The parameters values used in our simulations are as follows. M secondary users are randomly distributed within the 5 km radius of the fusion center. During the sensing time, each secondary keeps sampling its energy detector output at a rate of W samples per second and reporting its LLR value, if necessary, to the fusion center until it is told by the center to stop sensing. The path loss exponent factor α, in (3.1) is set to be 3.5, and P1are set at a value such that the SNR of one sample in SPRT is -2 dB at the fusion center and the secondary BS (fusion center) is 59.7 km away from the primary user. The secondary users’ thresholds are determined by PM = PF = 0.01, and ¯Kf 0 = ¯Kf 1 = ¯Kf. It is also assumed that each secondary user can estimate either di or γi perfectly.
Fig. 5.1 compares the designed ¯Kf i and the true ¯Kf i for various sensing methods with 20 secondary users. The fact that all curves but that corresponds to the FSST-FSST and FSST-FSST-SPRT scheme are not straight lines with slop 1 is due obviously to the nonzero excesses that occur when the fusion center performs the SPRT.
Fig. 5.2 shows the total sensing time of all 4 distributed sensing methods for different K¯f. These curves do exhibit some discontinuities as we applied the SPRT method at the FC. In order to explain these discontinuities, we take the SPRT-SPRT approach as an example. (4.24) shows that the average sensing time for the SPRT-SPRT method is a function of E[Lf|Hi], E[Lf e|Hi], E[Lse|Hi], and E[Lsu|Hi]. But it is only E[Lf e|Hi] that results in the discontinuities. The reason has been given in Section 4.3. Fig. 5.3
and Fig. 5.4 plot the detection and false alarm probabilities of the 4 proposed sensing methods which all meet the performance requirement.
Fig. 5.5 plots these four terms in (4.24) with 20 secondary users and reveals that E[Lf|Hi] remains constant, E[Lse|Hi] is small and insensitive to the threshold while E[Lsu|Hi] is a decreasing function of the threshold but E[Lf e|Hi] is not a continuous function of ¯Kf i. We can find that the excess at FC leads to the discontinuous in the Fig. 5.2 for the SPRT-SPRT method.
Fig. 5.6 shows the SPRT-SPRT predicted ¯Kf 0 and ¯Kf 1 versus the true ¯Kf 0 and K¯f 1 with 20 secondary users in AWGN, slow Rayleigh and Rician fading channels. Slow fading represents the case that in one sensing process the channel state (signal am-plitude) remain the same. That is, the H in (2.44) is unchanged in one sensing try.
The corresponding average sensing time and the detection and false alarm probabilities performance are given in Figs. 5.7, 5.8 and 5.9, respectively. For the Rician fading case, as the Ricain factor becomes larger, the channel becomes more and more close to a AWGN channel while as the Ricain factor becomes small, it converges to Rayleigh fading. As expected, the SPRT-SPRT method suffers from some performance loss in fading channels.
Fig. 5.10 compares the designed ¯Kf 0 and the true ¯Kf 0 and ¯Kf 1 with 20 secondary users in correlated Rayleigh fading channel; both slow and fast fadings are considered.
Fast fading assumes that the channel state changes for each sample. Fig. 5.11 shows the average sensing time and Figs. 5.12 and 5.13 show the detection and false alarm probabilities performance. The channel is based on Jakes’ model and the parameter values are derived from the IEEE 802.22 standard. The carrier frequency is 57 MHz and the bandwidth is 1 MHz. As the speed becomes higher, the average sensing time under H1 becomes shorter and the discrepancy between the true average reporting bits and that predicted by theory becomes smaller. This because when the speed becomes larger, the correlation is smaller, the PDF of output of ED meet the result we get in
section 2.2. For slow fading and jakes model, we have some performance loss because the signals are not i.i.d random variable in one sensing process.
Fig. 5.14 plots the SPRT-SPRT based ¯Kf 0and ¯Kf 1versus the true ¯Kf 0and ¯Kf 1with 20 secondary users in slow and fast Rician fadings when Rician factor is 2. Similarly, shown in Fig. 5.15 is the corresponding sensing time, Figs. 5.16 and 5.17 the detection and false alarm probabilities performance. Similar to the Rayleigh fading case, some performance degradation has been observed in slow fading channels.
The performance comparison in Nakagami-m fading channels is shown in Fig. 5.18, 5.19, 5.20 and 5.21. A larger m implies smaller average sensing time as the degree of fading becomes less severe. Other performance trends are similar to the other fading cases.
We examine the AWGN performance when a advanced adjustment method like ECUBE or ECRE is used in conjunction with the SPRT-FSST or SPRT-SPRT schemes in Figs. 5.22–5.24 ( ¯Kf 0 and ¯Kf 1 comparison), Figs. 5.23–5.25 (the average sensing time), and Figs. 5.26–5.27 (false alarm and detection probabilities with noise-level un-certainty). The discrepancy between the designed ¯Kf 0 and ¯Kf 1 and the true ¯Kf 0 and K¯f 1 is smaller and remains almost constant. The reason why ECRE needs more sens-ing time is that its thresholds at FC are larger than or equal to the thresholds of the other two method under the same ¯Kf. The SPRT, as expected, offers a more robust performance against noise-level uncertainty.
Figs. 5.28-5.31 plots the sensing time, false alarm and detection probabilities and the true ¯Kf 0 and ¯Kf 1 using 20 secondary users in an AWGN channel with noise-level uncertainty. One can see that the average sensing time is not very sensitive to the noise-level uncertainty but the true ¯Kf 0 and ¯Kf 1 is proportional to the noise-level uncertainty which also cause the false alarm and detection probabilities performance degradation.
Figs. 5.32–5.33 shows false alarm and detection probabilities performance of the SPRT-SPRT method with 20 secondary users in AWGN channels with noise-level
un-2 4 6 8 10 12 14 16 18 20 0
5 10 15 20 25
Theoretical E[K
f|H
i] E[K f] in simulation
FSST−FSST SPRT−FSST H
0
SPRT−FSST H
1
FSST−SPRT SPRT−SPRT H
0
SPRT−SPRT H
1
Figure 5.1: ¯Kf i used in simulation for the four distributed sensing schemes.
certainty. SNR at the secondary BS is -5 dB. The effect of noise-level uncertainty becomes more apparent as SNR decreases. Similar behaviors are observed in Figs. 5.34–
5.37 where different sampling intervals are assumed. It is reasonable that increasing sampling interval leads to enhanced system performance. The price we paid is longer average sensing time and larger control channel bandwidth.
2 4 6 8 10 12 14 16 18 20
Figure 5.2: Normalized sensing time as a function of ¯Kf 0for the four distributed sensing schemes.
Figure 5.3: False alarm probability as a function of ¯Kf 0 for the four distributed sensing schemes.
2 4 6 8 10 12 14 16 18 20
Figure 5.4: Detection probability as a function of ¯Kf 0 for the four distributed sensing schemes.
Figure 5.5: The average LR of the four terms in (4.24) of SPRT-SPRT for different K¯f 0’s.
2 4 6 8 10 12 14 16 18 20 E[K fi] in simulation
AWGN H0
0 (Rician factor=1) Rician H
1 (Rician factor=1)
Figure 5.6: ¯Kf i used in simulation in different fading and AWGN channels.
2 4 6 8 10 12 14 16 18 20
total sensing time (TW)
AWGN − H
Figure 5.7: Normalized sensing time as a function of ¯Kf 0in different fading and AWGN channels.
2 4 6 8 10 12 14 16 18 20 0
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
Theoretical E[K
fi]
False alarm probability
AWGN Rayleigh fading
Rician fading (Rician factor=1) Global requirement
Figure 5.8: False alarm probability as a function of ¯Kf 0 in different fading and AWGN channels.
2 4 6 8 10 12 14 16 18 20
0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 1.005
Theoretical E[K
fi]
Detection probability
AWGN Rayleigh fading
Rician fading (Rician factor=1) Global requirement
Figure 5.9: Detection probability as a function of ¯Kf 0 in different fading channel and AWGN.
2 4 6 8 10 12 14 16 18 20 E[K fi] in simulation
v=10 H fast fading H0 fast fading H1 slow fading H0 slow fading H
1
Figure 5.10: ¯Kf i used in simulation in Jakes’ fading, slow and fast Rayleigh fading channel.
Figure 5.11: Normalized sensing time as a function of ¯Kf 0 in Jakes’ fading, slow and fast Rayleigh fading channels.
2 4 6 8 10 12 14 16 18 20 0
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
Theoretical E[K
fi]
False alarm probability
v=10 v=40 v=100 slow fading fast fading
Figure 5.12: False alarm probability as a function of ¯Kf 0 in Jakes’ fading, slow and fast Rayleigh fading channels.
2 4 6 8 10 12 14 16 18 20
0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 1.005
Theoretical E[K
fi]
Detection probability
v=10 v=40 v=100 slow fading fast fading
Figure 5.13: Detection probability as a function of ¯Kf 0 in Jakes’ fading and slow and fast Rayleigh fading channels.
2 4 6 8 10 12 14 16 18 20 E[K fi] in simulation
slow fading H
Figure 5.14: ¯Kf i used in simulation in slow and fast Rician fading channels.
2 4 6 8 10 12 14 16 18 20
Figure 5.15: Normalized sensing time as a function of ¯Kf 0in slow and fast Rician fading channels.
2 4 6 8 10 12 14 16 18 20 0
0.005 0.01 0.015
Theoretical E[K
fi]
False alarm probability
slow fading fast fading
Figure 5.16: False alarm probability as a function of ¯Kf 0 in slow and fast Rician fading channels.
2 4 6 8 10 12 14 16 18 20
0.98 0.985 0.99 0.995 1 1.005
Theoretical E[K
fi]
Detection probability
slow fading fast fading
Figure 5.17: Detection probability as a function of ¯Kf 0 in slow and fast Rician fading channels.
2 4 6 8 10 12 14 16 18 20
Theoretical E[Kfi] E[K fi] in simulation
fast fading H0 m=2
Figure 5.18: ¯Kf i used in simulation in slow and fast Nakagami-m fading channels.
2 4 6 8 10 12 14 16 18 20
Theoretical E[Kfi]
Sensing time
Figure 5.19: Normalized sensing time as a function of ¯Kf 0in slow and fast Nakagami-m fading channels.
2 4 6 8 10 12 14 16 18 20 0
0.005 0.01 0.015 0.02 0.025 0.03
Theoretical E[K
fi]
False alarm probability
fast fading m=2 fast fading m=0.1 slow fading m=2 slow fading m=0.1
Figure 5.20: False alarm probability as a function of ¯Kf 0in Nakagami-m fading channel.
2 4 6 8 10 12 14 16 18 20
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Theoretical E[K
fi]
Detection probability
fast fading m=2 fast fading m=0.1 slow fading m=2 slow fading m=0.1
Figure 5.21: Detection probability as a function of ¯Kf 0 in Nakagami-m fading channel.
2 4 6 8 10 12 14 16 18 20 22 E[K fi] in simulation
SPRT−FSST H
Figure 5.22: ¯Kf i used in simulation for SPRT-FSST scheme with advanced methods of control channel adjustment.
Theoretical E[Kfi]
sensing time (TW)
SPRT−FSST H
0
SPRT−FSST H1 SPRT−FSST ECUBE H1 SPRT−FSST ECUBE H
0
SPRT−FSST ECRE H
0
SPRT−FSST ECRE H1
Figure 5.23: Normalized sensing time as a function of ¯Kf 0 for SPRT-FSST scheme with advanced methods of control channel adjustment.
0 5 10 15 20
Theoretical E[Kfi] E[K fi] in simulation
SPRT−SPRT H0 SPRT−SPRT H1 SPRT−SPRT ECUBE H
0
SPRT−SPRT ECUBE H1 SPRT−SPRT ECRE H
0
SPRT−SPRT ECRE H1 theorerical line
Figure 5.24: ¯Kf i used in simulation for SPRT-SPRT scheme with advanced methods of control channel adjustment.
Theoretical E[Kfi]
SPRT−SPRT H 0 SPRT−SPRT H
1 SPRT−SPRT ECUBE H0 SPRT−SPRT ECUBE H
1 SPRT−SPRT ECRE H
0 SPRT−SPRT ECRE H1
Figure 5.25: Normalized sensing time as a function of ¯Kf 0for SPRT-SPRT scheme with advanced methods of control channel adjustment.
2 4 6 8 10 12 14 16 18 20 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Theoretical E[K
fi]
False alarm probability
FSST−FSST FSST−SPRT SPRT−FSST SPRT−SPRT
Figure 5.26: False alarm probability as a function of ¯Kf 0 for various sensing schemes with noise uncertainty.
2 4 6 8 10 12 14 16 18 20
0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
Theoretical E[K
fi]
Detection probability
FSST−FSST FSST−SPRT SPRT−FSST SPRT−SPRT
Figure 5.27: Detection probability as a function of ¯Kf 0 for various sensing schemes with noise uncertainty.
2 4 6 8 10 12 14 16 18 20
Theoretical E[Kfi] E[Kfi] in simulation
noise uncetainty free H0 noise uncetainty free H1 noise uncetainty 0.05dB H0 noise uncetainty 0.05dB H
1 noise uncetainty 1dB H0 noise uncetainty 1dB H1 noise uncetainty 2dB H
0 noise uncetainty 2dB H1
Figure 5.28: ¯Kf i used in simulation for SPRT-SPRT scheme with noise uncertainty.
2 4 6 8 10 12 14 16 18 20
noise uncetainty free H0 noise uncetainty free H
1 noise uncetainty 0.05dB H0 noise uncetainty 0.05dB H
1 noise uncetainty 1dB H0 noise uncetainty 1dB H1 noise uncetainty 2dB H
0 noise uncetainty 2dB H1
Figure 5.29: Normalized sensing time as a function of ¯Kf 0for SPRT-SPRT scheme with noise level uncertainty.
2 4 6 8 10 12 14 16 18 20 0
0.05 0.1 0.15 0.2 0.25
Theoretical E[K
fi]
False alarm probability
noise uncetainty free noise uncetainty 0.05dB noise uncetainty 1dB noise uncetainty 2dB
Figure 5.30: False alarm probability as a function of ¯Kf 0 for SPRT-SPRT scheme with noise uncertainty.
2 4 6 8 10 12 14 16 18 20
0.99 0.992 0.994 0.996 0.998 1 1.002
Theoretical E[K
fi]
Detection probability
noise uncetainty free noise uncetainty 0.05dB noise uncetainty 1dB noise uncetainty 2dB
Figure 5.31: Detection probability as a function of ¯Kf 0 for SPRT-SPRT scheme with noise uncertainty.
2 4 6 8 10 12 14 16 18 20 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Theoretical E[K
fi]
False alarm probability
noise uncetainty 0.05dB noise uncetainty 1dB noise uncetainty 2dB
Figure 5.32: False alarm probability as a function of ¯Kf 0 for SPRT-SPRT scheme with noise uncertainty; SNR at the secondary BS (fusion center) = -5 dB.
2 4 6 8 10 12 14 16 18 20
0.94 0.95 0.96 0.97 0.98 0.99 1
Theoretical E[K
fi]
Detection probability
noise uncetainty 0.05dB noise uncetainty 1dB noise uncetainty 2dB
Figure 5.33: Detection probability as a function of ¯Kf 0 for SPRT-SPRT scheme with noise uncertainty; SNR at the secondary BS (fusion center) = -5 dB.
2 4 6 8 10 12 14 16 18 20
Theoretical E[Kfi] E[K fi] in simulation
Ts=1 H0
Figure 5.34: ¯Kf i used in simulation in scheme SPRT-SPRT with noise uncertainty and different sampling intervals; SNR at the secondary BS (fusion center) = -5 dB.
2 4 6 8 10 12 14 16 18 20
Figure 5.35: Normalized sensing time as a function of ¯Kf 0for SPRT-SPRT scheme with noise uncertainty and different sampling intervals in the sensors; SNR at the secondary BS (fusion center) = -5 dB.
2 4 6 8 10 12 14 16 18 20 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14
Theoretical E[K
fi]
False alarm probability
Ts=1 Ts=4 Ts=8 Ts=16
Figure 5.36: False alarm probability as a function of ¯Kf 0 for SPRT-SPRT scheme with noise uncertainty and different sampling intervals in the sensors; SNR at the secondary BS (fusion center) = -5 dB.
2 4 6 8 10 12 14 16 18 20
0.982 0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 1.002
Theoretical E[K
fi]
Detection probability
Ts=1 Ts=4 Ts=8 Ts=16
Figure 5.37: Detection probability as a function of ¯Kf 0 for SPRT-SPRT scheme with noise uncertainty and different sampling intervals in the sensors; SNR at the secondary BS (fusion center) = -5 dB.