be mathematically unwieldy.
However, a simple yet general technique which accounts for the
various devices that typically occupy the common spectrum would be
useful and also provide an attractive way to address the issue.
Finally, building on the above two efforts, effective waveforms
that are robust to interference need to be developed. The focus of
this paper is to evaluate the impact of interference from
coexisting users. Towards this end, we attempt to profile the
observed emissions in terms of various representative interference
types.
\begin{table*}[p]
\renewcommand{\arraystretch}{1.3}
\centering \caption{2.4 GHz ISM band interference characteristics
based on measurements.} \label{table:1}
\begin{tabular}{| l || l || c || c || c || c |}
  \hline \hline
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
  \bfseries{Emissions} & \bfseries{Interference Type} &
  \bfseries{Power Level} & \bfseries{Duty Cycle} & \bfseries{Periodicity} & \bfseries{Bandwidth} \\
  \hline \hline
  802.11b packets & Barrage & $Y_{1,j}$ & $\rho_{1,j}$ & No & $\gg B$\\
  \hline
  Microwave oven & Barrage & $Y_{1,j}$ & $\rho_{1,j}$
  & Yes & $\gg B$ \\ \hline
  Bluetooth packets & Partial-band & $Y_{2,j}$ &
  $\rho_{2,j}$ & No & $\approx B$ \\ \hline
  Others & Tone & $Y_{3,j}$ & $\rho_{3,j}$ & Yes &
  $\ll B$ \\ \hline
  \hline
\end{tabular}
\end{table*}

Table \ref{table:1} summarizes the interference types of measured
ISM band emissions. It is noted that the profiling presented here
is derived from the temporal and spectral measurements conducted
in a specific office environment with the usual set of devices
currently typical in the 2.4 GHz ISM band. The significance of
this interference modeling approach is that it showcases a simple
and sufficiently accurate methodology for profiling emissions in
an unlicensed band that can be used for different interference
scenarios. The spectrum occupancy of the ISM band can be
characterized in terms of transmission bandwidth and time.
Transmissions from devices operating in this band can cover the
entire available bandwidth or a part of it. This operational
characteristic is considered with respect to the bandwidth of the
signal of interest, which is denoted as $B$ and represents the
size of a frequency slot within the total available bandwidth. The
number of frequency slots that make up the total bandwidth is
denoted as $N$. Thus, the entire ISM band is considered to be
divided into $N$ frequency slots, each with bandwidth $B$.
Emissions from different sources also have temporal
characteristics such as periodicity and duty cycle. To
parameterize these in terms of the signal of interest, a duration
of $T_h$ time units is defined. As a result, the periodicity and
duty on-off times of interfering signals can be described with
reference to $T_h$. As listed in the table, the observed emissions
are categorized into three broad interference types based on their
transmission bandwidth -- barrage, partial-band and tone. Barrage
type interferers are those whose transmission bandwidth covers the
entire signal bandwidth $B$. Partial-band interference is a
generic grouping of interference sources characterized by
transmission bandwidth that occupies part of the desired signal
bandwidth $B$. Devices transmitting single frequency impulses are
grouped under the tone interference type. Based on the spectral
characteristics of the emissions as obtained from measured data,
they have been matched to their corresponding device type.
Furthermore, to capture the effect of spatial and temporal
variation on the emission characteristics of the different
interference sources, albeit of the same interference type, it is
required that they be classified in terms of additional parameters
that would facilitate a more accurate modeling of the
interference. Hence, emissions are also parameterized in terms of
their received power levels as $Y_{i,j}$, where the first
subscript corresponds to the interference type (i.e., $i \in
\{1,2,3\}$ denoting barrage interference, partial-band
interference and tone interference respectively) and the second
subscript, $j$, denotes a specific source of the given
interference type. Similarly, the duty cycle of each source is
parameterized as $\rho_{i,j}$. Periodicity refers to whether or
not an interference type exhibits periodic emission
characteristics with respect to the specified reference time
duration $T_h$. It is noted that the various emissions are
referenced under the listed interference types based on the
bandwidth they occupy with respect to $B$.

\section{Signal Model}\label{sec:sig model}
In order to evaluate the coexistence environment and quantify the
interference and its impact, an appropriate signal model has to be
adopted. Hybrid spread spectrum (SS), where a direct-sequence (DS)
SS modulated signal is frequency hopped according to a specific
frequency-hopping (FH) pattern, is an attractive choice. This
paper considers such a DS/FH signal for the reasons that it is has
an inherent capacity to mitigate interference caused to others, by
virtue of its low power spectrum density, and interference
resistance across multiple bands through FH. Additional
interference reduction is provided due to the DS spreading gain.
It is noted that in keeping with a more generalized treatment of
the approach presented in this paper and for lucidity of
presentation, specifics have been avoided. For instance, explicit
details of the signature sequence used for spreading, the
frequency hopping pattern and the signal processing aspects of
multipath fading have been ignored. Just enough detail is
furnished so as to account for the concerned phenomena for our
purposes.

For the analysis presented here, a single desired user is
considered and all the other transmitting sources occupying the
frequency band are taken to be interferers. Noncoherent reception
is considered so as to reflect situations where the receiver can
acquire time synchronization with the desired signal but it cannot
acquire a phase reference. This might be the case because of high
levels of interference in the frequency band. Also, modulation
schemes that allow for noncoherent reception need to be employed.
The modulation scheme of interest here is differential phase shift
keying (DPSK). Let us denote the DPSK modulated DS spread signal
as $c(t)$, and is given by
\begin{equation}\label{ds-sig}
    c(t) = 2\sqrt{2X}b(t) \Psi(t) a(t)
    \cos (2\pi f_ct+\theta)
\end{equation}
where $f_c$ is the carrier frequency, $X$ is the power of the
transmitted signal, $\theta$ is the phase introduced by the DPSK
modulator and $\Psi(t)$ is a chip waveform with duration $T_c$.
The data signal, $b(t)$, (which is a differentially encoded
version of the information signal) is a sequence of rectangular
pulses with amplitude equal to either $+1$ or $-1$. The code
waveform, $a(t)$ is a periodic sequence of positive and negative
rectangular pulses of unit amplitude and duration $T_c$. The DS-SS
signal, $c(t)$, is then frequency-hopped according to the hopping
pattern $f(t)$. The transmitted signal takes the form
\begin{equation}\label{tx-sig}
    s(t) = \sqrt{2X}b(t)\Psi(t) a(t)
    \cos \{ 2\pi[f_c+f(t)]t+\theta+\phi(t)\}
\end{equation}
where $\phi(t)$ represents the phase waveform introduced by the
frequency hopper.

We consider that the propagation channel for the desired signal is
characterized by Rayleigh fading with impulse response $h(t)$.
Thus, the signal at the input of the receiver is given by
\begin{equation}\label{faded-sig}
\begin{split}
    r(t) &= s(t)\otimes h(t) + y(t) + n(t) \\
    &= x(t) + y(t) + n(t)
\end{split}
\end{equation}
where $\otimes$ denotes the convolution operation, $y(t)$ denotes
the total interference and $n(t)$ is the additive white Gaussian
noise (AWGN) with two-sided spectrum density $N_0/2$. The receiver
is assumed capable of acquiring the frequency-hopping pattern,
signature sequence and time synchronization of the user. The
output of the frequency de-hopper in the receiver enters the
de-spreader and then the DPSK demodulator.

\section{Derivation of Capture Probabilities}\label{sec:capture}
In wireless communications, adequate
signal-to-interference-plus-noise ratio (SINR) is essential for
successful communications \cite{Yao92,Yao90}. Therefore, the
capture probability, defined as the probability of achieving a
SINR sufficient to give satisfactory reception, is an important
measure in the evaluation of performance of wireless systems.
Mathematically the capture probability $P_{cap}$ is given by
\begin{equation}\label{pdfcapconcept01}
\begin{split}
P_{cap}&= \int_{R_N}^{\infty}p_\gamma(\gamma)d\gamma
\\&=\Pr\left\{\frac{x}{y+n}>R_N\right\}
\end{split}
\end{equation}
in which $\gamma$ is the instantaneous SINR, $p_{\gamma}(\gamma)$
is the probability density function (pdf) of $\gamma$, and $R_N$
is a required threshold. The variable $\gamma$ is a function of
$x$, $y$ and $n$, with $x$ denoting the desired signal power, $y$
denoting the total interference power and $n$ denoting the noise
power. For the derivations presented here, the interfering signals
(if any) are considered to be either from one of the above
discussed three types (namely, barrage, partial-band or tone
interferers), or a combination of these different types.
Correspondingly, the capture probabilities are obtained as
discussed below.

\subsection{Capture Probabilities without Interference (AWGN Case)}
If there is no interferer in the same band while the desired
signal is transmitting, AWGN will affect the reception quality. If
the AWGN one-sided power spectrum density (psd) is denoted by
$N_0$, the power of the noise is $N_0B$. The instantaneous SINR is
\begin{equation}\label{gamma01}
\begin{aligned}
\gamma=\frac{1.5Gx}{N_0B}
\end{aligned}
\end{equation}
in which $G$ is the spreading gain and $1.5$ is due to the
waveform of spreading sequence. The capture probability is
\begin{equation}\label{Pcap01}
\begin{aligned}
P_{cap}&=\Pr\left\{\frac{1.5Gx}{N_0B}>R_N\right\}
\\&=\Pr\left\{x>\frac{N_0BR_N}{1.5G}\right\}
\\&=\int_{\frac{N_0BR_N}{1.5G}}^{\infty}p_x(x)dx
%\\&=\int_{\frac{N_0BR_N}{1.5G}}^{\infty}\frac{1}{X}\exp{(-\frac{x}{X})}dx
%\\&=\exp{(-\frac{N_0BR_N}{1.5GX})}
%\\&=\exp{(-\frac{1}{1.5G\frac{X}{N_0BR_N}})}
\\&=\exp{\left(-\frac{1}{1.5G \Gamma_N}\right)}
\end{aligned}
\end{equation}
in which $p_x(x) = \frac{1}{X}\exp{(-\frac{x}{X})}$ is the pdf of
the Rayleigh faded signal power and $\Gamma_N=\frac{X}{N_0BR_N}$
is the normalized average signal to noise ratio (SNR).

\subsection{Capture Probabilities with Barrage Interference}
In this subsection we derive the capture probabilities when the
interference is of the first type, i.e., barrage interference.
Barrage interferers transmit bandlimited signals at high power.
The performance of a spread spectrum signal is the same in the
scenario of either AWGN or barrage interferers \cite{Pet95}. We
begin with analyzing the capture probability under a single
barrage interferer, then we will derive the capture probability
under multiple barrage interferers.
\subsubsection{Capture Probabilities with One Barrage Interferer}
The instantaneous interference power after despreading is denoted
as $y_{1}$. Then, the instantaneous SINR is
\begin{equation}\label{gamma02}
\begin{aligned}
\gamma=\frac{1.5Gx}{y_{1}+N_0B}
\end{aligned}
\end{equation}

Assuming that the fading statistics of the desired signal and the
interferer are independent, the capture probability is calculated
by
\begin{equation}\label{Pcap02}
\begin{aligned}
P_{cap}&=Pr\left\{\frac{1.5Gx}{y_1+N_0B}>R_N \right \}
\\&=Pr\left\{x>\frac{(y_1+N_0B)R_N}{1.5G}\right\}
\\&=\int_{0}^{\infty}p_{y_1}(y_1)\int_{\frac{(y_1+N_0B)R_N}{1.5G}}^{\infty}p_x(x)dxd{y_1}
%\\&=\int_{0}^{\infty}p_{y_1}(y_1)\int_{\frac{(y_1+N_0B)R_N}{1.5G}}^{\infty}\frac{1}{X}\exp{(-\frac{x}{X})}dxd{y_1}
%\\&=\int_{0}^{\infty}p_{y_1}(y_1)\exp{(-\frac{(y_1+N_0B)R_N}{1.5GX})}d{y_1}
%\\&=\int_{0}^{\infty}\exp{(-\frac{(y_1+N_0B)R_N}{1.5GX})}\frac{1}{Y_1}\exp{(-\frac{y_1}{Y_1})}d{y_1}
%\\&=\exp{(-\frac{R_NN_0B}{1.5GX})}\int_{0}^{\infty}\exp{(-\frac{R_Ny_1}{1.5GX})}\frac{1}{Y_1}\exp{(-\frac{y_1}{Y_1})}d{y_1}
\\&=\exp{(-\frac{1}{1.5G\Gamma_N})} \times \frac{1}{1+\frac{1}{1.5G\Gamma_1}}
\end{aligned}
\end{equation}
in which $p_{y_1}(y_1)$ is the pdf of the Rayleigh faded
interference power and $\Gamma_1=\frac{X}{Y_1R_N}$ is the
normalized average signal-to-interference ratio (SIR). It is noted
that the received interference is considered to have experienced
Rayleigh fading. This is a valid assumption considering the
environment scenario in which the source of the perceived
interference is operating. The capture probability $P_{cap}$
versus SNR $\Gamma_N$ and SIR $\Gamma_1$ is plotted in Figure
\ref{Pcap2D}.
\subsubsection{Capture Probability with Multiple Barrage Interferers}
If there are $I$ barrage interferers present, and for the $j$th
interferer, denoting its average power as $Y_{1,j}$ and its
instantaneous power after despreading as $y_{1,j}$, the
instantaneous SINR is
\begin{equation}\label{gamma03}
\begin{aligned}
\gamma=\frac{1.5Gx}{\sum_{j=1}^{I}y_{1,j}+N_0B}
\end{aligned}
\end{equation}

Assuming that the fading statistics of the desired signal and the
interferers are mutually independent, the capture probability is
derived as
\begin{equation}\label{Pcap03}
\begin{split}
P_{cap}&=Pr\left\{\frac{1.5Gx}{\sum_{j=1}^{I}y_{1,j}+N_0B}>R_N\right\}
\\&=Pr\left\{x>\frac{(\sum_{j=1}^{I}y_{1,j}+N_0B)R_N}{1.5G}\right\}
\\&=\int_{0}^{\infty}\cdot\cdot\cdot\int_{0}^{\infty}p_{y_{1,1}}(y_{1,1})\cdot\cdot\cdot p_{y_{1,I}}(y_{1,I})\left(\int_{\frac{(\sum_{j=1}^{I}y_{1,j}+N_0B)R_N}{1.5G}}^{\infty}p_x(x)\right)dxd{y_{1,1}}\cdot\cdot\cdot d{y_{1,I}}
\\&=\int_{0}^{\infty}\cdot\cdot\cdot\int_{0}^{\infty}p_{y_{1,1}}(y_{1,1})\cdot\cdot\cdot p_{y_{1,I}}(y_{1,I})\exp{(-\frac{(\sum_{j=1}^{I}y_{1,j}+N_0B)R_N}{1.5GX})}dxd{y_{1,1}}\cdot\cdot\cdot d{y_{1,I}}
\\&=\exp{\left(-\frac{R_NN_0B}{1.5GX}\right)}\times \prod_{j=1}^I\int_{0}^{\infty}\exp{\left(-\frac{R_Ny_{1,j}}{1.5GX}\right)}\frac{1}{Y_{1,j}}\exp{\left(-\frac{y_{1,j}}{Y_{1,j}}\right)}d{y_{1,j}}
\\&=\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\times\prod_{j=1}^I\frac{1}{1+\frac{1}{1.5G\Gamma_{1,j}}}
\end{split}
\end{equation}
in which $\Gamma_{1,j}=\frac{X}{Y_{1,j}R_N}$ is the average SIR
accounting for the $j$th interferer.

\subsection{Capture Probabilities with Partial-band Interference}
In this subsection we derive the capture probabilities for the
case where the interference occupies part of the hop bandwidth. We
begin by considering the received power of a single partial-band
interferer and get its corresponding capture probability. We  then
derive the capture probability with multiple partial-band
interferers.
\subsubsection{Capture Probability with One Partial-band Interferer}
%A partial-band interferer to a DS/FH signal is the same as a
%partial-band jammer to a spread spectrum signal.
The received power of a partial-band interferer, with transmit
power $y_2$, after despreading is given by \cite{Pet95}
\begin{equation}\label{partial01}
\begin{aligned}
\tilde{y}_2 = y_2\times \frac{1}{B_2}
\int_{-B_{2}}^{B_{2}}\frac{\sin^2\left[(\triangle
f-f)\frac{2}{B}\right]}{\left[(\triangle
f-f)\frac{2}{B}\right]^2}df
\end{aligned}
\end{equation}
In the above equation $B_{2}$ is the bandwidth of the partial-band
interferer, and $\triangle f$ is the frequency offset between the
interferer and signal. Following the same procedure as in the
barrage interferer case, we get the capture probability with one
partial-band interferer as
\begin{equation}\label{Pcap21}
\begin{aligned}
P_{cap}=\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\times
\frac{1}{1+\frac{\alpha_2}{1.5G\Gamma_1}}
\end{aligned}
\end{equation}
in which $\alpha_2$ is a coefficient and it can numerically
calculated as
\begin{equation}\label{alpha2}
\begin{aligned}
\alpha_2=\frac{1}{B_2}
\int_{-B_{2}}^{B_{2}}\frac{\sin^2\left[(\triangle
f-f)\frac{2}{B}\right]}{\left[(\triangle
f-f)\frac{2}{B}\right]^2}df
\end{aligned}
\end{equation}
\subsubsection{Capture Probabilities with Multiple Partial-band Interferers}
If there are $J$ partial-band interferers, with the average
transmit power of the $j$th interferer denoted as $Y_{2,j}$ and
its instantaneous power after despreading denoted as $y_{2,j}$,
the instantaneous SINR is
\begin{equation}\label{gamma22}
\begin{aligned}
\gamma=\frac{1.5Gx}{\sum_{j=1}^{J}y_{2,j}+N_0B}
\end{aligned}
\end{equation}
After simplification the capture probability is obtained as
\begin{equation}\label{Pcap22}
\begin{aligned}
P_{cap}=\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\times
\prod_{j=1}^J\frac{1}{1+\frac{\alpha_{2,j}}{1.5G\Gamma_{2,j}}}
\end{aligned}
\end{equation}
in which $\Gamma_{2,j}=\frac{X}{Y_{2,j}R_N}$ is the normalized
average SIR in the presence of the $j$th interferer and
$\alpha_{2,j}$ is its corresponding coefficient in the form of
(\ref{alpha2}).

\subsection{Capture Probabilities with Tone Interference}
In this subsection we consider the capture probabilities in the
presence of tone interferers.
\subsubsection{Capture Probabilities with a Single Tone Interferer}
The received power of a single tone interferer, with transmit
power $y_3$, after despreading is found to be \cite{Pet95}
\begin{equation}\label{tone01}
\begin{aligned}
\tilde{y}_3 = y_3\times \frac{\sin^2\left(\frac{\triangle
W}{2B}\right)}{2\left(\frac{\triangle
W}{2B}\right)^2}\left(1+\frac{\cos\left(2\triangle
\phi+G\frac{\triangle W}{B}\right)\sin\left(G\frac{\triangle
W}{B}\right)}{G\sin\left(\frac{\triangle W}{B}\right)}\right)
\end{aligned}
\end{equation}
In which $\triangle W$ and $\triangle \phi$ are the frequency and
phase offset between the interferer and the signal. Following the
same procedure as in the barrage interferer case we get the
capture probability under a lone tone interferer as
\begin{equation}\label{Pcap31}
\begin{aligned}
P_{cap}=\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\times
\frac{\alpha_3}{1+\frac{1}{1.5G\Gamma_1}}
\end{aligned}
\end{equation}
in which $\alpha_3$ is a coefficient and
\begin{equation}\label{alpha3}
\begin{aligned}
\alpha_3=\frac{\sin^2\left(\frac{\triangle
W}{2B}\right)}{2\left(\frac{\triangle
W}{2B}\right)^2}\left(1+\frac{\cos\left(2\triangle
\phi+G\frac{\triangle W}{B}\right)\sin\left(G\frac{\triangle
W}{B}\right)}{G\sin\left(\frac{\triangle W}{B}\right)}\right)
\end{aligned}
\end{equation}
\subsubsection{Capture Probability with Multiple Tone Interferers}
Lets consider that there are $K$ tone interferers. For interferer
$j$, the average transmit power is denoted as $Y_{3,j}$ and the
instantaneous power after despreading is denoted as $y_{3,j}$.
Then, the instantaneous SINR is
\begin{equation}\label{gamma32}
\begin{aligned}
\gamma=\frac{1.5Gx}{\sum_{j=1}^{K}y_{3,j}+N_0B}
\end{aligned}
\end{equation}
And the capture probability is
\begin{equation}\label{Pcap32}
\begin{aligned}
P_{cap}=\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\times
\prod_{j=1}^K\frac{1}{1+\frac{\alpha_{3,j}}{1.5G\Gamma_{3,j}}}
\end{aligned}
\end{equation}
in which $\Gamma_{3,j}=\frac{X}{Y_{3,j}R_N}$ is the normalized
average SIR in the presence of the $j$th interferer and
$\alpha_{3,j}$ is its corresponding coefficient.

\subsection{Capture Probability with Different Interference Types}
There are three types of interferers. Our derivations above are
based on the assumption that only one type of interferer is
present at a given time.  Assuming there are $K_1$ barrage
interferers (each with average transmit power $Y_{1,j}$), $K_2$
partial-band interferers (each with average transmit power
$Y_{2,j}$) and $K_3$ tone interferers (each with average transmit
power $Y_{3,j}$) at the same time, the instantaneous SINR is
\begin{equation}\label{gamma51}
\begin{aligned}
\gamma=\frac{1.5Gx}{\sum_{i=1}^{3}\sum_{j=1}^{K_i}y_{i,j}+N_0B}
\end{aligned}
\end{equation}
If we assume that the fading statistics of the desired signal and
the interferers are mutually independent, the capture probability
is calculated as
\begin{equation}\label{Pcap51}
\begin{aligned}
P_{cap}&=Pr\left\{\frac{1.5Gx}{\sum_{i=1}^{3}\sum_{j=1}^{K_i}y_{i,j}+N_0B}>R_N\right\}
\\&=\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\times \prod_{i=1}^{3}\prod_{j=1}^{K_i}\frac{1}{1+\frac{\alpha_{i,j}}{1.5G\Gamma_{i,j}}}
\end{aligned}
\end{equation}
in which $\Gamma_{i,j}=\frac{X}{Y_{i,j}R_N}$ is the normalized
average SIR corresponding to interferer $j$ of the $i$th type.
$\alpha_{i,j}$ is its coefficient; $\alpha_{1,j}=1$,
$\alpha_{2,j}$ and $\alpha_{3,j}$ can be found by (\ref{alpha2})
and (\ref{alpha3}).

The above result (\ref{Pcap51}) is derived based on a specific set
values of interferer parameters $\alpha_{i,j}$. If the
$\alpha_{i,j}$ is a random variable, the average capture
probability becomes
\begin{equation}\label{PcapAVG52}
\begin{aligned}
P_{cap}&=\int\int...\int...\int\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\prod_{i=1}^{3}\prod_{j=1}^{K_i}\frac{1}{1+\frac{\alpha_{i,j}}{1.5G\Gamma_{i,j}}}
\\& \qquad \times\left[p_{\alpha_{1,1}}(\alpha_{1,1})p_{\alpha_{1,2}}(\alpha_{1,2})...p_{\alpha_{1,K_1}}(\alpha_{1,K_1})...p_{\alpha_{3,K_3}}(\alpha_{3,K_3})d\alpha_{1,1}d\alpha_{1,2}...d\alpha_{1,K_1}...d\alpha_{3,K_3}\right]
\\&=\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\left[\prod_{i=1}^{3}\prod_{j=1}^{K_i}\int\frac{1}{1+\frac{\alpha_{i,j}}{1.5G\Gamma_{i,j}}}p_{\alpha_{i,j}}(\alpha_{i,j})d\alpha_{i,j}\right]
\end{aligned}
\end{equation}
where $p_{\alpha_{i,j}}(\alpha_{i,j})$ is the pdf of
$\alpha_{i,j}$.

\section{Outage and Error Performance of a DS/FH Signal}\label{sec:outage}
Successful signal transmission in a wireless environment dictates
that the desired signal be received with adequate strength and
signal-to-interference-plus-noise ratio in the presence of
interfering emissions from other sources. Outage probability and
packet error rate are two widely used metrics for performance
analysis. In this paper, we are interested in quantifying the
performance of a DS/FH signal operating in an unlicensed band.
Therefore, utilizing the the closed-form capture probabilities of
a DS/FH signal derived in the previous section, we derive its
outage probability in this section. Packet error rate of a DS/FH
signal is also derived considering the use of forward error
correction (FEC) coding.

\subsection{Outage Probability}
Considering that the duty cycle of an interferer is $\rho_{i,j}$,
the probability that the interferer is present in a certain
channel (slot) is $\frac{\rho_{i,j}}{N}$, in which $N$ is the
total number of available channels. Let the total number of
interferers of the three interference types be denoted as $K_1$,
$K_2$, and $K_3$ respectively. The outage probability of a DS/FH
signal is obtained as
\begin{equation}\label{Pout51}
\begin{aligned}
%P_{out}&=1-\Pr\{Capture All\}
%\\&=1-\sum_{l_{1,1}=0}^{1}\sum_{l_{1,2}=0}^{1}\cdot\cdot\cdot\sum_{l_{3,K_3}=0}^{1}\left(\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\prod_{i=1}^{3}\prod_{j=1}^{K_i}\left(\frac{1}{1+\frac{1}{1.5G\alpha_{i,j}\Gamma_{i,j}}}\right)^{l_{i,j}}\right)\cdot
%\\&\left(\prod_{i=1}^{3}\prod_{j=1}^{K_i}\left(\frac{\rho_{i,j}}{N}\right)^{l_{i,j}}\left(1-\frac{\rho_{i,j}}{N}\right)^{1-l_{i,j}}\right)
P_{out}&=1-P_{cap}[All~Interferers] %\Prob\{Capture All\}
\\&=1-\sum_{l_{1,1}=0}^{1}\sum_{l_{1,2}=0}^{1}\cdots\sum_{l_{1,K_1}=0}^{1}\cdots\sum_{l_{3,K_3}=0}^{1}\left(\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\prod_{i=1}^{3}\prod_{j=1}^{K_i}\left(\int\frac{\alpha_{i,j}}{1+\frac{1}{1.5G\Gamma_{i,j}}}p_{\alpha_{i,j}}(\alpha_{i,j})d\alpha_{i,j}\right)^{l_{i,j}}\right)
\\& \qquad \times \left(\prod_{i=1}^{3}\prod_{j=1}^{K_i}\left(\frac{\rho_{i,j}}{N}\right)^{l_{i,j}}\left(1-\frac{\rho_{i,j}}{N}\right)^{1-l_{i,j}}\right)
\end{aligned}
\end{equation}
in which $l_{i,j}=\{0,1\}$ indicates the absence or presence of
the $j$th interferer of type $i$. If all the interferers of a
given type have the same transmitting power, duty cycle, and
coefficients, (\ref{Pout51}) reduces to
\begin{equation}\label{Pout52}
\begin{aligned}
P_{out}=1-\sum_{k_1=1}^{K_1}\sum_{k_2=1}^{K_2}\sum_{k_3=1}^{K_3}\left(\exp{\left(-\frac{1}{1.5G\Gamma_N}\right)}\prod_{i=1}^{3}\left(\frac{1}{1+\frac{\alpha_{i}}{1.5G\Gamma_{i}}}\right)^{k_i}\right)p({k_1})p({k_2})p({k_3})
\end{aligned}
\end{equation}
where
\begin{equation}\label{Pk1}
\begin{aligned}
p(k_1)={{K_1}\choose{k_1}}\left(\frac{\rho_1}{N}\right)^{k_1}\left(1-\frac{\rho_1}{N}\right)^{(K_1-k_1)} \\
p(k_2)={{K_2}\choose{k_2}}\left(\frac{\rho_2}{N}\right)^{k_2}\left(1-\frac{\rho_2}{N}\right)^{(K_2-k_2)} \\
p(k_3)={{K_3}\choose{k_3}}\left(\frac{\rho_3}{N}\right)^{k_3}\left(1-\frac{\rho_3}{N}\right)^{(K_3-k_3)}
\end{aligned}
\end{equation}

\subsection{Packet Error Probability}
Reed-Solomon (RS) code is an effective FEC coding scheme used in
packet transmissions. If the code length is $L$ and its symbol
error correction capability is $R_C$, the packet error probability
is
\begin{equation}\label{PacketE01}
\begin{aligned}
P_e=1-\sum_{i=0}^{R_C}{{L}\choose{i}}P_{out}^{i}\left(1-P_{out}\right)^{L-i}
\end{aligned}
\end{equation}
in which the outage probability $P_{out}$ can be calculated using
the results in (\ref{Pout51}) or (\ref{Pout52}).

\section{Numerical Analysis}\label{sec:num}
This section presents numerical results based on the derivations
in Sections \ref{sec:capture} and \ref{sec:outage}. For the
performance evaluation presented here, the desired signal is
considered to be a DS/FH signal with spreading gain $G=50$ and
occupying a 1 MHz bandwidth (i.e., $B=1$ MHz), which is reflective
of a typical transmission bandwidth of a frequency hopping signal.
The propagation of signals from the desired transmitter as well as
interfering sources to the receiver in a typical environment where
such devices as those that operate in the ISM band used, is well
modeled through Rayleigh fading. It is reasonable to assume that
the transmissions from the various sources are independent of each
other. Therefor, all the signals at the receiver are considered to
have undergone mutually independent Rayleigh fading.

Fig. \ref{Pcap2D} shows the capture probability of a DS/FH signal
in the presence of barrage interference. Here we use barrage
interference to model interfering emissions that occupy a
bandwidth much larger than that occupied by the desired signal
during a given hop duration. Capture probability of the desired
signal, evaluated using (\ref{Pcap02}), is plotted against
normalized SNR and normalized SIR for four scenarios, with
different number of interferers. An SINR threshold of 0 dB is
assumed. This can be understood as indicative of the upper
performance bound since a 0 dB threshold implies that the signal
can be successfully detected even when its received power is only
equal to the total interference plus noise power. In Fig.
\ref{Pcap2D}(a) one barrage interferer is considered in the
evaluation. Sub-plots (b), (c) and (d) in Fig. \ref{Pcap2D}
correspond to the cases with 3, 10 and 100 interferes
respectively. The variation in the transmitted powers of different
sources of interference, and that of the desired signal, can be
understood in terms of different SIR values; the effect this has
on the capture probability of the DS/FH signal is seen in the
plots. The plots also show how SNR affects the capture
probability. It is evident that the capture probability can be
limited by either SNR or SIR.
\begin{figure}[p]
  \centering
  \includegraphics[width=\columnwidth]{figure2.pdf}
  \centering
  \caption{Capture probability in the presence of barrage interference. (a) 1 interferer (b) 3 interferers (c) 10 interferers (d) 100 interferers. SIR and SNR are expressed in dB.}
\label{Pcap2D}
\end{figure}
\begin{figure}[p]
  \centering
  \includegraphics[width=\columnwidth]{figure3.pdf}
  \centering
  \caption{Capture probability of a DS/FH signal. 5 barrage interferers, 5 partial-band interferers and 8 tone interferers. SIR is expressed in dB.}
\label{Pcap1D}
\end{figure}
%% When using eps figures instead of pdf
%\begin{figure}[p]
%  \centering
%  \includegraphics[width=\columnwidth]{figure2.eps}
%  \centering
%  \caption{Capture probability in the presence of barrage interference. (a) 1 interferer (b) 3 interferers (c) 10 interferers (d) 100 interferers. SIR and SNR are expressed in dB.}
%\label{Pcap2D}
%\end{figure}
%\begin{figure}[p]
%  \centering
%  \includegraphics[width=\columnwidth]{figure3.eps}
%  \centering
%  \caption{Capture probability of a DS/FH signal. 5 barrage interferers, 5 partial-band interferers and 8 tone interferers. SIR is expressed in dB.}
%\label{Pcap1D}
%\end{figure}
%%

Fig. \ref{Pcap1D} illustrates the capture probability under a
combination of interference types. In the evaluations, multiple
interferers from each interference type are considered to be
transmitting in the shared frequency band -- five barrage
interferers, five partial-band interferers and eight tone
interferers. To analyze the effect of temporal and spatial
dynamics of the interferers (in terms of power levels and emission
activity) on receiver capture, a wide range of SIR values are
plotted against the capture probability, for a fixed SNR value of
0 dB. The number of channels available for the DS/FH signal to hop
is denoted as $N$. A higher value of $N$ means a larger shared
bandwidth. For a given number of interferers operating in the
frequency band, this translates into increased diversity in terms
of frequency channels utilized for transmission. This, in turn,
would lead to improved performance. Three different values of $N$
are considered here ($N=\{10,40,80\}$). The duty cycle, $\rho$,
characterizes the on-off operation of an interferer. For the
evaluation presented here, two scenarios are considered, namely,
`low $\rho$'(with the values $\rho_1 = \rho_2 = \rho_3 = 0.2$ for
the three types of interferers) and  `high $\rho$' ($\rho_1 =
\rho_2 = \rho_3 = 1$). Two sets of curves are plotted in Fig.
\ref{Pcap1D}, corresponding to `low $\rho$' and `high $\rho$', for
the three values of $N$. The bandwidth $B_2$ and frequency offset
$\triangle f$ of the partial-band interferer are chosen as zero
and $B/2$ respectively. The two parameters, $\triangle W$ and
$\triangle \phi$, characterizing frequency and phase offset of
tone interferers with respect to the desired signal, are both set
to zero. These values imply the case where the interferers have
the most impact on the desired signal.

The outage probability and packet error probability of a DS/FH
signal under multiple interferers are plotted in Fig. \ref{Pout}
and Fig. \ref{PacketError} respectively. Outage probability in a
hop duration is evaluated using Eq. (\ref{Pout51}). The number of
hopping channels is assumed to be 80 (i.e., $N=80$) and the
interference bandwidth and offsets are set to the same values as
for Fig. \ref{Pcap1D}. Outage probability is plotted against a
range of SNR values for three different SIR scenarios (-20 dB, -10
dB and 0 dB) and two set of values for the number of interferers
-- (i) 12 interferers (3 barrage, 3 partial-band and 6 tone
interferers) (ii) 50 interferers (15 barrage, 15 partial-band and
20 tone interferers). The duty cycle of the interferers belonging
to the same interference type is assumed to be the same. This
considers the scenario where multiple transmitting devices
belonging to a given type of system and conforming to the same set
of protocols are operating. The chosen values are $\rho_1=0.5$,
$\rho_2=0.7$ and $\rho_3=0.9$ for the three corresponding
interference types. It is observed from the plots that at high SIR
and SNR values, the actual number of interferers does not have a
very strong bearing on the outage probability. At lower SIR
values, however, even with high SNR the difference in the outage
probability of a DS/FH signal in the presence of a few interferers
and many interferers in more pronounced. This suggests that a
fewer number of interferers transmitting at high power levels
would be more damaging to the desired signal than numerous
interferers emitting at low power levels. In Fig.
\ref{PacketError}, we have plotted the packet error probability
against SNR for fixed SIR values. The packet error probability has
been evaluated using Eq. (\ref{PacketE01}). All the parameters
related to interferers are the same as in Fig. \ref{Pcap1D}. The
FEC code for the packet is assumed to be a Reed-Solomon code with
parameters values $L=16$ and $R_C=4$. Therefore, the RS code can
correct up to 4 symbol errors. For the given code length and
coding scheme, it is seen that to achieve an adequate packet error
rate (of the order of $10^{-5}$), a normalized SNR of about -2 dB
is required with a normalized SIR of 0 dB.
\begin{figure}[p]
  \centering
  \includegraphics[width=\columnwidth]{figure4.pdf}
  \centering
  \caption{Outage probability of a DS/FH signal with multiple interferers. $\rho_1=0.5$, $\rho_2=0.7$ and $\rho_3=0.9$. SIR and SNR are expressed in dB.}
\label{Pout}
\end{figure}
\begin{figure}[p]
  \centering
  \includegraphics[width=\columnwidth]{figure5.pdf}
  \centering
  \caption{Packet error probability of a DS/FH signal with multiple interferers. $\rho_1=0.5$, $\rho_2=0.7$ and $\rho_3=0.9$. (16,8) RS code is used. SIR and SNR are expressed in dB.}
\label{PacketError}
\end{figure}
%% When using eps figures instead of pdf
%\begin{figure}[p]
%  \centering
%  \includegraphics[width=\columnwidth]{figure4.eps}
%  \centering
%  \caption{Outage probability of a DS/FH signal with multiple interferers. $\rho_1=0.5$, $\rho_2=0.7$ and $\rho_3=0.9$. SIR and SNR are expressed in dB.}
%\label{Pout}
%\end{figure}
%\begin{figure}[p]
%  \centering
%  \includegraphics[width=\columnwidth]{figure5.eps}
%  \centering
%  \caption{Packet error probability of a DS/FH signal with multiple interferers. $\rho_1=0.5$, $\rho_2=0.7$ and $\rho_3=0.9$. (16,8) RS code is used. SIR and SNR are expressed in dB.}
%\label{PacketError}
%\end{figure}
%%

\section{Conclusion}\label{sec:concl}
In this paper, we have investigated the issue of coexistence
interference in unlicensed bands. Motivation for the work
presented here stems from the widespread use of the 2.4 GHz ISM
band for varied services and the growing realization of the
inadequacy of the licensing methodology for spectrum management
that is currently in place. The paradigm for operation in an
unlicensed band is to minimize the interference caused to other
users in the band and to avoid and suppress the interference from
coexisting devices. Towards this, interference modeling is often
used to get a better understanding of the operating environment
and is central to the evaluation of radio link performance. Based
on measurement studies in the 2.4 GHz ISM band, an analytical
interference model has been developed in this paper. The modeling
approach adopted involves profiling the observed emissions under
different interference types based on their spectral
characteristics. This offers the convenience of statistically
characterizing interference with satisfactory accuracy and in
terms of generic models rather than requiring that each source of
interference be specifically defined to reflect its temporal and
spectral emission characteristics. Such an approach to
interference modeling can be utilized in the development of
interference management strategies for unlicensed bands. The
effect of coexistence interference in the 2.4 GHz ISM band on the
performance of a hybrid DS/FH signal, in terms of outage
probability and packet error probability, has also been evaluated
in this paper. For this, the capture probabilities of a Rayleigh
faded signal in the presence of interfering transmissions from
coexisting sources were derived; considering both scenarios, when
interferers of the same type are present as well as the one with a
combination of the different interference types. Expressions for
the outage probability and the packet error probability (in terms
of the capture probabilities) were then derived. Results of
numerical evaluation of the derived performance measures have been
presented. This study is part of our ongoing effort to develop an
analytical framework for interference modeling and performance
evaluation in the unlicensed band, as it is seen an invaluable
tool for interference management.

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