Spectrum sensing is a key technical challenge for cognitive radio (CR). It is well known that multicycle cyclostationarity (MC) detector is a powerful method for spectrum sensing. However, conventional MC detector is difficult to implement due to its high computational complexity. This paper pays attention to the fact that the computation complexity can be reduced by simplifying the test statistic of conventional MC detector. Based on this simplification process, an improved MC detector is proposed. Compared with the conventional one, the proposed detector has the lowcomputational complexity and sufficientaccuracy on sensing performance. Subsequently, the sensing performances are further investigated for the cases of Rayleigh, Nakagami
m
, Rician, composite Rayleigh fadinglognormal shadowing and composite Nakagami fadinglognormal shadowing channels, respectively. Furthermore, the squarelaw combining (SLC) is introduced to improve the detection capability over fadingshadowing channels. The corresponding closedform expressions of average detection probability are derived for each case by the moment generation function (MGF) approach. Finally, illustrative and analytical results show that the efficiency and reliability of proposed detector and the improvement on sensing performance by SLC over composite fadingshadowing channels.
1. Introduction
R
adio spectrum, an expensive and limited resource, is surprisingly underutilized by licensed users (primary users, PU)
[1]
. Such spectral underutilization has motivated cognitive radio (CR) technology which has builtin radio environment awareness and spectrum intelligence
[2]
[3]
. Cognitive radio enables opportunistic access to unused licensed bands. For instance, CR users first sense the activities of primary users and access the spectrum holes if no primary activities are detected. While sensing accuracy is important for avoiding interference to the PU, reliable spectrum sensing is not always guaranteed, due to the multipath fading and shadowing
[4

7]
.
Different approaches for spectrum sensing for CR applications have been proposed, e.g.,
[8

11]
. The commonly considered approaches are based on power spectrum estimation, energy detection and multicycle cyclostationary (MC) detection. Power spectrum estimation may not work reliably in the low signaltonoise ratio (SNR) regime. Energy detection, on the other hand, is subject to uncertainty in noise and interference statistics. The MC detection, which are inherent properties of digital modulated signals, has been proposed in the literature to overcome the problems above
[11

13]
. In fact, it is well known that if a signal has strong cyclotationary properties, it can be detected at low SNRs. In addition, MC detector can inherently distinguish PUs from secondary users as well as interferers, if they have dissimilar cyclic features.
Although the conventional MC detectors operate much better than energy and power ones, it requires extensive computation to provide sufficiently low error probability, which causing high computational complexity
[12]
[13]
. The high computational complexity leads to a long detection time, which seriously degrades spectrum efficiency of the CRs, since all communications should be stopped during detection. Therefore, it's necessary to reduce the computational complexity of conventional MC detector. An improved MC detector with lowcomputational complexity and sufficientsensing accuracy is needed.
Furthermore, since the channel from PU to CR is multipath fading and shadowing in practical CR networks, the PU’s signal might be severely attenuated before it reaches to CR
[14]
. That is to say, the analysis of sensing performance in fading and shadowing environments could helps quantify the exact sensing performance of detector. However, the existing researches are mainly restricted to Rayleigh fading, which is not comprehensive enough
[15]
[16]
. To our best knowledge, none of the previous work has addressed the issue of MC detector over Nakagami
m
and Rician fading channels. Despite focus on the sensing performance over fading channels, all the previous researches omitted the effect of shadowing. In practical implementation, many detectors will be found either in stationary positions or with low mobility in environment with varying degree of vegetation covers
[17]
. These foliage and temporal stationary behavior introduce another degradation known as shadowing which cannot be mitigated by averaging the received signal strength
[18]
. Hence, if we wish to examine the exact sensing performance of MC detector, the effect of shadowing needs to be investigated.
To improve the sensing performance in the severe fading and shadowing conditions, multiple antennas for spectrum sensing can be deployed
[17]
[19]
. The diversity reception technology are then used to improve the detection reliability. The need for multiple antennas is also driven by the promise of a high data rate and high efficiency broadband services by standards such as the Long Term Evolution (LTE), WiMax and IMTAdvanced. The notion of using a multiple antenna CR for detecting the spectrum holes has thus attracted much interest.
In order to reduce the compexity of conventional MC detector, we proposed an improved MC detector, which is lowcomputational complexity and high accuracy. Subsequently, the sensing performance is further investigated by employing multiple antennas in fading and shadowing environments. The remainder of this paper is organized as follows. In Section 2, we present related work. The improved MC detector with low computational complexity and high accuracy on sensing performance is proposed in Section 3. In Section 4, the SLC diversity reception is empoyed to improve detection capability. Subsequently, the sensing performance is investigated over Rayleigh, Nakagami
m
, Rician fading, composite Rayleigh fadinglognormal shdowing and the composite Nakagami fadinglognormal shdowing channels, respectively. The closedform expressions of detection probability are then derived by MGF approach, correspondingly. Section 5 presents simulation results and the concluding remarks are given in Section 6.
2. Related Work
Many papers focus on reducing the computation complexity of conventional MC detector. In
[20]
, a cooperative MC detector was proposed in which the detector combines distinct singlecycle (SC) detectors for different cyclefrequencies (CFs) and the final decision is obtained by ORrule (a hard decision rule of cooperative sensing) the primitive decisions of the SC detectors. An adaptive cooperative cyclostationary beamformingbased spectrum sensing method with affordable complexity was introduced for multiple antenna CR in
[21]
. In
[22]
, a sequential framework was proposed to apply for collaborative MC detector in order to reduce the average detection time. In
[23

25]
, cooperative cyclostationary methods were proposed to improve the performance. The cooperative and collaborative MC detectors above were both introduced to reduce computation complexity. In the proposed cooperative and collaborative schemes, the computation of test statistics is done by a group of cooperative CRs with a distributed manner, so that the complexity burden on a single CR is reduced. Nevertheless, a major drawback of these schemes is the need for a very large number of cooperative CRs to make MC detection reliable, which is impractical in CR system sometimes. Therefore, it's still necessary to improve the conventional MC detector for single CR in local spectrum sensing scheme.
For multiple antenna spectrum sensing, the diversity reception technology of multiple antenna such as maximum ratio combiner (MRC) and squarelaw combining (SLC) are used to improve the detection reliability
[16]
[19]
. The detection performance of MRC has been analyzed in
[16]
. Study
[19]
deduces the exact detection performance analysis by utilizing SLC. When MRC is used, channel state information (CSI) from the primary user to CRs and from each CR to the fusion center is needed. When SLC is used with fixed amplification factor at each secondary user, only CSI from CRs to the fusion center is needed. Since MRC requires complete knowledge of the channel state information (CSI), a simpler technique is the SLC, which does not require the complete knowledge of CSI.
Moreover, previous performance analysis studies
[17

25]
primarily utilize the probability density function (PDF) based approach, i.e. the conditional detection probability is integrated over the PDF of the output signaltonoiseratio (SNR). In the processing of derivation of detection performance, the MGF approach in
[16]
provides a more flexible, general framework for analyzing the sensing performance than PDF approach. This new approach can avoid some difficulties of the PDF method by using a contour integral representation of the MarcumQ function
[26]
.
In summary, the major contributions of this paper include: 1) an improved MC detector with low computational complexity and high accuracy is proposed. In the proposed method, we present a reliable simplification for the test statistic of conventional MC detector. The benefits of the simplification are able to reduce the computational complexity which caused by compute the test statistic. From analysis, we find that the test statistic of proposed detector follows a chisquare distribution. The closedform expressions of detection probability and falsealarm probability are then derived; 2) since complete knowledge of CSI is hard to obtain in practical CR, SLC is introduced to improve detection capability and its contribution is demonstrated by comparing it with the case without SLC; 3) based on the SLC, the sensing performance of improved MC detector is investigated over Rayleigh, Nakagami
m
, Rician fading channels, composite Rayleigh fadinglognormal shdowing and the composite Nakagami fadinglognormal shdowing channels, respectively. The corresponding closedform average detection probability is derived by using the moment generation function (MGF) approach. The effect of fading and shadowing on sensing performance is then demonstrated by simulation results. These results help quantify and understand the achievable detection performance of the proposed detector.
3. Multicyclostationary (MC) Detector
In this section, we start by analyzing the conventional MC detector. Then, we propose an improved MC detector to reduce the computational complexity based on simplifying the test statistic of the conventional one. Subsequently, the closedform expressions of detection probability and false alarm probability of proposed detector are derived.
A typical signal detection problem is usually formulated as a binary hypothesis testing problem:
where
H
_{0}
denotes the absence of a primary user (PU) and
H
_{1}
denotes its presence,
r(t)
is the CR user’s received signal,
h
is the gain of channel between the PU and the CR user,
n(t)
is the additive white Gaussian noise (AWGN),
s(t)
is the PU’s transmitted signal. In spectrum sensing, CR user measures the sufficient statistics at first, and then compares it with a threshold which is determined with a desirable false alarm probability in order to decide between two hypotheses.
 3.1 Conventional MC Detector
The sufficient statistic of Maximum Likelihood (ML) detector in an AWGN channel is given by
[27, Ch. 4]
where
r(t)
is the received signal,
R_{s}
(
u,v
)=
E
[
s
(
u
)
s
^{*}
(
v
)] is the autocorrelation function of the PU’s transmitted signal
s(t)
and
T
is the observation interval. If the signal is cyclostationary, then the sufficient statistic of ML detector can be expressed as
where
S_{s}^{αk}
(
f
) and
S_{r}^{αk}
(
f
) are the Spectral Correlation Functions (SCFs) of
s(t)
and
r(t)
, respectively, at the
k
th cycle frequency (CF)
α_{k}
=
k/T_{c}, N_{α}
is the number of CFs,
S_{r}^{αk}
(
f
) is the Fourier transform of the cyclic autocorrelation function
R_{r}^{αk}
(
τ
), and the time independent function
R_{r}^{αk}
(
τ
) is calculated as the Fourierseries coefficient of the periodic autocorrelation function
R_{r}
(
t,τ
)=∑
_{k}R_{r}^{αk}
(
τ
)
e^{j2παkt}
,
α_{k}
=
k/T_{c}
, with period
T_{c}
≠0
[28]
[29]
. Since the PU’s transmitted signal is unknown, he
S_{s}^{αk}
(
f
) can be assumed as a rectangular function with Δ
f
bandwidth. Thus,
Consequently, the sum of cyclostationary signal powers at all CFs is the sufficient statistic for the optimum detector in ML criterion for cyclostationary signals. This detector is called the MC detector. Moreover, the total power can be calculated as
[25]
where
R_{r}^{αk}
(
τ
=0) represents the cyclostationary signal power at the
k
th CF.
 3.2 Improved MC Detector
 3.2.1 Simplified Test Statistic
Since
Y_{MC}
=Re(
Y_{MC}
)+
j
Im(
Y_{MC}
) is a complex random variable, the test statistic of conventional MC detector is given by
In order to reduce computational complexity, we will make a reliable approximation to
T_{MC}
. Since the second term of
T_{MC}
brings a large of computation, here is omitted. The first term of
T_{MC}
is used as the test statistic of the proposed detector. Thus, the simplified test statistic of proposed detector can be defined as
 3.2.2 Computational complexity analysis
In order to investigate the computational complexity of
T_{MC}
and
T_{sim}
, we make analysis as follows:

Computational complexity analysis of computingTMC: sincethe conventional detector needs to do complex multiplication forNα2times and complex addition forNα2–1 times to computeTMC, therefore, the total computation ofTMCis 2Nα2–1. Denoteas the computational complexity of conventional detector.

Computational complexity analysis of computingTsim: sincethe proposed detector needs to do complex multiplication forNαtimes and complex addition forNα–1 times to computeTsim, thus, the total computation ofTsimis 2Nα–1. Denoteas the computational complexity of proposed detector.
Since
N_{α}
≫1, it can be clearly seen that the computational complexity is greatly reduced by the simplification of test statistic, which means that proposed MC detector is of lower computational complexity than the conventional one.
Next, the sensing performance of proposed detector will be analyzed. By simulation results in Section IV, we will show that there is no appreciable difference in detection performance between proposed detector and conventional detector.
 3.2.3 Sensing Performance Analysis
Since
T_{sim}
is the test statistic of proposed detector, the structure of detector can be defined as
where
λ
is detection threshold.
Consequently, the falsealarm and detection probability of proposed detector can be represented as
In order to obtain the conditional probability density functions (pdfs)
P
(
T_{sim}

H
_{0}
) and
P
(
T_{sim}

H
_{1}
), we should focus on the
R_{r}^{αk}
(
τ
=0), in (7) first. Denote
The cyclic autocorrelation function of
Y
_{1}
can be shown as
From (11) and (12), the discretetime counterpart of
Y
_{1}
is given by
Hence, the
H_{0}
and
H_{1}
in an AWGN channel according to (1) can be represented as
where
n
[
w
]=Re(
n
[
w
])+
j
Im(
n
[
w
]) a complex AWGN sample with zero mean and variance is
σ
_{0}
^{2}
,
s
[
w
] is a PU signal sample. The proposed detector can be viewed as measures the signal power for single cycle frequency
α_{k}
, then the structure of detector can be defined as
Therefore,
The conditional probability density functions (pdf)
P
(
T
_{1}

H
_{0}
) and
P
(
T
_{1}

H
_{1}
) of
T
_{1}
can be derived from the conditional pdfs
P
(
Y
_{1}

H
_{0}
) and
P
(
Y
_{1}

H
_{1}
) of
Y
_{1}
, where (
T
_{1}

H
_{0}
), (
T
_{1}

H
_{1}
), (
Y
_{1}

H
_{0}
) and (
Y
_{1}

H
_{1}
) means that the
T
_{1}
and
Y
_{1}
are a random variables under hypothese
H
_{0}
and
H
_{1}
, respectively. Base on the central limit theorem for large
N_{s},N_{α}
≫1,
Y
_{1}
is approximately Gaussian.
Since var{
n
[
w
]
^{2}
}=var{Re
^{2}
(
n
[
w
])+Im
^{2}
(
n
[
w
])}=
σ
_{0}
^{4}
and E{
n
[
w
]
^{2}
}=
σ
_{0}
^{2}
, the expectation and variance of {
Y
_{1}

H
_{0}
}
Y
_{1}
under
H
_{0}
are
where
E
{·} and var{·} are the expectation and variance, respectively. Since {
Y
_{1}

H
_{0}
}=Re(
Y
_{1}

H
_{0}
)+
j
Im(
Y
_{1}

H
_{0}
) is a complex Gaussian random variable, then (
T
_{1}

H
_{0}
)={
Y
_{1}

H
_{0}
}
^{2}
=[Re(
Y
_{1}

H
_{0}
)]
^{2}
+[Im(
Y
_{1}

H
_{0}
)]
^{2}
follows a central chisquare distribution with two degrees of freedom, which probability density function can be obtained by
Thus, the false alarm probability is
Since
E
{
s
[
w
]+
n
[
w
]
^{2}
}=
s
[
w
]
^{2}
+
σ
_{0}
^{2}
and
the mean of (
Y
_{1}

H
_{1}
) is
where the complexvalued
P_{αk}
is the signal power at the
k
th of CF
α_{k}
. Since noise samples are statistically independent, the variance of (
Y
_{1}

H
_{1}
) is
where
. Therefore, the (
T
_{1}

H
_{1}
)={
Y
_{1}

H
_{1}
}
^{2}
+[Re(
Y
_{1}

H
_{1}
)]
^{2}
+[Im(
Y
_{1}

H
_{1}
)]
^{2}
follows a non central chisquare distribution with two degrees of freedom, its pdf is given by
where
and
σ
^{2}
_{2}
=2
σ
^{2}
_{0}
P
/
N_{s}
+
σ
_{0}
^{4}
/
N_{s}
. Then the detection probability can be given by
where
Q
_{1}
(·,·) is the first degree generalized MarcumQ function.
The derivation of above shows that the (
T
_{1}

H
_{0}
) follows a central chisquare distribution with two degrees of freedom and (
T
_{1}

H
_{1}
) follows a noncentral chisquare distribution with two degrees of freedom. Since
it can be easily conclude that (
T_{sim}

H
_{0}
) follows a central chisquare distribution with 2
N_{α}
degrees of freedom and (
T_{sim}

H
_{1}
) follows a non central chisquare distribution with 2
N_{α}
degrees of freedom. Then, the conditional pdfs of
T_{sim}
can be represented as
where
is the
v
thorder modified Bessel function of the first kind and
is the gamma function. The corresponding false alarm and detection probabilities are
where
Q_{Nα}
(·,·) is the generalized MarcumQ function with
N_{α}
degrees. Since MC detector is in AWGN channel (i.e.
h_{i}
= 1),
is signal power and
σ
_{0}
^{2}
is noise power,
r
=
u
_{2}
/
σ
_{0}
^{2}
denotes SNR. Then, (29) can be rewritten as
4. Sensing Performance of Proposed Detector with SLC diversity over Composite Multipath FadingLognormal Shadowing Channels
As shown in
Fig. 1
, we consider a CR with
L
antennas and assume that the PU is of a single antenna. Here, Squarelaw combining (SLC) is employed to combine the outputs of
L
antennas. We also assume a narrowband wireless system
[30]
.
L antennas based spectrum sensing.
In this section, we first derive the detection probability and false alarm probability of the proposed MC detector with SLC over AWGN channel, then, the multipath fading channels such as Rayleigh, Nakagami
m
and Rician are considered. Finally, the sensing performance over composite Rayleigh fadinglognormal shadowing and composite Nakagami fadinglognormal shadowing channels are investigated.
 4.1 Squarelaw Combining
In SLC scheme, the outputs of the squarelaw devices (squareandintegrate operation per antenna) are added to yield a new test statistic
T_{sim}
_{_∑}
. Therefore, the new test statistic
T_{sim}
_{_∑}
can be written by
where
T_{sim}
_{_i}
denotes the test statistic from the
i
th squarelaw device.
If SLC is employed to improve the sensing performance of proposed detector over AGWN channel, the test statistic
T_{sim}
_{_∑}
follows a centrality chisquare distribution with
LN_{α}
degrees of freedom under
H
_{0}
. Therefore, the false alarm probability becomes
Similarly, under
H
_{1}
,
T_{sim}
_{_∑}
has a noncentral chisquare distribution with
LN_{α}
degree of freedom, and noncentrality parameter
Thus, the detection probability can be obtained as
 4.2 Average Detection Probability over Composite Fadingshadowing Channels
In composite fadingshadowing fading channels,
P_{f}
of (32) will remain the same since
P_{f}
is considered for the case of no signal transmission and as such is independent of SNR. Therefore, we will derive the average detection probability
by using MGF over Rayleigh, Nakagami
m
, Rician, composite Rayleigh fadinglognormal shadowing and composite Nakagami fadinglognormal shadowing channels, respectively.
 4.2.1 Average Detection Probability  MGF Approach
The generalized MarcumQ function in (33) can be written as a circular contour integral within the contour radius
r
∈[0,1). Then, the expression (33) can be represented as
where Δ is a circular contour of radius
r
∈[0,1).
The moment generating function (MGF) of average SNR
is
Thus, the average detection probability,
is given by
where
In our study, the expression in (35) is fairly general and holds for any case where the MGF is available in suitable form.
 4.2.2 Average Detection Probability over Rayleigh Fading Channels
The MGF of Rayleigh fading combined SLC is
After substituting this MGF in (35), the average detection probability,
over Rayleigh channel can be written in the form of expression (35) with
where
and
β
_{1}
=
L
(
N_{α}
–1).
In radius
r
∈[0,1), there are
β
_{1}
poles at the origin
z
=0 and
L
pole at
z
=
θ
_{1}
. By applying the residue theorem to (35), the detection probability over Rayleigh fading is obtained as follows
where
and
D^{n}
(
f
(
z
)) denotes the
n
th derivative of
f
(
z
) with respect to
z
.
 4.2.3 Average Detection Probability over Nakagamim Fading Channels
The MGF of Nakagami
m
fading combined SLC is
Using (35), detection probability over Nakagami
m
channel,
can be written in the form of expression (35) with
where
and
β
_{2}
=
L
(
N_{α}
–
m
).
Following the similarly procedure as in Rayleigh fading case, the detection probability over Nakagami
m
channel is given by
where
and
From analysis above, we notice that the results are limited to an integer of
Lm
and allow us to compute
for certain noninteger values of
m
. For example,
over 2 branches can be computed for 1/2 multiples of
m
values. We also notice that Rayleigh fading channel is a special case of Nakagami
m
channel, which
can be obtained by substitute
m
=1 in (38).
 4.2.4 Average Detection Probability over Racian Fading Channels
The MGF of Rician fading combined SLC is given by
where
K
is the Rice factor. Notice that for the special case of
K
= 0 (Rayleigh fading), (40) reduces to Rayleigh in (36). Hence, using (40), the average detection probability over Rician fading can be obtained as
where
and
θ
_{3}
=
µ
_{2}
/(
µ
_{2}
+
K
+1).
Applying Laurent series expansion for
in (41) when
K
≠0 and using the residue theorem to integrate term by term, the average detection probability over Rician fading can be derived as
where
and
.
 4.2.5 Average Detection Probability over Composite Rayleigh Fadinglognormal Shadowing Channels
Since shadowing process is typically modeled as a lognormal distribution, the composite Rayleigh fadinglognormal shadowing channels model follows a mixture Rayleigh fadinglognormal distribution as
[17]
where
N
is the number of terms in the mixture,
η_{i}
and
ρ_{i}
are abscissas and weight factors for the GaussianLaguerre integration.
η_{i}
and
ρ_{i}
for different
N
values can be calculated by a simple MATLAB program.
ψ
and
δ
are the mean and the standard deviation of the lognormal distribution, respectively. Therefore, the MGF of composite Rayleigh fadinglognormal shadowing combined SLC can be expressed as
Thus, the average detection probability over composite Rayleigh fadinglognormal shadowing channels,
can be evaluated in closedform as
where
and
Following a similar procedure as in Rayleigh fading case, the residues Re
s
(
f
_{1}
_{i}
;0,
β
_{1}
) and Re
s
(
f
_{1}
_{i}
;
θ
_{1}
_{i}
,
L
) can be obtained.
 4.2.6 Average Detection Probability over Composite Nakagami Fadinglognormal Shadowing Channels
For a composite Nakagami fadinglognormal shadowing channel, the composite Nakagamilognormal distribution is a mixture of Nakagami fading and lognormal shadowing. Then, the SNR distribution
f_{nl}
(
x
) can be expressed as
[31]
where
and
f_{nl}
(
x
) is equal to a
mixture of gamma distributions
[32]
. Since the SNR’s possibility density function in this case follows gammalognormal distribution, we can accurately approximate the gammalogmormal distribution through a mixtures of gamma distribution. Then, the MGF of composite Nakagami fadinglognormal shadowing combined SLC can be expressed as
Thus, the average detection probability over composite Nakagami fadinglognormal shadowing channels,
can be evaluated in closedform as
where
and
β
_{2}
=
L
(
N_{α}
–
m
). Following a similar procedure as in Nakagami fading case, the residues Re
s
(
f
_{2}
_{i}
;0,
β
_{2}
)and Re
s
(
f
_{2}
_{i}
;
θ
_{2}
_{i}
,
Lm
) can be obtained, we omit the expressions here for brevity. We also notice that composite Rayleigh fadinglognormal shadowing channel is a special case of composite Nakagami fadinglognormal shadowing channel, which
can be obtained by substitute
m
=1 in (48).
5. Numerical Results and Analysis
In this section, we provide analytical and simulation results to verify the analytical framework. In order to show the sensing performance of the proposed detector, we plot receiver operating characteristic (ROC) curves (plots of detection probability
P_{d}
versus false alarm probability
P_{f}
) , complementary ROC curves (plots of miss detection probability
P_{m}
versus false alarm probability
P_{f}
),
P_{d}
vs. average SNR curves and
P_{d}
vs.
L
curves. Note that each of the following figures contains both analytical result and simulation result, which are represented by lines and discrete marks, respectively.
To verify the reliability and efficiency of proposed detector, we make analysis as follows: 1) reliability analysis:
Fig. 2
shows the ROC curves of proposed MC detector and conventional MC detector over AWGN channel, for different SNR
r
={5,10,15}
dB
and
N_{α}
=10. It can be observed that the analytical results match well with their simulation analysis, confirming the accuracy of the analysis. It can be seen that there is a slight difference on sensing performance between the proposed and conventional detector. Although this simplification slightly degrades sensing accuracy, it still maintains a satisfactory sensing capability; 2) computational complexity analysis: based on the analysis in Section 3.2, the computational complexity of conventional MC detector is
while the proposed detector is
(10) for
N_{α}
=10. The analysis above clearly illustrated that the proposed detector is more efficient than the conventional one. In all,
Fig.2
and the computational complexity analysis verify that the proposed MC detector can reduce the computational complexity while still maintains enough detection sensitivity.
ROC curves for the proposed and conventional MC detector with different SNRs (r={5,10,15dB}, N_{α}=10).
For further investigated the sensing performance of proposed detector, we plot the ROC curves with different numbers of
N_{α}
, for
N_{α}
=2,4,6,8,10. In
Fig. 3
, with the increase value of
N_{α}
, the sensing performance improves, which is similar to conventional detector. Since MC detector measures the sum of
N_{α}
spectral correlation function, the more
N_{α}
is, the more accurate binary decision the detector makes.
ROC curves for the proposed detector with different N_{α} (N_{α}={2,4,6,8,10}, r=10dB).
Fig. 4
depicts the complementary ROC curves of the proposed detector over Rayleigh, Nakagami
m
and Rician fading channels. Rayleigh and Rician K=0 curves coincide with the Nakagami
m
=1 curve and therefore not shown. The fading index quantifies the detection performance for several Nakagami parameter
m
and Rician factor K values. As expected, higher values of the fading index
m
and K imply a relatively less degraded received signal and thus lead to a lower miss detection probability
P_{m}
.
Complementary ROC curves over Rayleigh, Nakagamim and Rician fading channels ( average SNR , L = 2, N_{α} = 10).
Fig. 5
shows the complementary ROC curves of the proposed detector over Rician fading channel with SLC, for K=4 and
. The number of diversity branches varies for one to four. It can be observed that the sensing capability of 1branch (
L
=1, nonSLC) is the lowest bound of the sensing ability of the proposed detector. With increasing in the number of SLC diversity branch, there is much improvement on sensing capability of the proposed detector.
Complementary ROC curves with SLC diversity over Rician fading channel ( average SNR , L={1,2,3,4}, K=4, N_{α}=10).
Fig. 6
depicts the ROC curves of proposed detector over composite Rayleigh fading lognormal shadowing channel without SLC diversity. The shadowing effects considered here were introduced by Loo in
[11]
and are (i) light shadowing (
ψ
=0.115 and
δ
=0.115), which corresponds to sparse tree cover and (ii) heavy shadowing (
ψ
=3.914 and
δ
= 0.806), which corresponds to dense tree cover. We take
N
= 10 in (44), which makes the mean square error (MSE) between the exact gammalognormal channel model and the approximated mixture gamma channel model in (44) less than 10
^{4}
. The numerical results match well with their theoretical analysis, confirming the accuracy of the analysis. It can be seen that the performance of proposed detector degrades with increase in composite Rayleigh fading lognormal shadowing environments, and improves at higher SNR. Further, for Rayleigh and composite Rayleigh fading
light
shadowing environments, there is a slight difference in detector’s performance. However, there is a significant performance degradation of the proposed detector due to the
heavy
shadowing effect in lower average SNR (e.g. 5dB).
ROC curves of proposed detector without SLC over Rayleigh and composite Rayleigh fadinglognormal shadowing channels (average SNR N_{α}=10).
Next, we consider the effect of SLC diversity in alleviating the composite Rayleigh fadinglognormal shadowing.
Fig. 7
demonstrated that SLC diversity improves the detection performance, even in the serious composite Rayleigh fadinglognormal shadowing environments. For example, in composite Rayleigh
heavy
shadowing environment with the low average SNR (e.g. 5dB), we find that the detection probability for 4branches SLC case (
L
=4) is almost six times than that for 1branch nonSLC case (
L
=1). Furthermore, for 2branches SLC case (
L
=2), there is approximately 3 dB performance gain compared with nonSLC case. Therefore, the SLC diversity mitigates the impact of fadingshadowing and introduces a significant improvement in the detection probability.
P_{d} vs. average SNR with SLC diversity over Rayleigh and composite Rayleigh fadinglognormal shadowing channels (P_{f}=0.01, L={1,2,4}, N_{α}=10).
Fig. 8
generate the
Pd
vs.
L
curves for Nakagami and composite Nakagami fadinglognormal shadowing conditions with different Nakagami fading parameter
m
, for
m
=2,4,6. The shadowing effects considered here are (i) light shadowing (
ψ
=0.115 and
δ
=0.115) and (ii) heavy shadowing (
ψ
=3.914 and
δ
= 0.806). We also take
N
= 10 in (47). As we can see, there is an improvement in the performance with increasing in
m
. A higher value of
m
means better conditions of the sensing channels between the CR and PU. Furthermore, there is a great improvement in detection performance by employing SLC diversity, even in heavy shadowing environments. Clearly the greater is the number of SLC diversity branch, the better the detector works. That is to say, the SLC diversity mitigate the impact of composite Nakagami fadingshadowing, and introduce a significant improvement on sensing performance.
P_{d} vs. L over Nakagami and composite Nakagami fadinglognormal shadowing channels (P_{f}=0.01, m={2,4,6}, N_{α}=10).
6. Conclusion
In this paper, the sensing performance of improved MC detector based SLC diversity in fading and shadowing environments has been studied. We first proposed an improved MC detector, which is computational efficiency and sufficient accuracy on sensing performance. Subsequently, the sensing performance of proposed detector is investigated in fading and shadowing environments. For Rayleigh, Nakagami
m
, Rician, composite Rayleigh fadinglognormal shadowing and composite Nakagami fadinglognormal shadowing channels, the closedform expressions of detection probability by employing SLC are derived by using MGF approach, respectively. Illustrative and analytical results show that although the composite fading and shadowing degrade the sensing performance of proposed detector, the gain is significantly improved by SLC. Although we inverstigate the sensing performance over composite Rayleigh fadinglognormal shadowing channel and composite Nakagami fadinglognormal shadowing channel in this paper, the same analytical framework can be extended to composite Rician fadinglognormal shadowing channel in future work. These results help quantify the performance gains for MC detection with deversity reception over composite fading and shadowing channels, which can help emerging applications such as cognitive radio and ultra wideband radio.
BIO
Ying Zhu, is currently a Ph.D. candidate at Beijing University of Posts and Telecommunications (BUPT), Beijing, China. Her research interests include spectrum sensing, resource sharing and resource management in cognitive radio networks.
Jia Liu, received the M.S. degree in electronics and information engineering from Guilin University of Electronic Technology, Guilin, China, in 2008. He is currently working toward the Ph.D. degree in Communication and Information System at the School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing, China. His research interests include cognitive radio, relay, wireless networkcoding, et al.
Zhiyong Feng, received her M.S. and Ph.D. degrees from BUPT, China. She is a professor at BUPT, and is currently leading the Ubiquitous Network Lab in the Wireless Technology Innovation (WTI) Institute. She is a member of IEEE and active in standards development such as ITUR WP5A/WP5D, IEEE 1900, ETSI and CCSA. Her main research interests include the cognitive wireless networks, convergence of heterogeneous wireless networks, spectrum sensing, dynamic spectrum management, crosslayer design,.
Ping Zhang, is a professor at BUPT. Prof. Zhang is the director of Key Lab. of Universal Wireless Communication (BUPT) of Ministry of Education and currently leading the Wireless Technology Innovation (WTI) Institute. He is a member of IEEE and vice director of SinoGermany Joint Software Institute. He is one of three draftsmen of National Key Program, Member of Experts and Consultants Committee of NSFC. His major research interests include cognitive radio network, wireless communication theory and signal processing.
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