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Optimal Multiband Joint Detection for Spectrum

Sensing in Cognitive Radio Networks
Zhi Quan, Student Member, IEEE, Shuguang Cui, Member, IEEE, Ali H. Sayed, Fellow, IEEE, and
H. Vincent Poor, Fellow, IEEE

AbstractSpectrum sensing is an essential functionality that

enables cognitive radios to detect spectral holes and to opportunistically use under-utilized frequency bands without causing
harmful interference to legacy (primary) networks. In this paper,
a novel wideband spectrum sensing technique referred to as
multiband joint detection is introduced, which jointly detects the
primary signals over multiple frequency bands rather than over
one band at a time. Specifically, the spectrum sensing problem is
formulated as a class of optimization problems, which maximize
the aggregated opportunistic throughput of a cognitive radio system
under some constraints on the interference to the primary users.
By exploiting the hidden convexity in the seemingly nonconvex
problems, optimal solutions can be obtained for multiband joint
detection under practical conditions. The situation in which
individual cognitive radios might not be able to reliably detect
weak primary signals due to channel fading/shadowing is also
considered. To address this issue by exploiting the spatial diversity,
a cooperative wideband spectrum sensing scheme refereed to as
spatial-spectral joint detection is proposed, which is based on a
linear combination of the local statistics from multiple spatially
distributed cognitive radios. The cooperative sensing problem is
also mapped into an optimization problem, for which suboptimal
solutions can be obtained through mathematical transformation
under conditions of practical interest. Simulation results show that
the proposed spectrum sensing schemes can considerably improve
system performance. This paper establishes useful principles for
the design of distributed wideband spectrum sensing algorithms
in cognitive radio networks.

Index TermsCognitive radio, cooperative sensing, hypothesis

testing, multiband joint detection, nonlinear optimization, spectrum sensing.

Manuscript received November 04, 2007; accepted October 02, 2008. First
published November 07, 2008; current version published February 13, 2009.
This research was supported in part by the National Science Foundation
under Grants ANI-0338807, CNS-06-25637, ECS-0601266, ECS-0725441,
CNS-0721935, CCF-0726740, and by the Department of Defense under Grant
HDTRA-07-1-0037. Part of this work was presented at the IEEE Conference
on Communications, Beijing, China, May 2008, and at the IEEE International
Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas,
NV, April 2008.
Z. Quan and A. H. Sayed are with the Department of Electrical Engineering,
University of California, Los Angeles, CA 90095 USA (e-mail: quan@ee.ucla.
edu; sayed@ee.ucla.edu).
S. Cui is with the Department of Electrical and Computer Engineering, Texas
A&M University, College Station, TX 77843 USA (e-mail: cui@ece.tamu.edu).
H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: poor@princeton.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2008.2008540

I. INTRODUCTION

RADITIONAL wireless networks are regulated by fixed

spectrum allocation policies to operate in certain time
frames, over certain frequency bands, and within certain geographical regions. This regulation results in situations in which
some radio bands are overcrowded while other bands remain
moderately or rarely occupied. In order to improve spectral
utilization, cognitive radio (CR) technology has been proposed
as a potential communication paradigm [1]. Cognitive radios
are defined by the Federal Communications Commission
(FCC) [2] as radio systems that continuously perform spectrum
sensing, dynamically identify unused spectrum, and then operate in those spectral holes where the licensed (primary) radio
systems are idle. In CR networks, secondary users are allowed
to use some portions of licensed radio bands opportunistically
provided that they do not cause harmful interference to the
primary users in these frequency bands. CR is an important
component of the IEEE 802.22 standard being developed for
wireless regional area networks, which involves a cognitive
radio based air interface to operate in a licence-exempt way
over the TV broadcast bands. This new communication paradigm, also referred to as the dynamic spectrum access (DSA) or
neXt Generation (XG) network [3], can dramatically improve
spectral utilization.
Effective spectrum sensing needs to detect weak primary
radio signals of possibly-unknown formats reliably [4]. Generally, spectrum sensing techniques can be classified into three
broad categories: energy detection [5], [6], matched filtering
(coherent) detection [7], and feature detection [8]. Energy
detection has been shown to be optimal if the cognitive devices
have no a priori information about the features of the primary
signals except local noise statistics [9]. When the CRs have
some knowledge about the primary signal features such as
preambles, pilots, and synchronization symbols, the optimal
detector usually applies the matched filter structure to maximize the probability of detection. If the modulation schemes of
the primary signals are known, then the cyclostationary feature
detector can differentiate primary signals from the local noise
by exploiting certain periodicity exhibited by the mean and
autocorrelation of the corresponding modulated signals. Since
noncoherent energy detection is simple and able to determine
spectrum-occupancy information quickly, we will adopt it as
the building block for constructing the proposed wideband
spectrum sensing techniques.

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QUAN et al.: OPTIMAL MULTIBAND JOINT DETECTION FOR SPECTRUM SENSING

A. Related Work
Previous studies on spectrum sensing in CR networks have
focused primarily on cooperation among multiple secondary
users [4], [10], [11] using distributed detection approaches [12],
[13], but are limited to the detection of signals over a single frequency band. The scheme based on voting rules [14] is one of
the simplest suboptimal solutions, which counts the number of
nodes that vote for the presence of the signal and compares it
against a given threshold. In [15], a fusion rule known as the OR
logic operation was used to combine decisions from several secondary users. In [16], two decision-combining approaches were
studied: hard decision with the AND logic operation, and soft
decision using the likelihood ratio test [12]. It was shown that
the soft decision combination of spectrum sensing results yields
gains over hard decision combining. In [17], the authors exploited the fact that summing signals from two secondary users
can increase the signal-to-noise ratio (SNR) and detection reliability if the signals are correlated. In [18], a generalized likelihood ratio test for detecting the presence of cyclostationarity
using multiple cyclic frequencies was proposed and evaluated
using Monte Carlo simulations. Another two cooperative spectrum sensing algorithms based on the likelihood ratio test can
be found in [19] and [20]. Along with these works, we have developed an optimal cooperation strategy [21] based on a linear
combination of local statistics from multiple cognitive radios.
Other suboptimal solutions for linear cooperation such as maximal radio combining and maximal deflection coefficient combining can be found respectively in [21][23].
On the other hand, the literature of wideband spectrum
sensing for cognitive radio networks is rather limited at this
time. An existing approach is to use a tunable narrowband
bandpass filter at the radio frequency (RF) front-end to search
one narrow frequency band at a time [24], over which existing
narrowband spectrum sensing techniques can be applied. In
order to search over multiple frequency bands at a time, the
RF front-end needs a wideband architecture and spectrum
sensing usually operates over an estimate of the power spectral
density (PSD) of the wideband signal. In [25] and [26], the
multiresolution features of the wavelet transform were used to
estimate the PSD over a wide frequency range. However, no
prior work attempts to make decisions over multiple frequency
bands jointly, which is essential for implementing efficient cognitive radio networks. A survey of existing spectrum sensing
techniques can be found in [35].
B. Contribution
The contribution of this paper is twofold. First, we introduce
the multiband joint detection framework for wideband spectrum
sensing in a single CR. Within this framework, we jointly optimize a bank of multiple narrowband detectors to improve the
aggregate opportunistic throughput of a cognitive radio system
while limiting the interference to the primary communication
system. In particular, we formulate the design of wideband spectrum sensing into a class of optimization problems. The objective is to maximize the aggregate opportunistic throughput in
an interference-limited cognitive radio network. By exploiting
the hidden convexity of the seemingly nonconvex problems, we
show that the optimization problem can be reformulated into

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a convex program under practical conditions. The multiband

joint detection strategy allows cognitive radios to best take advantage of the unused frequency bands and limit the resulting
interference.
In addition, we consider the situation in which spectrum
sensing is compromised by destructive channel conditions
between the target-under-detection and the detecting cognitive
radios, where it is hard to distinguish between a white spectrum
and a weak signal attenuated by deep fading. We propose a
cooperative wideband spectrum sensing scheme that exploits
the spatial diversity among multiple CRs to improve the sensing
reliability. The cooperation is based on a linear combination
of local statistics from spatially distributed cognitive radios
[21], [23], where these signals are assigned different weights
according to their individual positive contributions to the joint
sensing. In such a scenario, we view the design of distributed
wideband spectrum sensing as a spatialspectral joint detection
(SSJD) problem, which is further formulated into an optimization problem with the objective of maximizing the aggregate
opportunistic throughput under constraints on the interference
to primary users. Through mathematical reformulation, we
derive two sets of suboptimal but efficient solutions for the
optimization problem, which can considerably improve the
sensing performance.
The rest of the paper is organized as follows. In Section II,
we describe the system model for wideband spectrum sensing.
In Section III, we develop multiband joint detection algorithms
for spectrum sensing, which seek to maximize the aggregate
opportunistic throughput of a CR system. The spatialspectral
joint detection strategy is formulated in Section IV, where we
derive two efficient solutions to optimize the cooperation among
a network of CRs. The advantages of the proposed spectrum
sensing algorithms are illustrated by simulations in Section V,
and conclusions are drawn in Section VI.
II. SYSTEM MODEL
A. Wideband Spectrum Sensing
Consider a primary communication system (e.g., multicarrier
modulation based) operating over a wideband channel that is
divided into nonoverlapping narrowband subbands. In a particular geographical region and within a particular time interval,
some of the subbands might not be used by the primary users
and are available for opportunistic spectrum access.
We model the detection problem on subband as one of
(0), which represents the
choosing between a hypothesis
absence of primary signals, and an alternative hypothesis
(1), which represents the presence of primary signals. An example where only some of the bands are occupied by primary
users is depicted in Fig. 1. The underlying hypothesis vector is
a binary representation of the subbands that are allowed for or
prohibited from opportunistic spectrum access.
The crucial task of spectrum sensing is to sense the
subbands and identify spectral holes for opportunistic use. For
simplicity, we assume that the upper-layer protocols, e.g., the
medium access control (MAC) layer, can guarantee that all
cognitive radios stay silent during the detection interval such
that the only spectral power remaining in the air is emitted by

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Fig. 1. Schematic illustration of the occupancy of a multiband channel.

the primary users. In this paper, instead of considering a single

band at a time, we propose to use a multiband joint detection
technique, which jointly takes into account the detection of
primary users across multiple frequency bands.
B. Received Signal
,
Consider a multipath fading environment, where
, represents the discrete-time channel impulse response between the primary transmitter and a CR receiver with
denoting the number of resolvable paths. The received baseband signal at the RF front-end can be written as

[28], it is reasonable to assume that the channels between the

primary transmitter and secondary receivers change slowly such
that they can be assumed to be constant during each operation
period of interest. Our sensing algorithm needs to know only the
noise power and the squared values of the channel frequency
, which can be estimated in practice. Specifiresponses
cally, can be calibrated in a given band that is known for sure
to be idle (e.g., TV channel 37 is currently always empty) [29].
can be learned a priori during a period
Accordingly,
when the primary transmitter was known for sure to be working.
This a priori information is obtainable since most current TV
stations transmit pilot signals periodically at a fixed power level.
C. Signal Detection in Individual Bands
We start from signal detection in a single narrowband subband, which will constitute a building block for our multiband
joint detection procedure. Following [21], [23], to decide
whether the th subband is occupied or not, we test the following binary hypotheses:

(5)

(1)
represents the primary transmitted signal (with
where
and
is additive complex
cyclic prefix) at time
, i.e.,
white Gaussian noise with zero mean and variance
. In a multipath fading environment, the
wideband channel exhibits frequency-selective features [27]
and its discrete frequency response can be obtained through a
-point fast Fourier transform (FFT)
:
(2)

is the secondary received signal,

is the primary
where
is the channel gain between the pritransmitted signal, and
mary transmitter and the secondary receiver.
For each subband , we compute the summary statistic as the
samples,
sum of received signal energy over an interval of
i.e.,
(6)
and the decision rule is chosen as

In the frequency domain, the received signal at each subchannel

can be represented by its discrete Fourier transform (DFT):

(3)
where

is the primary transmitted signal at subchannel

and
(4)

is the received noise represented in the frequency domain.

are independent and norThe random variables
, i.e.,
mally distributed with zero means and variances
, since
and the DFT is a
linear unitary operation. Without loss of generality, we assume
, and additive
that the transmitted signal , channel gain
are independent of each other.
noise
Since the IEEE 802.22 consumer premise equipment (CPE)
is generally used in a fixed wireless network in the TV bands

(7)
where is the decision threshold of subband . For simplicity,
we assume that the transmitted signal in each subband has unit
; this assumption holds when primary
power, i.e.,
radios adopt uniform power transmission strategies given no
channel knowledge at the transmitter side. However, the development of the multiband joint detection algorithm does not rely
on this assumption while only the knowledge of the received
signal power and noise power is needed.
According to the central limit theorem [30], for large , the
are approximately normally distributed [5]
statistics
with means
(8)
and variances
(9)

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Fig. 2. Schematic representation of multiband joint detection for wideband spectrum sensing in cognitive radio systems.

Thus, we can write these approximate statistics compactly as

.
Using the decision rule in (7), the probabilities of false
alarm and detection in the th subband can be approximately
expressed as
(10)

causing harmful interference to the primary users. For a given

threshold vector , the probabilities of false alarm and detection
can be compactly represented as
(12)
and
(13)

and

Similarly, the probabilities of missed detection can be written in

a vector as
(11)
Note that the SNR of such an energy detector is defined as
, which plays an important role in determining the detection performance. The choice of the threshold
leads to a tradeoff between the probability of false alarm and
the probability of missed detection,
.
Specifically, a higher threshold will result in a smaller probability of false alarm, but a larger probability of miss, and vice
versa.
The probabilities of false alarm and miss have unique implications for CR networks. Low probabilities of false alarm are
necessary to maintain high spectral utilization in CR systems,
since a false alarm would prevent the unused spectral segments
from being accessed by secondary users. On the other hand, the
probability of missed detection measures the interference of secondary users to the primary users, which should be limited in
opportunistic spectrum access. These implications are based on
a typical assumption that if primary signals are detected, the secondary users will not use the corresponding channel, and if no
primary signals are detected, then the corresponding frequency
band will be used by secondary users.
III. MULTIBAND JOINT DETECTION
In this section, we present the multiband joint detection
framework for wideband spectrum sensing [31], as illustrated
in Fig. 2. The design objective is to find the optimal threshold
so that the cognitive radio system
vector
can make efficient use of the unused spectral segments without

(14)
The vector
can be expressed as
, where
denotes the all-one vector.
narrowband subbands
Consider a CR device sensing the
to take use of the unused ones for opportunistic transmission.
denote the throughput achievable over the th subband
Let
. If the
if used by secondary users, and
transmit power and the channel gains between secondary users
are known, can be estimated using the Shannon capacity formula [27]. Since
measures the opportunistic spectral
utilization of subband , the aggregate opportunistic throughput
of the CR system can be defined as
(15)
which is a function of the threshold vector . Due to the inherent
tradeoff between
and
, maximizing the sum
will result in large
, hence causing harmful
rate
interference to primary users.
However, the interference to primary users should be limited in a CR network. For a wideband primary communication system, the effect of interference induced by CR devices
can be characterized by a relative priority factor for each primary user transmitting over the corresponding subbands, i.e.,
, where
indicates the cost incurred if
the primary user in subband is interfered with. In a special
case where the th primary user is equally important, we may
. Suppose that primary users share a portion of
have

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the subbands and each primary user occupies a subset

of
subbands. The aggregate interference to primary user can be
expressed as

Substituting (11) into (19) gives

(20)
where

(16)
This expression models, for example, the situation arising in a
multiuser orthogonal frequency division multiplexing (OFDM)
system, where various primary users have different levels of priority. Alternatively, can be defined as a function of the bandwidth of subband since in some applications each particular
subband does not have to occupy an equal amount of bandwidth.
To summarize, our objective is to find the optimal thresholds
for the subbands in order to collectively maximize
the aggregate opportunistic throughput subject to some interference constraints for each primary user. As such, the opportunistic rate optimization problem in the context of a multiuser
primary system can be formulated as
P1

(21)
Similarly, the combination of (10) and (18) leads to
(22)
where
(23)
Consequently, the original problem (P1) has the following
equivalent form
P2
(24)

(17)
(18)
where the constraint (17) limits the interference in each subband
, and the constraint (18) dictates
with
that each subband should be able to achieve a minimum opportunistic spectral utilization given by
. For a single-user primary system where all the subbands
.
are used by one primary user, we have
Intuitively, some factors need to be considered in the multiband joint detection. First, the subband with a higher opportunistic rate should have a higher threshold (i.e., a smaller
probability of false alarm) such that it can be best used by CRs.
Second, the subband that carries a higher priority primary user
(i.e., a smaller probability of
should have a lower threshold
missed detection) in order to prevent opportunistic access by
secondary users. Third, a little compromise on those subbands
carrying less important primary users might boost the opportunistic rate considerably. Thus, in the determination of the optimal threshold vector, it is necessary to strike a balance among
the channel conditions, the opportunistic throughput, and the
relative priority of each subband.
The objective and constraint functions in (P1) are generally
nonconvex, making it difficult to efficiently solve for the global
optimum. In most cases, suboptimal solutions or heuristics have
to be used. However, we find that this seemingly nonconvex
problem can be made convex by exploiting hidden convexity
properties and reformulating the problem.
The fact that the -function is monotonically nonincreasing
allows us to transform the constraints in (17) and (18) into linear
constraints. Specifically, from (17), we obtain
(19)

(25)
Although the constraint (25) is linear, the problem is still nonconvex. However, it can be transformed into a tractable convex
optimization problem in the regime of low probabilities of false
alarm and miss. To establish the transformation, we need the
following results.
Lemma 1: The function
is convex in
if
.
Proof: Refer to Appendix A.
Lemma 2: The function
is convex in
if
.
Proof: Refer to Appendix B.
Recall that the nonnegative weighted sum of a set of convex
functions is also convex [32]. The problem (P1) then becomes
a convex program if we introduce the following conditions:
(26)
This regime of probabilities of false alarm and missed detection is of practical interest for achieving rational opportunistic
throughput and interference levels in CR networks.
Under the conditions in (26), the feasible set of problem (P2)
is convex because the intersection of a convex set and a set of
halfspaces is also convex. The optimization problem takes the
form of minimizing a convex function subject to a convex constraint, and thus a local optimum is also the global optimum.
Efficient numerical search algorithms such as the interior-point
method can be used to find the optimal solution [32].
Alternatively, we can formulate the multiband joint detection
problem into another optimization problem that minimizes the
interference from CRs to the primary communication system

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1133

. The statistics across the

subbands can be compactly represented in matrix form as follows:

..
.

..
.

..

(27)

..
.

Fig. 3. Weighting cooperation for spectrum sensing in the k th subband.

subject to some constraints on the aggregate opportunistic

throughput, i.e.,

To exploit the spatial diversity, we linearly combine the summary statistics from spatially distributed CRs in each subband
to obtain a global test statistic
(28)

P3
are the combining
where
coefficients for subband , which can be compactly written as

where
is the minimum required aggregate opportunistic
throughput. Like (P1), this problem can be transformed into
a convex optimization problem by enforcing the conditions
in (26). The result will be illustrated numerically later in
Section V.

..
.

..
.

..

..
.

(29)

, for all
.
Note that
are normally distributed, the test
Since the elements in
statistics ,
, are also normally distributed with
means

IV. SPATIALSPECTRAL JOINT DETECTION

The detection performance of spectrum sensing is usually
compromised by destructive channel conditions between the
target-under-detection and the CRs, since it is hard to distinguish between a white spectrum and a weak signal attenuated
by deep fading. In such scenarios, a network of cooperative CR
devices, which experience different channel conditions from
the target, would have a better chance of detecting the primary
signal if they combine their sensing results. In this section,
we present a cooperation framework for wideband spectrum
sensing, within which CRs can exploit spatial diversity by
exchanging local sensing results in order for the secondary
network to obtain a more accurate estimate of the unused
frequency bands [33].
spatially distributed CRs collaboratively
Suppose that
sense a wide frequency band, aiming to find unused spectral
segments for opportunistic communication. By combining the
summary statistics from individual CRs, a fusion center, which
could be one of the CRs, makes the final decision on the presence or absence of primary signals in each of the subbands.
We propose a linear weighting fusion scheme as illustrated in
Fig. 3. It is assumed that there is a control channel, through
which the summary statistics of individual secondary users are
transmitted to the fusion center.
denote the summary statistic of the th secondary
Let
user in the th subband. For each subband, the statistics from
individual secondary users can be written in a vector as

(30)
and variances
(31)
where
(32)
is a diagonal matrix assuming that the elements of
pendent and

are inde-

are the squared magnitudes of the channel gains between the

primary transmitter and secondary receivers on each subband.
becomes a nondiagonal matrix if the elePlease note that
are not independent, but the following derivation
ments of
is a positive semidefinite matrix.
is still valid since
In order to decide the presence or absence of the primary
signal in subband , we use the following binary test:

(33)

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Accordingly, the detection performance in terms of the probabilities of false alarm and detection are given approximately by

since
where
(38) can be expressed as

. From (35), the constraint

(34)
(41)
and
(35)
In the design of an efficient distributed cooperative sensing
system, the goal is to maximize the system performance measure of interest by controlling the weight coefficient matrix
and the threshold vector . Just as we did in the previous section, we would like to maximize the opportunistic rate while
satisfying some constraints on the interference to the primary
communication system. Note that the aggregate opportunistic
subbands is now a function of both the
throughput of the
threshold vector and the weight coefficient matrix , i.e.,

for
, since
is implied by (40) and (41) that

and

. It

(42)
Observing that the left-hand side on the constraint (40) is a
convex function and the right-hand side is a linear function,
. Similarly, (41) is also
(40) defines a convex set for
a convex constraint.
Then, we reformulate problem (P4) by introducing a new
variable
(43)

(36)
Consequently, the spatialspectral joint detection problem is
formulated as
P4
(37)
(38)
(39)

Define
and
(41) can be written as

. The constraints (40) and

(44)
and
(45)

Note that the formulation in (P4) is in the context of a singleuser primary system and it can be easily extended to the case
of a multiuser primary system as (P1) does. Finding the exact
optimal solution for the above problem is difficult, since for any
subband, the probabilities of false alarm and miss are neither
and
convex nor concave functions of the weight coefficients
the test threshold
according to (34) and (35).
In the following, we will develop two efficient methods for
and the thresholds ,
solving for the weight coefficients
which lead to near-optimal solutions for (P4). For consistency,
we still assume the practical conditions in (26) unless explicitly
stated otherwise.
A. Joint Optimization
To jointly optimize
and , we show that (P4) can be reformulated into an equivalent form with a convex feasible set
and an objective function lower bounded by a concave function.
Through maximizing the lower bound of the objective function,
we are able to obtain a good approximation to the optimal solution of the original problem.
First, we show how to transform the nonconvex constraints in
(38) and (39) into convex constraints by exploiting the monotonicity of the -function. Substituting (34) into the constraint
(39), we have

(40)

Note that (45) is actually a linear constraint.

The constraint (37) now becomes

(46)
which can be shown to be convex through the following result.
, the function
Lemma 3: If
is concave in
.
Proof: Refer to Appendix C.
By changing the variables
and
, we can write the objective function in
(P4) as

(47)

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From the RayleighRitz theorem [34], we have

(48)
Now define a new function

1135

will result in a larger probability of detection if

is
and
.
normally distributed under both hypotheses
In the spatial optimization, we would like to choose the
in order to achieve the maximum modweight coefficients
ified deflection coefficient for each subband. Note that the
is unbounded. To obtain
feasible set for maximizing
a unique solution, we confine the weight vector to be on the
unit-norm ball and pose
P6

(49)
, the convexity of which is established
for
through the following result.
Lemma 4: If
, the function
is
convex in
.
Proof: The proof is similar to that of Lemma 3, and thus is
omitted.
Consequently, the aggregate opportunistic rate can be lower
bounded as

for
.
This problem can be solved as follows. First, apply the linear
transform
(53)
where

is the square root of the matrix

(50)

, i.e.,

(54)

..
.

An efficient suboptimal method to solve (P4) is to maximize the

lower bound of its objective function, i.e.,
In addition, we know that

is upper bounded by

P5

(55)
where
denotes the maximum eigenvalue of a matrix and
follows from the RayleighRitz theorem [34]. Note that the
is achieved if
equality in
(51)

(56)

Implied by the practical conditions in (26), this problem is a

convex optimization problem and can be efficiently solved.

which is the eigenvector corresponding to the maximal eigen.

value of the positive semidefinite matrix
Therefore, the optimal solution of (P6) is given by

B. Sequential Optimization
Here, we present another heuristic approach that divides the
optimization of the original problem (P4) into two stages. In
the first stage, referred to as spatial optimization, we choose the
weight coefficients in order to maximize a performance measure for signal detection. In the second phase, called spectral
obtained from spatial optioptimization, we fix the values of
mization and optimize the thresholds across all the subbands.
1) Spatial Optimization: A good measure for evaluating the
detection performance, called the modified deflection coefficient
[21], [23], is defined as

(57)
2) Spectral Optimization: Substituting the weight vectors
obtained in the first subproblem (P6) into (34) and
(35), the probabilities of false alarm and detection become functions of only the threshold . Following the procedure in (P1),
we can solve the following subproblem for the threshold vector
:
P7

(52)
can be interpreted as a signal-to-noise
The quantity
ratio. For any given probability of false alarm, a larger value of

(58)
(59)

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TABLE I
POWER DELAY PROFILE IN EXAMPLE 1

TABLE II
OTHER PARAMETERS USED IN EXAMPLE 1

where
(60)
and

(61)
As before, the problem is convex and can be solved efficiently.
As an alternative example, the spatialspectral joint detection problem can be reformulated to minimize the interference
subject to some constraint on the aggregate opportunistic
throughput , i.e.,
P8

Fig. 4. The aggregate opportunistic throughput versus the constraint on the aggregate interference to the primary communication system.

Near-optimal solutions can be obtained using the same techniques as in solving (P4).
V. SIMULATION RESULTS
In this section, we numerically evaluate the proposed spectrum sensing schemes. Consider a 48-MHz primary system
where the wideband channel is equally divided into eight
, we assume an
subbands. For each subband
achievable throughput rate if used by CRs and a cost coefficient indicating the penalty if the primary signal is interfered
with by secondary users. It is expected that the opportunistic
, and the
spectrum utilization is at least 50%, i.e.,
probability that the primary user is interfered with is at most
. For simplicity it is assumed that the noise power
level is
, and the length of each detection interval is
.
A. Example 1: Multiband Joint Detection for Individual CRs
This example studies multiband joint detection in a single
CR. The proposed spectrum sensing algorithms are examined
by comparing with an approach that searches for a uniform
threshold to maximize the aggregate opportunistic throughput.
We consider a power delay profile, as given in Table I, that
specifies the frequency selective channel between the primary
transmitter and the secondary receiver. The channel gain, opportunistic rate, and interference penalty on each subband are
given in Table II.
We would like to maximize the aggregate opportunistic
throughput over the eight subbands subject to the constraints
on the interference to the primary users, as formulated in

(P1). Fig. 4 plots the maximum aggregate opportunistic rates

against the aggregate interference to the primary communication system. It can be seen that the multiband joint detection
algorithm with optimized thresholds can achieve a much higher
opportunistic rate than that achieved by the uniform threshold
method. That is, the proposed multiband joint detection makes
better use of the wide frequency band by balancing the conflict between improving spectral utilization and reducing the
interference. In addition, it is observed that the aggregate
opportunistic rate increases as we relax the constraint on the
aggregate interference .
An alternative example is depicted in Fig. 5, showing the numerical results of minimizing the aggregate interference subject
to the constraints on the aggregate opportunistic throughput as
formulated in (P3). It can be observed that the multiband joint
detection strategy outperforms the one using uniform thresholds in terms of the induced interference to the primary users
for any given opportunistic throughput target. For illustration
purposes, the optimized thresholds and the associated probabilities of missed detection and false alarm are given in Fig. 6 for
(P1) and (P3).
B. Example 2: Cooperative Wideband Sensing among CRs
In this example, we consider the case in which two CRs cooperatively sense the eight subbands by exchanging the summary statistics of their sensed data. We compare the two proposed spatialspectral joint detection schemes with the multiband joint detection algorithms performed individually without
cooperation. The two CRs experience different channel models
as specified in Table III. Other parameters are given in Table IV.

QUAN et al.: OPTIMAL MULTIBAND JOINT DETECTION FOR SPECTRUM SENSING

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TABLE III
POWER DELAY PROFILES IN EXAMPLE 2

TABLE IV
OTHER PARAMETERS USED IN EXAMPLE 2

Fig. 5. The aggregate interference to the primary communication system versus

the constraint on the aggregate opportunistic throughput.

Fig. 7. The aggregate opportunistic throughput versus the constraint on the aggregate interference to the primary communication system.

Fig. 6. The optimized thresholds and the associated probabilities of missed

:
and (P3)
detection and false alarm on individual subbands: (P1) "

:
.

= 2 354 Mbps

= 1 474

The numerical results of solving (P4), which maximizes the

aggregate opportunistic throughput subject to the constraints on
the aggregate interference, are illustrated in Fig. 7. It is observed
that the spectrum sensing algorithms with cooperation result in
higher opportunistic rates than the sensing algorithms without
cooperation. In Fig. 8, we examine the problem (P8), which
minimizes the aggregate interference under the constraints on

the minimum aggregate opportunistic throughput. Similarly, the

algorithms with cooperation perform much better than those
without cooperation, and the joint optimization outperforms the
sequential optimization.
Generally speaking, these numerical results show that multiband joint detection can improve the spectral efficiency considerably by making better use of the spectral diversity, and the spatialspectral joint detection strategies can further enhance the
system performance by exploiting the spatial diversity.
VI. CONCLUSION
In this paper, we have proposed multiband joint detection for
wideband spectrum sensing in CR networks. The basic strategy
is to take into account the detection of primary users jointly
across a bank of narrowband subbands rather than considering
only one single band at a time. We have formulated the joint detection problem into a class of optimization problems to improve

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 3, MARCH 2009

APPENDIX C
PROOF OF LEMMA 3
Proof: To prove the lemma, we take the Hessian of
over
and obtain

..
.

..
.

Fig. 8. The aggregate interference to the primary communication system versus

the constraint on the aggregate opportunistic throughput.

the spectral efficiency and reduce the interference. By exploiting

the hidden convexity in the seemingly nonconvex problem formulations, we have obtained the optimal solution under practical conditions. In addition, we have presented a spatialspectral joint detection strategy for cooperative wideband spectrum
sensing, in which spatially distributed CRs can collaborate with
each other to improve the sensing reliability by exchanging the
individual sensing statistics. We have provided efficient suboptimal solutions for the problems that jointly optimize the cooperation among spatially distributed secondary users and the decision thresholds over multiple bands. The proposed spectrum
sensing algorithms have been examined numerically and have
been shown to perform well.
APPENDIX A
PROOF OF LEMMA 1
Proof: Taking the second derivative of
gives

from (10)

(62)

, we have
. Consequently, the
Since
second derivative of
is larger than or equal to zero,
which implies that
is convex in .
APPENDIX B
PROOF OF LEMMA 2
Proof: This can be proved using a similar technique to that
used in proving Lemma 1. By taking the second derivative of
is concave, and hence that
(11), we can show that
is a convex function.

Since
denoted by

, the
by
matrix
is a negative semidefinite matrix,
. Consequently,
is concave in
.
ACKNOWLEDGMENT

The authors would like to thank S. Shellhammer at

Qualcomm for his helpful discussions.
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Zhi Quan (S07) received the B.E. degree in communication engineering from the Beijing University of
Posts and Telecommunications, China, and the M.S.
degree in electrical engineering from Oklahoma State
University, Stillwater, OK. He is currently working
towards the Ph.D. degree in electrical engineering at
the University of California, Los Angeles (UCLA).
He was a visiting researcher with Princeton University, Princeton, NJ, during summer 2007, and was
an interim engineering intern with Qualcomm during
summer 2008. His current research interests include
statistical signal processing, wireless communication and networking, cognitive
radios, and multimedia.
Mr. Quan was the recipient of the UCLA Chancellors Dissertation Fellowship (20082009).

Shuguang Cui (S99M05) received the B.Eng. degree in radio engineering (with the highest distinction) from Beijing University of Posts and Telecommunications, Beijing, China, in 1997, the M.Eng. degree in electrical engineering from McMaster University, Hamilton, ON, Canada, in 2000, and the Ph.D.
degree in electrical engineering from Stanford University, Stanford, CA, in 2005.
From 1997 to 1998, he was a System Engineer at
Hewlett-Packard, Beijing, China. In summer 2003,
he worked at National Semiconductor, Santa Clara,
CA, on the ZigBee project. From 2005 to 2007, he worked as an Assistant Professor at the Department of Electrical and Computer Engineering, University of
Arizona, Tucson. He is now working as an Assistant Professor in Electrical and
Computer Engineering at the Texas A&M University, College Station. His current research interests include cross-layer energy minimization for low-power
sensor networks, hardware and system synergies for high-performance wireless
radios, statistical signal processing, and general communication theories.
Dr. Cui was a recipient of the NSERC graduate fellowship from the National
Science and Engineering Research Council of Canada and the Canadian Wireless Telecommunications Association (CWTA) graduate scholarship. He has
been serving as the TPC co-chair for the 2007 IEEE Communication Theory
Workshop and the ICC08 Communication Theory Symposium. He is currently
serving as an Associate Editor for the IEEE COMMUNICATION LETTERS and
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.

Ali H. Sayed (S90M92SM99F01) received

the Ph.D. degree from Stanford University, Stanford,
CA, in 1992.
He is Professor and Chairman of Electrical Engineering at the University of California, Los Angeles
(UCLA) and Principal Investigator of the Adaptive
Systems Laboratory. He has published widely, with
over 300 articles and five books, in the areas of statistical signal processing, estimation theory, adaptive
filtering, signal processing for communications and
wireless networking, and fast algorithms for large
structured problems. He is coauthor of the textbook Linear Estimation (Prentice-Hall, 2000), of the research monograph Indefinite Quadratic Estimation
and Control (SIAM, 1999), and co-editor of Fast Algorithms for Matrices with
Structure (SIAM, 1999). He is also the author of the textbooks Fundamentals
of Adaptive Filtering (Wiley, 2003) and Adaptive Filters (Wiley, 2008). He has
contributed several encyclopedia and handbook articles.

1140

Dr. Sayed has served on the editorial boards of the IEEE Signal Processing
Magazine, the European Signal Processing Journal, the International Journal
on Adaptive Control and Signal Processing, and the SIAM Journal on Matrix
Analysis and Applications. He also served as the Editor-in-Chief of the IEEE
TRANSACTIONS ON SIGNAL PROCESSING from 2003 to 2005 and the EURASIP
Journal on Advances in Signal Processing from 2006 to 2007. He is a member
of the Signal Processing for Communications and the Signal Processing Theory
and Methods technical committees of the IEEE Signal Processing Society. He
has served on the Publications (20032005), Awards (2005), and Conference
(2007present) Boards of the IEEE Signal Processing Society. He served on
the Board of Governors (20072008) of the same Society and is now serving as
Vice-President of Publications (2009present). His work has received several
recognitions, including the 1996 IEEE Donald G. Fink Award, the 2002 Best
Paper Award from the IEEE Signal Processing Society, the 2003 Kuwait Prize
in Basic Sciences, the 2005 Terman Award, the 2005 Young Author Best Paper
Award from the IEEE Signal Processing Society, and two Best Student Paper
Awards at international meetings (1999 and 2001). He has served as a 2005
Distinguished Lecturer of the IEEE Signal Processing Society and as General
Chairman of ICASSP 2008.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 3, MARCH 2009

H. Vincent Poor (S72M77, SM82F87) received the Ph.D. degree in electrical engineering
and computer science from Princeton University,
Princeton, NJ, in 1977.
From 1977 until 1990, he was on the faculty of
the University of Illinois at Urbana-Champaign.
Since 1990, he has been on the faculty at Princeton
University, where he is the Michael Henry Strater
University Professor of Electrical Engineering and
Dean of the School of Engineering and Applied
Science. His research interests are in the areas
of stochastic analysis, statistical signal processing and their applications in
wireless networks, and related fields. Among his publications in these areas
are the recent books MIMO Wireless Communications (Cambridge University
Press, 2007) and Quickest Detection (Cambridge University Press, 2009).
Dr. Poor is a member of the National Academy of Engineering, a Fellow of
the American Academy of Arts and Sciences, and a former Guggenheim Fellow.
He is also a Fellow of the Institute of Mathematical Statistics, the Optical Society of America (OSA), and other organizations. In 1990, he served as President of the IEEE Information Theory Society, and in 20042007, he served as
the Editor-in-Chief of the IEEE TRANSACTIONS ON INFORMATION THEORY. He
was the recipient of the 2005 IEEE Education Medal. Recent recognition of his
work includes the 2007 IEEE Marconi Prize Paper Award, the 2007 Technical
Achievement Award of the IEEE Signal Processing Society, and the 2008 Aaron
D. Wyner Award of the IEEE Information Theory Society.