# A Cluster-Based Cooperative Spectrum Sensing in Cognitive Radio Network Using Eigenvalue Detection Technique with Superposition Approach

Authors: Md. Sipon Miah; Heejung Yu; Tapan Kumar Godder; Md. Mahbubur Rahman
Journal: International Journal of Distributed Sensor Networks
 (2015)

---

## Abstract

Cognitive radio (CR) networks have been active area of research because of its ability to opportunistically share the spectrum. A cluster-based cooperative spectrum sensing (CCSS) has a tremendous impact on sensing reliability compared with cooperative spectrum sensing. The energy detection (ED) technique requires perfect knowledge of noise power. An eigenvalue-based spectrum sensing has mitigated the noise uncertainty problem. Sensing and reporting time slots are rigidly separated in the conventional ED and eigenvalue-based detection (EVD) schemes. In CCSS, more reporting time slots are required as the number of CR users (CRUs) increases. If the reporting time slots of other CRUs as sensing time slots with a superposition allocation, the more reliable channel sensing can be achieved. In this paper, we propose CCSS using EVD technique with a superposition approach scheme where the reporting time slot is properly utilized to sense the primary user’s (PU’s) signal more accurately by rescheduling the reporting time slot for CRUs and cluster heads (CHs). Simulation result shows that the proposed EVD scheme has better detection probability than the conventional CCSS using both ED and EVD techniques.


---

## Body

## 1. Introduction

CR [1, 2] as a revolutionary intelligent technology can maximize the utilization of the frequency resources by allowing CRUs to access the spectrum bands allocated to PU when they are idle temporally. This technology can enable CRU to opportunistically access an unused portion of the licensed bands. CR technologies can be categorized into the following two approaches [3, 4]: (i) theoverlay approach, where the CRUs identify an unused portion of the licensed bands through the spectrum sensing process and opportunistically access the available spectrum for data transmission; (ii) theunderlay approach, in which the communication of the CRUs with very low transmit power coexists with that of the PUs in the same channel, and the CRUs avoid interfacing with the communication of the PUs.An ED scheme can be used by single CRU for finding idle spectrum. However, it cannot deal with the hidden terminal problem [5] which arises due to multipath fading and shadow effects and results in performance degradation. Recently, cooperative spectrum sensing schemes were proposed to overcome the hidden terminal problem in single node sensing [6–8]. To implement the conventional cooperative spectrum sensing methods, each CRU makes a local decision and its decision is reported to a fusion center (FC) to make a final decision according to some fusion rules. These fusion rules can be classified as hard decision fusion (HDF) [7, 8] or soft decision fusion (SDF) [9, 10]. In HDF, a local CRU makes its own decision on the presence of PU and its corresponding resultant 1-bit decision is forwarded to the FC for fusion. Therefore, the traffic overhead is reduced because only single bit is reported to the FC from each CRU. On the other hand, in the SDF scheme, each CRU reports their measurements as raw data to the FC and these types of data will be fused to construct a final decision on the existence of the PU’s signal. The SDF scheme shows better detection performance than the HDF scheme. However, more reporting data are required for each CRU to report its local decision to FC as the number of cooperative CRU’s increases.An alternative way to improve the sensing performance is to group the CRUs into clusters and send their 1-bit hard decisions to CHs which will then forward their judgement to the FC [11–13]. These methods based on clustering work well even when reporting channels experience Rayleigh fading and shadowing effects.There are several spectrum sensing techniques, including ED, matched filter detection, and cyclostationary detection [14, 15]. The ED has lower complexity and is widely used as a signal detection method in practical systems. However, ED relies on knowledge of noise power, and the inaccurate estimation of the noise power will lead to high probability of false alarm as well as misdetection. Thus, ED is vulnerable to the noise uncertainty. Moreover, ED is not optimal for real environment due to the fact that the received signals from PU for different time are correlated. Matched filter is known as the optimum method for detection of the primary users when the transmitted signal is known to receivers. The main advantage of the matched filtering is that it takes short time to achieve the spectrum sensing performance under a certain value of the probabilities of false alarm and misdetection, compared to the other methods. However, it requires the perfect knowledge of the PU’s signaling features such as bandwidth, operating frequency, the modulation type and order, pulse shaping, and the packet format. Cyclostationary detection has good performance but requires partial knowledge of the PU’s signal such as properties of cyclic frequencies.Recently, many papers have investigated the application of eigenvalue-based cooperative spectrum sensing which outperforms the ED techniques [16–19]. In eigenvalue-based cooperative spectrum sensing, covariance matrix of the received signal was used for spectrum detection. In [20, 21], Ratnarajah et al. performed asymptotic analysis of exact decision thresholds for maximum eigenvalue detector (MED) and maximum-minimum eigenvalue (MME) detector. They also pointed out that MED has better spectrum detection probability.Both conventional ED and EVD schemes cannot achieve reliable sensing performance with a longer sensing time slot. In this paper, we propose a CCSS in CRN using EVD with a superposition approach scheme. By rescheduling the reporting time slots, a longer sensing time can be obtained and better sensing performance can also be achieved. We investigate how to achieve better detection performance within an extended sensing time slot when a superposition approach [22] is applied to the conventional EVD scheme [19]. In the proposed EVD scheme, each CRU will utilize other CRUs reporting time and CHs reporting time by rescheduling them. In this scheme, we can easily achieve the reliable sensing performance because CRU reporting time and CH reporting time merge to a sensing duration to detect the PU’s signal more accurately without decreasing system throughput.The remainder of the paper is structured as follows. In Section2, we introduce the cognitive radio network system model: firstly, the characterization of primary user-cognitive radio user (PU-CRU) link and, secondly, the characterization of cognitive radio user-cluster header (CRU-CH) link. In this section, we explain the conventional ED scheme, the conventional EVD scheme, and the proposed EVD scheme. Finally, the characterization of cluster header-fusion center (CH-FC) link is explained. In Section 3, simulation results are shown, and conclusion of the paper is drawn in Section 4. Additionally, parameters used in this paper are summarized as follows.Parameters and Their Definition Are as Follows:K: the number of clusters.
M: the number of CRUs in each cluster.
N
x: total number of signal samples for sensing decision.
N
x
p
r
o
p
(
i
,
k
): total number of signal samples for sensing decision of the ith CRU in the kth cluster in the proposed approach.
T
s: duration of sensing time slot.
T
r: duration of reporting time slot.
l
(
λ
): sensing threshold for EVD (ED).
C
o
v
z: covariance matrix for the received signal vector.
C
o
v
¯
z
N: sample covariance matrix for the N received signal samples.
ζ
m
a
x
(
ζ
m
i
n
): maximum (minimum) eigenvalue of the sample covariance.
R
m
t
m
p
r
o
p
(
R
m
t
m
c
o
n
): ratio between maximum and minimum eigenvalues in the proposed (conventional) approach.
P
d
,
k, P
f
,
k, and P
m
,
k: detection, false alarm, and misdetection probabilities for kth cluster’s decision in the conventional ED scheme.
Q
d, Q
f, and Q
m: detection, false alarm, and misdetection probabilities for global decision in the conventional ED scheme.


## 2. Cognitive Radio Network System Model

In CRN, the detection performance of PU signal might be degraded when the sensing decisions are forwarded to FC through fading channels. Figure1 shows the CRN deployment where M CRUs are grouped into a cluster governed by a CH based on low energy adaptive clustering hierarchy-centralized (LEACH-C) protocol [23] and the K CHs of the K clusters report their decisions to FC through a common control channel. Here, HDF will be applied to obtain a final decision on the presence of the PU activities. The process of LEACH-C protocol is made up of rounds, and each round consists of two phases: a setup phase when the CHs and clusters are organized and a steady state phase when the cluster members begin to send their measurements to CH and CHs send their decision to the FC. In setup phase, each CRU sends information about its current location and SNR of reporting channel to the FC. Based on this information, the FC determines CHs among all CRUs, while the remaining CRUs will act as cluster members. After the CHs are determined, the FC broadcasts a message that contains not only the CH ID for each CRU but also the information of time synchronization. If CRU’s CH ID matches its own ID, the CRU is a CH; otherwise, the CRU is a cluster member and goes to sleep. In steady state phase, the CRUs start to forward the measurement of the received PU’s signal to the CH, and then the CH collects the measurements from the cluster members and makes the cluster decision about the presence of the PU and sends it to the FC during their allocated reporting time slots. Afterword, the FC combines the received clustering decision to make the final decision and then broadcasts it back to all CHs and the CHs send it to their cluster members. Figure 1 shows the general deployment of the CRN with three main sequential links: the primary user-cognitive radio user (PU-CRU) link, the cognitive radio user-cluster header (CRU-CH) link, and, finally, the cluster header-fusion center (CH-FC) link. For the simplicity of illustration, we assume that each cluster contains the same number of CRUs.Figure 1
A cluster-based cooperative spectrum sensing in CRN.


### 2.1. Characterization of Primary User-Cognitive Radio User (PU-CRU) Link

In the PU-CRU link, the CRUs perform local spectrum sensing independently to detect the PU’s activities. For CRUi  i
=
1
,
2
,
…
,
M, the spectrum sensing is given by a binary hypothesis test that is formulated as follows [24–26]:(1)
When  PU  is  absent
⟹
H
0
:
y
i
t
=
η
i
t
,
When  PU  is  present
⟹
H
1
:
y
i
t
=
h
i
x
t
+
η
i
t
,where t
=
1,2
,
…
,
N
x, is the sample index, N
x is the total number of samples of the received signal which is defined as N
x
=
2
T
s
B where T
s is the sensing time, and B is a prior known bandwidth. Here, y
i
[
t
] is the received signal at the ith CRU receiver, x
[
t
] is the PU’s transmitted signal which is assumed to be a Gaussian random process with zero mean and variance σ
x
2, that is, x
[
t
]
~
N
0
,
σ
x
2, and η
i
[
t
] is noise of the ith sensing channel which is assumed to be additive white Gaussian with zero mean and variance σ
η
,
i
2, that is, η
i
[
t
]
~
N
0
,
σ
η
,
i
2. The channel gain of the PU-CRU, h
i, is assumed to be constant over each sensing period.


### 2.2. Characterization of Cognitive Radio User-Cluster Header (CRU-CH) Link

In the CRU-CH link, theM CRUs relay their individual measurements of the PU’s signal, y
i
[
t
], to the kth corresponding CH through a dedicated control channel in a sequential manner. Each CRU will simply act in an amplify-and-forward (AF) manner. The channel noise ϕ
i of the CRU-CH link is considered to be zero mean and an additive white Gaussian noise with variances ϑ
i
2. Then, the signal received by the corresponding CH from the ith CRU will be (2)
z
i
t
=
P
T
i
g
i
y
i
t
+
ϕ
i
t
,where P
T
i is the transmit power of the ith CRU relay and g
i is the amplitude of channel gain of the ith CRU-CH link. Using the two hypotheses in (1), the received signal at the kth CH can be written as (3)
z
i
t
∣
H
0
=
P
T
i
g
i
η
i
t
+
ϕ
i
t
=
δ
0
,
i
t
,
(4)
z
i
t
∣
H
1
=
P
T
i
g
i
h
i
x
t
+
P
T
i
g
i
η
i
t
+
ϕ
i
t
=
P
T
i
g
i
h
i
x
t
+
δ
0
,
i
twhose statistical properties for hypothesis H
0 and H
1 are given by(5)
z
i
t
∣
H
0
~
ℵ
0
,
σ
0
,
i
2
~
ℵ
0
,
P
T
i
g
i
2
σ
η
,
i
2
+
ϑ
i
2
,
z
i
t
∣
H
1
~
ℵ
0
,
σ
1
,
i
2
~
ℵ
0
,
P
T
i
h
i
2
g
i
2
σ
x
2
+
σ
0
,
i
2
,where ℵ
0
,
σ
0
,
i
2 denotes the distribution of hypothesis. The details about this distribution are not needed because they are approximated by using the central limit theorem (CLT) in the next subsection.


#### 2.2.1. Conventional Energy Detection (ED) Scheme

At the corresponding CH withM CRUs, each received sequence z
i
[
t
] will be individually averaged and squared using a separate energy detector to estimate its own energy [26] as(6)
Ζ
i
=
1
N
x
∑
t
=
0
N
x
-
1
z
i
t
2
.For a large number of samples, CLT approximates the decision statistic for hypothesis H
0 as a Gaussian distribution with mean and variance given by (7)
E
Z
i
∣
H
0
=
N
x
σ
0
,
i
2
=
N
x
P
T
i
g
i
2
σ
η
,
i
2
+
ϑ
i
2
,
V
a
r
Z
i
∣
H
0
=
2
N
x
σ
0
,
i
4
=
2
N
x
P
T
i
g
i
2
σ
η
,
i
2
+
ϑ
i
2
2
.Similarly, the mean and variance for hypothesis H
1 are given by(8)
E
Z
i
∣
H
1
=
N
x
σ
1
,
i
2
=
N
x
P
T
i
h
i
2
g
i
2
σ
x
2
+
σ
0
,
i
2
,
V
a
r
Z
i
∣
H
1
=
2
N
x
σ
1
,
i
4
=
2
N
x
P
T
i
h
i
2
g
i
2
σ
x
2
+
σ
i
2
2
.Now, all the individual test statistics Z
1
,
Z
2
,
…
,
Z
M are used to linearly formulate the resultant test statistic of the kth cluster, C
k, which can be expressed as (9)
C
k
∣
H
0
=
∑
i
=
1
M
ω
i
Z
i
∣
H
0
,
C
k
∣
H
1
=
∑
i
=
1
M
ω
i
Z
i
∣
H
1
.Simply, (9) can be rewritten as (10)
C
k
=
∑
i
=
1
M
ω
i
Z
i
,where k
=
1
,
2
,
…
,
K and ω
i denotes the weighting coefficient for ith CRU in the cluster.Considering that the cluster decision threshold at thekth CH is λ
k, the overall probability of false alarm, P
f, and probability of detection, P
d, for the M CRUs of the kth cluster can be written as follows:(11)
P
f
,
k
=
P
C
k
>
λ
k
∣
H
o
=
Q
λ
k
-
E
C
k
∣
H
0
V
a
r
C
k
∣
H
0
,
P
d
,
k
=
P
C
k
>
λ
k
∣
H
1
=
Q
λ
k
-
E
C
k
∣
H
1
V
a
r
C
k
∣
H
1
.


#### 2.2.2. Conventional Eigenvalue-Based Detection (EVD) Scheme

Let us consider a consecutive smoothing factorF. From (3), a set of the received signals z
i
[
t
] at the kth CH can be written in the following matrix form under H
1 hypothesis:(12)
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
=
P
T
i
h
i
g
i
x
t
x
t
-
1
⋯
⋯
x
t
-
F
+
1
+
δ
0
,
i
t
δ
0
,
i
t
-
1
⋯
⋯
δ
0
,
i
t
-
F
+
1
.With (4), a set of the received signal z
i
[
t
] at the kth CH under H
0 hypothesis can be expressed as(13)
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
=
P
T
i
g
i
η
i
t
η
i
t
-
1
⋯
⋯
η
i
t
-
F
+
1
+
ϕ
i
t
ϕ
i
t
-
1
⋯
⋯
ϕ
i
t
-
F
+
1
=
δ
0
,
i
t
δ
0
,
i
t
-
1
⋯
⋯
δ
0
,
i
t
-
F
+
1
.It is known that any signal can be identified easily using the statistical covariance matrix.Now, we write the statistical covariance matrices of the received signalz
i
[
t
], the transmitted signal x
[
t
], and the noise signal δ
i
[
t
], respectively, as follows [16]: (14)
C
o
v
z
=
E
z
→
i
t
z
→
i
t
T
,
(15)
C
o
v
x
=
E
x
→
t
x
→
t
T
,
(16)
C
o
v
δ
=
E
δ
→
i
t
δ
→
i
t
T
,where(17)
z
→
i
t
=
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
T
,
x
→
t
=
x
t
x
t
-
1
⋯
⋯
x
t
-
F
+
1
T
,
δ
→
i
t
=
δ
i
t
δ
i
t
-
1
⋯
⋯
δ
i
t
-
F
+
1
T
,the superscript ·
T and E
· denote a transpose and expectation operations, respectively. We can verify the statistical covariance matrix C
o
v
z under the two hypotheses that is following [16]:(18)
C
o
v
z
=
C
o
v
x
+
σ
0
2
I
F
,
H
1
σ
0
2
I
F
,
H
0
,where σ
0
2
=
σ
0
,
i
2, for all i, is the variance of the channel noise at the CRU-CH link and I
F is an identity matrix with size of F.The eigenvalues of the covariance matrix of the received signal vector have the following characteristics: firstly, the statistical covariance matrixC
o
v
z is a symmetrical matrix. Then all eigenvalues are the real numbers, and the sum of all eigenvalues is related to the smoothing factor F. Secondly, if the samples of CRUs are completely uncorrelated, matrix C
o
v
z is a unitary matrix and all eigenvalues are equal to σ
0
2. This situation corresponds to the assumption H
0. Finally, if the samples of CRUs are completely correlated, all the elements of matrix C
o
v
z are greater than σ
0
2. This situation corresponds to the assumption H
1.In an ideal case, the correlations of the received signal samples at different times would be larger than zero if the time separations are smaller than the data symbol duration whereas the correlations of the received noise samples at different times should be zero due to the AWGN channel. Let us define the sample autocorrelations of the received signal as follows [27]: (19)
A
j
=
1
N
x
∑
n
=
0
N
x
-
1
z
i
n
z
i
n
-
j
,
j
=
0,1
,
…
,
F
-
1
,where N
x is the total number of available samples and F is a positive integer called the smoothing factor.In practical environment, the sample covariance matrixC
o
v
¯
z
N
x
s
a can only be calculated by using a limited number of signal samples given by N
x
s
a. We can only obtain the sample covariance matrix other than the statistical covariance matrix C
o
v
z. The sample covariance matrix C
o
v
¯
z
N
x
s
a is formed as follows [28]:(20)
C
o
v
¯
z
N
x
s
a
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.Proposition 1.
In the conventional EVD scheme, each CRU has a fixed number of samples at thekth of the clusters which is denoted as N
x
s
a
≈
N
x
c
o
n.Proof.
We know that the number of samples(21)
N
x
~
N
x
c
o
n
=
2
T
s
B
.In the CRUs, the sensing time slot T
s is a constant and let us assume the bandwidth of PU signal B is also a constant; then the number of samples N
x
~
N
x
c
o
n is also constant. In this conventional scheme, the reporting time slot T
r cannot be utilized as a sensing time. The main diagram of the conventional EVD scheme is shown in Figure 2. In this figure, each CRU’s reporting time slot T
r is not utilized for sensing the PU’s signal spectrum. Therefore, the sensing time slot T
s is a constant; then the number of samples N
x
c
o
n is also constant.Figure 2
A conventional CCSS using an EVD technique in CRN [29].Equation (20) can be rewritten as follows:(22)
C
o
v
¯
z
N
x
c
o
n
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.We can compute the decision statistics as follows:(i)
Select appropriate smoothing factorsF and calculate the sample covariance matrix C
o
v
¯
z
N
x
c
o
n from (22) for fixed samples N
x
c
o
n.
(ii)
Calculate the eigenvalueζ
=
e
i
g
n
[
C
o
v
¯
z
N
x
c
o
n
] sorting the eigenvalues in ascending order form the eigenvalues ζ
=
s
o
r
t
i
n
g
[
ζ
1
<
ζ
2
<
ζ
3
<
⋯
].
(iii)
Determine the maximum(
ζ
m
a
x
) and minimum (
ζ
m
i
n
) eigenvalues from the eigenvalues ζ and calculate their ratio, R
m
t
m
c
o
n
i
=
ζ
m
a
x
/
ζ
m
i
n for the ith CRU.
(iv)
Compare∑
i
=
1
M
R
m
t
m
c
o
n
(
i
) with the threshold value, l, at the CH. If ∑
i
=
1
M
R
m
t
m
c
o
n
(
i
)
>
l then the PU signal is present; otherwise, it is absent.Here, the conventional EVD threshold l is determined with the target probability of false alarm. The decision statistics at the kth CH is given by(23)
T
k
=
∑
i
=
1
M
R
m
t
m
c
o
n
i
.The cluster decision C
k at the kth cluster can be expressed as (24)
C
k
=
1
,
if
T
k
>
l
0
,
Otherwise.


#### 2.2.3. Proposed Eigenvalue-Based Detection (EVD) Scheme

In this paper, we investigate CCSS using an EVD technique with a superposition approach to provide more efficient spectrum sensing. In this proposed EVD scheme, we should utilize the CRU reporting time as a CRU sensing time to sense the PU lengthily. The CRU achieves a nonfixed and a longer sensing time slotT
s by allocating different report time slots for CRUs as in Figure 3. The sensing performance at FC is enhanced with the extended sensing duration and fusion rule combining the multiple forwarded decisions. The main diagram of the proposed EVD scheme is shown in Figure 3. In both conventional ED and EVD schemes, the CRU and CH reporting times are not utilized as a CRU sensing time to sense the PU’s signal. In the proposed approach, the CRU and CH reporting times are merged to the CRU sensing time as shown in Figure 3. Therefore, a sensing time slot in the proposed EVD scheme is greater than sensing time slot in both conventional ED and EVD schemes.Figure 3
The proposed CCSS using an EVD technique with a superposition approach in CRN.Proposition 2.
In the proposed EVD scheme, theith CRU in the kth cluster obtains a nonfixed number of samples which is denoted as N
x
p
r
o
p
(
i
,
k
)
=
k
T
s
+
{
(
k
-
1
)
M
+
i
-
1
}
T
r.Proof.
In this scheme, all the CRUs have a different waiting time for its reporting to the CH. As an example, the CRU21, the 2nd CRU in the 1st cluster, can utilize reporting time slot of CRU11 for sensing the PU’s signal, and 3rd CRU31 can utilize reporting time slots of CRU11 and CRU21 for spectrum sensing and so on. From Figure 3, it is observed that the ith CRU in the first cluster has a nonfixed number of samples to sense the PU’s signal as follows:(25)
N
x
p
r
o
p
i
,
1
=
T
s
+
i
-
1
T
r
.In addition, for the second cluster C
2, the sensing time for each CRU can be obtained by(26)
N
x
p
r
o
p
i
,
2
=
N
x
p
r
o
p
M
,
1
+
T
s
+
i
-
1
T
r
=
2
T
s
+
M
+
i
-
1
T
r
.Likewise, for the third cluster, we have(27)
N
x
p
r
o
p
i
,
3
=
N
x
p
r
o
p
M
,
2
+
T
s
+
i
-
1
T
r
=
3
T
s
+
2
M
+
i
-
1
T
r
.In summary, the ith CRU in the kth cluster has the sensing time of (28)
N
x
p
r
o
p
i
,
k
=
k
T
s
+
k
-
1
M
+
i
-
1
T
r
.Therefore, from (25), (26), (27), and (28), it is obvious that the proposed EVD scheme utilizes reporting time of previous CRU’s and we have (29)
N
x
p
r
o
p
i
,
1
<
N
x
p
r
o
p
i
,
2
<
N
x
p
r
o
p
i
,
3
<
⋯
<
N
x
p
r
o
p
i
,
K
.Comparing (21) and (29), we have(30)
N
x
c
o
n
i
,
1
~
N
x
c
o
n
i
,
2
~
⋯
~
N
x
c
o
n
i
,
K
<
N
x
p
r
o
p
i
,
1
<
N
x
p
r
o
p
i
,
2
<
⋯
<
N
x
p
r
o
p
i
,
K
.For simplicity, (31)
N
x
c
o
n
<
N
x
p
r
o
p
C
1
<
N
x
p
r
o
p
C
2
<
N
x
p
r
o
p
C
3
<
N
x
p
r
o
p
C
K
.From (20), we can rewrite the covariance matrix based on a nonfixed number of samples as follows:(32)
C
o
v
¯
z
N
x
p
r
o
p
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.We can compute the ratio of maximum and minimum eigenvalues as follows:(i)
Select appropriate smoothing factorsF and calculate the sample covariance matrix C
o
v
¯
z
N
x
p
r
o
p from (32) for nonfixed samples N
x
p
r
o
p.
(ii)
Calculate the eigenvalue,ζ
^
=
e
i
g
n
[
C
o
v
¯
z
N
x
p
r
o
p
], sorting the eigenvalues in ascending order form the eigenvalues [
ζ
^
]
=
s
o
r
t
i
n
g
[
ζ
^
1
<
ζ
^
2
<
ζ
^
3
<
⋯
].
(iii)
Determine the maximum(
ζ
^
m
a
x
) and minimum (
ζ
^
m
i
n
) eigenvalues from the eigenvalues [
ζ
^
] and calculate their ratio, R
^
m
t
m
p
r
o
p
i
=
ζ
^
m
a
x
/
ζ
^
m
i
n, for the ith CRU.
(iv)
Compare∑
i
=
1
M
R
^
m
t
m
p
r
o
p
(
i
) with the threshold value l
^. If ∑
i
=
1
M
R
^
m
t
m
p
r
o
p
(
i
)
>
l
^ then the primary signal is present; otherwise, it is absent.The proposed threshold l
^ is a function of the target false alarm. The decision statistics of the kth CH is given by (33)
T
^
k
=
∑
i
=
1
M
R
^
m
t
m
p
r
o
p
i
.The cluster decision C
^
k for the kth cluster can be expressed as (34)
C
^
k
=
1
,
if
T
^
k
>
l
^
,
0
,
Otherwise
.


### 2.3. Characterization of Cluster Header-Fusion Center (CH-FC) Link

In the CH-FC link, all CHs communicate with a fixed FC through a dedicated control channel. The aggregated cluster decisions will be forwarded from theK CHs to the FC at which a final global decision is made based on HDF. The HDF is used to reduce the reporting traffic overhead from the K CHs to the FC.


#### 2.3.1. Conventional Energy Detection (ED) Scheme

In this paper, it is assumed that the reporting channel of thekth CH-FC link is a binary symmetric channel with a probability of reporting error, p
e
,
k, and the HDF OR-rule and the HDF Majority (M)-rule are employed at the FC. From (11), it was shown that the global probability of detection, Q
d, the probability of false alarm, Q
f, and the probability of misdetection, Q
m, of the whole CRN at the FC are given by [10](35)
Q
d
=
1
-
∏
k
=
1
K
1
-
P
d
,
k
1
-
p
e
,
k
+
P
d
,
k
p
e
,
k
Q
f
=
1
-
∏
k
=
1
K
1
-
P
f
,
k
1
-
p
e
,
k
+
P
f
,
k
p
e
,
k
Q
m
=
∏
k
=
1
K
P
m
,
k
1
-
p
e
,
k
+
1
-
P
m
,
k
p
e
,
k
OR-rule
Q
d
=
1
-
∏
k
=
1
K
/
2
1
-
P
d
,
k
1
-
p
e
,
k
+
P
d
,
k
p
e
,
k
Q
f
=
1
-
∏
k
=
1
K
/
2
1
-
P
f
,
k
1
-
p
e
,
k
+
P
f
,
k
p
e
,
k
Q
m
=
∏
k
=
1
K
/
2
P
m
,
k
1
-
p
e
,
k
+
1
-
P
m
,
k
p
e
,
k
Majority  (M)-rule
,where P
d
,
k, P
f
,
k, and P
m
,
k are the probabilities of detection, false alarm, and misdetection of the kth cluster, respectively. · is a ceiling function.


#### 2.3.2. Conventional Eigenvalue-Based Detection (EVD) Scheme

In conventional EVD scheme, the final global decisionG of the whole CRN for the two data fusion rules are made upon the following simple forms: (36)
G
=
1
,
if
∑
k
=
1
K
C
k
≥
1
,
O
R
-
r
u
l
e
∑
k
=
1
K
/
2
C
k
≥
1
,
M
-
r
u
l
e
0
,
o
t
h
e
r
w
i
s
e
,where C
k stands for the cluster decision at the kth cluster in the conventional EVD scheme.


#### 2.3.3. Proposed Eigenvalue-Based Detection (EVD) Scheme

In this proposed EVD scheme, the final global decisionG
^ of the whole CRN for the two data fusion rules are made upon the following simple forms: (37)
G
^
=
1
,
if
∑
k
=
1
K
C
^
k
≥
1
,
O
R
-
r
u
l
e
∑
k
=
1
K
/
2
C
^
k
≥
1
,
M
-
r
u
l
e
0
,
o
t
h
e
r
w
i
s
e
,where C
^
k stands for the cluster decision at the kth cluster in the proposed EVD scheme.


## 2.1. Characterization of Primary User-Cognitive Radio User (PU-CRU) Link

In the PU-CRU link, the CRUs perform local spectrum sensing independently to detect the PU’s activities. For CRUi  i
=
1
,
2
,
…
,
M, the spectrum sensing is given by a binary hypothesis test that is formulated as follows [24–26]:(1)
When  PU  is  absent
⟹
H
0
:
y
i
t
=
η
i
t
,
When  PU  is  present
⟹
H
1
:
y
i
t
=
h
i
x
t
+
η
i
t
,where t
=
1,2
,
…
,
N
x, is the sample index, N
x is the total number of samples of the received signal which is defined as N
x
=
2
T
s
B where T
s is the sensing time, and B is a prior known bandwidth. Here, y
i
[
t
] is the received signal at the ith CRU receiver, x
[
t
] is the PU’s transmitted signal which is assumed to be a Gaussian random process with zero mean and variance σ
x
2, that is, x
[
t
]
~
N
0
,
σ
x
2, and η
i
[
t
] is noise of the ith sensing channel which is assumed to be additive white Gaussian with zero mean and variance σ
η
,
i
2, that is, η
i
[
t
]
~
N
0
,
σ
η
,
i
2. The channel gain of the PU-CRU, h
i, is assumed to be constant over each sensing period.


## 2.2. Characterization of Cognitive Radio User-Cluster Header (CRU-CH) Link

In the CRU-CH link, theM CRUs relay their individual measurements of the PU’s signal, y
i
[
t
], to the kth corresponding CH through a dedicated control channel in a sequential manner. Each CRU will simply act in an amplify-and-forward (AF) manner. The channel noise ϕ
i of the CRU-CH link is considered to be zero mean and an additive white Gaussian noise with variances ϑ
i
2. Then, the signal received by the corresponding CH from the ith CRU will be (2)
z
i
t
=
P
T
i
g
i
y
i
t
+
ϕ
i
t
,where P
T
i is the transmit power of the ith CRU relay and g
i is the amplitude of channel gain of the ith CRU-CH link. Using the two hypotheses in (1), the received signal at the kth CH can be written as (3)
z
i
t
∣
H
0
=
P
T
i
g
i
η
i
t
+
ϕ
i
t
=
δ
0
,
i
t
,
(4)
z
i
t
∣
H
1
=
P
T
i
g
i
h
i
x
t
+
P
T
i
g
i
η
i
t
+
ϕ
i
t
=
P
T
i
g
i
h
i
x
t
+
δ
0
,
i
twhose statistical properties for hypothesis H
0 and H
1 are given by(5)
z
i
t
∣
H
0
~
ℵ
0
,
σ
0
,
i
2
~
ℵ
0
,
P
T
i
g
i
2
σ
η
,
i
2
+
ϑ
i
2
,
z
i
t
∣
H
1
~
ℵ
0
,
σ
1
,
i
2
~
ℵ
0
,
P
T
i
h
i
2
g
i
2
σ
x
2
+
σ
0
,
i
2
,where ℵ
0
,
σ
0
,
i
2 denotes the distribution of hypothesis. The details about this distribution are not needed because they are approximated by using the central limit theorem (CLT) in the next subsection.


### 2.2.1. Conventional Energy Detection (ED) Scheme

At the corresponding CH withM CRUs, each received sequence z
i
[
t
] will be individually averaged and squared using a separate energy detector to estimate its own energy [26] as(6)
Ζ
i
=
1
N
x
∑
t
=
0
N
x
-
1
z
i
t
2
.For a large number of samples, CLT approximates the decision statistic for hypothesis H
0 as a Gaussian distribution with mean and variance given by (7)
E
Z
i
∣
H
0
=
N
x
σ
0
,
i
2
=
N
x
P
T
i
g
i
2
σ
η
,
i
2
+
ϑ
i
2
,
V
a
r
Z
i
∣
H
0
=
2
N
x
σ
0
,
i
4
=
2
N
x
P
T
i
g
i
2
σ
η
,
i
2
+
ϑ
i
2
2
.Similarly, the mean and variance for hypothesis H
1 are given by(8)
E
Z
i
∣
H
1
=
N
x
σ
1
,
i
2
=
N
x
P
T
i
h
i
2
g
i
2
σ
x
2
+
σ
0
,
i
2
,
V
a
r
Z
i
∣
H
1
=
2
N
x
σ
1
,
i
4
=
2
N
x
P
T
i
h
i
2
g
i
2
σ
x
2
+
σ
i
2
2
.Now, all the individual test statistics Z
1
,
Z
2
,
…
,
Z
M are used to linearly formulate the resultant test statistic of the kth cluster, C
k, which can be expressed as (9)
C
k
∣
H
0
=
∑
i
=
1
M
ω
i
Z
i
∣
H
0
,
C
k
∣
H
1
=
∑
i
=
1
M
ω
i
Z
i
∣
H
1
.Simply, (9) can be rewritten as (10)
C
k
=
∑
i
=
1
M
ω
i
Z
i
,where k
=
1
,
2
,
…
,
K and ω
i denotes the weighting coefficient for ith CRU in the cluster.Considering that the cluster decision threshold at thekth CH is λ
k, the overall probability of false alarm, P
f, and probability of detection, P
d, for the M CRUs of the kth cluster can be written as follows:(11)
P
f
,
k
=
P
C
k
>
λ
k
∣
H
o
=
Q
λ
k
-
E
C
k
∣
H
0
V
a
r
C
k
∣
H
0
,
P
d
,
k
=
P
C
k
>
λ
k
∣
H
1
=
Q
λ
k
-
E
C
k
∣
H
1
V
a
r
C
k
∣
H
1
.


### 2.2.2. Conventional Eigenvalue-Based Detection (EVD) Scheme

Let us consider a consecutive smoothing factorF. From (3), a set of the received signals z
i
[
t
] at the kth CH can be written in the following matrix form under H
1 hypothesis:(12)
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
=
P
T
i
h
i
g
i
x
t
x
t
-
1
⋯
⋯
x
t
-
F
+
1
+
δ
0
,
i
t
δ
0
,
i
t
-
1
⋯
⋯
δ
0
,
i
t
-
F
+
1
.With (4), a set of the received signal z
i
[
t
] at the kth CH under H
0 hypothesis can be expressed as(13)
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
=
P
T
i
g
i
η
i
t
η
i
t
-
1
⋯
⋯
η
i
t
-
F
+
1
+
ϕ
i
t
ϕ
i
t
-
1
⋯
⋯
ϕ
i
t
-
F
+
1
=
δ
0
,
i
t
δ
0
,
i
t
-
1
⋯
⋯
δ
0
,
i
t
-
F
+
1
.It is known that any signal can be identified easily using the statistical covariance matrix.Now, we write the statistical covariance matrices of the received signalz
i
[
t
], the transmitted signal x
[
t
], and the noise signal δ
i
[
t
], respectively, as follows [16]: (14)
C
o
v
z
=
E
z
→
i
t
z
→
i
t
T
,
(15)
C
o
v
x
=
E
x
→
t
x
→
t
T
,
(16)
C
o
v
δ
=
E
δ
→
i
t
δ
→
i
t
T
,where(17)
z
→
i
t
=
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
T
,
x
→
t
=
x
t
x
t
-
1
⋯
⋯
x
t
-
F
+
1
T
,
δ
→
i
t
=
δ
i
t
δ
i
t
-
1
⋯
⋯
δ
i
t
-
F
+
1
T
,the superscript ·
T and E
· denote a transpose and expectation operations, respectively. We can verify the statistical covariance matrix C
o
v
z under the two hypotheses that is following [16]:(18)
C
o
v
z
=
C
o
v
x
+
σ
0
2
I
F
,
H
1
σ
0
2
I
F
,
H
0
,where σ
0
2
=
σ
0
,
i
2, for all i, is the variance of the channel noise at the CRU-CH link and I
F is an identity matrix with size of F.The eigenvalues of the covariance matrix of the received signal vector have the following characteristics: firstly, the statistical covariance matrixC
o
v
z is a symmetrical matrix. Then all eigenvalues are the real numbers, and the sum of all eigenvalues is related to the smoothing factor F. Secondly, if the samples of CRUs are completely uncorrelated, matrix C
o
v
z is a unitary matrix and all eigenvalues are equal to σ
0
2. This situation corresponds to the assumption H
0. Finally, if the samples of CRUs are completely correlated, all the elements of matrix C
o
v
z are greater than σ
0
2. This situation corresponds to the assumption H
1.In an ideal case, the correlations of the received signal samples at different times would be larger than zero if the time separations are smaller than the data symbol duration whereas the correlations of the received noise samples at different times should be zero due to the AWGN channel. Let us define the sample autocorrelations of the received signal as follows [27]: (19)
A
j
=
1
N
x
∑
n
=
0
N
x
-
1
z
i
n
z
i
n
-
j
,
j
=
0,1
,
…
,
F
-
1
,where N
x is the total number of available samples and F is a positive integer called the smoothing factor.In practical environment, the sample covariance matrixC
o
v
¯
z
N
x
s
a can only be calculated by using a limited number of signal samples given by N
x
s
a. We can only obtain the sample covariance matrix other than the statistical covariance matrix C
o
v
z. The sample covariance matrix C
o
v
¯
z
N
x
s
a is formed as follows [28]:(20)
C
o
v
¯
z
N
x
s
a
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.Proposition 1.
In the conventional EVD scheme, each CRU has a fixed number of samples at thekth of the clusters which is denoted as N
x
s
a
≈
N
x
c
o
n.Proof.
We know that the number of samples(21)
N
x
~
N
x
c
o
n
=
2
T
s
B
.In the CRUs, the sensing time slot T
s is a constant and let us assume the bandwidth of PU signal B is also a constant; then the number of samples N
x
~
N
x
c
o
n is also constant. In this conventional scheme, the reporting time slot T
r cannot be utilized as a sensing time. The main diagram of the conventional EVD scheme is shown in Figure 2. In this figure, each CRU’s reporting time slot T
r is not utilized for sensing the PU’s signal spectrum. Therefore, the sensing time slot T
s is a constant; then the number of samples N
x
c
o
n is also constant.Figure 2
A conventional CCSS using an EVD technique in CRN [29].Equation (20) can be rewritten as follows:(22)
C
o
v
¯
z
N
x
c
o
n
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.We can compute the decision statistics as follows:(i)
Select appropriate smoothing factorsF and calculate the sample covariance matrix C
o
v
¯
z
N
x
c
o
n from (22) for fixed samples N
x
c
o
n.
(ii)
Calculate the eigenvalueζ
=
e
i
g
n
[
C
o
v
¯
z
N
x
c
o
n
] sorting the eigenvalues in ascending order form the eigenvalues ζ
=
s
o
r
t
i
n
g
[
ζ
1
<
ζ
2
<
ζ
3
<
⋯
].
(iii)
Determine the maximum(
ζ
m
a
x
) and minimum (
ζ
m
i
n
) eigenvalues from the eigenvalues ζ and calculate their ratio, R
m
t
m
c
o
n
i
=
ζ
m
a
x
/
ζ
m
i
n for the ith CRU.
(iv)
Compare∑
i
=
1
M
R
m
t
m
c
o
n
(
i
) with the threshold value, l, at the CH. If ∑
i
=
1
M
R
m
t
m
c
o
n
(
i
)
>
l then the PU signal is present; otherwise, it is absent.Here, the conventional EVD threshold l is determined with the target probability of false alarm. The decision statistics at the kth CH is given by(23)
T
k
=
∑
i
=
1
M
R
m
t
m
c
o
n
i
.The cluster decision C
k at the kth cluster can be expressed as (24)
C
k
=
1
,
if
T
k
>
l
0
,
Otherwise.


### 2.2.3. Proposed Eigenvalue-Based Detection (EVD) Scheme

In this paper, we investigate CCSS using an EVD technique with a superposition approach to provide more efficient spectrum sensing. In this proposed EVD scheme, we should utilize the CRU reporting time as a CRU sensing time to sense the PU lengthily. The CRU achieves a nonfixed and a longer sensing time slotT
s by allocating different report time slots for CRUs as in Figure 3. The sensing performance at FC is enhanced with the extended sensing duration and fusion rule combining the multiple forwarded decisions. The main diagram of the proposed EVD scheme is shown in Figure 3. In both conventional ED and EVD schemes, the CRU and CH reporting times are not utilized as a CRU sensing time to sense the PU’s signal. In the proposed approach, the CRU and CH reporting times are merged to the CRU sensing time as shown in Figure 3. Therefore, a sensing time slot in the proposed EVD scheme is greater than sensing time slot in both conventional ED and EVD schemes.Figure 3
The proposed CCSS using an EVD technique with a superposition approach in CRN.Proposition 2.
In the proposed EVD scheme, theith CRU in the kth cluster obtains a nonfixed number of samples which is denoted as N
x
p
r
o
p
(
i
,
k
)
=
k
T
s
+
{
(
k
-
1
)
M
+
i
-
1
}
T
r.Proof.
In this scheme, all the CRUs have a different waiting time for its reporting to the CH. As an example, the CRU21, the 2nd CRU in the 1st cluster, can utilize reporting time slot of CRU11 for sensing the PU’s signal, and 3rd CRU31 can utilize reporting time slots of CRU11 and CRU21 for spectrum sensing and so on. From Figure 3, it is observed that the ith CRU in the first cluster has a nonfixed number of samples to sense the PU’s signal as follows:(25)
N
x
p
r
o
p
i
,
1
=
T
s
+
i
-
1
T
r
.In addition, for the second cluster C
2, the sensing time for each CRU can be obtained by(26)
N
x
p
r
o
p
i
,
2
=
N
x
p
r
o
p
M
,
1
+
T
s
+
i
-
1
T
r
=
2
T
s
+
M
+
i
-
1
T
r
.Likewise, for the third cluster, we have(27)
N
x
p
r
o
p
i
,
3
=
N
x
p
r
o
p
M
,
2
+
T
s
+
i
-
1
T
r
=
3
T
s
+
2
M
+
i
-
1
T
r
.In summary, the ith CRU in the kth cluster has the sensing time of (28)
N
x
p
r
o
p
i
,
k
=
k
T
s
+
k
-
1
M
+
i
-
1
T
r
.Therefore, from (25), (26), (27), and (28), it is obvious that the proposed EVD scheme utilizes reporting time of previous CRU’s and we have (29)
N
x
p
r
o
p
i
,
1
<
N
x
p
r
o
p
i
,
2
<
N
x
p
r
o
p
i
,
3
<
⋯
<
N
x
p
r
o
p
i
,
K
.Comparing (21) and (29), we have(30)
N
x
c
o
n
i
,
1
~
N
x
c
o
n
i
,
2
~
⋯
~
N
x
c
o
n
i
,
K
<
N
x
p
r
o
p
i
,
1
<
N
x
p
r
o
p
i
,
2
<
⋯
<
N
x
p
r
o
p
i
,
K
.For simplicity, (31)
N
x
c
o
n
<
N
x
p
r
o
p
C
1
<
N
x
p
r
o
p
C
2
<
N
x
p
r
o
p
C
3
<
N
x
p
r
o
p
C
K
.From (20), we can rewrite the covariance matrix based on a nonfixed number of samples as follows:(32)
C
o
v
¯
z
N
x
p
r
o
p
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.We can compute the ratio of maximum and minimum eigenvalues as follows:(i)
Select appropriate smoothing factorsF and calculate the sample covariance matrix C
o
v
¯
z
N
x
p
r
o
p from (32) for nonfixed samples N
x
p
r
o
p.
(ii)
Calculate the eigenvalue,ζ
^
=
e
i
g
n
[
C
o
v
¯
z
N
x
p
r
o
p
], sorting the eigenvalues in ascending order form the eigenvalues [
ζ
^
]
=
s
o
r
t
i
n
g
[
ζ
^
1
<
ζ
^
2
<
ζ
^
3
<
⋯
].
(iii)
Determine the maximum(
ζ
^
m
a
x
) and minimum (
ζ
^
m
i
n
) eigenvalues from the eigenvalues [
ζ
^
] and calculate their ratio, R
^
m
t
m
p
r
o
p
i
=
ζ
^
m
a
x
/
ζ
^
m
i
n, for the ith CRU.
(iv)
Compare∑
i
=
1
M
R
^
m
t
m
p
r
o
p
(
i
) with the threshold value l
^. If ∑
i
=
1
M
R
^
m
t
m
p
r
o
p
(
i
)
>
l
^ then the primary signal is present; otherwise, it is absent.The proposed threshold l
^ is a function of the target false alarm. The decision statistics of the kth CH is given by (33)
T
^
k
=
∑
i
=
1
M
R
^
m
t
m
p
r
o
p
i
.The cluster decision C
^
k for the kth cluster can be expressed as (34)
C
^
k
=
1
,
if
T
^
k
>
l
^
,
0
,
Otherwise
.


## 2.2.1. Conventional Energy Detection (ED) Scheme

At the corresponding CH withM CRUs, each received sequence z
i
[
t
] will be individually averaged and squared using a separate energy detector to estimate its own energy [26] as(6)
Ζ
i
=
1
N
x
∑
t
=
0
N
x
-
1
z
i
t
2
.For a large number of samples, CLT approximates the decision statistic for hypothesis H
0 as a Gaussian distribution with mean and variance given by (7)
E
Z
i
∣
H
0
=
N
x
σ
0
,
i
2
=
N
x
P
T
i
g
i
2
σ
η
,
i
2
+
ϑ
i
2
,
V
a
r
Z
i
∣
H
0
=
2
N
x
σ
0
,
i
4
=
2
N
x
P
T
i
g
i
2
σ
η
,
i
2
+
ϑ
i
2
2
.Similarly, the mean and variance for hypothesis H
1 are given by(8)
E
Z
i
∣
H
1
=
N
x
σ
1
,
i
2
=
N
x
P
T
i
h
i
2
g
i
2
σ
x
2
+
σ
0
,
i
2
,
V
a
r
Z
i
∣
H
1
=
2
N
x
σ
1
,
i
4
=
2
N
x
P
T
i
h
i
2
g
i
2
σ
x
2
+
σ
i
2
2
.Now, all the individual test statistics Z
1
,
Z
2
,
…
,
Z
M are used to linearly formulate the resultant test statistic of the kth cluster, C
k, which can be expressed as (9)
C
k
∣
H
0
=
∑
i
=
1
M
ω
i
Z
i
∣
H
0
,
C
k
∣
H
1
=
∑
i
=
1
M
ω
i
Z
i
∣
H
1
.Simply, (9) can be rewritten as (10)
C
k
=
∑
i
=
1
M
ω
i
Z
i
,where k
=
1
,
2
,
…
,
K and ω
i denotes the weighting coefficient for ith CRU in the cluster.Considering that the cluster decision threshold at thekth CH is λ
k, the overall probability of false alarm, P
f, and probability of detection, P
d, for the M CRUs of the kth cluster can be written as follows:(11)
P
f
,
k
=
P
C
k
>
λ
k
∣
H
o
=
Q
λ
k
-
E
C
k
∣
H
0
V
a
r
C
k
∣
H
0
,
P
d
,
k
=
P
C
k
>
λ
k
∣
H
1
=
Q
λ
k
-
E
C
k
∣
H
1
V
a
r
C
k
∣
H
1
.


## 2.2.2. Conventional Eigenvalue-Based Detection (EVD) Scheme

Let us consider a consecutive smoothing factorF. From (3), a set of the received signals z
i
[
t
] at the kth CH can be written in the following matrix form under H
1 hypothesis:(12)
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
=
P
T
i
h
i
g
i
x
t
x
t
-
1
⋯
⋯
x
t
-
F
+
1
+
δ
0
,
i
t
δ
0
,
i
t
-
1
⋯
⋯
δ
0
,
i
t
-
F
+
1
.With (4), a set of the received signal z
i
[
t
] at the kth CH under H
0 hypothesis can be expressed as(13)
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
=
P
T
i
g
i
η
i
t
η
i
t
-
1
⋯
⋯
η
i
t
-
F
+
1
+
ϕ
i
t
ϕ
i
t
-
1
⋯
⋯
ϕ
i
t
-
F
+
1
=
δ
0
,
i
t
δ
0
,
i
t
-
1
⋯
⋯
δ
0
,
i
t
-
F
+
1
.It is known that any signal can be identified easily using the statistical covariance matrix.Now, we write the statistical covariance matrices of the received signalz
i
[
t
], the transmitted signal x
[
t
], and the noise signal δ
i
[
t
], respectively, as follows [16]: (14)
C
o
v
z
=
E
z
→
i
t
z
→
i
t
T
,
(15)
C
o
v
x
=
E
x
→
t
x
→
t
T
,
(16)
C
o
v
δ
=
E
δ
→
i
t
δ
→
i
t
T
,where(17)
z
→
i
t
=
z
i
t
z
i
t
-
1
⋯
⋯
z
i
t
-
F
+
1
T
,
x
→
t
=
x
t
x
t
-
1
⋯
⋯
x
t
-
F
+
1
T
,
δ
→
i
t
=
δ
i
t
δ
i
t
-
1
⋯
⋯
δ
i
t
-
F
+
1
T
,the superscript ·
T and E
· denote a transpose and expectation operations, respectively. We can verify the statistical covariance matrix C
o
v
z under the two hypotheses that is following [16]:(18)
C
o
v
z
=
C
o
v
x
+
σ
0
2
I
F
,
H
1
σ
0
2
I
F
,
H
0
,where σ
0
2
=
σ
0
,
i
2, for all i, is the variance of the channel noise at the CRU-CH link and I
F is an identity matrix with size of F.The eigenvalues of the covariance matrix of the received signal vector have the following characteristics: firstly, the statistical covariance matrixC
o
v
z is a symmetrical matrix. Then all eigenvalues are the real numbers, and the sum of all eigenvalues is related to the smoothing factor F. Secondly, if the samples of CRUs are completely uncorrelated, matrix C
o
v
z is a unitary matrix and all eigenvalues are equal to σ
0
2. This situation corresponds to the assumption H
0. Finally, if the samples of CRUs are completely correlated, all the elements of matrix C
o
v
z are greater than σ
0
2. This situation corresponds to the assumption H
1.In an ideal case, the correlations of the received signal samples at different times would be larger than zero if the time separations are smaller than the data symbol duration whereas the correlations of the received noise samples at different times should be zero due to the AWGN channel. Let us define the sample autocorrelations of the received signal as follows [27]: (19)
A
j
=
1
N
x
∑
n
=
0
N
x
-
1
z
i
n
z
i
n
-
j
,
j
=
0,1
,
…
,
F
-
1
,where N
x is the total number of available samples and F is a positive integer called the smoothing factor.In practical environment, the sample covariance matrixC
o
v
¯
z
N
x
s
a can only be calculated by using a limited number of signal samples given by N
x
s
a. We can only obtain the sample covariance matrix other than the statistical covariance matrix C
o
v
z. The sample covariance matrix C
o
v
¯
z
N
x
s
a is formed as follows [28]:(20)
C
o
v
¯
z
N
x
s
a
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.Proposition 1.
In the conventional EVD scheme, each CRU has a fixed number of samples at thekth of the clusters which is denoted as N
x
s
a
≈
N
x
c
o
n.Proof.
We know that the number of samples(21)
N
x
~
N
x
c
o
n
=
2
T
s
B
.In the CRUs, the sensing time slot T
s is a constant and let us assume the bandwidth of PU signal B is also a constant; then the number of samples N
x
~
N
x
c
o
n is also constant. In this conventional scheme, the reporting time slot T
r cannot be utilized as a sensing time. The main diagram of the conventional EVD scheme is shown in Figure 2. In this figure, each CRU’s reporting time slot T
r is not utilized for sensing the PU’s signal spectrum. Therefore, the sensing time slot T
s is a constant; then the number of samples N
x
c
o
n is also constant.Figure 2
A conventional CCSS using an EVD technique in CRN [29].Equation (20) can be rewritten as follows:(22)
C
o
v
¯
z
N
x
c
o
n
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.We can compute the decision statistics as follows:(i)
Select appropriate smoothing factorsF and calculate the sample covariance matrix C
o
v
¯
z
N
x
c
o
n from (22) for fixed samples N
x
c
o
n.
(ii)
Calculate the eigenvalueζ
=
e
i
g
n
[
C
o
v
¯
z
N
x
c
o
n
] sorting the eigenvalues in ascending order form the eigenvalues ζ
=
s
o
r
t
i
n
g
[
ζ
1
<
ζ
2
<
ζ
3
<
⋯
].
(iii)
Determine the maximum(
ζ
m
a
x
) and minimum (
ζ
m
i
n
) eigenvalues from the eigenvalues ζ and calculate their ratio, R
m
t
m
c
o
n
i
=
ζ
m
a
x
/
ζ
m
i
n for the ith CRU.
(iv)
Compare∑
i
=
1
M
R
m
t
m
c
o
n
(
i
) with the threshold value, l, at the CH. If ∑
i
=
1
M
R
m
t
m
c
o
n
(
i
)
>
l then the PU signal is present; otherwise, it is absent.Here, the conventional EVD threshold l is determined with the target probability of false alarm. The decision statistics at the kth CH is given by(23)
T
k
=
∑
i
=
1
M
R
m
t
m
c
o
n
i
.The cluster decision C
k at the kth cluster can be expressed as (24)
C
k
=
1
,
if
T
k
>
l
0
,
Otherwise.


## 2.2.3. Proposed Eigenvalue-Based Detection (EVD) Scheme

In this paper, we investigate CCSS using an EVD technique with a superposition approach to provide more efficient spectrum sensing. In this proposed EVD scheme, we should utilize the CRU reporting time as a CRU sensing time to sense the PU lengthily. The CRU achieves a nonfixed and a longer sensing time slotT
s by allocating different report time slots for CRUs as in Figure 3. The sensing performance at FC is enhanced with the extended sensing duration and fusion rule combining the multiple forwarded decisions. The main diagram of the proposed EVD scheme is shown in Figure 3. In both conventional ED and EVD schemes, the CRU and CH reporting times are not utilized as a CRU sensing time to sense the PU’s signal. In the proposed approach, the CRU and CH reporting times are merged to the CRU sensing time as shown in Figure 3. Therefore, a sensing time slot in the proposed EVD scheme is greater than sensing time slot in both conventional ED and EVD schemes.Figure 3
The proposed CCSS using an EVD technique with a superposition approach in CRN.Proposition 2.
In the proposed EVD scheme, theith CRU in the kth cluster obtains a nonfixed number of samples which is denoted as N
x
p
r
o
p
(
i
,
k
)
=
k
T
s
+
{
(
k
-
1
)
M
+
i
-
1
}
T
r.Proof.
In this scheme, all the CRUs have a different waiting time for its reporting to the CH. As an example, the CRU21, the 2nd CRU in the 1st cluster, can utilize reporting time slot of CRU11 for sensing the PU’s signal, and 3rd CRU31 can utilize reporting time slots of CRU11 and CRU21 for spectrum sensing and so on. From Figure 3, it is observed that the ith CRU in the first cluster has a nonfixed number of samples to sense the PU’s signal as follows:(25)
N
x
p
r
o
p
i
,
1
=
T
s
+
i
-
1
T
r
.In addition, for the second cluster C
2, the sensing time for each CRU can be obtained by(26)
N
x
p
r
o
p
i
,
2
=
N
x
p
r
o
p
M
,
1
+
T
s
+
i
-
1
T
r
=
2
T
s
+
M
+
i
-
1
T
r
.Likewise, for the third cluster, we have(27)
N
x
p
r
o
p
i
,
3
=
N
x
p
r
o
p
M
,
2
+
T
s
+
i
-
1
T
r
=
3
T
s
+
2
M
+
i
-
1
T
r
.In summary, the ith CRU in the kth cluster has the sensing time of (28)
N
x
p
r
o
p
i
,
k
=
k
T
s
+
k
-
1
M
+
i
-
1
T
r
.Therefore, from (25), (26), (27), and (28), it is obvious that the proposed EVD scheme utilizes reporting time of previous CRU’s and we have (29)
N
x
p
r
o
p
i
,
1
<
N
x
p
r
o
p
i
,
2
<
N
x
p
r
o
p
i
,
3
<
⋯
<
N
x
p
r
o
p
i
,
K
.Comparing (21) and (29), we have(30)
N
x
c
o
n
i
,
1
~
N
x
c
o
n
i
,
2
~
⋯
~
N
x
c
o
n
i
,
K
<
N
x
p
r
o
p
i
,
1
<
N
x
p
r
o
p
i
,
2
<
⋯
<
N
x
p
r
o
p
i
,
K
.For simplicity, (31)
N
x
c
o
n
<
N
x
p
r
o
p
C
1
<
N
x
p
r
o
p
C
2
<
N
x
p
r
o
p
C
3
<
N
x
p
r
o
p
C
K
.From (20), we can rewrite the covariance matrix based on a nonfixed number of samples as follows:(32)
C
o
v
¯
z
N
x
p
r
o
p
=
A
0
A
1
⋯
A
F
-
1
A
1
A
0
⋯
A
F
-
2
⋯
⋯
⋯
⋯
A
F
-
1
A
F
-
2
⋯
A
0
.We can compute the ratio of maximum and minimum eigenvalues as follows:(i)
Select appropriate smoothing factorsF and calculate the sample covariance matrix C
o
v
¯
z
N
x
p
r
o
p from (32) for nonfixed samples N
x
p
r
o
p.
(ii)
Calculate the eigenvalue,ζ
^
=
e
i
g
n
[
C
o
v
¯
z
N
x
p
r
o
p
], sorting the eigenvalues in ascending order form the eigenvalues [
ζ
^
]
=
s
o
r
t
i
n
g
[
ζ
^
1
<
ζ
^
2
<
ζ
^
3
<
⋯
].
(iii)
Determine the maximum(
ζ
^
m
a
x
) and minimum (
ζ
^
m
i
n
) eigenvalues from the eigenvalues [
ζ
^
] and calculate their ratio, R
^
m
t
m
p
r
o
p
i
=
ζ
^
m
a
x
/
ζ
^
m
i
n, for the ith CRU.
(iv)
Compare∑
i
=
1
M
R
^
m
t
m
p
r
o
p
(
i
) with the threshold value l
^. If ∑
i
=
1
M
R
^
m
t
m
p
r
o
p
(
i
)
>
l
^ then the primary signal is present; otherwise, it is absent.The proposed threshold l
^ is a function of the target false alarm. The decision statistics of the kth CH is given by (33)
T
^
k
=
∑
i
=
1
M
R
^
m
t
m
p
r
o
p
i
.The cluster decision C
^
k for the kth cluster can be expressed as (34)
C
^
k
=
1
,
if
T
^
k
>
l
^
,
0
,
Otherwise
.


## 2.3. Characterization of Cluster Header-Fusion Center (CH-FC) Link

In the CH-FC link, all CHs communicate with a fixed FC through a dedicated control channel. The aggregated cluster decisions will be forwarded from theK CHs to the FC at which a final global decision is made based on HDF. The HDF is used to reduce the reporting traffic overhead from the K CHs to the FC.


### 2.3.1. Conventional Energy Detection (ED) Scheme

In this paper, it is assumed that the reporting channel of thekth CH-FC link is a binary symmetric channel with a probability of reporting error, p
e
,
k, and the HDF OR-rule and the HDF Majority (M)-rule are employed at the FC. From (11), it was shown that the global probability of detection, Q
d, the probability of false alarm, Q
f, and the probability of misdetection, Q
m, of the whole CRN at the FC are given by [10](35)
Q
d
=
1
-
∏
k
=
1
K
1
-
P
d
,
k
1
-
p
e
,
k
+
P
d
,
k
p
e
,
k
Q
f
=
1
-
∏
k
=
1
K
1
-
P
f
,
k
1
-
p
e
,
k
+
P
f
,
k
p
e
,
k
Q
m
=
∏
k
=
1
K
P
m
,
k
1
-
p
e
,
k
+
1
-
P
m
,
k
p
e
,
k
OR-rule
Q
d
=
1
-
∏
k
=
1
K
/
2
1
-
P
d
,
k
1
-
p
e
,
k
+
P
d
,
k
p
e
,
k
Q
f
=
1
-
∏
k
=
1
K
/
2
1
-
P
f
,
k
1
-
p
e
,
k
+
P
f
,
k
p
e
,
k
Q
m
=
∏
k
=
1
K
/
2
P
m
,
k
1
-
p
e
,
k
+
1
-
P
m
,
k
p
e
,
k
Majority  (M)-rule
,where P
d
,
k, P
f
,
k, and P
m
,
k are the probabilities of detection, false alarm, and misdetection of the kth cluster, respectively. · is a ceiling function.


### 2.3.2. Conventional Eigenvalue-Based Detection (EVD) Scheme

In conventional EVD scheme, the final global decisionG of the whole CRN for the two data fusion rules are made upon the following simple forms: (36)
G
=
1
,
if
∑
k
=
1
K
C
k
≥
1
,
O
R
-
r
u
l
e
∑
k
=
1
K
/
2
C
k
≥
1
,
M
-
r
u
l
e
0
,
o
t
h
e
r
w
i
s
e
,where C
k stands for the cluster decision at the kth cluster in the conventional EVD scheme.


### 2.3.3. Proposed Eigenvalue-Based Detection (EVD) Scheme

In this proposed EVD scheme, the final global decisionG
^ of the whole CRN for the two data fusion rules are made upon the following simple forms: (37)
G
^
=
1
,
if
∑
k
=
1
K
C
^
k
≥
1
,
O
R
-
r
u
l
e
∑
k
=
1
K
/
2
C
^
k
≥
1
,
M
-
r
u
l
e
0
,
o
t
h
e
r
w
i
s
e
,where C
^
k stands for the cluster decision at the kth cluster in the proposed EVD scheme.


## 2.3.1. Conventional Energy Detection (ED) Scheme

In this paper, it is assumed that the reporting channel of thekth CH-FC link is a binary symmetric channel with a probability of reporting error, p
e
,
k, and the HDF OR-rule and the HDF Majority (M)-rule are employed at the FC. From (11), it was shown that the global probability of detection, Q
d, the probability of false alarm, Q
f, and the probability of misdetection, Q
m, of the whole CRN at the FC are given by [10](35)
Q
d
=
1
-
∏
k
=
1
K
1
-
P
d
,
k
1
-
p
e
,
k
+
P
d
,
k
p
e
,
k
Q
f
=
1
-
∏
k
=
1
K
1
-
P
f
,
k
1
-
p
e
,
k
+
P
f
,
k
p
e
,
k
Q
m
=
∏
k
=
1
K
P
m
,
k
1
-
p
e
,
k
+
1
-
P
m
,
k
p
e
,
k
OR-rule
Q
d
=
1
-
∏
k
=
1
K
/
2
1
-
P
d
,
k
1
-
p
e
,
k
+
P
d
,
k
p
e
,
k
Q
f
=
1
-
∏
k
=
1
K
/
2
1
-
P
f
,
k
1
-
p
e
,
k
+
P
f
,
k
p
e
,
k
Q
m
=
∏
k
=
1
K
/
2
P
m
,
k
1
-
p
e
,
k
+
1
-
P
m
,
k
p
e
,
k
Majority  (M)-rule
,where P
d
,
k, P
f
,
k, and P
m
,
k are the probabilities of detection, false alarm, and misdetection of the kth cluster, respectively. · is a ceiling function.


## 2.3.2. Conventional Eigenvalue-Based Detection (EVD) Scheme

In conventional EVD scheme, the final global decisionG of the whole CRN for the two data fusion rules are made upon the following simple forms: (36)
G
=
1
,
if
∑
k
=
1
K
C
k
≥
1
,
O
R
-
r
u
l
e
∑
k
=
1
K
/
2
C
k
≥
1
,
M
-
r
u
l
e
0
,
o
t
h
e
r
w
i
s
e
,where C
k stands for the cluster decision at the kth cluster in the conventional EVD scheme.


## 2.3.3. Proposed Eigenvalue-Based Detection (EVD) Scheme

In this proposed EVD scheme, the final global decisionG
^ of the whole CRN for the two data fusion rules are made upon the following simple forms: (37)
G
^
=
1
,
if
∑
k
=
1
K
C
^
k
≥
1
,
O
R
-
r
u
l
e
∑
k
=
1
K
/
2
C
^
k
≥
1
,
M
-
r
u
l
e
0
,
o
t
h
e
r
w
i
s
e
,where C
^
k stands for the cluster decision at the kth cluster in the proposed EVD scheme.


## 3. Simulation Results

In order to evaluate the proposed scheme, Monte-Carlo simulations were carried out under following conditions: the sampling frequencyf
c is 300 kHz, the local sensing time T
s is 1 ms, the local reporting time T
r is 0.125 ms, the number of samples N
x is 100, the PU signal is a BPSK signal, the noise in CRU is AWGN, the number CRUs is 12, the number clusters is 3, and the number of CRUs in each cluster is 4. Moreover, it is assumed that the signal is independent and identically distributed (i.i.d), and the channel is under different channels. The relay transmit power is set to 12 dBm and the channel gains of the PU-CRU and CRU-CH links, {
h
i
} and {
g
i
}, respectively, are normally distributed but remain constant within each sensing interval T
s. {
h
i
} and {
g
i
} are randomly generated so that a low SNR environment at CRU, CH, and FC stages is realized (SNR < −10 dB).Under such a condition, the curves of receiver operating characteristics (ROC) are illustrated in Figure4. It is shown that the proposed EVD scheme has the highest probability of detection compared with both the conventional EVD and ED schemes because the proposed superallocation can have longer sensing time than both conventional approaches. The larger smoothing factor (F) means the higher probability of detection. However, it is very difficult to determine the best F because it is related to signal property.Figure 4
The ROC curves of the conventional ED scheme, conventional EVD scheme, and the proposed EVD scheme for different smoothing factors.In an environment with noise uncertainty, it can assume that we need the probability of detection to be more than 0.9 and the probability of false alarm to be less than 0.1. In both conventional ED and EVD schemes, we can achieve the target sensing performance with a longer sensing time slot but the throughput of the CRN decreases. In the proposed EVD scheme, we can easily achieve more than 0.9 and less than 0.1 for the probabilities of detection and false alarm, respectively, because CRU reporting time and CH reporting time merge to sense the PU signal more accurately without decreasing system throughput.Finally, Figure5 shows ROC curves for the global decision at FC for the conventional cooperative ED scheme, the conventional cooperative EVD scheme, and the proposed cooperative EVD scheme with cluster reporting time. In each cluster, 4 CRUs are spread in the network to perform local sensing where distributed CRUs have different SNR values. We consider the condition in which the received signals of all four CRUs are given by −28 dB, −20 dB, −15 dB, and −10 dB, respectively. Also, we have considered both OR-rule and M-rule [30] as a data fusion rules at the FC. Under such a condition, the probability of detection of OR-rule is always larger than M-rule. The conventional EVD scheme has better detection performance than the conventional ED scheme for both OR-rule and M-rule. By utilizing OR-rule, it is shown that the proposed EVD scheme has better performance compared with both the conventional EVD scheme and ED scheme, for the M-rule as well. The simulation results proved that the propose EVD scheme has the ability to significantly improve the sensing detection performance in CRN.Figure 5
Global probabilities of detection and false alarm of the conventional cooperative ED, conventional cooperative EVD, and proposed cooperative EVD when four CRUs’ SNRs are given by −28 dB, −20 dB, −15 dB, and −10 dB, respectively.


## 4. Conclusion

In this paper, we have proposed a CCSS in CRN using an EVD technique with a superposition approach. The proposed EVD scheme can achieve better detection performance in comparison with both the conventional EVD scheme and ED scheme because of each CRU to sense the PU’s signal more accurately by rescheduling the reporting time slots of CRUs and CHs as obtain a longer sensing duration that are guaranteed for all CRUs. As further works, the other noise conditions including impulse noise can be considered.



---

*Source: 207935-2015-07-06.xml*