(38)
where i denotes ith lag time.
Therefore, for a speciﬁc value of p, if the diﬀerence between the means of null
and alternative hypotheses is maximized, it can be concluded that the performance has improved. Due to complex relations obtained for the means in (25)
and (37), diﬀerentiation and solve the result of its equation for this purpose
is not possible, however, with the help of numerical results, we can obtain the
optimal amount of fractional order, p.
In Fig. 1, diﬀerence of means under two hypotheses for a certain lag time is
2
. In this ﬁgure,
plotted versus changes of p for various value of noise variance, σw
12
H. Hashemi et al.
0
Probability of miss detection P
m
10
−1
10
−2
10
−3
10
p = 0.65
p = 0.75
p = 0.85
p = 0.90
p = 1.00
−4
10
−4
10
−3
10
−2
10
Probability of false alarm
−1
10
0
10
Fig. 2. The complementary ROC of proposed detector for average SN R = −3dB.
the values are normalized with respect to means diﬀerence value in p = 1 which
is used in cyclostationary detectors. As can be seen in Fig. 1, for example, the
2
= 1 and
diﬀerence of means increases about 0.75 percent in p = 0.75 for σw
also for other value of noise variance, we can found speciﬁc p that improves the
detector performance.
7
Simulation Results
In this section, we provide simulation results of cyclostationary-based detectors
performance in fractional order of observations Monte Carlo simulation and we
compared it with other detectors. For this purpose, we assume a linear modulation for PU signals which its pulse width for outgoing data is 1ms. This signal
has Gaussian distribution with unit variance which has been sampled in receiver.
To detect these signals that aﬀected by environmental additive Gaussian noise,
we have used cyclostationary detector in fractional order of observations. Also,
we assume the number of lag times is 16.
In Fig. 4, performance of this detector has been investigated in orders of
p = 0.65, 0.75, 0.85, 0.9 and 1, with the probability of detection Pd versus SN R
2
= 1 and ﬁxed probability of false alarm 0.01. As can be seen,
with assumption σw
by changing the fractional orders, the detector performance will changes and
when the value get close to 0.75, detector performance improves approximately
3dB compared to p = 1 has been used used in previous detectors. This change
and improvement is due to an increase in mean diﬀerence of observations under
the two hypotheses.
Fractional Low Order Cyclostationary-Based Detector
13
1
Probability of detection P
d
0.9
0.8
Tsub1
TCCA
Tsub2
0.7
Tsub3
0.6
Tratio
0.5
0.4
0.3
0.2
0.1
0
−15
−10
−5
SNR(dB)
0
5
Fig. 3. The probability of detection of diﬀerent detectors versus SN R for Pfa = 0.01.
1
0.9
Probability of detection P
d
0.8
0.7
p = 1,00
p = 0.90
p = 0.85
p = 0.75
p = 0.65
Performance Analysis
for p = 0.75
0.6
0.5
0.4
0.3
0.2
0.1
0
−15
−10
−5
SNR(dB)
0
5
Fig. 4. The probability of detection of Tsub1 versus SN R for Pfa = 0.01 and various
2
= 1.
fractional moment with assumption σw
Fig. 2 depicts the receiver operating characteristics (ROC) curve of proposed
cyclostationary detector for diﬀerent fractional order of observations. This ﬁgure
reveals of the detector behavior for diﬀerent values of the false alarm probability
Pfa .
In Fig. 3, performance of detectors has been investigated by the probability of
2
= 1 and ﬁxed probability of false
detection Pd versus SN R with assumption σw
alarm 0.01. This ﬁgure compares performance of obtained GLR-based detectors
with detectors that are mentioned in [15,16]. In [15], the ratio of CAF absolute
14
H. Hashemi et al.
value in cyclic frequency and another amount has been proposed as detector,
Tratio =
α
Rxx
∗ (τ )
α+δ
Rxx
∗ (τi )
, where δ is a frequency shift. In [16], authors by using canon-
ical correlation analysis to detect presence of PU signal for M antennas SU. If
λm is mth eigenvalue of canonical correlation analysis result, statistic is deﬁned
M
as, TCCA = m=1 ln(1 − λ2m ). As we expected, when noise and signal variance
are known, the best performance of the detector can be achieved.
8
Conclusion
In this paper, we investigated the problem of cyclostationary spectrum sensing
in cognitive radio networks based on cyclic properties of linear modulated signal. First, we derived GLR detector for the situation in which SU has knowledge
of cyclic frequency of signal. Then, we found the optimum value for fractional
moment of observations in additive Gaussian noise and the exact performance of
the GLR detector is evaluated analytically. Finally, we simulated and derived the
GLR detector performance for various values of fractional moment of observations. We revealed that GLR detector performance improves for Gaussian noise
if we use fractional moment of observation for any value of noise variance. We
found the optimum value for the fractional moment, p. Our results have been
conﬁrmed by simulation.
Acknowledgments. This publication was made possible by the National Priorities Research Program (NPRP) award NPRP 6-1326-2-532 from the Qatar National
Research Fund (QNRF) (a member of the Qatar Foundation). The statements made
herein are solely the responsibility of the authors.
Appendix
Covariance Matrices Estimation
According to [14], in order to calculate of correlation between two lag times mth
and nth of CAF, we need,
1
Sxτm xτn (2α, α) =
T
1
Sx∗τm xτn (0, −α) =
T
T −1
2
W (s)Fτn (α −
2πs
2πs
)Fτm (α +
),
N
N
(A-1)
W (s)Fτ∗n (α +
2πs
2πs
)Fτm (α +
).
N
N
(A-2)
s=− T −1
2
T −1
2
s=− T −1
2
Where Sxτm xτn (2α, α) and Sx∗τm xτn (0, −α), respectively are unconjugated and
conjugated cyclic-spectrum of observations and
1
Fτ (ω) = √
N
N −1
n=0
xp (n)x∗p (n + τ )e−jωn .
(A-3)
Fractional Low Order Cyclostationary-Based Detector
15
Thus, covariance matrix estimation of vector rα
xx∗ can be calculated as,
[Σ]i,j = Re{
Sxτi xτj (2α, α) + Sx∗τ
i
xτj (0, −α)
2
}, i, j = 1, 2, ..., M.
(A-4)
pth Moment of Gaussian Random Variable
Suppose N is a Gaussian random variable with mean μ and variance σn2 . Thus,
−
p
μ2
2 2 σnp e 2σn2
√ p
E[N ] =
πj
p
By assumption of β 2 = 1 and q =
calculated for p > −1 as follows,
−
p
μ2
2 2 σnp e 2σn2
√ p
E[N p ] =
πj
−p
2
2
(jt)p e
√
2μj
σn
−t2 −j
√
2μj
σn t
dt.
(B-1)
in section 3.462 of [16], (B-1) has been
μ2
√ 4σ
πe n2 Dp
jμ
σn
−
μ2
σ p e 4σn2
Dp
= n p
j
where Dp (.) is parabolic cylinder function,
√
√
p
−z 2
π
2πz
p 1 z2
Dp (z) =2 2 e 4
Φ
−
1−p Φ − 2 , 2 ; 2
Γ − p2
Γ 2
jμ
σn
1 − p 3 z2
, ;
2
2 2
,
(B-2)
, (B-3)
and also Φ(., .; .) is Kummer conﬂuent hypergeometric function, Φ(a, b; c) =
∞ ak ck
k=0 bk k! .Where,
ak is rising factorial function, ak =
Γ (a+k)
Γ (a) .
Mean and Variance of (12)
Mean of (12) under two hypotheses is,
μTsub3 |Hν =
1
M
M
μν (m), ν = 0, 1,
(C-1)
m=1
and variance of (12) can be calculated as follows,
2
σT
=
sub3 |Hν
1
M2
M
M
α
α
E[rxx
(C-2)
∗ (m1 )rxx∗ (m2 )|Hν ] − μν (m1 )μν (m2 ).
m1 =1 m2 =1
Therefore, variance of (12) is sum of (A-4) entries.
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Achievable Rate of Multi-relay Cognitive Radio
MIMO Channel with Space Alignment
Lokman Sboui(B) , Hakim Ghazzai, Zouheir Rezki, and Mohamed-Slim Alouini
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division,
King Abdullah University of Science and Technology (KAUST),
Thuwal, Makkah Province, Saudi Arabia
{lokman.sboui,hakim.ghazzai,zouheir.rezki,slim.alouini}@kaust.edu.sa
Abstract. We study the impact of multiple relays on the primary user
(PU) and secondary user (SU) rates of underlay MIMO cognitive radio.
Both users exploit amplify-and-forward relays to communicate with the
destination. A space alignment technique and a special linear precoding
and decoding scheme are applied to allow the SU to use the resulting
free eigenmodes. In addition, the SU can communicate over the used
eigenmodes under the condition of respecting an interference constraint
tolerated by the PU. At the destination, a successive interference cancellation (SIC) is performed to estimate the secondary signal. We present
the explicit expressions of the optimal PU and SU powers that maximize
their achievable rates. In the numerical results, we show that our scheme
provides cognitive rate gain even in absence of tolerated interference.
In addition, we show that increasing the number of relays enhances the
PU and SU rates at low power regime and/or when the relays power is
suﬃciently high.
Keywords: Underlay cognitive radio
Amplify-and-forward multiple-relay
1
·
MIMO space alignment
·
Introduction
In order to cope with the continuous growth of wireless networks, new emerging systems need to oﬀer higher data rate and to overcome bandwidth shortage. Consequently, many techniques have been presented to enhance the network performances and spectrum scarcity [1]. From one side, the cognitive radio
(CR) paradigm was introduced to avoid spectrum ineﬃcient allocation. In this
paradigm, unlicensed secondary users (SU’s) are allowed to share the spectrum
with licensed primary users (PU’s) under the condition of maintaining the PU
quality of service (QoS). One of the CR modes is the underlay mode in which
the PU tolerates a certain level of interference coming from the SU [2]. From
the other side, relay-assisted communications [3], was introduced as a solution
to considerably enhance distant and non-line of sight communications. The relying was ﬁrst intended to enhance single-antenna communications. Nevertheless,
relaying in MIMO systems was shown to improve performances as well [4]. In
addition, adopting MIMO power allocation within a CR framework has been
c Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 2015
M. Weichold et al. (Eds.): CROWNCOM 2015, LNICST 156, pp. 17–29, 2015.
DOI: 10.1007/978-3-319-24540-9 2
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