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Spectrum Sensing Improvement by SNR
Maximization in Cognitive Two-way Relay
Networks
Ardalan Alizadeh and Seyed Mohammad-Sajad Sadough
Cognitive Telecommunication Research Group
Faculty of Electrical and Computer Engineering
Shahid Beheshti University
1983963113, Tehran, Iran
Email: [email protected], s [email protected]
Abstract—In this paper, we consider a cognitive two-way relaynetwork consisting of two primary transceivers and some cog-nitive radio (CR) terminals forming the secondary network. As-suming no direct link between the two primary transceivers, CRterminals play the relaying role when the primary transceiversare in operation to provide a two-step two-way relaying schemefor primary transmission. A cognitive base station (CBS) usesthe received signal in the second step of relaying to detect thepresence of the primary transmission. With the aim of increasingthe accuracy of spectrum sensing at the CBS (trough signal-to-noise ratio maximization), we perform beamforming at CR
terminals while satisfying the quality of service requirementsat primary transceivers. This optimization problem is firstlyrelaxed to the semidefinite form and then solved iteratively.Simulation results demonstrate the superiority in terms of sensingperformance for the proposed technique in comparison withconventional beamforming methods.
I. INTRODUCTION
Spectrum sensing is one of the main parts in each cognitive
radio (CR) system [1]. Classically, it is assumed that a single
secondary user may not be able to reliably distinguish between
a white space and a weak primary user signal in the spectrum
sensing process. Thus, to improve the detection reliability, co-
operative spectrum sensing involving several secondary users
is usually deployed [2].
The idea of using a single-antenna relay terminal to imple-
ment the one-way relaying in CR networks, sometimes also
called cognitive relaying, has recently attracted a great deal
of attention (see e.g., [3] and references therein). In order
to further increase the spectrum efficiency, the multi-antenna
and/or two-way relaying technology can be applied. The main
idea of these schemes is based on beamforming techniques
which apply a weighting vector to each antenna or to some
distributed terminals [4], [5], [6].
In this paper, relay nodes of the primary network are
assumed to have a CR capability in the sense that they are able
to communicate with a cognitive base station (CBS) when the
primary transceivers are not in operation [7], [8]. This scenario
requires a spectrum sensing scheme which is performed at
the CBS. We consider the signal-to-noise ratio (SNR) at the
CBS as a criterion for characterizing the spectrum sensing
Fig. 1. System model of the considered cognitive two-way relay network.
performance. We aim at deriving the optimal beamforming
weights for the two-way relay network so as to maximize the
SNR at the CBS subject to some quality of service (QoS)
constraints at the primary transceivers. We show that the
formulated beamforming optimization problem can be written
as a semidefinite form [9]. Then, semidefinite relaxation (SDR)
is applied to change the problem into a convex form and obtain
an efficient solution for the convex problem. This problem is
solved efficiently by an iterative algorithm by using efficient
semidefinite programming (SDP) tools.
II. SYSTEM MODEL AND MAIN ASSUMPTIONS
The block diagram of the considered system model is
shown in Fig. 1. As shown, the CR network consists of
nr terminals and one CBS. The distance between the two
primary transceivers is very large and thus they are not able
to communicate via a direct link.
In our considered scenario, CR nodes are able to sense
the presence/absence of primary transceivers and at the same
time, they are able to play the relay role for establishing the
connection between the primary transceivers. We assume that
the beginning time of each frame are available for all CR nodes
and the we consider this synchronization for sensing process
ICEE2012 1569546155
1
at each frame. Under the assumption of a two-step two-way
amplify-and-forward relaying protocol, the received complex
signal vector x of size nr × 1 at CR terminals during the first
stage of relaying can be written as:
x =√
P1f1s1 +√
P2f2s2 + ν, (1)
where P1 and P2 are the transmit power of two transceivers
(assumed constant), s1 and s2 are the information symbols
transmitted by transceivers 1 and 2, respectively, ν is the
nr × 1 complex noise vector at CR terminals, and fk∆=
[f1k f2k ... fnrk]Tis the vectors of channel coefficients be-
tween the k-th (k = 1, 2) transceiver and CR terminals, and
(.)T is the transpose operator. We assume that the channel
state information (CSI) is known at the two transceivers.
In the second relaying step, each CR node multiplies its
received signal by a complex weight w∗i and broadcasts it
toward the network. Then, the nr×1 complex vector t denotingthe transmitted signal from the i-th CR terminal is written as:
t =Wx, (2)
where the weighting matrixW = diag{[
w∗1 w∗
2 ... w∗nr
]}
and
diag{a} is a diagonal matrix with diagonal elements equal to
the vector a.
A. Received Signals at Primary Transceivers
In the second relaying step, the received signals at the two
transceivers, denoted by y1 and y2, are respectively equal to:
y1 = fT1Wx+ n1
= fT1W(
√
P1f1s1 +√
P2f2s2 + ν
)
+ n1, (3)
and
y2 = fT2Wx+ n2
= fT2W(
√
P1f1s1 +√
P2f2s2 + ν
)
+ n2, (4)
where nk is the received noise at the k-th transceiver during
the second step, for k = 1, 2. Noting that aT diag (b) =bT diag (a), we can rewrite (3) and (4) respectively as:
y1 =√
P1wHF1f1s1 +
√
P2wHF1f2s2
+ wHF1ν + n1, (5)
and
y2 =√
P1wHF2f1s1 +
√
P2wHF2f2s2
+ wHF2ν + n2, (6)
where Fk∆= diag (fk) for k = 1, 2, and w = diag{WH},
and (.)H denotes the Hermitian transpose. Also, diag{A} is avector formed by the diagonal elements of the square matrix
A. Note that the beamforming weight vector w is obtained
from an optimization problem solved at the CBS and then
transmitted among CR/relays via a control channel. Since s1is known at transceiver 1,
√P1w
HF1f1s1 is also known at
the primary transceiver 1 and thus the first term in (5) is
known. Hence, this term can be subtracted from y1 and the
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Fig. 2. Energy detection metric evaluation by data fusion at the CBS.
residual signal can be processed at transceiver 1 to extract
the information s2. Similarly, the second term in (6) can be
subtracted from y2 and the residual signal can be processed
at transceiver 2 to extract the information s1. Therefore, wedefine the residual signals y1 = y1 − √
P1wHF1f1s1 and
y2 = y2 − √P2w
HF2f2s2, as the observation signals used
at their corresponding transceivers to extract the symbol of
the other transceiver.
B. Received Signals at the CBS
In our model, we assume that the cognitive radio network
does not impose any interference on the primary network.
Thus, the primary network operates as a two-way relay net-
work and the CBS senses the channel for eventual opportunis-
tic transmission. The signal y3 received at the CBS during the
second step of relaying is written as:
y3 = fT3Wx+ n3
= fT3W(
√
P1f1s1 +√
P2f2s2 + ν
)
+ n3, (7)
where
f3 = [f13 f23 ... fnr3]T
(8)
is the vector of channel coefficients between the CR users and
the CBS, and n3 is the noise present at the CBS. Similar to
(5) and (6), we can rewrite y3 as:
y3 =√
P1wHF3f1s1 +
√
P2wHF3f2s2
+ wHF3ν + n3, (9)
where F3∆= diag (f3).
Different samples of y3 are gathered at the CBS to make
a decision about the absence or the presence of the primary
transmission during the second step of relaying, as explained
in the next Section.
III. PERFORMING SPECTRUM SENSING AT THE CBS
Here, we explain the energy detection scheme adopted for
spectrum sensing [10], [11] at the CBS. As illustrated in Fig. 2,
spectrum sensing can be viewed as a binary hypothesis testing
2
problem. At the i-th time instant, based on the received signalat the CBS denoted by yi3, from (9) we have [11]:
yi3 =
wHFi3νi + ni
3, H0,√P1w
HFi3fi1s
i1 +
√P2w
HFi3fi2s
i2
+wHFi3νi + ni
3, H1,
(10)
where hypotheses H0 and H1 denote that the primary signal
is absent or present, respectively.
We assume that ni3 and ν
i are zero-mean complex additive
white Gaussian noise (AWGN) with equal variance, i.e., dis-
tributed as CN (0, σ2) and CN ([0]nr×1, σ2Inr
), respectively.Furthermore, the signal samples si1, s
i2 are assumed to be zero-
mean complex Gaussian random processes with unit variances,
i.e., E[|si1|2] = E[|si2|2] = 1. Without loss of generality, si1,si2, ν
i and ni3 are assumed to be independent of each other. We
also assume that channel gains fk for k = 1, 2, 3 are constant
during the detection interval. Under these assumptions, the
received signal at the CBS, i.e., yi3, is a Gaussian random
variable (RV) distributed as:
yi3 ∼{
CN (0, σ21) H0,
CN (0, σ22) H1,
(11)
where σ21 and σ2
2 are the variances of each sample in the two
hypotheses and are respectively equal to:
σ2
1 = E[∣
∣wHF3νi + ni
3
∣
∣
2]
= E[(wHF3νi + ni
3)(νiHFH3 w+ ni
3)]
= σ2(wHF3FH3 w+ 1), (12)
and
σ2
2 = E [ |√
P1wHF3f1s
i1 +
√
P2wHF3f2s
i2
+ wHF3νi + ni
3
∣
∣
2
]
= P1wHF3f1f
H1 F
H3 w+ P2w
HF3f2fH2 F
H3 w
+ σ2(wHF3FH3 w+ 1), (13)
where E[.] denotes expectation.The measured energy summation at the CBS over a detec-
tion interval composed of N samples yi3 (i = 1, 2, ..., N ) is
thus given by:
Z =
N∑
i=1
|yi3|2
=
∑N
i=1
∣
∣wHF3νi + ni
2
∣
∣
2, H0,
∑N
i=1
∣
∣
√P1w
HF3f1si1 +
√P2w
HF3f2si2
+wHF3νi + ni
3
∣
∣
2, H1.
(14)
The test statistic Z is the sum of the squares of N Normal
random variables. Then, Z has a central chi-square X 2 dis-
tribution with N degrees of freedom for hypothesis H0 and
H1.
According to the central limit theorem (CLT), if the number
of samples N is large enough, the test statistic follows normal
distributions [10]. Then, we can write the test statistic Z as
[12]:
Z ∼{
N (Nσ21 , 2Nσ4
1), H0,
N (Nσ22 , 2Nσ4
2), H1.(15)
Let λ be the decision threshold used at the CBS to decide
between the two above hypothesis. Therefore, the false-alarm
probability PF , and the detection probability PD , usually
defined in CR systems can be obtained from (15) as:
PF = Pr(Z > λ|H0) = Q
(
λ−Nσ21
√
2Nσ41
)
, (16)
and
PD = Pr(Z > λ|H1) = Q
(
λ−Nσ22
√
2Nσ42
)
, (17)
where Q(.) is the standard Gaussian complementary cu-
mulative distribution function (CDF), defined as Q(x) =1√2π
∫∞x
exp(−u2
2) du. By setting the threshold λ at the CBS
so as to have a desired probability of false-alarm equal to PF ,
we get:
λ =√
2Nσ41Q−1(PF ) +Nσ2
1 , (18)
Substituting λ from (18) in (17), we can rewrite the probability
of detection as follows:
PD = Q
(
√
2Nσ41Q−1(PF ) +Nσ2
1 −Nσ22
√
2Nσ42
)
= Q
√2NQ−1(PF ) +N −N
(
σ2
σ1
)2
(
σ2
σ1
)2 √2N
. (19)
Let us define SNR3 as the ratio of the two transceivers’
received energy to the energy of the additive noise at the CBS.
This parameter can be formulated as follows:
SNR3 =
P1wHF3f1f
H1 F
H3 w+ P2w
HF3f2fH2 F
H3 w
σ2wHF3FH3 w+ σ2
. (20)
From (12) and (13), we can rewrite(
σ2
σ1
)2
involved in (19)
as:(
σ2
σ1
)2
=
1 +P1w
HF3f1fH1 F
H3 w+ P2w
HF3f2fH2 F
H3 w
σ2wHF3FH3 w+ σ2
,
= 1 + SNR3, (21)
and consequently, the probability of detection in our proposed
two-way cognitive relay model can be rewritten as:
PD = Q
(√2NQ−1(PF )− SNR3N
(1 + SNR3)√2N
)
. (22)
3
We observe from (22) that increasing SNR3 increases the
Q-function. Thus, the probability of detection is increased
when SNR3 increases.
We propose here to use SNR3 as a detection criterion
in the fusion center. More precisely, instead of using the
energy summation as in (14), we propose to use SNR3 as
discriminator between the two hypothesis. In this way, the
binary hypothesis involved in spectrum sensing at the CBS
can be written as:{
SNR3 ≤ γ3, H0,
SNR3 > γ3, H1,(23)
where γ3 is a predefined SNR threshold ensuring a target
detection probability. In next section, we set this decision
threshold as a constraint of the optimization problem.
IV. OPTIMAL SPECTRUM SENSING
In this section, we formulate our optimization problem in
order to increase the spectrum sensing performance in the CR
network while both transceivers meet their minimum SNR
requirements. Thereafter, we change the form of the problem
to a convex form and an algorithm is proposed to provide the
solution. First, we formulate our optimization problem as:
maxw
SNR3
s.t. SNR1 ≥ γ1,
SNR2 ≥ γ2,
PT ≤ PmaxT (24)
where PT = P1+P2+Pr, and Pr is the sum of transmit power
of CR nodes and PmaxT is the maximum transmit power of
the total network. Note that P1 and P2 are assumed constant
in this problem because generally, the secondary network is
not allowed to control the power of the primary transmitter,
and here our optimization problem is solved at the secondary
network. The relay transmit power Pr is given by:
Pr = E{
tH t}
= wH(
P1F1FH1 + P2F2F
H2 + σ2
I)
w
= wH(
P1D1 + P2D2 + σ2I)
w
= wHDw, (25)
where D1
∆= F1F
H1 , D2
∆= F2F
H2 and D =
(
P1D1 + P2D2 + σ2I)
are diagonal matrices with real and
positive elements.
The received SNRs in the second transmission step at
transceivers 1 and 2 can be expressed respectively as:
SNR1 =P2w
HhhHw
σ2 + σ2wHD1w, (26)
and
SNR2 =P1w
HhhHw
σ2 + σ2wHD2w(27)
where h∆= F1f2 = F2f1. Similarly, we can rewrite (20) as
follows:
SNR3 =P1w
Hh′h′Hw+ P2wHh′′h′′Hw
σ2 + σ2wHD3w, (28)
where h′∆= F3f1, h
′′ ∆= F3f2 and D3
∆= F3F
H3 .
We define X = wwH and note that X = wwH is equivalent
to X which is a rank one symmetric positive semidefinite
(PSD) matrix. We change the form of the problem as:
maxX
Tr[(
P1H′ + P2H
′′)X]
σ2 + Tr [σ2D3X]
s.t. Tr((P2H− γ1D1)X) ≥ γ1,
Tr((P1H− γ2D2)X) ≥ γ2,
P1 + P2 + Tr[(
P1D1 + P2D2 + σ2)
X]
≤ PmaxT ,
X = wwH , X � 0 and rank(X) = 1, (29)
where wHAw = Tr[
A(wwH)]
, and H = hhH , H′ = h′h′H
and H′′ = h′′h′′H . By using the idea of semidefinite relaxationand dropping the non-convex rank-one constraint, the problem
can be relaxed as:
maxX,t
t
s.t. Tr[(
P1H′ + P2H
′′ − tσ2D3
)
X]
≥ σ2t
Tr((P2H− γ1D1)X) ≥ γ1,
Tr((P1H− γ2D2)X) ≥ γ2,
P1 + P2 + Tr[(
P1D1 + P2D2 + σ2)
X]
≤ PmaxT ,
X � 0. (30)
In (30), if the value of t is kept fixed, the set of feasible Xis convex. By solving (30), the maximum achievable SNR at
the CBS can be obtained where the maximum value of t, isdenoted as tmax.
To solve (30), we use an iterative algorithm provided in [13].
For some given SNR value t, the convex feasibility problem
is:
find X
s.t. Tr[(
P1H′ + P2H
′′ − tσ2D3
)
X]
≥ σ2t
Tr((P2H− γ1D1)X) ≥ γ1,
Tr((P1H− γ2D2)X) ≥ γ2,
P1 + P2 + Tr[(
P1D1 + P2D2 + σ2)
X]
≤ PmaxT ,
X � 0. (31)
which is feasible when tmax ≥ t and if tmax < t, then(31) is not feasible. Based on this observation, one can check
whether the optimal value tmax of the quasi-convex problem
(30) is smaller or greater than any given value of t. To find themaximum value of t while holding the problem feasible, we
can use a simple bisection algorithm provided in Table. I. Let
us start with some preselected interval [tl tu] that is knownto contain the optimal value tmax, where the problem (31) is
solved at the midpoint t = (tl + tu)/2. If (31) is feasible forthis value of t , then tl = t is set; otherwise tu = t is chosen.
4
This procedure is repeated until the difference between tuand tl is smaller than some preselected threshold. To find the
maximum value of t while holding the feasibility of problem,we can use a simple bisection algorithm provided in Table. I.
TABLE IITERATIVE ALGORITHM FOR FINDING THE OPTIMAL POINT t IN (30) [13]
Step 1 Properly set the initial values of tl and tu.Step 2 Set t := (tl + tu)/2 and solve (31).
Step 3 If (31) is feasible, then tl := t;otherwise tu := t.
Step 4 If tu − tl < δ , then go to Step 5;otherwise go to Step 2.
Step 5 Find the weight vector from the principal eigenvectorof the resulting matrix Xopt.
V. NUMERICAL RESULTS
We consider three channel vectors f1, f2 and f3 with
elements distributed as complex zero-mean Gaussian random
variables with variance respectively equal to σ2
f1, σ2
f2and σ2
f3.
We assume that noise power for the three channels (denoted
by σ2) is also equal to one and the transmit power of two
transceivers are 10 dB (P1 = P2 = 10 dB). We have used
a number of 200 random channel realizations for averaging
the solution of our optimization problem. The optimization
problem is solved by using the convex optimization toolbox
CVX [14]. For performance comparison, we consider two
competitive beamforming technique. We first consider a uni-
form beamforming weighting vector for relay nodes in which,
to ensure a fair comparison, the total transmit power of relays
is equal to the power achieved by our proposed method,
but this power is uniformly distributed among relays. Then,
we consider a conventional beamforming, i.e., without any
cognitive radio capabilities where the weighting vector is set
so as to minimize the total power dissipated in the two-way
relay network.
This comparison is shown in Fig. 3, where we compare
the receiver operating characteristic (ROC) of our proposed
beamforming method with ROCs obtained with uniform beam-
forming and the conventional method. This figure illustrates
that considering the proposed method reduces significantly the
miss-detection probability while providing the requirements at
the primary network, compared to uniform weight beamform-
ing and conventional methods.
In Fig. 4, the effect of number of CR nodes in the per-
formance of spectrum sensing is depicted. By increasing the
number of CR nodes, the cooperative diversity provides higher
SNR value in CBS while this performance decreases by bad
conditions of CR-CBS channels.
The effect of channel variations in the performance of spec-
trum sensing in terms of the SNR at the CBS is depicted in Fig.
5. This figure depicts the SNR at the CBS versus the channel
variances σ2
f1and σ2
f2, achieved by using the proposed and
conventional beamforming methods, respectively. It is clearly
seen from this figure that the proposed method increases the
SNR and as a consequence increases the sensing accuracy (in
10−6
10−4
10−2
100
10−4
10−3
10−2
10−1
100
Pf
Pm
Uniform Beamforming
Conventional Beamforming
Proposed Method
Fig. 3. Probability of miss-detection versus the probability of false-alarm(ROC) for proposed distributed beamforming, uniform beamforming andconventional beamforming; nr = 10 and γ1 = γ2 = 10 dB.
5 7 9 11 13 1514.5
15
15.5
16
16.5
17
17.5
18
18.5
19
nr
SNR3 (dB)
σf3
2 = 0 (dB)
σf3
2 = 10 (dB)
Fig. 4. The performance of spectrum sensing versus the number of CRs, fordifferent channel variances σ2
f3and γ1 = γ2 = 10 dB.
terms of detection probability) at the CBS compared to the
conventional beamforming method.
Figure. 6 illustrates the effect of the minimum requirements
of two transceivers in spectrum sensing. In this figure, since the
constant transmit power of two transceivers are high enough,
the two SNR constraints for the optimization problem are
satisfied and the SNR value is not changed for different
γ1 = γ2. Besides, the channel conditions of relaying links
changes the performance of the spectrum sensing. When the
variance of relaying links increases, CR nodes needs lower
power for satisfying the constraints and the remained power
leads to an increase in the SNR value at the CBS.
VI. CONCLUSION
In this work, we formulated the problem of distributed
beamforming in a two-step two-way cognitive radio network.
For a given set of minimum required SNR at the transceivers,
5
0 2 4 6 8 1012
14
16
18
20
22
24
26
28
σf1
2 = σ
f2
2 (dB)
SNR3 (dB)
Proposed Method: σf3
2 = 0 (dB)
Proposed Method: σf3
2 = 10 (dB)
Conventional Method: σf3
2 = 0 (dB)
Conventional Method: σf3
2 = 10 (dB)
Fig. 5. The performance of spectrum sensing versus the channel variancesσ2
f1= σ2
f2for different channel variances σ2
f3; nr = 10 and γ1 = γ2 =
10 dB.
4 6 8 10 12 14 16
5
10
15
20
25
γ1=γ
2 (dB)
SN
R3 (
dB
)
Proposed Method: σf1
2 = σ
f2
2 = 0 (dB)
Proposed Method: σf1
2 = σ
f2
2 = 10 (dB)
Conventional: σf1
2 = σ
f2
2 = 0 (dB)
Conventional:= σf1
2 = σ
f2
2 = 10 (dB)
Fig. 6. The performance of spectrum sensing at the CBS versus differentvalues of minimum SNR in two transceivers, for σ2
f1= σ2
f2= 0 and 10 dB,
nr = 10 and σ2
f3= 0 dB.
an optimal beamforming weight vector was obtained to max-
imize the SNR at the CBS. The optimization problem was
relaxed to semidefinite form and solved efficiently by an
iterative algorithm. Simulation results demonstrated that the
performance of spectrum sensing is increased for large number
of CR terminals and our proposed SNR maximization method
improves the performance of spectrum sensing in comparison
with uniform and conventional two-way relay beamforming
methods.
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