Recall from [15] that Qχ2N(γ) is the detection probability for ED when SNR = σ2s/σv2 ≈ 0.
In this case, the performance of ED can be very poor since the energy of the received signal in either hypothesis is very close to the noise floor. To further enhance the detection performance when SNR is low and the signal timing mismatch is present, robust ED schemes based on the Bayesian principle and the GLRT principle will be proposed in next two chapters.
2.3 Simulation Results
In the following simulations we consider the hypothesis signal model (1.2), in which the total number of samples is set to be N = 200 and the primary signal arrival time n0 is uniformly distributed within 0 ≤ n0 ≤ 199. Note that the simulated results are obtained from 5000 Monte-Carlo runs. Figure 2.1 plots the ROC curves of ED (2.1), with SNR set to be −5 dB;
Figure 2.2 plots the probability of detection PD at various SNR levels, assuming that the false-alarm probability PF A = 0.05. As can be seen from the figures, the derived analytic formula (2.25) closely matches the simulated results.
0 0.2 0.4 0.6 0.8 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PFA
P D
Experimental Analytic
Figure 2.1: Analytic and experimental ROC curves of ED. (N = 200, SNR = −5 dB)
−150 −10 −5 0 5 10 15 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SNR (dB)
P D
Experimental Analytic
Figure 2.2: Detection probability PD versus SNR. (N = 200, PF A = 0.05)
Chapter 3
Bayesian Based Detection
3.1 The Test Statistic of Bayesian Detection
To exploit the prior statistical knowledge of n0 for enhancing the detection performance, a typ-ical approach is the Bayesian philosophy [15]. The conditional joint PDF of the data samples under two hypotheses H0 and H1 are
p(x, H0) = 1
The Bayesian test decides H1 if [15]
p(x; n0, H1)
After some manipulations of (3.3), the test statistic of Bayesian detection can be represented as
1 N
N −1X
n0=0
σv2 σ2v + σs2
(N −n0)/2
exp
"
1
2σv2 − 1 2(σv2+ σs2)
N −1
X
n=n0
|x[n]|2
#
> γ. (3.4)
3.2 Simulation Results
The following simulation results are obtained from 5000 Monte-Carlo runs under the hypothesis signal model (1.2), in which the total number of samples is set to be N = 200 and the primary signal arrival time n0 is uniformly distributed within 0 ≤ n0 ≤ 199. Figure 3.1 compares the ROC curves of ED (2.1) and the Bayesian based detection rule (3.4). Figures 3.2 and 3.3, respectively, compare PD and 1 − PF A curves (as a function of SNR) of the ED (2.1) and the Bayesian based solution (3.4); note that large values of 1 − PF A mean better channel utilization efficiency of secondary users [17]. The figures show that the Bayesian based solution (3.4), which takes into account the statistical knowledge of the primary signal arrival time, not only improves PD but also leads to larger 1 − PF A, especially when SNR is low.
0 0.2 0.4 0.6 0.8 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PFA
P D
Bayesian (3.4) ED (2.1)
Figure 3.1: Experimental ROC curves of ED and Bayesian ED. (N = 200, SNR = −5 dB)
−150 −10 −5 0 5 10 15 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SNR (dB)
P D
Bayesian (3.4) ED (2.1)
Figure 3.2: Detection probabilities PD of ED and Bayesian ED versus SNR. (N = 200, PF A = 0.05)
−150 −10 −5 0 5 10 15 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SNR (dB) 1 − P FA
Bayesian (3.4) ED (2.1)
Figure 3.3: 1 − PF A of ED and Bayesian ED versus SNR. (N = 200, PD = 0.95)
Chapter 4
GLRT Based Detection
4.1 The Test Statistic of GLRT Based Detection
In chapter 3, we consider n0 as a uniform random variable, and then propose a Bayesian based detection rule to deal with the timing mismatch. However, the prior statistic knowledge of n0
is not always available at the receiver. Hence, instead of modeling n0 as a random variable, an alternative approach is to consider time delay n0 as a deterministic unknown, and resort to the GLRT based test rule for spectrum sensing. Recall the joint PDF of the data samples under the two hypotheses H0 and H1 are
According to [15] and after some straightforward manipulations, the GLRT decides H1 if the test statistic exceeds a threshold γ
LG(x) := max
Since the test statistic is maximized over all possible value of n0, the primary user arrival time can also be estimated.
4.2 Performance Analysis
The probability of false-alarm of the test rule (4.3) is by definition given by
PF A = P r and the probability of detection is
PD = P r However, neither the exact form of PF A nor the exact form of PD exist. We then try to derive a lower bound of PD and that of PF A.
The probability of false-alarm PF A in (4.4) can be expressed as
PF A = 1 − P r and it will be lower bounded by
PF A ≥ 1 − 1
Since σv2, σs2, and n0 are known, (4.7) can be further rewritten as where (d) follows sincePN −1
n=n0
|x[n]|2
σ2v ∼ χ2N and P (·, ·) is the regular Gamma function.
On the other hand, the probability of detection PD in (4.5) will be similarly lower bounded by
PD ≥ 1 − 1
4.3 Simulation Results
In the following simulations the total number of samples is set to be N = 100 and the Monte-Carlo run is 5000. For SNR = 5 dB, Figure 4.1 compares the ROC curves of the ED (2.1) and the GLRT (4.3) for two arrival time n0 = 56, 96. It is seen from the figure that the performance of ED is poor for n0 = 96, and, in this case, the GLRT (4.3) does significantly improve the detection probability. With fixed n0 = 96 and PF A = 0.1, Figure 4.2 plots the detection probability of ED (2.1) and the GLRT (4.3) as a function of SNR. As expected, the GLRT performs better over a wide range of SNR. By setting PD = 0.9, Figure 4.3 plots 1 − PF A versus SNR (with n0 = 96), whereas Figure 4.4 depicts 1 − PF A versus n0 (with SNR= 0 dB) for ED (2.1) and GLRT (4.3). The figures show that the GLRT does enhance the spectrum utilization efficiency, especially when SNR is small to moderate and is large. Figure 4.5, Figure 4.6, and Figure 4.7 examine the tightness of the lower bound of PD (4.9) by plotting ROC curves and PD versus SNR respectively. As we can see, the lower bound is close to the simulated PD when SNR is large.
0 0.2 0.4 0.6 0.8 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PFA
P D
GLRT (4.3), n
0 = 56 ED (2.1), n
0 = 56 GLRT (4.3), n
0 = 96 ED (2.1), n0 = 96
Figure 4.1: Experimental ROC curves of ED and GLRT ED with two different n0. (N = 100, SNR = 5 dB)
−100 −5 0 5 10 15 20 25 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SNR (dB)
P D
GLRT (4.3) ED (2.1)
Figure 4.2: Detection probability PD of ED and GLRT ED versus SNR. (N = 100, n0 = 96, PF A = 0.1)
−100 −5 0 5 10 15 20 25 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SNR (dB) 1 − P FA
GLRT (4.3) ED (2.1)
Figure 4.3: 1 − PF A of ED and GLRT ED versus SNR. (N = 100, n0 = 96, PD = 0.9)
0 10 20 30 40 50 60 70 80 90 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
n0
1 − P FA
GLRT (4.3) ED (2.1)
Figure 4.4: 1 − PF A of ED and GLRT ED versus n0. (N = 100, SNR = 5 dB, PD = 0.9)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PFA
P D
GLRT (4.3)
GLRT Lower Bound (4.9)
Figure 4.5: Experimental ROC curve and the lower bound of PD of GLRT ED. (N = 100, SNR
= 5 dB)
0 0.2 0.4 0.6 0.8 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
PFA
P D
GLRT (4.3)
GLRT Lower Bound (4.9)
Figure 4.6: Experimental ROC curve and the lower bound of PD of GLRT ED. (N = 100, SNR
= −5 dB)
−150 −10 −5 0 5 10 15 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SNR (dB)
P D
GLRT (4.3)
GLRT Lower Bound (4.9)
Figure 4.7: PD and the lower bound of PD of GLRT ED versus n0. (N = 100, PF A = 0.1)
Chapter 5 Conclusion
Spectrum sensing in the presence of unknown arrival time of the primary signal finds appli-cations in many practical system scenarios and is thus an important issue in the study of CR networks. In this thesis we derive the exact formula of conditional detection probability given the primary signal arrival time for ED. when the primary signal arrival time is modeled as a uniform random variable over the observation interval, the exact detection probability for ED can be obtained by averaging the conditional detection probability over all possible arrival time. To further improve the detection performance against the timing uncertainty, we then propose a Bayesian based detection scheme. Moreover, when the prior statistical knowledge of the primary signal arrival time is not available, we consider the time delay as a deterministic unknown, and then proposed a GLRT based detection rule. Simulation results show that the Bayesian ED and the GLRT ED not only improve the detection probability but also reduce the false-alarm probability, thus enhancing the spectrum utilization in the considered asyn-chronous scenario. Future research will be dedicated to characterizing the ROC performance of the Bayesian scheme and extending the current results to the cooperative sensing scenario.
Appendix A
Proof of Lemma 2.5
We first observe that p(x) in (2.19) satisfies e−x/2× e−SN Rx/2
By taking the inverse Laplace transform of both sides of (A.3) we have Z x where the last equality holds due to Lemma 2.1. With the aid of (A.4), (A.1) becomes
Γ((N − n0)/2)Γ(n0/2)
Γ(N/2) xN/2−1e−(1+SN R)x/2
≤ p(x) ≤ Γ((N − n0)/2)Γ(n0/2)
Γ(N/2) xN/2−1e−x/2. (A.5)
Based on (A.5), we have
PD(n0) = (1 + SNR)−(N −n0)/2
√2NΓ(n0/2)Γ ((N − n0)/2) Z ∞
γ
p(x)dx
≥ (1 + SNR)−(N −n0)/2
√2NΓ(N/2)
Z ∞ γ
xN/2−1e−(1+SN R)x/2dx
(a)= (1 + SNR)−(N −n0)/2
√2NΓ(N/2)
1 + SNR 2
−N/2
Γ N
2, γ1 + SNR 2
= Γ N2, γ1+SN R2
(1 + SNR)n0/2+1Γ(N/2), (A.6)
where (a) follows sinceR∞
γ xν−1e−µxdx = µ−νΓ(ν, µγ) [16]. Similarly we have PD(n0) ≤ (1 + SNR)(N −n0)/2−1
Γ(N/2) Γ N
2,γ 2
. (A.7)
The assertion follows from (A.6) and (A.7). 2
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