Wideband Spectrum Sensing in Cognitive Radio
Networks
Zhi Quan†, Shuguang Cui‡ , Ali H. Sayed†, and H. Vincent Poor§
†
arXiv:0802.4130v1 [cs.IT] 28 Feb 2008
‡
Department of Electrical Engineering, University of California, Los Angeles, CA 90095
Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843
§
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544
Email: {quan, sayed}@ee.ucla.edu; cui@ece.tamu.edu; poor@princeton.edu
Abstract— Spectrum sensing is an essential enabling functionality for cognitive radio networks to detect spectrum holes and
opportunistically use the under-utilized frequency bands without
causing harmful interference to legacy networks. This paper
introduces a novel wideband spectrum sensing technique, called
multiband joint detection, which jointly detects the signal energy
levels over multiple frequency bands rather than consider one
band at a time. The proposed strategy is efficient in improving
the dynamic spectrum utilization and reducing interference to the
primary users. The spectrum sensing problem is formulated as
a class of optimization problems in interference limited cognitive
radio networks. By exploiting the hidden convexity in the
seemingly non-convex problem formulations, optimal solutions
for multiband joint detection are obtained under practical conditions. Simulation results show that the proposed spectrum sensing
schemes can considerably improve the system performance. This
paper establishes important principles for the design of wideband
spectrum sensing algorithms in cognitive radio networks.
I. I NTRODUCTION
Spectrum sensing is an essential functionality of cognitive
radios since the devices need to reliably detect weak primary
signals of possibly-unknown types [1]. In general, spectrum
sensing techniques can be classified into three categories:
energy detection [2], matched filter coherent detection [3],
and cyclostationary feature detection [4]. Since non-coherent
energy detection is simple and is able to locate spectrumoccupancy information quickly, we will adopt it as a building block for constructing the proposed wideband spectrum
sensing scheme.
There are previous studies on spectrum sensing in cognitive
radio networks with focus on cooperation among multiple
cognitive radios [1] [5] [6] via distributed detection approaches
[7] [8]. However, they are limited to the detection of signals
on a single frequency band. In [9], two decision-combining
approaches were studied: hard decision with the AND logic
operation and soft decision using the likelihood ratio test [7].
It was shown that the soft decision combination of spectrum
sensing results yields gains over hard decision combining.
In [10], 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 [11], a generalized likelihood ratio test for detecting the
presence of cyclostationarity over multiple cyclic frequencies
was proposed and evaluated through Monte Carlo simulations.
Along with these works, we have developed a linear cooperation strategy [12] [13] based on the optimal combination of
the local statistics from spatially distributed cognitive radios.
Generally speaking, the quality of the detector depends on the
level of cooperation and the bandwidth of the control channel.
The literature of wideband spectrum sensing for cognitive
radio networks is very limited. An early approach is to use
a tunable narrowband bandpss filter at the RF front-end to
sense one narrow frequency band at a time [14], over which
the existing narrowband spectrum sensing techniques can be
applied. In order to operate over multiple frequency bands at
a time, the RF front-end requires a wideband architecture and
the spectrum sensing usually involves the estimation of the
power spectral density (PSD) of the wideband signal. In [15]
and [16], the wavelet transform was used to estimate the PSD
over a wide frequency range given its multi-resolution features.
However, none of the previous works considers making joint
decisions over multiple frequency bands, which is essential for
implementing efficient cognitive radios networks.
In this paper, we introduce the multiband joint detection
framework for wideband spectrum sensing in individual cognitive radios. Within this framework, we jointly optimize a
bank of multiple narrowband detectors in order to improve the
opportunistic throughput capacity of cognitive radios and reduce their interference to the primary communication systems.
In particular, we formulate wideband spectrum sensing into a
class of optimization problems. The objective is to maximize
the opportunistic throughput in an interference limited cognitive radio network. By exploiting the hidden convexity of the
seemingly non-convex problems, we show that the optimization problems can be reformulated into convex programs under
practical conditions. The multiband joint detection strategy
allows cognitive radios to efficiently take advantage of the
unused frequency bands and limit the resulting interference.
The rest of this paper is organized as follows. In Section II,
we describe the system model for wideband spectrum sensing.
In Section III, we develop the multiband joint detection algorithms, which seek to maximize the opportunistic throughput.
The proposed spectrum sensing algorithms are examined by
numerical examples in Section IV and conclusions are drawn
in Section V.
Subbands occupied by
primary users
Spectrum holes
0
1
Fig. 1.
1
0
1
0
0
0
1
1
A schematic illustration of a multiband channel.
II. S YSTEM M ODELS
A. Wideband Spectrum Sensing
Consider a primary communication system (e.g., a multicarrier modulation based system) over a wideband channel that
is divided into K non-overlapping narrowband subchannels.
In a particular geographical region and time, some of the
K subchannels might not be utilized by the primary users
and are available for opportunistic spectrum access. Multiuser
orthogonal frequency division multiplexing (OFDM) is an
ideal candidate for such a scenario since it makes the subband
manipulation easy and flexible.
We model the occupancy detection problem on subchannel
k as one of choosing between H0,k (“0”), which represents the
absence of primary signals, and H1,k (“1”), which represents
the presence of primary signals. An illustrative example where
only some of the K bands are occupied by primary users
is depicted in Fig. 1. The underlying hypothesis vector is a
binary representation of the subchannels that are allowed for
or prohibited from opportunistic spectrum access.
The crucial task of spectrum sensing is to sense the K
narrowband subchannels and identify spectral holes for opportunistic use. For simplicity, we assume that the high-layer
protocols, e.g., the medium access control (MAC) layer, can
guarantee that all cognitive radios keep quiet during the detection interval such that the only spectral power remaining in the
air is emitted by the primary users in addition to background
noises. In this paper, instead of considering a single subband
at a time, we propose to use a multiband detection technique,
which jointly takes into account the detection of primary users
across multiple frequency bands. We next present the system
model.
B. Received Signal
Consider a multi-path fading environment, where h(l), l =
0, 1, . . . , L − 1, denotes the discrete-time channel impulse
response between the primary transmitter and cognitive radio
receiver, with L as the number of resolvable paths. The received baseband signal at the CR front-end can be represented
as
r(n) =
L−1
X
l=0
h (l) s (n − l) + v(n), n = 0, 1, . . . , N − 1 (1)
where s(n) is the primary transmitted signal at time n (after
the cyclic prefix has been removed) and v(n) is additive complex white Gaussian noise
with zero mean and variance σv2 ,
2
i.e., v(n) ∼ CN 0, σv . In a multi-path fading environment,
the wideband channel exhibits frequency-selective features
[17] [18] [19] and its discrete frequency response is given
by
L−1
1 X
h(n)e−j2πnk/N ,
Hk = √
N n=0
k = 0, 1, . . . , K − 1 (2)
where L ≤ N . We assume that the channel is slowly varying
such that the channel frequency responses {Hk }K−1
k=0 remain
constant during a detection interval. In the frequency domain,
the received signal at each subchannel can be estimated by
first computing its discrete Fourier transform (DFT):
N −1
1 X
r(n)e−j2πnk/N
Rk = √
N n=0
k = 0, 1, . . . , K − 1
= Hk Sk + Vk ,
(3)
where Sk is the primary transmitted signal at subchannel k
and
L−1
1 X
Vk = √
v(n)e−j2πnk/N ,
N n=0
k = 0, 1, . . . , K − 1 (4)
is the received noise in frequency domain. The random
variable Vk is independently and normally distributed
with
2
2
zero mean and variance
σ
,
i.e.,
V
∼
CN
0,
σ
k
v
v , since
v(n) ∼ CN 0, σv2 and the DFT is a linear operation. Without
loss of generality, we assume that the transmitted signal Sk ,
the channel gain Hk , and the additive noise Vk are independent
of each other.
C. Signal Detection in Individual Bands
Here, we consider signal detection in a single narrowband
subchannel, which will constitute a building block for multiband joint detection. To decide whether the k-th subchannel
is occupied or not, we test the following binary hypotheses
H0,k : Rk = Vk
H1,k : Rk = Hk Sk + Vk ,
k = 0, 1, . . . , K − 1
(5)
where H0,k and H1,k indicate, respectively, the absence and
presence of the primary signal in the k-th subchannel. For each
subchannel k, we compute the summary statistic as the sum
of received signal energy over an interval of M samples, i.e.,
Yk =
M−1
X
m=0
|Rk (m)|2 ,
k = 0, 1, . . . , K − 1
(6)
and the decision rule is given by
Yk
H1,k
R γk ,
H0,k
k = 0, 1, . . . , K − 1
where γk is the corresponding decision threshold.
(7)
RF Front-End
R0 (m)
Down
Conversion
R1 (m)
Remove Prefix,
Var (Yk ) =
2M σv4
2M σv2 + 2|Hk |2 σv2
H0,k
H1,k
PM1
| · |2
m=0
Y0
0
Y1
1
YK1
K1
H0 /H1
H0 /H1
FFT
PM1
m=0
| · |2
H0 /H1
RK1 (m)
Joint Detection
(9)
for k = 0, 1, . . . , K − 1. Thus, we write these statistics
compactly as Yk ∼ N (E (Yk ) , Var (Yk )), k = 0, 1, . . . , K −1.
Using the decision rule in (7), the probabilities of false
alarm and detection at subchannel k can be respectively
calculated as
γk − M σv2
(k)
√
(10)
Pf (γk ) = Pr (Yk > γk |H0,k ) = Q
σv2 2M
and
| · |2
…...
h(n) s(n) + w(n)
and variance
and Serial-toParallel Convert
m=0
…...
A/D
PM1
…...
For simplicity, we assume that the transmitted
signal at
each subchannel has unit power, i.e., E |Sk |2 = 1. This
assumption holds when primary radios deploy uniform power
transmission strategies given no channel knowledge at the
transmitter side. According to the central limit theorem [20],
Yk is asymptotically in M normally distributed with mean
M σv2
H0,k
E (Yk ) =
(8)
M σv2 + |Hk |2
H1,k
Fig. 2. A schematic representation of multiband joint detection for wideband
spectrum sensing in cognitive radio networks.
users. For a given threshold vector γ, the probabilities of false
alarm and detection can be compactly represented as
h
iT
(0)
(1)
(K−1)
P f (γ) = Pf (γ0 ), Pf (γ1 ), . . . , Pf
(γK−1 )
(12)
and
h
iT
! P d (γ) = Pd(0) (γ0 ), Pd(1) (γ1 ), . . . , Pd(K−1) (γK−1 )
(13)
γk − M σv2 + |Hk |2
(k)
p
Pd (γk ) = Pr (Yk > γk |H1,k ) = Q
σv 2M (σv2 + 2|Hk |2 ) respectively. Similarly, the probabilities of miss can be written
(11) in a vector as
h
iT
where Q(·) denotes the complementary distribution function
(0)
(1)
(K−1)
P m (γ) = Pm
(γ0 ), Pm
(γ1 ), . . . , Pm
(γK−1 )
(14)
of the standard normal distribution.
The choice of the threshold γk leads to a tradeoff between
(k)
(k)
where Pm (γk ) = 1 − Pd (γk ), k = 0, 1, . . . , K − 1,
the probability of false alarm and the probability of miss1 ,
compactly written as P m (γ) = 1− P d (γ), with 1 the all-one
Pm = 1 − Pd . Specifically, a higher threshold will result in a
vector.
smaller probability of false alarm and a larger probability of
Consider a cognitive radio sensing the K narrowband submiss, and vice versa.
channels in order to opportunistically utilize the unused ones
The probabilities of false alarm and miss have unique
for transmission. Let rk denote the throughput achievable over
implications for cognitive radio networks. Low probabilities
the k-th subchannel if used by cognitive radios, and r =
of false alarm are necessary in order to maintain possible
(k)
T
high throughput in cognitive radio systems, since a false alarm [r0 , r1 , . . . , rK−1 ] . Since 1−Pf measures the opportunistic
spectrum utilization of subchannel k, we define the aggregate
would prevent the unused spectral segments from being acopportunistic throughput capacity as
cessed by cognitive radios. On the other hand, the probability
of miss measures the interference from cognitive radios to
R (γ) = rT [1 − P f (γ)]
(15)
the primary users, which should be limited in opportunistic
threshold vector γ. Due to the inspectrum access. These implications are based on a typical which is a function of the (k)
(k)
herent
trade-off
between
P
assumption that if primary signals are detected, the secondary
f (γk ) and Pm (γk ), maximizing
users should not use the corresponding channel and that if no the sum rate R(γ) will result in large P m (γ), hence causing
primary signals are detected, then the corresponding frequency harmful interference to primary users.
The interference to primary users should be limited in a
band will be occupied by secondary users.
cognitive radio network. For a widband primary communicaIII. M ULTIBAND J OINT D ETECTION
tion system, the impact of interference induced by cognitive
In this section, we present the multiband joint detection devices can be characterized by a relative priorityT vector over
framework for wideband spectrum sensing, as illustrated in the K subchannels, i.e., c = [c0 , c1 , . . . , cK−1 ] , where ck
Fig. 2. The design objective is to find the optimal threshold indicates the cost incurred if the primary user at subchannel
vector γ = [γ0 , γ1 , . . . , γK−1 ]T so that the cognitive radio k is interfered with. Suppose that J primary users share a
system can make efficient use of the unoccupied spectral portion of the K subchannels and each primary user occupies a
we define the aggregate interference
segments without causing harmful interference to the primary subset Sj . Consequently,
P
(i)
to primary user j as i∈Sj ci Pm (γi ). In special cases where
1 The subscript k is omitted whenever we refer to a generic frequency band.
each primary user is equally important, we may have c = 1.
To summarize, our objective is to find the optimal thresholds
{γk }K−1
k=0 of these K subchannels, collectively maximizing the
aggregate opportunistic throughput subject to constraints on
the aggregate interference for each primary user and individual
constraints on the subbands. As such, the optimization problem
for a multi-user primary system can be formulated as
max
s.t.
R (γ)
X
(i)
ci Pm
(γi ) ≤ εj , j = 0, 1, . . . , J − 1
(P1)
with the optimization variables γ = [γ0 , γ1 , . . . , γK−1 ] . The
constraint (16) limits the interference on each subchannel with
α = [α0 , α1 , . . . , αK−1 ]T , and the last constraint in (17) dictates that each subchannel should achieve at least a minimum
opportunistic spectrum utilization that is proportional to 1−βk .
For the single-user primary system where all the subchannels
are used by one primary user, we have J = 1.
Intuitively, we could make some observations on the multiband joint detection. First, the subchannel with a higher
opportunistic rate rk should have a higher threshold γk (i.e.,
a smaller probability of false alarm) so that it can be highly
used by cognitive radios. Second, the subchannel that carries a
higher priority primary user should have a lower threshold γk
(i.e., a smaller probability of miss) in order to prevent harmful
interference by secondary users. Third, a little compromise
on those subchannels carrying less important primary users
might boost the aggregate rate considerably. Thus, in the
determination of the optimal threshold vector, it is necessary to
strike a balance among the channel condition, the opportunistic
throughput, and the relative priority of each subchannel.
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 reformulating the problem
and exploiting the hidden convexity.
We observe the fact that the Q-function is monotonically
non-increasing allows us to transform the constraints in (16)
and (17) into linear constraints. From (16), we have
k = 0, 1, . . . , K − 1.
k = 0, 1, . . . , K − 1
2
2M σv2 + 2 |Hk |
(19)
Q−1 (1 − αk ) .
(20)
Similarly, the combination of (10) and (17) leads to
γk ≥ γmin,k
k = 0, 1, . . . , K − 1
(k)
rk Pf (γk )
X
i∈Sj
(P2)
(21)
(i)
ci Pm
(γi ) ≤ εj , j = 0, 1, . . . , J − 1
γmin,k ≤ γk ≤ γmax,k , k = 0, 1, . . . , K − 1.
(23)
(24)
Although the constraint (24) is linear, the problem is still
nonconvex. However, it can be furthermore 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.
(k)
Lemma 1: The function Pf (γk ) is convex in γk if
(k)
Pf (γk ) ≤ 21 .
(k)
Proof: Taking the second derivative of Pf (γk ) from
(10) gives
"
2 #
(k)
d2 Pf (γk )
γk − M σv2
−1 d
exp −
= √
dγk2
4M σv4
2π dγk
"
2 #
γk − M σv2
γk − M σv2
√ exp −
=
. (25)
4M σv4
2M σv2 2π
(k)
Since Pf (γk ) ≤ 21 , we have γk ≥ M σv2 . Consequently, the
(k)
second derivative of Pf (γk ) is greater than or equal to zero,
(k)
which implies that Pf (γk ) is convex in γk .
(k)
Lemma 2: The function Pm (γk ) is convex in γk if
(γk ) ≤ 12 .
Proof: This result can be proved using a similar technique to that used to prove Lemma 1. By taking the second
(k)
derivative of (11), we can show that Pd (γk ) is concave, and
(k)
(k)
hence Pm (γk ) = 1 − Pd (γk ) is a convex function.
Recall that the nonnegative weighted sum of a set of convex
functions is also convex [21]. The problem (P1) becomes a
convex program if we enforce the following conditions:
(k)
Pm
1
1
and 0 < βk ≤ ,
2
2
k = 0, 1, 2, . . . , K − 1.
(26)
This regime of probabilities of false alarm and miss is that of
practical interest in cognitive radio networks.
With the conditions in (26), the feasible set of problem
(P2) is convex. The optimization problem takes the form of
minimizing a convex function subject to a convex constraint,
and thus a local maximum is also the global maximum.
Efficient numerical search algorithms such as the interior-point
method can be used to solve for the optimal solutions [21].
Alternatively, we can formulate the multiband joint detection problem into another optimization problem that minimizes
0 < αk ≤
where
σv
K−1
X
(18)
Substituting (11) into (18) gives
∆
2
γmax,k = M σv2 + |Hk | +
r
min
(16)
(17)
T
γk ≤ γmax,k
(22)
Consequently, the original problem (P1) has the following
equivalent form
s.t.
P m (γ) α
P f (γ) β
(k)
h
i
√
γmin,k = σv2 M + 2M Q−1 (βk ) .
k=0
i∈Sj
1 − Pd (γk ) ≤ αk ,
where
the interference from cognitive radios to the primary communication system, subject to some constraints on the aggregate
opportunistic throughput, i.e.,
minimize
(P3)
|Hk |2
.50
.30
.45
.65
.25
.60
.40
.70
r (kbps)
612
524
623
139
451
409
909
401
c
1.91
8.17
4.23
3.86
7.16
6.05
0.82
1.30
T
r [1 − P f (γ)] ≥ δ
P m (γ) α
with δ the required minimum aggregated rate and γ the
optimization variables. Like problem (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 IV.
IV. S IMULATION R ESULTS
In this section, we numerically evaluate the proposed
spectrum sensing schemes. Consider a multiband single-user
OFDM system in which a wideband channel is equally divided
into 8 subchannels. Each subchannel has a channel gain Hk
between the primary user and the cognitive radio, a throughput
rate rk if used by cognitive radios, and a cost coefficient
ck indicating a penalty incurred when the primary signal is
interfered with by the cognitive radio. For each subchannel k
(0 ≤ k ≤ 7), it is expected that the opportunistic spectrum
utilization is at least 50%, i.e., βk = 0.5, and the probability
that the primary user is interfered with is at most αk = 0.1. For
simplicity, it is assumed that the noise power level is σv2 = 1
and the length of each detection interval is M = 100. This
example studies multiband joint detection in a single cognitive
radio. The proposed spectrum sensing algorithms are examined
by comparing with an approach that searches a uniform threshold to maximize the aggregate opportunistic throughput. We
randomly generate the channel condition between the primary
user and the cognitive radio, the opportunistic throughput
over each subchannel, and the cost of interference of each
subchannel. One realization example is given in Table I.
We maximize the aggregate opportunistic throughput over
the 8 subchannels subject to some constraints on the interference to the primary users, as formulated in (P1). Fig. 3
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 one with uniform threshold.
Note that in the reference algorithm, the uniform threshold is
searched to maximize the achievable rate for a fair comparison.
That is, the proposed multiband joint detection algorithm
makes better use of the wide spectrum by balancing the
conflict between improving spectrum 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 ε.
An alternative example is depicted in Fig. 4, showing the
numerical results of minimizing the aggregate interference
subject to the constraints on the opportunistic throughput as
3500
Aggregate Opportunistic Throughput (kbps)
P f (γ) β
Multiband Joint Detection
Uniform Threshold
3000
2500
2000
0.12
0.13
0.14
Aggregate Interference
0.15
0.16
Fig. 3. The aggregate opportunistic throughput capacity vs. the constraint
on the aggregate interference to the primary communication system.
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. For illustration purposes, the optimized thresholds and the associated
probabilities of miss and false alarm are given in Fig. 5
for (P1) and (P3). To summarize, these numerical results
show that multiband joint detection can considerably improve
the spectrum efficiency by making more efficient use of the
spectral diversity.
2.8
2.6
Multiband Joint Detection
Uniform Threshold
2.4
Aggregate Interference
st.
cT P m (γ)
TABLE I
PARAMETERS U SED IN S IMULATIONS
2.2
2
1.8
1.6
1.4
1.2
1
2400
2500
2600
2700
2800
2900
3000
Aggregate Opportunistic Throughput (kbps)
3100
Fig. 4. The aggregate interference to the primary communication system vs.
the constraint on the aggregate opportunistic throughput.
Thresholds
130
120
110
100
Prob. Miss Det.
90
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
0.05
0
Prob. False Alarm
0
0.1
0.6
P1
P3
0.4
0.2
0
0
1
2
3
4
5
6
7
Fig. 5. The optimized thresholds and the associated probabilities of miss
and false alarm: (P1) ε = 1.25 and (P3) δ = 3224 kbps.
V. C ONCLUSION
In this paper, we have proposed a multiband joint detection
approach for wideband spectrum sensing in cognitive radio
networks. The basic strategy is to take into account the detection of primary users across a bank of narrowband subchannels
jointly rather than to consider only one single band at a time.
We have formulated the joint detection problem into a class of
optimization problems to improve the spectral efficiency and
reduce the interference. By exploiting the hidden convexity
in the seemingly nonconvex problems, we have obtained the
optimal solution under practical conditions. The proposed
spectrum sensing algorithms have been examined numerically
and shown to be able to perform well.
ACKNOWLEDGMENT
This research was supported in part by the National Science Foundation under Grants ANI-03-38807, CNS-06-25637,
ECS-0601266, ECS-0725441, CNS-0721935, CCF-0726740,
and by the Department of Defense under Grant HDTRA-071-0037.
R EFERENCES
[1] D. Cabric, S. M. Mishra, and R. Brodersen, “Implementation issues
in spectrum sensing for cognitive radios,” in Proc. 38th Asilomar
Conference on Signals, Systems and Computers, Pacific Grove, CA, Nov.
2004.
[2] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection
Theory. Prentice Hall, Upper Saddle River, NJ, 1998.
[3] H. V. Poor, An Introduction to Signal Detection and Estimation.
Springer-Verlag, New York, 1994.
[4] S. Enserink and D. Cochran, “A cyclostationary feature detector,” in
Proc. 28th Asilomar Conference on Signals, Systems, and Computers,
Pacific Grove, CA, Oct. 1994.
[5] D. Cabric, A. Tkachenko, and R. W. Brodersen, “Experimental study of
spectrum sensing based on energy detection and network cooperation,”
in Proc. ACM Int. Workshop on Technology and Policy for Accessing
Spectrum (TAPAS), Boston, MA, Aug. 2006.
[6] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Select. Areas Commun., vol. 23, no. 2, pp. 201–220,
Feb. 2005.
[7] R. S. Blum, S. A. Kassam, and H. V. Poor, “Distributed detection with
multiple sensors: Part II - Advanced topics,” Proc. IEEE, vol. 85, no. 1,
pp. 64–79, Jan. 1997.
[8] P. K. Varshney, Distributed Detection and Data Fusion. SpringerVerlag, New York, 1997.
[9] E. Vistotsky, S. Kuffner, and R. Peterson, “On collaborative detection
of TV transmissions in support of dynamic spectrum sharing,” in
Proc. IEEE Symposium on New Frontiers in Dynamic Spectrum Access
Networks (DySPAN), Baltimore, MD, Nov. 2005.
[10] G. Ghurumuruhan and Y. Li, “Agility improvement through cooperative
diversity in cognitive radio,” in Proc. IEEE Global Commun. Conf., St.
Louis, MO, Nov. 2005.
[11] J. Lunden, V. Koivunen, A. Huttunen, and H. V. Poor, “Spectrum sensing
in cognitive radios based on multiple cyclic frequencies,” in Proc.
2nd International Conference on Cognitive Radio Oriented Wireless
Networks and Communications (CROWNCOM), Orlando, FL, July 2007,
(invited paper).
[12] Z. Quan, S. Cui, and A. H. Sayed, “An optimal strategy for cooperative
spectrum sensing in cognitive radio networks,” in Proc. IEEE Global
Commun. Conf., Washington D.C., Nov., pp. 2947–2951.
[13] Z. Quan, S. Cui, and A. H. Sayed, “Optimal linear cooperation for
spectrum sensing in cognitive radio networks,” IEEE J. Select. Topics
Signal Processing, vol. 2, no. 1, June 2008.
[14] A. Sahai and D. Cabric, “A tutorial on spectrum sensing: Fundamental
limits and practical challenges,” in Proc. IEEE Symposium on New
Frontiers in Dynamic Spectrum Access Networks (DySPAN), Baltimore,
MD, Nov. 2005.
[15] Z. Tian and G. B. Giannakis, “A wavelet approach to wideband
spectrum sensing for cognitive radios,” in Proc. 1st Int. Conference
on Cognitive Radio Oriented Wireless Networks and Communications
(CROWNCOM), Mykonos Island, Greece, June 2006.
[16] Y. Hur, J. Park, W. Woo, K. Lim, C.-H. Lee, H. S. Kim, and J. Laskar,
“A wideband analog multi-resolution spectrum sensing technique for
cognitive radio systems,” in Proc. IEEE International Symposium on
Circuits and Systems (ISCAS), Island of Kos, Greece, May 2006, pp.
4090–4093.
[17] J. G. Proakis, Digital Communications, 4th ed. McGraw-Hill, New
York, 2001.
[18] A. Goldsmith, Wireless Communications. Cambridge University Press,
Cambridge, UK, 2006.
[19] A. H. Sayed, Fundamentals of Adaptive Filtering. Wiley, NY, 2003.
[20] B. V. Gendenko and A. N. Kolmogorov, Limit Distributions for Sums of
Independent Random Variables. Addison-Wesley, Reading, MA, 1954.
[21] S. Boyd and L. Vandenberghe, Convex Optimization.
Cambridge
University Press, Cambridge, UK, 2003.