1
MaxMin SNR Signal Energy based SpectrumSensing Algorithms for Cognitive Radio Networkswith Noise Variance Uncertainty
Tadilo Endeshaw Bogale,
Student Member, IEEE
and Luc Vandendorpe,
Fellow, IEEE
Abstract
—This paper proposes novel spectrum sensing algorithms for cognitive radio networks. By assuming knowntransmitter pulse shaping ﬁlter, synchronous and asynchronousreceiver scenarios have been considered. For each of thesescenarios, the proposed algorithm is explained as follows: First,by introducing a combiner vector, an oversampled signal of totalduration equal to the symbol period is combined linearly. Second,for this combined signal, the SignaltoNoise ratio (SNR) maximization and minimization problems are formulated as Rayleighquotient optimization problems. Third, by using the solutions of these problems, the ratio of the signal energy corresponding to themaximum and minimum SNRs are proposed as a test statistics.For this test statistics, analytical probability of false alarm (
P
f
)and detection (
P
d
) expressions are derived for additive whiteGaussian noise (AWGN) channel. The proposed algorithms arerobust against noise variance uncertainty. The generalization of the proposed algorithms for unknown transmitter pulse shapingﬁlter has also been discussed. Simulation results demonstratethat the proposed algorithms achieve better
P
d
than that of the Eigenvalue decomposition and energy detection algorithmsin AWGN and Rayleigh fading channels with noise varianceuncertainty. The proposed algorithms also guarantee the desired
P
f
(
P
d
)
in the presence of adjacent channel interference signals.
I. I
NTRODUCTION
The current wireless communication networks adopt ﬁxedspectrum access strategy. The Federal Communications Commission have found that this ﬁxed spectrum access strategyutilizes the available frequency bands inefﬁciently [1], [2]. Apromising approach of addressing this problem is to deploy acognitive radio (CR) network. One of the key characteristicsof a CR network is its ability to discern the nature of thesurrounding radio environment. This is performed by thespectrum sensing (signal detection) part of a CR network.The most common spectrum sensing algorithms for CRnetworks are matched ﬁlter, energy and cyclostationary basedalgorithms. If the characteristics of the primary user such asmodulation scheme, pulse shaping ﬁlter and packet format areknown perfectly, matched ﬁlter is the optimal signal detectionalgorithm as it maximizes the received SignaltoNoise Ratio
The authors would like to thank SES for the ﬁnancial support of thiswork, the french community of Belgium for funding the ARC SCOOPand BELSPO for funding the IAP BESTCOM project. Part of this work has been published in the 8th International Conference on Cognitive RadioOriented Wireless Networks (CROWNCOM), Washington DC, Jul. 8  10,2013. Tadilo E. Bogale and Luc Vandendorpe are with the ICTEAM Institute,Universit´e catholique de Louvain, Place du Levant 2, 1348  Louvain LaNeuve, Belgium. Email:
{
tadilo
.
bogale
,
luc
.
vandendorpe
}
@uclouvain.be,Phone: +3210478071, Fax: +3210472089.
(SNR). This algorithm has two major drawbacks: The ﬁrstdrawback is it needs dedicated receiver to detect each signalcharacteristics of a primary user [3]. The second drawback is it requires perfect synchronization between the transmitterand receiver which is impossible to achieve. This is due to thefact that, in general, the primary and secondary networks areadministered by different operators. Energy detector does notneed any information about the primary user and it is simple toimplement. However, energy detector is very sensitive to noisevariance uncertainty, and there is an SNR wall below whichthis detector can not guarantee a certain detection performance[3]–[5]. Cyclostationary based detection algorithm is robustagainst noise variance uncertainty and it can reject the effectof adjacent channel interference. However, the computationalcomplexity of this detection algorithm is high, and largenumber of samples are required to exploit the cyclostationaritybehavior of the received signal [5], [6]. On the other hand, thisalgorithm is not robust against cyclic frequency offset whichcan occur due to clock and timing mismatch between thetransmitter and receiver [7]. In [8], Eigenvalue decomposition(EVD)based spectrum sensing algorithm has been proposed.This algorithm is robust against noise variance uncertaintybut its computational complexity is high. Furthermore, forsingle antenna receiver, this algorithm is sensitive to adjacentchannel interference signal, and for multiantenna receiver, thisalgorithm requires a channel covariance matrix different froma scaled identity [9].This paper proposes novel spectrum sensing algorithmsfor cognitive radio networks. It is well known that a digitalcommunication signal is constructed by passing an oversampled signal through a transmitter pulse shaping ﬁlter. Inmost primary networks, for a given frequency band, as thispulse shaping ﬁlter is designed at the time of frequencyplanning stage and it is kept ﬁxed, it is assumed to beknown to the cognitive receiver. We consider synchronous andasynchronous receiver scenarios. For each of these scenarios,the proposed detection algorithm is explained as follows: First,by introducing a combiner vector, an oversampled signal of total duration equal to the symbol period is combined linearly.Second, for this combined signal, the SNR maximization andminimization problems are formulated as Rayleigh quotientoptimization problems. Third, by using the solutions of theseproblems, the ratio of the signal energy corresponding to themaximum and minimum SNRs are proposed as a test statistics.For this test statistics, analytical probability of false alarm (
P
f
)and probability of detection (
P
d
) expressions are derived for
additive white Gaussian noise (AWGN) channel. The generalization of the proposed algorithms for unknown transmitterpulse shaping ﬁlter scenarios has also been discussed. It isshown that these detection algorithms (i.e., synchronous andasynchronous receiver scenarios) are robust against noise variance uncertainty. Moreover, under noise variance uncertainty,simulation results demonstrate that the proposed detectionalgorithms achieve better detection performance compared tothat of the EVDbased and energy detection algorithms inAWGN and Rayleigh fading channels. The proposed detectionalgorithms also guarantee the desired
P
f
(
P
d
)
in the presenceof low (moderate) adjacent channel interference (ACI) signals.The remaining part of this paper is organized as follows:Section II discusses the hypothesis test problem. Section IIIpresents the proposed spectrum sensing algorithms for a narrow band signal with synchronous and asynchronous receiverscenarios. In Section IV, the extension of the proposed spectrum sensing algorithms for a wide band signal is discussed.In Section V, computer simulations are used to compare theperformance of the proposed and existing spectrum sensingalgorithms. Conclusions are presented in Section VI.
Notations:
The following notations are used throughoutthis paper. Upper/lower case boldface letters denote matrices/column vectors. The
X
(
n,n
)
,
X
(
n,
:)
,
X
T
and
X
H
denotethe
(
n,n
)
element,
n
th row, transpose and conjugate transpose of
X
, respectively.
I
n
(
I
)
is an identity matrix of size
n
×
n
(appropriate size) and,
(
.
)
⋆
,
E
{
.
}
,

.

and
(
.
)
∗
denoteoptimal, expectation, absolute value and conjugate operators,respectively.II. P
ROBLEM FORMULATION
Assume that the transmitted symbols
s
n
,
∀
n
are pulseshaped by a ﬁlter
g
(
t
)
. After the digital to analog converter,the baseband transmitted signal is given by
x
(
t
) =
∞
k
=
−∞
s
k
g
(
t
−
kT
s
)
(1)where
T
s
is the symbol period. We assume that
x
(
t
)
is narrowband signal
1
. In an AWGN channel, the baseband receivedsignal is expressed as
r
(
t
) =
∞−∞
f
∗
(
τ
)(
x
(
t
−
τ
) +
w
(
t
−
τ
))
dτ
=
∞−∞
f
∗
(
τ
)(
∞
k
=
−∞
s
k
g
(
t
−
kT
s
−
τ
) +
w
(
t
−
τ
))
dτ
=
∞
k
=
−∞
s
k
h
(
t
−
kT
s
) +
∞−∞
f
∗
(
τ
)
w
(
t
−
τ
)
dτ
where
f
∗
(
t
)
is the receiver ﬁlter,
w
(
t
)
is the additive whiteGaussian noise and
h
(
t
) =
∞−∞
f
∗
(
τ
)
g
(
t
−
τ
)
dτ
. The ob jective of spectrum sensing is to decide between
H
0
and
H
1
1
The extension of the proposed spectrum sensing algorithms for a wideband signal will be discussed later.
from
r
(
t
)
, where
r
(
t
) =
∞−∞
f
⋆
(
τ
)
w
(
t
−
τ
)
dτ, H
0
(2)
=
∞
k
=
−∞
s
k
h
(
t
−
kT
s
) +
∞−∞
f
⋆
(
τ
)
w
(
t
−
τ
)
dτ, H
1
.
Without loss of generality, we assume that
r
(
t
)
is a zero meansignal. Note that when
r
(
t
)
has a nonzero mean, its meancan be removed before examined by the proposed spectrumsensing algorithms.III. P
ROPOSED SPECTRUM SENSING ALGORITHMS
We deﬁne the
n
th discrete signal
{
˜
y
[
n
]
}
N n
=1
as follows:
˜
y
[
n
]
L
−
1
i
=0
α
i
r
((
n
−
1)
T
s
+
t
i
)
(3)
=
∞
k
=
−∞
s
kL
−
1
i
=0
α
i
h
((
n
−
1)
T
s
+
t
i
−
kT
s
)+
L
−
1
i
=0
α
i
∞−∞
f
⋆
(
τ
)
w
((
n
−
1)
T
s
+
t
i
−
τ
)
dτ
where
{
t
i
}
L
−
1
i
=0
are chosen such that
t
L
−
t
0
=
T
s
and
{
α
i
}
L
−
1
i
=0
are the introduced variables. By assuming that the signal andnoise (i.e.,
x
(
t
)
and
w
(
t
)
) are independent, the power of
˜
y
[
n
]
can be expressed as
E
{
˜
y
[
n
]

2
}
=E
{
∞
k
=
−∞
s
kL
−
1
i
=0
α
i
h
((
n
−
1)
T
s
+
t
i
−
kT
s
)

2
}
+E
{
L
−
1
i
=0
α
i
∞−∞
f
⋆
(
τ
)
w
((
n
−
1)
T
s
+
t
i
−
τ
)
dτ

2
}
=
σ
2
s
∞
k
=
−∞
α
H
A
nk
α
+
σ
2
w
α
H
B
n
α
=
σ
2
s
α
H
A
n
α
+
σ
2
w
α
H
B
n
α
(4)where
σ
2
s
and
σ
2
w
are the variances of the signal andnoise, respectively,
α
= [
α
0
,α
1
,
···
,α
L
−
1
]
T
,
A
nk
=
a
nk
a
H nk
,
A
n
=
∞
k
=
−∞
A
nk
and
B
n
=
1
σ
2
w
E
{
b
n
b
H n
}
with
a
nk
= [
h
((
n
−
1)
T
s
+
t
0
−
kT
s
)
,h
((
n
−
1)
T
s
+
t
1
−
kT
s
)
,
···
,h
((
n
−
1)
T
s
+
t
L
−
1
−
kT
s
)]
T
and
b
n
=[
∞−∞
f
⋆
(
τ
)
w
((
n
−
1)
T
s
+
t
0
−
τ
)
dτ,
∞−∞
f
⋆
(
τ
)
w
((
n
−
1)
T
s
+
t
1
−
τ
)
dτ,
···
,
∞−∞
f
⋆
(
τ
)
w
((
n
−
1)
T
s
+
t
L
−
1
−
τ
)
dτ
]
T
.The entries of
A
n
and
B
n
can further be expressed as
(
A
n
)
(
i
+1
,j
+1)
=
∞
k
=
−∞
h
((
n
−
1
−
k
)
T
s
+
t
i
)
h
⋆
((
n
−
1
−
k
)
T
s
+
t
j
) =
∞
k
′
=
−∞
h
(
k
′
T
s
+
t
i
)
h
⋆
(
k
′
T
s
+
t
j
)
A
(
i
+1
,j
+1)
and
(
B
n
)
(
i
+1
,j
+1)
=
∞−∞
f
⋆
(
τ
)
f
(
t
i
−
t
j
+
τ
)
dτ
B
(
i
+1
,j
+1)
. It follows
E
{
˜
y
[
n
]

2
}
=
σ
2
s
α
H
A
α
+
σ
2
w
α
H
B
α
.
(5)
For given
A
and
B
, the SNR minimization and maximizationproblems of
E
{
˜
y
[
n
]

2
}
can be expressed as
min
α
min
σ
2
s
α
H min
A
α
min
σ
2
w
α
H min
B
α
min
≡
min
α
min
α
H min
A
α
min
α
H min
B
α
min
(6)
≡
min
α
min
α
H min
(
A
+
B
)
α
min
α
H min
B
α
min
max
α
max
α
H max
(
A
+
B
)
α
max
α
H max
B
α
max
.
(7)These optimization problems are Rayleigh quotient problems.Since
A
and
B
are positive semideﬁnite matrices, the Generalized eigenvalue solution approach can be applied to get theoptimal solutions of these problems which is summarized asfollows [10], [11]:As
B
is a positive semideﬁnite matrix, applying eigenvaluedecomposition gives us
B
=
U
Σ 00 0
U
H
UDDU
H
(8)where
Σ
is a diagonal matrix containing nonzero eigenvaluesof
B
,
U
is a unitary matrix and
D
=
Σ
12
00 0
.
(9)The pseudoinverse of
B
is given by
B
†
=
U
Σ
−
1
00 0
U
H
=
U˜D˜DU
H
(10)where
˜D
=
Σ
−
12
00 0
.
(11)By employing (8)  (11), and deﬁning
˜
α
DU
H
α
min
for(6) and
˜˜
α
DU
H
α
max
for (7), we can rewrite the problems(6) and (7) as
min
˜
α
˜
α
H
˜A˜
α
˜
α
H
˜
α
(12)
max
˜˜
α
˜˜
α
H
˜A˜˜
α
˜˜
α
H
˜˜
α
(13)where
˜A
= (
U˜D
)
H
(
A
+
B
)(
U˜D
) = [
I0
;
00
]+
˜DU
H
AU˜D
.The optimal
˜
α
and
˜˜
α
of these problems are given by theeigenvectors corresponding to the minimum and maximumnonzero eigenvalues of
˜A
, respectively. Since
˜A
is alsoa positive semideﬁnite matrix, its minimum and maximumnonzero eigenvalues are always positive. The optimal solutionsof the srcinal problems (6) and (7) are thus given by
λ
α
⋆min
=
U˜D˜
α
⋆
and
τ
α
⋆max
=
U˜D
(
˜˜
α
)
⋆
.At optimality, the denominator terms of the above problemsare equal to unity (or any other positive value). Thus, under
H
0
hypothesis, the optimal values of (12) and (13) are thesame and equal to unity. However, under
H
1
hypothesis, theoptimal value of (13) is higher than that of (12)
2
. Due to this
2
Note that under
H
1
hypothesis, the optimal values of (12) and (13) areequal if and only if
A
=
ρ
B
, where
ρ
is any real number, which will neverhappen in a practical scenario.
fact, we propose the following test statistics:
T
=
N n
=1

˜
y
[
n
]

2
α
max
N n
=1

˜
y
[
n
]

2
α
min
N n
=1

z
[
n
]

2
N n
=1

e
[
n
]

2
M
a
2
z
M
a
2
e
(14)where
M
a
2
z
= 1
N
N
n
=1

z
[
n
]

2
,
M
a
2
e
= 1
N
N
n
=1

e
[
n
]

2
.
The authors of [8] propose oversampling along with prewhitening method to apply the EVDbased detection algorithmfor the single receiver antenna case. However, in practice, thereis always a nonzero (with very small power) adjacent channelinterference signal. And, as will be clear in the simulationsection, the algorithm of [8] can not ensure a predeﬁned
P
f
when there is an adjacent channel interference signal.However, as we can see from (14), the proposed test statisticscan guarantee a predeﬁned
P
f
(
P
d
)
when the adjacent channelinterference signal power is very small compared to that of the desired signal and noise powers (see also the simulationsection).For sufﬁciently large
N
(which is the case in a CR), byapplying central limit theorem, we can interpret
z
[
n
]
and
e
[
n
]
as ﬁltered and downsampled versions of
{
w
[
i
]
}
LN i
=1
, wherethe ﬁlters are
η
R
+
L
−
1
×
1
2(1 +
γ
max
)
diag(
Υ
,k
)
and
θ
R
+
L
−
1
×
1
2(1 +
γ
min
)
diag(
Ψ
,k
)
for
z
[
n
]
and
e
[
n
]
,respectively,
k
= [
−
(
L
−
1)
,
−
(
L
−
2)
,
···
,R
−
1]
and
γ
max
and
γ
min
denote the SNRs obtained by solving the problems(6) and (7), respectively, with
w
[
i
]
,
∀
i
are independent andidentically distributed (i.i.d) zero mean circularly symmetriccomplex Gaussian (ZMCSCG) random variables all with unitvariance
3
,
Υ
=
τ
T
⊗
f
,
Ψ
=
λ
T
⊗
f
,
⊗
denotes a kroneckerproduct,
f
= [
f
0
,f
1
,
···
,f
R
]
is the sampled version of thereceiver ﬁlter
f
(
t
)
with sampling period
T
s
L
,
R
is the ﬁlterlength and
diag(
X
,k
)
denotes the sum of the
k
th (
k
=0
,k >
0
and
k <
0
, denote the main diagonal, above the maindiagonal and below the main diagonal, respectively) diagonalelements of
X
.For better exposition, let us introduce a new variable
T
T
=lim
N
→∞
1
N
N n
=1

z
[
n
]

2
lim
N
→∞
1
N
N n
=1

e
[
n
]

2
M
a
2
z
M
a
2
e
.
(15)By deﬁning
σ
2
z
1+
γ
max
,
σ
2
e
1+
γ
min
and
γ
d
γ
max
−
γ
min
,
T
can be expressed as
T
=2
σ
2
z
2
σ
2
e
= 1
,
H
0
=2
σ
2
z
2
σ
2
e
= 1 +
γ
d
1 +
γ
min
,
H
1
.
(16)From this equation, one can notice that our problem turns toexamining whether
T
= 1
or
T >
1
for sufﬁciently large
N
. To get the
P
d
and
P
f
of the proposed test statistics, weexamine the following Theorem [12].
3
This is due to the fact that the noise power does not have any effect onthe test statistics under
H
0
hypothesis, and the effect of the signal power isincorporated by the ﬁlters
η
and
θ
under
H
1
hypothesis.
Theorem 1
: Given a real valued function
T
=
M
a
2
z
M
a
2
e
, theasymptotic distribution of
√
N
(
T
−
T
)
is given by
√
N
(
T
−
T
)
∼N
(0
,
˜
σ
2
)
(17)where
˜
σ
2
=
v
T
Φv
,
v
=
∂
T ∂
M
a
2
z
, ∂
T ∂
M
a
2
e
T
M
a
2
z
=
M
a
2
z
,
M
a
2
e
=
M
a
2
e
=
1
M
a
2
e
,
−
M
a
2
z
M
2
a
2
e
T
(18)and
Φ
is the asymptotic covariance matrix of a multivariaterandom variable
√
N
([
M
a
2
z
,
M
a
2
e
]
T
−
[
M
a
2
z
,M
a
2
e
]
T
)
∼ N
(
0
,
Φ
)
.
Proof:
See
Theorem
3. 3. A on page 122 of [12].Substituting
Φ
into (17) gives
˜
σ
2
=
M
2
a
2
e
Φ
(1
,
1)
−
2
M
a
2
e
M
a
2
z
Φ
(1
,
2)
+
M
2
a
2
z
Φ
(2
,
2)
M
4
a
2
e
.
(19)The coefﬁcients of
Φ
can be computed numerically (seeAppendix A).As
T
= 1
under
H
0
hypothesis, we modify the teststatistics
T
to
T
=
√
N
(
T
−
1)
.
(20)The
P
f
of this test statistics is expressed as
P
f
(
λ
) =
Pr
{
T > λ

H
0
}
.
(21)Under
H
0
hypothesis, as
T
∼N
(0
,
˜
σ
2
H
0
)
, the
P
f
is given by
P
f
=
∞
λ
1
2
π
˜
σ
2
H
0
exp
−
x
22˜
σ
2
H
0
dx
=
Q
λ
˜
σ
H
0
(22)where
Q
(
.
)
is the Qfunction which is deﬁned as [13]
Q
(
λ
) = 1
√
2
π
∞
λ
exp
−
x
22
dx
and
˜
σ
2
H
0
is
˜
σ
2
of (19) under
H
0
hypothesis.Mathematically,
P
d
(
λ
)
is expressed as
P
d
(
λ
) =
Pr
{
T > λ

H
1
}
(23)
=
∞
λ
1
2
π
˜
σ
2
H
1
exp
−
(
x
−
µ
)22˜
σ
2
H
1
dx
=
Q
λ
−
µ
˜
σ
H
1
where
µ
=
√
N
(
T
−
1) =
√
N
γ
d
1+
γ
min
and
˜
σ
2
H
1
is
˜
σ
2
of (19) under
H
1
hypothesis. From the above expression, we canunderstand that for given
γ
d
>
0
and
λ
, increasing
N
increases
P
d
. This is due to the fact that
Q
(
.
)
is a decreasing function.Thus, the proposed detection algorithm is consistent (i.e., forany given
P
f
>
0
and SNR, as
N
→∞
,P
d
→
1
).As can be seen from (6) and (7), for a given
g
(
t
)
, theachievable maximum and minimum SNRs depend on theselection of
f
(
t
)
,
L
and
{
t
i
}
L
−
1
i
=0
. For a given
g
(
t
)
, getting theoptimal
f
(
t
)
,
L
and
{
t
i
}
L
−
1
i
=0
ensuring the highest detectionperformance is an open research topic. In our simulation, we
00.5TsTs1.5Ts2Ts2.5Ts3Ts3.5Ts4Ts−0.2−0.100.10.20.30.40.50.60.70.80.911.11.2time (t)t
0
for SRRCF
t
0
t
0
t
0
t
0
. . .z[1],e[1]z[2],e[2]z[3],e[3]
Fig. 1. Description of
t
0
for SRRCF.
have observed better detection performance when we select
f
(
t
) =
g
(
t
)
(i.e., matched ﬁlter),
L
≥
8
and
{
t
i
=
T
s
(
12
+
iL
)
}
L
−
1
i
=0
. For example, if
f
(
t
)
is square root raised cosine ﬁlter(SRRCF), the initial timing (
t
0
) will be as in Fig. 1.From (23) we can also notice that
P
d
increases as
γ
d
1+
γ
min
increases. This is achieved when
γ
d
>
0
. To get
γ
d
=
γ
max
−
γ
min
>
0
, the ranks of
A
and
B
must be at least 2. When
f
(
t
)
is SRRCF with rolloff factors
0
.
2
,
0
.
25
and
0
.
35
, we haveobserved that
γ
d
1+
γ
min
increases as the rolloff factor increases,whereas, the ranks of
A
and
B
are the same (which is equalto 4) for these rolloff factors. From this explanation, we canunderstand that the quality of the proposed detector can notbe determined just from the ranks of
A
and
B
.From Fig. 1, one can realize that to get the
P
d
of (23),
t
0
must be known perfectly. The exact
t
0
is known whenthe receiver is synchronized perfectly with the transmitter.However, in general, since the transmitters and receivers areadministered by different operators, perfect synchronization isnot possible. Even in some scenario, the pulse shaping ﬁltermay not be known to the cognitive receiver. In the following,we generalize the aforementioned detector for known andunknown pulse shaping ﬁlters with asynchronous receiverscenarios.
A. Asynchronous receiver and known transmitter pulse shaping ﬁlter scenario
In this subsection, we generalize the aforementioned detection algorithm for known pulse shaping with asynchronousreceiver scenario. This algorithm is designed based on theestimation of
t
0
. As can be seen from (3), there are
L
possiblevalues of
t
0
. Thus, from the received signal
r
(
t
)
, we canobtain
L
possible values of
˜
T
(
T
)
. Under
H
0
hypothesis, allvalues of
{
T
i
}
Li
=1
are almost the same, whereas, under
H
1
hypothesis, the values of
{
T
i
}
Li
=1
are not the same. And,the
t
0
corresponding to
T
max
= max[
T
1
,T
2
,
···
,T
L
]
can beconsidered as the best estimate of the true
t
0
. Due to this
reason, we propose the following test statistics:
T
max
= max[
T
1
,T
2
,
···
,T
L
]
.
(24)The cumulative distribution function (CDF) of
T
max
is givenby [14]
F
T
max
(˜
λ
) =
˜
λ
−∞
˜
λ
−∞
···
˜
λ
−∞
1(2
π
)
L/
2

Σ

1
/
2
×
exp
−
12
(
ω
−
µ
)
T
Σ
−
1
(
ω
−
µ
)
dω
1
dω
2
···
dω
L
(25)where
µ
and
Σ
are the mean vector and covariancematrix of the random variables
[
T
1
,T
2
,
···
,T
L
]
, respectively and
ω
= [
ω
1
,ω
2
,
···
,ω
L
]
T
. By deﬁning
{
t
k
[
T
k
,T
k
+1
,
···
,T
L
,T
1
,
···
,T
k
−
1
]
∼
(
µ
k
,
Σ
k
)
}
Lk
=1
, it can beeasily seen that
{
µ
k
,
Σ
k
}
Lk
=1
are not necessarily the same if the mean and variances of
{
T
i
}
Li
=1
are not the same.When the receiver is not synchronized with the transmitter, we can obtain
L
possible values of
µ
and
Σ
(i.e.,
{
(
µ
k
,
Σ
k
)
}
Lk
=1
) all with equal probability of occurrence.Thus, the
P
f
and
P
d
of the test statistics
T
max
can beexpressed as
P
f
= 1
L
L
k
=1
Pr
{
T
max
>
˜
λ

(
H
0
,
t
k
)
}
= 1
L
L
k
=1
(1
−
F
T
max

(
H
0
,
t
k
)(˜
λ
))=1
−
F
T
max

(
H
0
,
t
1
)(˜
λ
)
P
d
= 1
L
L
k
=1
Pr
{
T
max
>
˜
λ

(
H
1
,
t
k
)
}
= 1
L
L
k
=1
(1
−
F
T
max

(
H
1
,
t
k
)(˜
λ
))=1
−
1
L
L
k
=1
F
T
max

(
H
1
,
t
k
)(˜
λ
)
(26)where
F
T
max

(
H
0
,
t
k
)(˜
λ
)
and
F
T
max

(
H
1
,
t
k
)(˜
λ
)
are (25)under
H
0
and
H
1
hypothesis with the statistics of
t
k
, respectively. The third equality follows from the fact that
{
t
k
∼
(
µ
,
Σ
)
}
Lk
=1
under
H
0
hypothesis. For a given
t
k
,
µ
k
can becomputed like
µ
of (23) and
Σ
k
can be computed numerically(see Appendix B).To the best of our knowledge, there is no any closed formsolution for the integral (25). Due to this reason, this papersolves (25) using ”mvncdf” matlab numerical package
4
. Thispackage computes
F
T
max
(˜
λ
)
for a given
˜
λ
.In practice, however, we are required to get
˜
λ
for a desired
P
f
(i.e., a constant false alarm rate detector). In this paper, weapply a bisection search method to get
˜
λ
satisfying the
P
f
of (26) [15]. To apply the bisection search, the lower and upperbounds of
˜
λ
are required which can be computed as follows:Since
T
max
is the supreme of all
{
T
i
}
Li
=1
, for ﬁxed
P
f
,one can notice that
˜
λ
≥
λ
= ˜
λ
min
, where
λ
is as given in
4
Note that this package is designed to compute the CDF of a multivariateGaussian random variable.
(22). On the other hand,
˜
λ
of (26) becomes maximum when
{
T
i
}
Li
=1
are independent. In such a case, the exact closed formexpression of
F
T
max
(˜
λ
)
is given as [14]
F
T
max
(˜
λ
) = (
F
T
(˜
λ
))
L
(27)where
F
T
(˜
λ
) = 1
−
Q
˜
λ
˜
σ
H
0
is the CDF of
T
(20). Thus,the maximum possible value of
˜
λ
is given by
P
f
= 1
−
F
T
max
(˜
λ
max
) = 1
−
(
F
T
1
(˜
λ
max
))
L
⇒
Q
λ
˜
σ
H
0
= 1
−
1
−
Q
˜
λ
max
˜
σ
H
0
L
⇒
˜
λ
max
= ˜
σ
H
0
Q
−
1
1
−
1
−
Q
λ
˜
σ
H
0
1
/L
(28)where the third equality is due to (22).The bisection search method for computing the exact
˜
λ
issummarized in
Algorithm I
.
Algorithm I
Initialization: Set
˜
λ
min
and
˜
λ
max
as in (22) and (28),respectively and
ε
= 10
−
3
.
Repeat
:Set
¯
λ
=
12
(˜
λ
min
+ ˜
λ
max
)
.1) Compute
¯
P
f
= 1
−
F
T
max
(¯
λ
)
by employing ”mvncdf”matlab package.2) If
¯
P
f
≤
P
f
, set
˜
λ
min
= ¯
λ
else set
˜
λ
max
= ¯
λ
Until

¯
P
f
−
P
f
≤
ε
.Set
˜
λ
= ¯
λ
as the threshold.
B. Asynchronous receiver and unknown transmitter pulseshaping ﬁlter scenario
In this subsection, the generalization of the proposedalgorithm for the detection of a signal with unknown pulseshaping ﬁlter is discussed. For this scenario, there are twoobvious questions: The ﬁrst question is how can we select thereceiver ﬁlter. The second question is how can we optimize
α
to achieve the best detection performance in asynchronousreceiver scenario
5
. To address these questions, let us examinethe detection of DVBS2 signals. According to [16], a DVBS2 signal employs a SRRCF with rolloff factor
0
.
2
,
0
.
25
or
0
.
35
.
Selection of the receiver ﬁlter (
B
)
: We design
B
based onthe smallest rolloff factor (i.e., a SRRCF with rolloff factor0.2). This is due to the fact that if we design
B
with a rolloff factor
>
0
.
2
, the examined band contains strong and unknownadjacent channel interference signal when the rolloff factor of the transmitter ﬁlter is smaller than that of the receiver ﬁlter.Consequently, a predeﬁned
P
f
cannot be ensured under
H
0
hypothesis.
Optimization of
α
: We optimize
α
by considering allpossible pulse shaping ﬁlters and taking into account theprobability of each pulse shaping ﬁlter. The optimal
α
can
5
As the transmitted signal pulse shaping ﬁlter is not known, perfectsynchronization between the transmitter and receiver can never be achieved.