Blind Spectrum Sensing for OFDM Based cognitive Radio Systems

1. 858
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
Blind Spectrum Sensing for OFDM-Based
Cognitive Radio Systems
Simin Bokharaiee, Student Member, IEEE, Ha H. Nguyen, Senior Member, IEEE, and Ed Shwedyk
Abstract—Given the ever-growing demand for radio spectrum,
cognitive radio has recently emerged as an attractive wireless technology. Since orthogonal frequency-division multiplexing (OFDM)
is one of the major wideband transmission techniques, detection
of OFDM signals in low-signal-to-noise-ratio scenario is an important research problem. In this paper, it is shown that cyclic prefix (CP) correlation coefficient (CPCC)-based spectrum sensing,
which was previously introduced as a simple and computationally
efficient spectrum-sensing method for OFDM signals, is a special
case of the constrained generalized likelihood ratio test (GLRT) in
the absence of multipath. As such, the performance of this algorithm degrades in a multipath scenario, where OFDM is usually
implemented. Furthermore, by considering multipath correlation
in the GLRT algorithm and employing the inherent structure of
OFDM signals, a simple and low-complexity algorithm called the
multipath-based constrained-GLRT (MP-based C-GLRT) algorithm is obtained. The MP-based C-GLRT algorithm is shown
to outperform the CPCC-based algorithm in a rich multipath
environment. Further performance improvement can be achieved
by simply combining both the CPCC- and MP-based C-GLRT
algorithms. A simple GLRT-based detection algorithm is also
developed for unsynchronized OFDM signals, whose performance
is only slightly degraded when compared with the synchronized
detection in a rich multipath environment.
Index Terms—Cognitive radio (CR), generalized likelihood ratio test (GLRT), likelihood ratio test (LRT), orthogonal frequencydivision multiplexing (OFDM), spectrum sensing.
I. I NTRODUCTION
W
ITH THE recent rapid growth in wireless communication applications and systems, the problem of spectrum
utilization has become more critical than ever before. As an
emerging solution, cognitive radio (CR) technology has been
introduced to take advantage of the licensed spectrum when
it is not in use by its primary users. The temporary use of
idle licensed frequency bands by unlicensed users provides an
efficient exploitation of limited spectrum bands. However, it is
Manuscript received February 8, 2010; revised July 6, 2010 and October 27,
2010; accepted November 29, 2010. Date of publication January 6, 2011; date
of current version March 21, 2011. This work was supported by Discovery
Grants from Natural Sciences and Engineering Council of Canada (NSERC).
The review of this paper was coordinated by Dr. Y.-C. Liang.
S. Bokharaiee is with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada (e-mail:
simin@ee.umanitoba.ca).
H. H. Nguyen is with the Department of Electrical and Computer Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada (e-mail:
ha.nguyen@usask.ca).
E. Shwedyk, retired, was with the Department of Electrical and Computer
Engineering, University of Manitoba, Winnipeg, MB R3M 2G7, Canada.
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/TVT.2010.2103095
crucial for the CR system to continually monitor the spectrum
to prevent any interference with the primary users.
Orthogonal frequency-division multiplexing (OFDM), which
has been known to be one of the most effective multicarrier
techniques for broadband wireless communications due to its
inherent capability to combat multipath fading and avoid intersymbol interference (ISI), has attracted significant attention
in the development of CR technology [1], [2]. In an OFDMbased system, CR spectrum sensing can be performed in either
the time or the frequency domain. In the frequency domain,
a subcarrierwise detection can be performed [3], [4], which
becomes important when the available bandwidth is divided
into subbands, which are allocated to different primary users.
In [3], a frequency-domain spectrum-sensing method has been
proposed, which optimizes sensing parameters over each subcarrier, whereas the algorithm in [4] is proposed to detect available portions of the spectrum. Although some OFDM-based
technologies, such as digital audio broadcasting or IEEE 802.16
wireless metropolitan area networks, employ data multiplexing
for different primary users, single-transmission OFDM systems
are also common in practice. Among them are digital video
broadcasting, IEEE 802.11 wireless local area networks, and
IEEE 802.16e mobile WiMAX. For such wireless systems,
the CR receiver needs to detect the primary OFDM signals to
monitor the licensed bandwidth.
The likelihood ratio test (LRT) is known to be the optimal
algorithm for spectrum sensing [5] if the exact knowledge
of channel state information and noise variance can be made
available at the receiver. Unfortunately, such knowledge is
generally difficult to obtain at the CR receiver. On the other
hand, the energy detection algorithm does not require channel
information and performs well if the noise variance is known.
However, even a slight uncertainty, i.e., around 0.5 dB in noise
variance, causes its performance to severely drop below the
performance with known noise variance [6]. It should also
be pointed out that the energy detector (ED) is the optimal
spectrum-sensing method for independent and identically distributed (i.i.d.) primary signals, and its performance is degraded
if the received primary signals are correlated [7]. Therefore,
it is of great importance to develop efficient spectrum-sensing
algorithms for the detection of signals, which are not necessarily i.i.d. To alleviate the requirement on precise knowledge of
channel and noise variance, which are generally unavailable at
the CR receiver, a suitable detection algorithm is expected to
extract and use some particular features of the received primary
signals. In fact, a number of techniques have been proposed and
evaluated; among them are the statistical covariance algorithm
0018-9545/$26.00 © 2011 IEEE

2. BOKHARAIEE et al.: BLIND SPECTRUM SENSING FOR OFDM-BASED CR SYSTEMS
[8], eigenvalue-based algorithm [9], [10], and cyclostationary
algorithm [11], [12].
As OFDM is an effective technique to combat multipath
fading, it is expected that OFDM is employed in rich scattering environments. With advanced compression techniques,
the information signals (in discrete-time complex baseband
representation) can be considered as i.i.d. random variables.
However, when these signals are passed through a multipath
fading channel, the received signals at the receiver are correlated versions of the transmitted signals, which typically have
different statistics from the background noise. On the other
hand, a very important feature of OFDM transmission is the
use of cyclic prefix (CP), which results in nonzero correlation
of the received primary signal samples at certain delays. In [11],
a method based on the cyclic feature of OFDM blocks in the
time domain has been proposed. Although the algorithm shows
good performance, the signal correlation induced by multipath
propagation is not exploited in such an algorithm.
The aim of a detection algorithm is to decide between the two
hypotheses of whether the primary signal is present or absent. In
the case when some parameters (e.g., channel state information
and noise variance) are not known, the hypothesis is called a
composite hypothesis [13]. For a composite hypothesis, one
approach is to perform estimations of the unknown parameters
(typically, the maximum-likelihood estimates). The estimated
parameters are then used in the LRT as if they are the correct
values. This results in the so-called generalized LRT (GLRT)
[5], [13]. GLRT has been widely employed in many hypothesistesting problems, e.g., [12], [14], and [15], including spectrumsensing applications [16], [17].
In this paper, it is first shown that the GLRT algorithm can
exploit both multipath and cyclic correlations to yield a novel
blind spectrum-sensing algorithm. It is then verified that the
cyclic-prefix correlation coefficient (CPCC)-based detection algorithm is a special case of the constrained GLRT algorithm in
the absence of multipath fading channel. It is further shown that,
when multipath fading is present, which is the case for OFDM
applications, performance of the detection based on the CPCC
degrades. Furthermore, by exploiting the known structure of
the OFDM channel matrix in a constrained GLRT algorithm,
a detection algorithm that is solely based on the multipath
correlation coefficients (MPCCs) is obtained. By combining
the CPCC- and MPCC-based algorithms, an even more reliable
spectrum-sensing method can be realized.
The spectrum-sensing algorithms developed in the first part
of this paper (Sections IV and V) assume that perfect synchronization can be obtained at the CR receiver. As such,
the detection performance of sensing algorithms in this case
serves as an upper bound for situations where synchronization
has to be actually performed at the CR receiver. In [18], a
blind synchronization algorithm was proposed for maximumlikelihood estimations of time and carrier frequency offsets
(CFOs) of OFDM signals. While such a blind synchronization
algorithm can be conveniently incorporated in our spectrumsensing framework, there are two main drawbacks: 1) Using
a synchronization algorithm adds more complexity and may
cause delay to the sensing task. 2) In the low-signal-to-noiseratio (SNR) region where the CR is operating, the synchro-
859
nization is far from perfect; hence, the sensing performance is
significantly degraded when compared with the case of perfect
synchronization. These drawbacks motivate us to develop a
simple GLRT-based algorithm in Section VI that does not
require timing synchronization to be established between the
primary and secondary users.
The rest of the paper is organized as follows: Section II
reviews the system model and describes the spectrum-sensing
problem. Section III presents a general framework of the GLRT
detection scheme and shows how to exploit CP and multipath
correlation features. In Sections IV and V, two constrained
GLRT algorithms are presented, where the data and cyclic parts
of the OFDM signal are separately considered to exploit the
structure of the covariance matrix in enhancing the performance
of the GLRT algorithm. It is shown in both cases that the constrained GLRT algorithms lead to detection algorithms, which
solely depend on the sample correlation coefficients. A combined detection scheme is also proposed. The spectrum-sensing
algorithm for unsynchronized OFDM signals is proposed in
Section VI. Simulation results are presented in Section VII, and
Section VIII concludes the paper.
II. S YSTEM M ODEL
The OFDM signal model considered in our work is the same
as that in [11] and [19], which assumes that the primary OFDM
system employs L subcarriers and the CR and primary users
can be perfectly synchronized. The case of no timing synchronization is separately discussed and treated in Section VI. Let
2
{Sn,k }L−1 , with E{|Sn,k |2 } = σS , be the complex symbols
k=0
to be transmitted in the nth OFDM block. Then, the baseband
OFDM modulated signal can be expressed as
1
sn (m) = √
L
L−1
Sn,k e
j2πmk
L
,
m = 0, . . . , L − 1.
(1)
k=0
For a large number of subcarriers L [i.e., the size of discrete
Fourier transform (DFT)/inverse DFT (IDFT)], sn (m) can be
approximately modeled as a zero-mean circularly symmet2
ric complex Gaussian random variable of variance σS , i.e.,
2
sn (m) ∼ CN (0, σS ).
Represent the length-(L + Lp ) vector of the nth transmitted
OFDM block as
⎤
⎡
⎥
⎢
sn = ⎣sn (L − 1) . . . sn (0) sn (L − 1) . . . sn (L − Lp )⎦
(2)
Cyclic Prefix
where Lp denotes the number of samples in the guard interval,
i.e., the length of the CP. The corresponding received signal and
noise vectors are denoted by
xn = [xn (L−1) xn (L−2) . . . xn (0)xn (−1) . . . xn (−Lp )]
(3)
vn = [vn (L−1) vn (L−2) . . . vn (0)vn (−1) . . . vn (−Lp )]
(4)
2
where the noise samples vn (l)’s are i.i.d. CN (0, σv ) random
variables.

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
The primary user signal is received through a wireless multipath fading channel whose discrete-time baseband model is
represented by channel filter taps hi , i = 1, . . . , Lc , where Lc
denotes the number of multipath components. It is also assumed
that the fading process remains static during the interval of
spectrum sensing. This implies that the channel filter taps can
be treated as unknown constants during the period of spectrum
sensing. The relationship of xn , sn , and vn can be expressed in
matrix form as follows:
s
xn = h¯n + vn
h1
⎢ 0
⎢ .
⎢ .
⎢ .
h=⎢ .
⎢ .
⎢ .
⎢ .
⎣ .
.
···
h1
0
···
···
hLc
···
..
.
0
hLc
0
h1
..
.
···
···
···
0
..
.
···
···
···
..
.
0
hLc
h1
···
···
0
..
.
···
0
0
.
.
.
⎤
⎥
⎥
⎥
⎥
⎥.
0 ⎥
⎥
⎥
0 ⎦
hLc
(y|H1 , Ry ) H1
≷η
2
fy|H0 ,σ2 (y|H0 , σz ) H0
fy|H
1 , Ry
if T ≤
if T >
(10)
z
in which y = [y1 , . . . , yN ] is a collection of N received blocks.
2
In the preceding test, Ry and σz are the maximum-likelihood
2
estimates of Ry and σz under hypotheses H1 and H0 ,
respectively.
2
The maximum-likelihood estimate of σz can be obtained as
2
2
2
σz = max ln fy|H0 ,σz y|H0 , σz
2
(11)
σz
(6)
where
N
2
2
fy|H0 ,σz y|H0 , σz =
Note that the last Lp samples of xn is the ISI part.
Since xn (l)’s are linear combinations of zero-mean complex
Gaussian random variables, they are also zero-mean complex
Gaussian random variables. Based on (5), the variance of xn (l)
2
2
2
is σx = σS Lc |hi |2 + σv when the primary user’s signal is
i=1
2
2
present; otherwise, σx = σv . It is also of interest to define the
SNR in the presence of the primary user’s signal as SNR =
2
2
σS Lc |hi |2 /σv .
i=1
Two binary hypotheses H0 and H1 are defined in spectrum
sensing, in which H0 denotes the idle state of the primary
user and H1 represents the active state of the primary user. To
classify the observations into H0 or H1 , a test statistics T is
formulated, and a general test decision is given as follows:
Decide H0 ,
Decide H1 ,
LG (y) =
(5)
where ¯n = [sn , sn−1 (L − 1), . . . , sn−1 (L − Lc + 1)] , and h
s
is (L + Lp ) × (L + Lp + Lc − 1) Toeplitz matrix constructed
from the channel filter taps as
⎡
for the detection of OFDM signals in the succeeding sections.
Let yn ∼ CN (0, Ry ) denote the length-M column vector of
2
the nth received signal block, and let zn ∼ CN (0, σz I) denote
the length-M column vector containing the noise samples.
2
In the scenario in which the noise variance σz and signal
covariance matrix Ry are unknown, the GLRT is given as
follows [17]:
(7)
1
2 M
n=1 (πσz )
exp −
1
yn
2
σz
2
(12)
and ·
that [17]
denotes the vector Euclidean norm. It follows
2
σz =
1
1
¯
tr(yyH ) =
tr(Ry ).
NM
M
(13)
¯
where Ry = 1/N yyH denotes the sample covariance matrix,
and H represents the Hermitian transpose.
On the other hand, the maximum-likelihood estimate of Ry
can be obtained as
Ry = max ln fy|H1 ,Ry (y|H1 , Ry )
Ry ∈SRy
(14)
where SRy specifies the set of Ry having certain structures,
and
N
where is some threshold value. Two probabilities of interest
are given as follows: 1) the probability of detection Pd , which
is the probability that the primary user is correctly detected in
its active mode, and 2) the probability of false alarm Pf , which
represents the probability of a false detection of the primary
user when it is in the idle state. Mathematically
Pf = Pr {T > |H0 }
(8)
Pd = Pr {T > |H1 } .
fy|H1 ,Ry (y|H1 , Ry ) =
To obtain a more explicit expression of the test in (10),
2
2
first rewrite fy|H0 ,σz (y|H0 , σz ) and fy|H1 ,Ry (y|H1 , Ry ) as
follows:
(9)
2
2
fy|H0 ,σz y|H0 , σz
=
III. G ENERALIZED L IKELIHOOD R ATIO T EST
As mentioned before, spectrum sensing based on GLRT has
been presented in [17], in which different tests are obtained
under different parameter assumptions, i.e., unknown noise
variance and/or signal covariance matrix. In the sequel, the
GLRT is reviewed in its general form, and it will be employed
1
H
exp −yn R−1 yn .
y
π M det(Ry )
n=1
(15)
=
=
1
N
exp −
2
(πσz )M
1
2
(πσz )M
N
exp −
1
σ2
exp − z
2
2
πσz
σz
1
2
σz
N
yn
2
n=1
1
tr(yyH )
2
σz
NM
(16)

4. BOKHARAIEE et al.: BLIND SPECTRUM SENSING FOR OFDM-BASED CR SYSTEMS
fy|H1 ,Ry (y|H1 , Ry )
N N
1
= M
π det(Ry )
N
1
= M
π det(Ry )
N
¯
yyH =N Ry
=
Since the covariance matrices involved in our study meet the
aforementioned structural condition, the test in (22) shall be
employed for spectrum sensing throughout the paper.
On the other hand, for the special case when there is no constraint on the structure of the covariance matrix, the maximum¯
likelihood estimate of Ry is Ry = Ry , and the test is the same
as that developed in [17] and has the following form:
H
exp −yn R−1 yn
y
n=1
1
M det(R )
π
y
=
exp −tr yH R−1 y
y
¯
TG (y) =
exp −tr R−1 yyH
y
1
π M det(Ry )
N
exp −N tr
¯
R−1 Ry
y
1
1
¯
=
exp − tr R−1 Ry
1
y
M
(π) det M (Ry )
NM
.
(17)
2
After substituting Ry and σz into (16) and (17), we have
2
fy|H0 ,σ2 y|H0 , σz
z
=
=
fy|H
1
exp(−1)
(18)
(y|H1 , Ry )
1
1
M
π det (Ry )
×exp −
1
¯
tr R−1 Ry
y
M
NM
.
(19)
Finally, substituting (18) and (19) into (10) yields the following GLRT algorithm:
1
TG (y) = {LG (y)} M N =
1
¯
H1
M tr(Ry )
≷ .
1
¯
det M (Ry ) H0
(23)
The preceding unconstrained GLRT (U-GLRT) is described
for a general observation y = [y1 , . . . , yN ], where yn ∼
CN (0, Ry ), n = 1, . . . , N . When applied to the OFDM-based
CR system in Section II, one can simply replace yn by xn in
(5). Furthermore, it is also possible to use only a portion of
the complete observation y in the GLRT algorithm. Although
this appears to be counterproductive, the key advantage is that,
by concentrating on a certain part of the observation, one can
exploit structural properties of the covariance matrix to improve
its estimation in the GLRT. This is presented in detail in the
succeeding sections for an OFDM-based CR system.
IV. G ENERALIZED L IKELIHOOD R ATIO T EST BASED ON
C YCLIC P REFIX C ORRELATION
NM
π
¯
M tr(Ry )
1 , Ry
=
NM
1
exp(−1)
2
πσz
861
1
¯
M tr(Ry )
1
M
det (Ry )
H1
β(Ry ) ≷
H0
(20)
One way to exploit a strong structural correlation property of
the observation is to use only the head and tail of each received
OFDM block in the GLRT algorithm. To this end, define
˙
xn = [xn (L − 1) . . . xn (L − Lp ), xn (0) . . . xn (−Lp )]
(24)
as the vector containing the 2Lp samples, i.e., the Lp samples
in the first part and the Lp samples in the last part, of the
nth OFDM block. The corresponding transmitted signal and
additive white Gaussian noise (AWGN) vectors are defined as
˙
sn = [sn (L − 1) . . . sn (L − (Lp + Lc − 1)) sn (L − 1) . . .
sn (L − Lp )sn−1 (L − 1) . . . sn−1 (L − Lc + 1)]
˙
vn = [vn (L − 1) . . . vn (L − Lp ), vn (−1) . . . vn (−Lp )] .
where
β(Ry ) =
Then, one has
1
¯
exp − M tr R−1 Ry
y
(21)
exp(−1)
1/M N
=η
is a fixed threshold value selected to meet a requirement on the probability of false alarm, and it is independent of
the received primary signal characteristics.
It is important to point out that the form in (20) is still very
general and encompasses a large class of GLRT. What it says
is that the test itself and, hence, its performance depend on
how one estimates the covariance matrix of the received signal
block under H1 . According to [20], under the mild condition
that both Ry and its variation δ(Ry ) have the same structure
(e.g., Ry does not have any constant entries), the structured
¯
maximum-likelihood estimate of Ry satisfies tr(R−1 Ry ) =
y
M . This implies that β(Ry ) = 1, and the test becomes
TG (y) =
1
¯
M tr(Ry )
1
det M (Ry )
=
1
H H1
M N tr(yy )
1
det M (Ry )
≷ .
H0
(22)
˙s
˙
˙
xn = h˙ n + vn
(25)
˙
where h is the following 2Lp × 2(Lp + Lc − 1) block diagonal
channel matrix:
˙
h
0
˙
h= A ˙
,
0 hA
⎤
⎡
h1 . . . . . . hLc
0
...
0
⎥
⎢
..
⎢
.
0 ⎥
˙ A =⎢ 0 h1 . . . . . . hLc
⎥
(26)
h
⎥
⎢
..
..
⎣
.
.
0 ⎦
0 ... 0
h1
...
hLc Lp ×(Lp +Lc −1).
˙
˙
˙
˙
˙
s
Let x = [x1 , . . . , xN ] and s = [˙ 1 , . . . , sN ]. The covariance
˙ ˙n
matrix Rx = E{xn xH } under H1 can be shown to be
˙
˙ ˙
˙ ˙
hA hH hB hH
2
2
2
B
˙ ˙˙
Rx = hRs hH + σv ILp = ˙ ˙ A ˙ ˙ H σS + σv ILp
˙
hB hH hA hA
B
(27)

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˙
˙
where hB is the first Lp × Lp block of matrix hA , and ILp
denotes the identity matrix of size Lp .
A CP correlation coefficient (CPCC)-based spectrumsensing algorithm was, in fact, proposed in [11] with the focus
on AWGN channels. Next, it is shown that this CPCC-based
sensing algorithm is exactly the constrained version of the
˙
GLRT algorithm based on observation x and in the absence of
multipath environment. It will be shown in Section VII that the
constrained GLRT algorithm provides a substantial improve˙
ment over the U-GLRT algorithm when only the observation x
is used. Furthermore, since the CPCC-based algorithm is only
equivalent to the constrained GLRT (C-GLRT) algorithm when
there is no multipath, it shall also be explicitly shown that its
performance is degraded in a multipath environment.
In the absence of the multipath propagation effect, one has
Lc = 1 and the covariance matrix Rx in (27) has the following
˙
simpler form:
ILp
ρILp
ρILp
ILp
(28)
where ρ = E{xn (k)x∗ (k − L)}/E{xn (k)x∗ (k)}, k = L −
n
n
1, . . . , L − Lp is the correlation coefficient among the corresponding Lp samples in the head and tail of the OFDM block.
It is simple to show that this correlation coefficient is given as
H0
H1 .
0,
ρ=
f (ρ, a) = −N L ln(2π) + 2N Lp ln(a) − N Lp ln(1 − ρ2 )
a
˙
˙
(g1 (x) − ρg2 (x)) . (33)
−
(1 − ρ2 )
The first derivatives of (33) with respect to ρ and a can be
obtained as
∂f (ρ, a) 2N Lp
1
˙
˙
=
−
(g1 (x)−ρg2 (x))
∂a
a
(1−ρ2 )
(34)
˙
∂f (ρ, a) 2N Lp ρ
2aρ
ρag2 (x)
˙
˙
=
−
(g1 (x)−ρg2 (x))+
.
∂ρ
1−ρ2
(1−ρ2 )2
1−ρ2
A. No-Multipath Propagation
2
Rx = σx
˙
˙
˙ 2,n ˙
˙
where g1 (x) = N (xH x1,n + xH x2,n ) and g2 (x) =
n=1 ˙ 1,n ˙
N
H ˙
H ˙
˙
˙
˙
˙
n=1 (x1,n x2,n + x2,n x1,n ), and x1,n and x2,n are the vectors
˙
containing the first and last Lp components of xn , respectively.
2
Let a = 1/σx . Then, (31) can be rewritten as
2
σS |h1 |2
2
2
σS |h1 |2 +σv
=
SNR
1+SNR ,
(29)
Given the structure of the covariance matrix in (28), estimat2
ing Rx is equivalent to estimating ρ and σx . Their maximum˙
likelihood estimates are given as follows:
2
2
2
˙
ρ, σx = max ln fx|H1 ,ρ,σx x|H1 , ρ, σx .
˙
2
ρ,σx
(30)
˙
Substituting y = x, M = 2Lp , and Ry = Rx in (15) gives
˙
(35)
By simultaneously solving ∂f (ρ, a)/∂a = 0 and ∂f (ρ, a)/
∂ρ = 0, one obtains
2
σx =
ρ=
N
n=1
˙
g1 (x)
1
=
=
a
2N Lp
˙
g1 (x)
=
˙
g2 (x)
˙ 1,n ˙
˙ 2,n ˙
xH x1,n + xH x2,n
2N Lp
N
˙H ˙
˙H ˙
n=1 (x1,n x2,n + x2,n x1,n )
N
˙H ˙
˙H ˙
n=1 x1,n x1,n + x2,n x2,n
(36)
(37)
which are exactly the sample variance and sample correlation
coefficient, respectively.
Next, (22) can be employed to obtain the test statistics by
setting M = 2Lp and Ry = Rx . With the aid of (28), the
˙
test is
√
1
˙ ˙H
H1
H1
1
˙2 − 1
2N Lp tr(xx )
˙
.
≷ ˙⇒ρ≷
TG (x) =
1
2 H
˙
H0
1−ρ 0
det 2Lp (Rx )
˙
(38)
As can be seen, the test statistics in (38) simply compares
the cyclic correlation coefficient with a threshold. It is therefore
identical to the detection algorithm proposed in [11].
˙
ln fx|H1 ,Rx (x|H1 , Rx )
˙
˙
˙
B. Multipath Channel Propagation
N
˙n x ˙
xH R−1 xn .
˙
= −N L ln(2π) − N ln (det(Rx )) −
˙
(31)
n=1
Using the identities det(rAM ×M ) = (rM ) det(AM ×M )
A B
and det
= det(A − BD−1 C), one has det(Rx ) =
˙
C D
2
2
(σx )2Lp (1 − ρ2 )Lp . In addition, R−1 = (1/σx (1 − ρ2 )) ×
˙
x
ILp
−ρILp
. Substituting these expressions in (31)
−ρILp
ILp
evaluates to
2
= −N L ln(2π) − N ln (σx )2Lp (1 − ρ2 )Lp
1
˙
˙
(g1 (x) − ρg2 (x))
2 (1 − ρ2 )
σx
lim ρ =
N −→∞
SNR
.
1 + SNR
(39)
¯˙
The sample covariance matrix Rx can be decomposed into
four Lp × Lp block matrices as follows:
¯
R˙
¯˙
Rx = ¯ x,11
Rx,21
˙
2
2
˙
ln fx|H1 ,ρ,σx x|H1 , ρ, σx
˙
=−
In this part, the asymptotic behavior of ρ under H1 is
analyzed to illustrate the performance degradation of the test in
(38) in the multipath scenario. First, observe that, in an AWGN
channel, we have the following limit:
¯˙
Rx,12
¯˙
Rx,22
(40)
¯˙
¯
where Rx,21 = RH . Thus, (37) can be expressed as
˙
x,12
(32)
ρ=
¯˙
¯˙
tr(Rx,12 + Rx,21 )
.
¯˙
¯˙
tr(Rx,11 + Rx,22 )
(41)

6. BOKHARAIEE et al.: BLIND SPECTRUM SENSING FOR OFDM-BASED CR SYSTEMS
Employing (27), the denominator of (41) has the following
asymptotic behavior under H1 :
2
2
˙ ˙
¯˙
¯˙
Dh = lim tr(Rx,11 + Rx,22 ) = 2 tr hA hH σS +Lp σv
A
N −→∞
Lc
2
2
2
|hi |2 σS + σv = 2Lp σv (1 + SNR).
= 2Lp
(42)
i=1
For the nominator of (41), we have
2
˙ ˙
¯¯
¯¯
Nh = lim tr(Rx,12 + Rx,21 ) = 2tr hB hH σS
B
N −→∞
Lp
|hi |2
j=1 i=1
Lp
Lc
Lc
2
|hi |2 = 2Lp σS
2
< 2σS
j=1 i=1
path also introduces strong correlation to the received OFDM
samples, which could also be exploited in the constrained
GLRT algorithm. This is precisely the motivation and objective
of this section. The developed algorithm shall use the portion of
the received OFDM symbol that does not include the ISI part.
In this way, the known structure of the observation can be taken
into account to improve the estimation of the signal covariance
matrix. Furthermore, a simplified test statistics is derived as a
function of the received signal correlation coefficients.
A. Constrained GLRT Algorithm
j
2
= 2σS
863
2
|hi |2 = 2Lc σv SNR.
The portion of the received OFDM block without the ISI
part and the corresponding transmitted signal and noise vectors
¨
s
can be represented as xn = [xn (L − 1), . . . , xn (0)] , ¨n =
¨
[sn (L − 1), . . . , sn (0)] , and vn = [vn (L − 1), . . . , vn (0)] ,
respectively. They are related according to
i=1
¨s
¨
¨
xn = h¨n + vn
(43)
It then follows that the CP correlation coefficient in the
multipath channel ρh is equal to
˜
Nh
˜
=
H1 : ρh = lim ρ =
N −→∞
Dh
Lp
Lp
j=1
j
i=1
Lc
i=1
2
|hi |2 σS
2
¨¨ 2
Rx = hhH σS + σv I.
¨
SNR
= ρ (44)
1 + SNR
where the subscript h indicates the dependence on channel
realization h = [h1 , . . . , hLc ]T . Note that, for a given Pf , the
threshold in (38) is fixed and does not depend on the correlation coefficient. Therefore, (39) and (44) can be compared
to observe the degradation of the detection performance of
CPCC-based detection algorithm in the presence of multipath
propagation. To underestand the role of Lp and Lc in the
detection performance, one simple way is to obtain the expected
value of ρh with respect to all channel realizations with a fixed
˜
SNR. In Appendix A, it is shown that
˜
ρ
H1 : ρ = E{˜h } =
=ρ
2Lp − Lc + 1
SNR
1 + SNR
2Lp
2Lp − Lc + 1
.
2Lp
¨
where h is the L × L circulant channel matrix, whose first row
is [ h1 · · · hLc −1 hLc 0 · · · 0 ].
¨ ¨n
Let Rx ≡ [rk,j ]k,j=1,...,L = E{xn xH } be the signal covari¨
ance matrix. Using (46), under H1 , we have
2
2
|hi |2 σS + σv
<
(45)
From (45), it is clearly seen that, for a fixed Lp and SNR, the
average CP correlation coefficient decreases with an increase in
the number of multipath delay components.
(46)
(47)
The matrix Rx has the following properties:
¨
∗
1) Rx is Hermitian: rk,j = rj,k .
¨
2) All the diagonal elements are equal: rk,k =
Lc
2 2
2
i=1 |hi | σS + σv , k = 0, . . . , L − 1
¨¨
3) Since hhH is circularly symmetric, Rx is also circu¨
larly symmetric. This last property is specific to OFDM
transmission.
From the preceding properties, one has the following
proposition.
Proposition: Let Lp denote the length of the CP. To obtain an
estimate of the covariance matrix Rx , it is sufficient to estimate
¨
Lp values.
¨
Proof: Since Lc ≤ Lp , the circulant channel matrix h has
¨ hH has at
¨
at most Lp nonzero values in each row. Therefore, h
most 2Lp − 1 nonzero values in each row that also appear in all
¨¨
¨¨
the other rows of hhH due its circularity. Since hhH is also a
Hermitian matrix, there are Lp − 1 conjugate pairs in each row,
excluding the diagonal element. Therefore, only Lp values are
needed to completely define Rx .
¨
The first row of the covariance matrix can be expressed as
[r1,0 r1,1 . . . r1,L−1 ]
V. G ENERALIZED L IKELIHOOD R ATIO T EST A LGORITHM
BASED ON M ULTIPATH C ORRELATION
As shown in the previous section, the CPCC-based detection algorithm suffers a performance degradation in a multipath channel. On the one hand, this is expected, because the
CPCC-based algorithm only uses observation in the head and
tail of an OFDM block to exploit the correlation structure,
which results from the use of the CP. On the other hand, multi-
∗
∗
= γ0 γ1 . . . γLp −1 0 . . . 0 γLp −1 . . . γ1 .
(48)
The consecutive rows are obtained through a right circular
shift of the previous row. To employ the GLRT algorithm, the
vector γ = [γ0 , γ1 , . . . , γLp −1 ] can be estimated based on the
criterion in (14), which is equivalent to
x
γ = max ln fx|H1 ,γ (¨ |H1 , γ).
¨
γ
(49)

7. 864
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
Solving the preceding problem is quite challenging since the
term R−1 cannot be easily differentiated with respect to γ.
¨
x
Instead, we consider an equivalent optimization problem as
described here.
Since Rx is a circulant matrix, all the vectors wk =
¨
√
(1/ L)[1, e−j2πk/L , . . . , e−j2πk(L−1)/L ] , k = 0, . . . , L − 1,
are its eigenvectors with the corresponding eigenvalues, i.e.,
L−1
ψ(λ0 , . . . , λL−1 ) = −N L ln(2π) − N ln
r1,i e−
j2πki
L
Re γm e−
= γ0 +
j2πkm
L
. (50)
m=1
i=0
Let W = [w0 , . . . , wL−1 ] denote the matrix of eigenvectors
(which is also an IDFT matrix), and let Λ = diag(λk ) denote
the diagonal matrix of the eigenvalues. Then, Rx = WH ΛW
¨
is the eigenvalue decomposition of Rx , and we have
¨
H
H
H
W Rx W = W WΛW W = Λ.
¨
¨
Let Xn = WH xn = [X0,n , . . . , XL−1,n ]
¨
xn . It is obvious that
N L−1
−
n=1 k=0
L−1
φ(α0 , . . . , αL−1 ) = −N L ln(2π) + N ln
N L−1
−
n=1 k=0
It is simple to verify that
N
j2πkm
L
,
m = 0, . . . , Lp − 1.
λk
(54)
then the maximum-likelihood estimate of γm ’s can be obtained as
1
=
L
λk =
k=0
x
λk = max ln fx|H1 ,Rx (¨ |H1 , Rx )
¨
¨
¨
λk e
,
m = 0, . . . , Lp − 1.
(55)
γm =
=
x
ln fx|H1 ,Rx (¨ |H1 , Rx )
¨
¨
¨
=
¨n x ¨
xH R−1 xn .
¨
(56)
n=1
With the aid of (52), it is observed that
¨n
¨n x ¨
¨
xH R−1 xn = xH WΛ−1 WH xn = XH Λ−1 Xn
n
¨
|Xn,k |2
λk
(57)
λk .
k=0
1
NL
1
NL
L−1 N
|Xn,k |2 e
k=0 n=1
L−1 N
∗
Xn,k e
k=0
j2πkm
L
k=0 n=1
1
1
xn (i) √
N L n=1 i=0
L
1
√
L
L−1
xn (i)e−
j2πki
L
i=0
L−1
∗
Xn,k e−
j2π(i−m)k
L
k=0
1
1
=
xn (i) √
N L n=1 i=0
L
∗
L−1
Xn,k e
j2π(i−m)k
L
k=0
(58)
1
xn (i)x∗ ([i − m] mod L)
n
N L n=1 i=0
m = 0, . . . , Lp − 1
(64)
which represent the sample correlation values up to the maximum delay of Lp − 1 samples.
Next, the covariance matrix estimate Rx can be constructed
¨
from the sample correlation values obtained in (64). The resulting test statistics can then be established by exploiting (22) with
¨
y = x, Ry = Rx , and M = L. In the time domain, one has
¨
L−1
det(Rx ) =
¨
j2πkm
L
N L−1
=
N
L−1
(63)
n=1
N L−1
= −N L ln(2π) − N ln det(Rx ) −
¨
=
|Xn,k |2 .
N L−1
k=0
x
To express ln fx|H1 ,Rx (¨ |H1 , Rx ) as a function of λk ’s, first,
¨
¨
¨
¨
substitute y = x and M = L in (15) to yield
N
1
1
=
αk
N
Employing (63) in (55) gives
L−1
j2πkm
L
(62)
Therefore, φ(α0 , . . . , αL−1 ) is a concave function of
α0 , . . . , αL−1 . Using (61), we have
(53)
Therefore, if we can find the maximum-likelihood estimate
of λk ’s by solving
(61)
∂ 2 φ(α0 , . . . , αL−1 )
N
= − 2 < 0.
2
∂αk
αk
(52)
L−1
λk e
αk |Xn,k |2 . (60)
N
∂φ(α0 , . . . , αL−1 )
=
−
|Xn,k |2
∂αk
αk n=1
The preceding means that λk ’s represent the average energy
per each subcarrier, and hence, they are positive. From (50), it
is observed that
1
=
L
αk
k=0
(51)
be the DFT of
¨ ¨n
= WH E xn xH W = Λ.
γm
|Xn,k |2
. (59)
λk
For convenience, define αk = (1/λk ) > 0, and rewrite
(59) as
¨ ¨n
RX = E Xn XH = E WH xn xH W
n
γm
λk
k=0
Lp −1
L−1
λk =
Using (57) and (58), (56) can be equivalently expressed as
x
TG (¨ ) =
1
x ¨ H H1
LN tr(¨ x )
1
det L (Rx )
¨
≷ ¨.
H0
(65)

8. BOKHARAIEE et al.: BLIND SPECTRUM SENSING FOR OFDM-BASED CR SYSTEMS
865
B. MPCC-Based Test
The spectrum-sensing framework elaborated so far in this
section makes use of the correlation property of the primary
signal to identify it from the background noise. It is of interest to
establish an approximated but simpler test that can still capture
the multipath correlation of the primary signal. Some recent
works [21], [22] have also intuitively developed test statistics as
functions of the received signal in the time domain by exploiting the multipath correlation property. Appendix B shows that,
by making appropriate approximation in the low-SNR region,
the test in (65) can also be simplified as a function of the sample
correlation coefficients. In particular, the simplified test is
Lp −1
|
T (¨ ) =
x
2
m|
(66)
m=1
where m represents the sample correlation coefficient corresponding to a delay of m samples, which is given as
m
=
N
n=1
L−1
∗
i=0 xn (i)xn ((i − m)
N
L−1
2
n=1
i=0 |xn (i)|
mod L)
.
(67)
It is demonstrated in Section VII that the performance of the
preceding simplified test is very close to that of the constrained
GLRT in Section V.
C. Combination of CPCC- and MPCC-Based
Detection Algorithms
As discussed at the end of Section III, the full multipath and
cyclic correlations can be jointly considered in one covariance
matrix. However, the success of the constrained GLRT algorithms introduced in Sections IV and V with a finite sample
size and at low-SNR value lies in the structural constraints
of their covariance matrices. A simple but effective approach
to combine multipath and cyclic correlations is to decide H1
whenever one of the two constrained GLRT algorithms detects
the presence of the primary user. The combined test always
yields the best performance between the two detection algorithms in each channel realization. It should be noted that the
threshold values have to be selected in such a way that the
overall probability of false alarm meets the required constraint.
VI. D ETECTION A LGORITHM FOR U NSYNCHRONIZED
O RTHOGONAL F REQUENCY D IVISION
M ULTIPLEXING S IGNALS
As pointed out before, the spectrum-sensing algorithms
presented in Sections IV and V require symbol timing synchronization between the secondary and primary users. In the absence of symbol timing synchronization, the cyclic correlation
is taken into account by considering samples located within the
lags of ±L [11]. The consequence of this approach is a decrease
in the correlation coefficient to ρ = (Lp /L + Lp )ρ [11], which
˘
causes a drop in the performance of the detection algorithm.
At the secondary user’s receiver, the received signal samples
are divided into blocks of L samples each. This means that
the corresponding samples in adjacent blocks are correlated
due to the CP [11]. Compared with the received signal model
Fig. 1. Timing relation between transmitter and receiver in the unsynchronized case.
in the synchronized case [see, e.g., (5)], the receiver in the
unsynchronized case does not know exactly when an OFDM
block will start. As such, the timing index in this section is
with reference to the time the secondary user’s receiver starts
to collect the receive signal samples. In general, the timing
origin at the secondary user’s receiver can be lead or lag over
the timing origin at the transmitter by τ samples, as shown in
Fig. 1. For N transmitted OFDM blocks, the number of sample
˘
blocks processed by the receiver is N = N (L + Lp − τ )/L .
To develop an efficient GLRT-based detection algorithm for
unsynchronized OFDM signals, consider only the last portion
˘
of L = L − Lp + 1 samples in each block of L samples (see
Fig. 1), which can be represented as
˘s
˘
˘
xn = [˘n (L − 1), . . . , xn (Lp − 1)] = h˘n + vn
x
˘
˘
n = 1, . . . , N .
(68)
˘
˘
In the preceding expression, h is the L × L Toeplitz
channel matrix with the first row [h1 , . . . , hLc , 0, . . . , 0] and
first column [h1 , 0, . . . , 0] . The corresponding length-L
s
transmitted signal vector is denoted by ˘n = [˘n (L − 1),
s
˘
. . . , sn (0)] , n = 1, . . . , N . It is important to emphasize
˘
that, due to unsynchronization, ˘n does not necessarily
s
align with the transmitted OFDM blocks, and it generally
contains data symbols from two consecutive transmitted
OFDM blocks. It also has to be noted that the reason for
employing the signal model in (68) in our analysis is to
efficiently exploit the correlation among the transmitted
signals due to the presence of CP. Based on our model, all the
corresponding samples in the neighboring transmitted vectors,
s
i.e., ˘n and ˘n−1 , are correlated with the average correlation
s
coefficient ϑ = E{˘n (i)˘∗ (i)}/E{|˘n (i)|2 } = Lp /L + Lp .
s
sn−1
s
Consequently, the corresponding samples in the neighboring
˘
˘
received vectors, i.e., xn and xn−1 , are correlated with the
correlation coefficient ρ = E{˘n (i)˘∗ (i)}/E{|˘n (i)|2 } =
˘
x
xn−1
x
Lc
Lc
2
2
2
2
2
ϑ(σS i=1 |hi | /σS i=1 |hi | + σv ) = ϑ(SNR/SNR + 1).
˘
To take the correlation between the neighboring vectors xn
˘
˜
into account, define the length-2L vectors xn = [˘ n , xn−1 ] ,
x ˘
˘
˜
n = 2, . . . , N . The covariance matrix of xn is expressed and
approximated as
˘ 2
˜ ˜n
Rx = E xn xH = hσS
˜
∼
=
Rx
˘
ρRx
˘ ˘
ρRx
˘ ˘
.
Rx
˘
IL
ϑIL
ϑIL ˘ H
2
h + σv I2L
IL
(69)

9. 866
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
˘ ˘
The matrix Rx is an L × L Hermitian Toeplitz matrix, and
˘
˘
it can be described by its first row [˘0 , . . . , γLp −1 , 0, . . . , 0]
γ
˜
˘∗
and its first column [˘0 , γ1 , . . . , γLp −1 , 0, . . . 0] . Let x =
γ ∗ ˘∗
˜˘
˘
˘˘
x
[˜ 2 , . . . , xN ] and x = [˘ 1 , . . . , xN ]. To facilitate matrix max
nipulations and parameter estimations, the correlation among
˜
adjacent xn ’s is ignored. This allows one to apply the general
˜
˜
GLRT test in (23) to the observation x by substituting y = x,
˘
˘
M = 2L, and N = N − 1. The resulting test is
˘ x =
TG (˜ ) ∼
1
tr(˜ xH )
x˜
˘ ˘
2L(N −1)
=
1
˘
det 2L (Rx )
˜
1
x˘H
˘ ˘ tr(˘ x )
LN
2
˘
1/L
(1 − ρ ) det
˘
≷˘
(Rx )
˘
H0
where Rx , Rx , and ρ are estimates of Rx , Rx , and ρ, respec˘
˘
˜
˜
˘
˘
˘
tively. Based on (70), both Rx and ρ need to be estimated.
˘
Since obtaining the ML estimate of Rx appears to be very
˘
cumbersome, the entries of Rx shall be estimated by the sample
˘
correlation coefficients as follows:
⎡
˘
N
L
1 ⎣
γm =
˘
xn (i)˘∗ (i−m)
˘
xn
˘˘
N L n=1 i=m
⎤
˘
N −1 m−1
n=2
xn (i)˘∗ (L+i−m)⎦, m = 0, . . . , Lp −1.
˘
xn−1
(71)
n=1 i=1
⎡
⎛
=
H1
(70)
+
To simplify (75), we can make use of the following approxi˘
mation under the assumption of sufficiently large N :
⎤
⎡
˘
N
1
⎣
˘n x ˘
˘ n−1 x ˘
xH R−1 xn + xH R−1 xn−1 ⎦
˘
˘
˘ − 1)L
˘
2(N
=
1
˘
˘
2(N − 1)L
tr ⎝R−1 ⎣
˘
x
˘
N
˘
N
˘ ˘n
xn xH +
n=2
⎤⎞
˘
˘ n−1
xn−1 xH ⎦⎠
n=2
1
¯˘ =
tr R−1 Rx ∼ 1
˘
x
˘
L
(76)
˘
˘
¯˘
˘
˘ n−1
where Rx = (1/2(N − 1))( N xn xH + N xn−1 xH )
n=2 ˘ ˘ n
n=2 ˘
˘
represents the sample covariance matrix of x’s. Using (76), (75)
can be simplified to
2˘
ρ
(1− ρ2 )
˘
⎛
ρ+
˘
× ⎝−1+
+
˘
N
ρ
˘
˘
˘
2(N −1)L n=2
˘
N
1
˘
˘
2(N −1)L n=2
⎞
˘ n−1 x ˘
˘n x ˘
xH R−1 xn + xH R−1 xn−1 ⎠
˘
˘
˘ n−1 x ˘
˘n x ˘
xH R−1 xn + xH R−1 xn−1 ∼ 0.
=
˘
˘
(77)
To obtain the ML estimate of ρ, the log-likelihood function
˘
˜
of x is approximately expressed as
It can be shown that the following solution satisfies (77):
˘
˘
ln fx|H1 ,Rx (˜ |H1 , Rx ) ∼ −(N − 1) · L · ln(2π)
x
˜ =
˜
˜
ρ=
˘
˘
N
˜n x ˜
xH R−1 xn .
˜
˘
˘
− (N − 1) · L · ln (det(Rx )) −
˜
(72)
n=1
It can be easily verified that
1
1 − ρ2
˘
R−1 =
˜
x
R−1
˘
x
−˘R−1
ρ x
˘
−˘R−1
ρ x
˘
.
R−1
˘
x
˘
N−1
˘n x ˘
˘˘ n−1 x ˘
xH R−1 xn − ρxH R−1 xn
˘
˘
˘ n−1 x ˘
−˘ H R−1 xn−1 + xH R−1 xn−1 .
xn x ˘
˘
˘
(74)
˘ρ ˘
Solving ∂ ψ(˘, Rx )/∂ ρ = 0 gives
˘
˘
2˘(N −1)L
ρ ˘
2)
(1− ρ
˘
2˘
ρ
−
(1− ρ2 )2
˘
˘
N
˘n x ˘
˘˘ n−1 x ˘
xH R−1 xn − ρxH R−1 xn
˘
˘
n=2
˘ n−1 x ˘
− ρxH R−1 xn−1 + xH R−1 xn−1
˘˘ n x ˘
˘
˘
˘
N
+
1
˘
˘n x ˘
xH R−1 xn + xH R−1 xn−1 = 0.
˘ ˘
˘
(1− ρ2 ) n=2 n−1 x
˘
˘
˘
2(N − 1)L n=2
H
H
˘
˘
xn−1 R−1 xn + xn R−1 xn−1 . (78)
˘
˘
x ˘
x ˘
In summary, after finding Rx based on (71), ρ can be
˘
˘
obtained from (78). The results are then used in (70) to realize
the test.
VII. S IMULATION R ESULTS
˘ρ ˘
˘
˘
˘
˘
˘
ψ(˘, Rx ) = −(N −1)L ln(2π)−(N −1)L ln(1− ρ2 )
1
˘
−2(N −1) ln(Rx )−
˘
1− ρ2
˘
n=2
˘
N
(73)
Using the preceding expression and det(Rx ) = (1 −
˜
˘
ρ2 )L det2 (Rx ), (72) can be equivalently expressed as
˘
˘
·
1
(75)
The simulation parameters are chosen similarly to those
in [11]. In particular, the primary user’s OFDM system has
L = 32 subcarriers and transmits i.i.d. 16-QAM symbols with
normalized unit power. The detection period is taken to be
equal to N = 100 OFDM blocks, and the results are averaged
over 1000 random realizations of a Rayleigh multipath fading
channel. Except for Fig. 9, the channel coefficients are i.i.d.
complex Gaussian random variables. The case of correlated
channels is considered for Fig. 9. Note that, for an OFDM
system having a bandwidth of 5 MHz, 32 subcarriers, and a
CP length of 8 (similar to [11]), the sensing time is roughly
((32 + 8)/5 × 106 ) × 100 = 8 × 10−4 s or 0.8 ms. The performance of different spectrum-sensing algorithms is evaluated
and compared via the probability of detection Pd for a constant
false alarm rate of Pf = 0.05.
First, Fig. 2 compares the detection performance of the
energy detector (ED) and three spectrum-sensing algorithms
developed and analyzed in this paper under perfect synchronization assumption, i.e., CPCC-based algorithm (Section V-A
and [11]), multipath correlation-based constrained GLRT
(MP-based C-GLRT algorithm, Section V-A), and the simpler

10. BOKHARAIEE et al.: BLIND SPECTRUM SENSING FOR OFDM-BASED CR SYSTEMS
Fig. 2. Performance comparison of C-GLRT and U-GLRT spectrum-sensing
algorithms (Lc = Lp = 8).
Fig. 3. Performance comparison of C-GLRT and U-GLRT spectrum-sensing
algorithms (Lc = Lp = 8).
MPCC-based algorithm (Section V-B). For this particular figure, Lp = Lc = 8 is used. As pointed out before, the ED algorithm requires a precise knowledge of the noise variance, and a
small noise uncertainty, e.g., 0.5 and 1.0 dB, causes huge performance degradation, as can be seen from the figure. In contrast, the three other algorithms are completely blind, and their
performances are impressive. Note that the simplified MPCCbased algorithm performs closely to the MP-based C-GLRT
algorithm, and both of them clearly outperform the CPCCbased algorithm. This superior performance is expected since,
with a large number of channel taps (Lc = 8), it would be more
beneficial to exploit multipath correlation than CP correlation.
Next, Fig. 3 shows performance improvement of the constrained GLRT algorithms over their unconstrained counterparts, both with multipath correlation and CP correlation. For
this figure, we also set Lp = Lc = 8. Recall that the U-GLRT
algorithm is basically (23), except that it uses only CP portions of the observations in the CP-based algorithm or ISI-free
867
Fig. 4. Effect of Lp and Lc on the performance of spectrum-sensing
algorithms.
portions of the observations in the multipath-based algorithm.
In both cases, the improvements in detection performance are
very large.
Fig. 4 shows the detection performance of both MP- and
CPCC-based C-GLRT algorithms under different values of Lp
and Lc . Observe that the performance of MP-based C-GLRT
changes very little between Lp = Lc = 4 and Lp = Lc = 8.
This can be explained as follows: While a bigger value of Lc
is desirable in terms of having stronger correlation property,
there is a larger number of quantities to estimate with the
same size of observations. These opposing effects appear to
cancel out in the scenarios considered in Fig. 4. In contrast,
the performance of CPCC-based algorithm can be significantly
improved by increasing the length of the CP (comparing the
settings of Lp = Lc = 4 with those of Lp = 8, Lc = 4). Such
a performance improvement is obviously expected but comes
at the expense of additional resources since, as far as ISI
avoidance is concerned, it is desirable to use the minimum CP
length of Lp = Lc = 4. Furthermore, performance degradation
of the CPCC-based C-GLRT algorithm in an environment with
higher channel taps (rich multipath environment) can also be
observed by comparing the curves with Lp = 8, Lc = 4 and
Lp = Lc = 8. Such an observation agrees with the analysis in
(45) in Section IV-B.
Fig. 5 shows the convergence behavior of the probability of
detection of the proposed MP-based C-GLRT algorithm as a
function of N . As can be seen from the figure, for every SNR
value, the probability of detection can be made arbitrarily close
to 1 by allowing sufficient observations.
Fig. 6 shows the performance of the combined algorithm
(discussed in Section V-C) for the case Lc = Lp = 8. It is
seen that the combined sensing algorithm outperforms both the
MP- and CPCC-based C-GLRT algorithms. Such a result is
as expected since the combined algorithm takes into account
all the observations (both ISI-free and CP portions of received
blocks) and, at the same time, benefits from the covariance
structures exploited in MP- and CPCC-based algorithms.

11. 868
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
Fig. 5. Performance of MP-based C-GLRT algorithm with respect to observation size N .
Fig. 7. Performance of the spectrum-sensing algorithms in the presence
of residual time and CFOs (Lc = Lp = 8). Note that the types of sensing
algorithm are distinguished by different line styles, whereas the combinations of imperfect/perfect timing synchronization (indicated as “I-Synch” and
“P-Synch”) and CFO values are identified by different markers.
Fig. 6. Performance of the combined algorithm (Lp = Lc = 8).
Fig. 7 shows the performance of the MP-based C-GLRT and
CPCC-based algorithms when the synchronization algorithm in
[18] is performed first over the sensing interval. The normalized
residual CFO is set to 0.5, which is the worst case. Note that the
types of sensing algorithm are distinguished by different line
styles, whereas the combinations of imperfect/perfect timing
synchronization (indicated as “I-Syn” and “P-Syn”) and CFO
values are identified by different markers. The CFO introduces
a phase shift to the time-domain samples of an OFDM signal
[18]. Nevertheless, the circularity of the received signal covariance matrix is preserved in this case, and the MP-based
C-GLRT algorithm is not affected by a CFO. However, the
CP correlation coefficient becomes a complex value when
frequency offset is present. The effect of CFO on the CPCCbased spectrum-sensing algorithm can be compensated by considering the magnitude of the sample correlation coefficient
in the CPCC-based algorithm [18]. As shown in Fig. 7, the
performance of sensing algorithms remains unchanged in the
presence of CFO. On the other hand, it is seen that the imperfect
Fig. 8. Performance comparison of spectrum-sensing algorithms in synchronized, imperfectly synchronized, and unsynchronized transmission scenarios.
symbol timing causes performance degradation in both algorithms. For the MP-based C-GLRT in the presence of timing
offset, the degradation is due to the fact that the ISI part
cannot be perfectly removed; therefore, the covariance matrix
of the received signal is not truly circulant. The time offset
also degrades the performance of the CPCC-based algorithm
since the identical head and tail of the OFDM signal cannot be
perfectly retrieved at the receiver.
Fig. 8 compares the performance of the proposed sensing
algorithms for perfectly synchronized, imperfectly synchronized, and unsynchronized OFDM signals, which are labeled
in the figure as “P-Syn,” “I-Syn,” and “Unsyn,” respectively. In
particular, the combined algorithm in Section V-C is compared
against the algorithm in (70) for two different numbers of
channel taps, i.e., Lc = 8 and Lc = 1, as well as when the
CP length is set to Lp = 8. Observe the opposite performance

12. BOKHARAIEE et al.: BLIND SPECTRUM SENSING FOR OFDM-BASED CR SYSTEMS
Fig. 9. Performance comparison of MP-based C-GLRT in a time-varying
Rayleigh fading channel (Lc = Lp = 8).
behaviors when the number of channel taps reduces from Lc =
8 to Lc = 1; the performance gets better for the synchronized
and imperfectly synchronized cases, whereas it gets worse for
the unsynchronized case. The latter behavior is, in fact, expected since performance degradation of the proposed detection
algorithm for unsynchronized OFDM signals is mainly due to
the reduction in the cyclic correlation coefficient. Because the
CP correlation strongly determines the performance of the detection algorithm in a fading environment with a fewer channel
taps, the unsynchronized detection algorithm performs quite
poorly in such an environment. In contrast, in a rich multipath
environment, for example, with Lc = 8, there is only a small
performance loss when detecting unsynchronized OFDM signals. In addition, in such an environment, the imperfectly synchronized algorithm performs worse than the unsynchronized
algorithm. Considering the added complexity due to performing
synchronization, the algorithm developed for unsynchronized
OFDM signals in Section VI is a better candidate for spectrum
sensing in a rich multipath environment.
Finally, Fig. 9 shows the performance of MP-based C-GLRT
algorithm over time-varying Rayleigh fading channels when the
Doppler frequency fd is set to different values. Typical Doppler
shifts correspond to the mobile velocities of about 3–60 km/h.
If the system is operated in frequency bands of 2–4 GHz
(e.g., IEEE 802.11, IEEE 802.16, and IEEE 802.20), the
Doppler shifts are about 5–200 Hz. It can be seen that, for the
typical Doppler frequencies, the multipath correlation is still
very beneficial, and the sensing performance is not substantially
degraded, even for fast changes in channel taps.
VIII. C ONCLUSION
In this paper, a spectrum-sensing method for OFDM-based
cognitive radio systems has been developed based on the
GLRT framework. The key feature in our development is to
explicitly take into account the structure (constraint) of the
covariance matrix of the underlying OFDM signal so that the
ML estimations of unknown parameters are improved, which
869
leads to robust and efficient spectrum-sensing tests. In particular, it has been shown that the CPCC-based test, which was
recently proposed in [11], can be obtained as a constrained
GLRT for an AWGN channel. It has also been shown that
the performance of CPCC-based test degrades in a multipath
environment. Moreover, by exploiting the multipath correlation
in the GLRT framework, an efficient test has been obtained,
which can be sequentially updated with any new reception
of OFDM symbol. A simplified MPCC-based test has also
been presented. Simulation results verify that both the CPCCand MP-based C-GLRT algorithms greatly outperform energy
detection in an environment with noise uncertainty. The MPbased C-GLRT algorithm performs better than the CPCC-based
algorithm in a rich multipath environment. Furthermore, a
simple algorithm that combines both the CPCC- and MP-based
C-GLRT algorithms is suggested, which can further improve
the detection performance in a multipath environment.
While our studies have mainly focused on the detection of
synchronized OFDM signals, the developed algorithms can be
applied together with the synchronization algorithm in [18], and
they experience only a small performance loss due to imperfect
synchronization. Lastly, a simple GRLT-based algorithm has
also been proposed for the detection of unsynchronized OFDM
signals from the primary user. Simulation results demonstrated
satisfactory performance of such an algorithm in a rich multipath fading environment.
A PPENDIX A
P ROOF OF (45)
2
For a fixed SNR and an i.i.d. channel with E{|hi |2 } = σh ,
we have
2
σS
2
σv
SNR = E
Lc
|hi |2
=
i=1
2 2
Lc σS σh
.
2
σv
(79)
First, rewrite Nh as
⎡
Lc −1
2
Nh = 2σS ⎣
Lp
j
j
|hi |2 +
j=1 i=1
⎤
|hi |2 ⎦ .
(80)
j=Lc i=1
It can be easily seen that
2 2
E{Nh } = 2σS σh
Lc (Lc − 1)
+ (Lp − Lc )Lc
2
2Lp − Lc + 1
2
(81)
2
E{Dh } = Dh = 2Lp σv [SNR + 1].
(82)
2
= 2σv SNR
Therefore
˜
ρ
H1 : ρ = E{˜h } =
2Lp − Lc + 1
E{Nh }
SNR
=
.
Dh
1 + SNR
2Lp
(83)

13. 870
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 3, MARCH 2011
A PPENDIX B
D EVELOPING THE M ULTIPATH C ORRELATION
C OEFFICIENT T EST
Lp −1
m=1 Re
≈
γm e
⎛
1⎝
−
2
j2πkm
L
γ0
Lp −1
m=1 Re
γm e
j2πkm
L
γ0
First, observe from (65) that
det(Rx )
¨
ln det(Rx )
¨
1
L
1
L
ln
=e
Therefore
(84)
1
1
= ln det(Rx ) =
¨
L
L
L
˜
ln(λk )
L
1
L
(85)
k=1
k=1
⎡
1
˜ =
ln(λk ) ∼ ln γ0 +
L
⎛
˜
where λk ’s are the eigenvalues of Rx . From (50), we have
¨
−
Lp −1
Re γm e−
j2πkm
L
.
(86)
m=1
Hence
⎛
˜
ln(λk ) = ln⎝γ0 +
⎛ ⎛
Re γm e
Lp −1
m=1
⎠
j2πkm
L
m=1
Lp −1
m=1
⎛
= ln(γ0 )+ln⎝1+
Re γm e
j2πkm
L
γ0
Lp −1
m=1
Re γm e
L
k=1
⎠⎠
⎝
⎞
j2πkm
L
⎠.
L
=
⎛
Re γm e−
Lp −1 1
m=1 2
⎝
1
L
γm e−
Lp −1 1
q=1 4
Lp −1
m=1
+
k=1
Lp −1
m=1
+
Re γm e
⎞2 ⎤
⎠ ⎥ . (89)
⎦
j2πkm
L
Re γm
L
− j2πkm
L
k=1 e
= 0.
γ0
(90)
L
˜ =
ln(λk ) ∼ ln γ0 −
k=1
1
4
Lp −1
m=1
|γm |2
2 .
γ0
(92)
det(Rx )
¨
1
L
−1
4
∼ γ0 e
=
Lp −1
m=1
γm 2
γ2
0
.
(93)
⎠
j2πkm
L
∗
+ γm e
j2πkm
L
γm γq e−
j2π(k+q)m
L
⎞2
⎠
∗ ∗
+ γm γq e
j2π(k+q)m
L
)
2
γ0
k=1
L
=
− j2πkm
L
Consequently
γ0
Lp −1
m=1
=
γ0
k=1
⎞2
j2πkm
L
k=1
L
j2πkm
L
The second term can be rewritten as (91), shown at the
bottom of the page.
Therefore, (89) can be approximated by
(87)
γ0
k=1
Re γm e−
γ0
Lp −1
m=1
=
L −1
Lp −1
m=1
1⎝
2
Lp −1
m=1
γ0
p
1 for low SNR valSince ( m=1 Re(γm ej2πkm/L )/γ0 )
ues, the second term of (87) can be expanded by keeping the
first two terms of the Taylor series. That is
⎛
⎞
j2πkm
Lp −1
L
m=1 Re γm e
⎠
ln⎝1+
γ0
⎛
⎢
⎣
Lp −1
m=1
Re γm e−
⎞⎞
γ0
L
L
The first term inside the square brackets of (89) can be
simplified as follows:
⎞
Lp −1
= ln⎝γ0⎝1+
⎠ .
(88)
1
det(Rx ) L
¨
˜
λk = γ0 +
⎞2
Lp −1 1
q=1 4
Lp −1
m=1
1
=0+
4
j2π(k−q)m
Lp −1 1
∗ −
L
)
q=1 2 Re(γm γq e
2
γ0
L
k=1
Lp −1 1
q=1 2 Re
Lp −1
2
m=1 |γm
2
γ0
γm γq e−
j2π(k+q)m
L
∗ ∗
+ γm γq e
j2π(k+q)m
L
2
γ0
∗
γm γq
2
γ0
1
=
4
j2π(k−q)m
L
−
L
k=1 e
Lp −1
2
m=1 |γm
2
γ0
(91)

14. BOKHARAIEE et al.: BLIND SPECTRUM SENSING FOR OFDM-BASED CR SYSTEMS
By noting that (1/L)(tr(Rx )) = γ0 , the test statistics given
¨
in (65) can be closely approximated as
x
TG (¨ ) =
1
L tr(Rx )
1
det(Rx ) L
γ0
∼
=
−1
4
γ0 e
Lp −1 |γm |2
2
m=1
γ0
=e
1
4
Lp −1
m=1
γm 2
γ0
.
(94)
Taking the logarithm of the preceding test yields the following equivalent test:
T (¨ ) =
x
The ratio
is given by
m
=
m
Lp −1
m=1
γm
2
γ0
2
.
(95)
= γm /γ0 can be easily obtained from (55) and
N
n=1
L−1
∗
i=0 xn (i)xn ((i − m)
N
L−1
2
n=1
i=0 |xn (i)|
mod L)
.
871
[16] A. Taherpour, S. Gazor, and M. Nasiri-Kenari, “Invariant wideband
spectrum-sensing under unknown variances,” IEEE Trans. Wireless
Commun., vol. 8, no. 5, pp. 2182–2186, May 2009.
[17] T. J. Lim, R. Zhang, Y. C. Liang, and Y. Zeng, “GLRT—Based spectrumsensing for cognitive radio,” in Proc. IEEE Global Telecommun. Conf.,
Nov. 2008, pp. 1–5.
[18] J.-J. van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time
and frequency offsets in OFDM systems,” IEEE Trans. Signal Process.,
vol. 45, no. 7, pp. 1800–1805, Jul. 1997.
[19] T. Hwang, C. Yang, G. Wu, S. Li, and G. Y. Li, “OFDM and its wireless applications: A survey,” IEEE Trans. Veh. Technol., vol. 58, no. 4,
pp. 1673–1694, May 2009.
[20] J. P. Burg, D. G. Luenberger, and D. L. Wegner, “Estimation of structured
covariance matrices,” Proc. IEEE, vol. 70, no. 9, pp. 963–974, Sep. 1982.
[21] W. Zeng and G. Bi, “Exploiting the multi-path diversity and multi-user
cooperation to detect OFDM signals for cognitive radio in low SNR with
noise uncertainty,” in Proc. IEEE Global Telecommun. Conf., Dec. 2009,
pp. 1–6.
[22] W. Zeng and G. Bi, “Robust detection of OFDM signals for cognitive
UWB in low SNR with noise uncertainty,” in Proc. IEEE Int. Symp. Pers.,
Indoor, Mobile Radio Commun., Sep. 2008, pp. 1–5.
(96)
Recall that m = E{xn (k)x∗ (k − m)}/E{xn (k)x∗ (k)},
n
n
m = 0, . . . , Lp − 1 is the correlation coefficients between the
ISI-free portions of the OFDM block. On the other hand, m
represents the sample correlation coefficient corresponding to a
delay of m samples.
R EFERENCES
[1] J. Ma, G. Y. Li, and B. H. Juang, “Signal processing in cognitive radio,”
Proc. IEEE, vol. 97, no. 5, pp. 805–823, May 2009.
[2] H. A. Mahmoud, T. Yucek, and H. Arslan, “OFDM for cognitive radio:
Merits and challenges,” IEEE Wireless Commun. Mag., vol. 16, no. 2,
pp. 6–15, Apr. 2009.
[3] Z. Quan, S. Cui, A. H. Sayed, and H. V. Poor, “Optimal multiband joint
detection for spectrum-sensing in cognitive radio networks,” IEEE Trans.
Signal Process., vol. 57, no. 3, pp. 1128–1140, Mar. 2009.
[4] C.-H. Hwang, G.-L. Lai, and S.-C. Chen, “Spectrum-sensing in wideband
OFDM cognitive radios,” IEEE Trans. Signal Process., vol. 58, no. 2,
pp. 709–719, Feb. 2010.
[5] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection
Theory. Englewood Cliffs, NJ: Prentice-Hall, 1998.
[6] R. Tandra and A. Sahai, “SNR walls for signal detection,” IEEE J. Sel.
Topics Signal Process., vol. 2, no. 1, pp. 4–17, Feb. 2008.
[7] S. Kim, J. Lee, H. Wang, and D. Hong, “Sensing performance of energy
detector with correlated multiple antennas,” IEEE Signal Process. Lett.,
vol. 16, no. 8, pp. 671–674, Aug. 2009.
[8] Y. Zeng and Y. C. Liang, “Spectrum-sensing algorithms for cognitive
radio based on statistical covariances,” IEEE Trans. Veh. Technol., vol. 58,
no. 4, pp. 1804–1815, May 2009.
[9] Y. Zeng and Y. C. Liang, “Eigenvalue-based spectrum-sensing algorithms
for cognitive radio,” IEEE Trans. Commun., vol. 57, no. 6, pp. 1784–1793,
Jun. 2009.
[10] F. Penna, R. Garello, and M. A. Spirito, “Cooperative spectrum-sensing
based on the limiting eigenvalue ratio distribution in Wishart matrices,”
IEEE Commun. Lett., vol. 13, no. 7, pp. 507–509, Jul. 2009.
[11] S. Chaudhari, V. Koivunen, and H. V. Poor, “Autocorrelation-based decentralized sequential detection of OFDM signals in cognitive radios,” IEEE
Trans. Signal Process., vol. 57, no. 7, pp. 2690–2700, Jul. 2009.
[12] J. Lunden, V. Koivunen, A. Huttunen, and H. V. Poor, “Spectrum-sensing
in cognitive radios based on multiple cyclic frequencies,” in Proc. Cogn.
Radio Oriented Wireless Netw. Commun., Aug. 2007, pp. 37–43.
[13] M. Barkat, Signal Detection and Estimation. Norwood, MA: Artech
House, 1991.
[14] O. Bolkhovskaya and A. Maltsev, “The performance of the GLRT for
the spatial signals detection with a small number of observations,” IEEE
Signal Process. Lett., vol. 11, no. 10, pp. 841–844, Oct. 2004.
[15] M. Novey, T. Adalý, and A. Roy, “Circularity and Gaussianity detection using the complex generalized Gaussian distribution,” IEEE Signal
Process. Lett., vol. 16, no. 11, pp. 993–996, Nov. 2009.
Simin Bokharaiee (S’09) was born in Mashhad,
Iran, in 1980. She received the B.Sc. degree from
Sharif University of Technology, Tehran, Iran, and
the M.Sc. degree from Ferdowsi University of
Mashhad, both in electrical engineering. She is currently working toward the Ph.D. degree with the
Department of Electrical and Computer Engineering,
University of Manitoba, Winnipeg, MB, Canada.
Her current research interests include detection
and spectrum sensing in cognitive radio networks.
Ha H. Nguyen (M’01–SM’05) received the B.Eng.
degree from Hanoi University of Technology, Hanoi,
Vietnam, in 1995, the M.Eng. degree from the
Asian Institute of Technology, Bangkok, Thailand, in
1997, and the Ph.D. degree from the University of
Manitoba, Winnipeg, MB, Canada, in 2001, all in
electrical engineering.
Since 2001, he has been with the Department of
Electrical Engineering, University of Saskatchewan,
Saskatoon, SK, Canada, where he is currently a
Full Professor. He holds adjunct appointments at the
Department of Electrical and Computer Engineering, University of Manitoba,
and TRLabs, Saskatoon, and was a Senior Visiting Fellow with the School
of Electrical Engineering and Telecommunications, University of New South
Wales, Sydney, Australia, during October 2007–June 2008. He is a coauthor,
with E. Shwedyk, of the textbook A First Course in Digital Communications
(Cambridge University Press). His research interests include digital communications, spread spectrum systems, and error-control coding.
Dr. Nguyen is a Registered Member of the Association of Professional Engineers and Geoscientists of Saskatchewan. He currently serves as an Associate
Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and
the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.
Ed Shwedyk received the B.Sc. (E.E.) and M.Sc.
(E.E.) degrees from the University of Manitoba,
Winnipeg, MB, Canada, in 1965 and 1968, respectively, and the Ph.D. degree from the University of
New Brunswick, Saint John, NB, Canada, in 1974.
He was with the Department of Electrical and
Computer Engineering, University of Manitoba,
from 1974 to 2005, retiring as a full Professor.
His research interests are digital communications,
principally estimation and detection, and biosignal
processing.