In this sub-section, the performance of the K-means and EMGMM algo- rithms for spectrum sensing is investigated. In particular, for simplicity a scenario whereby the SUs experience near LOS propagation from the PU is assumed such that the magnitude of the PU-SU channel coefficient,|ϕ(xmsu)|, is considered to be fairly the same for the sensors. It is also assumed that the PU-SU channel is quasi-static throughout the learning and testing dura- tion. Furthermore, a two-SU, single-PU network is considered and the data set,S ={xi}Di=1 ∈ {H0, H1},xi ∈R2 is assumed to be collected across the
active and idle states of the PU. The PU is also assumed to switch states in a predetermined manner known to the SUs so that there is no overlapping in the data collection process. Under this setting, the underlying distribution of S may be characterized by GMM, essentially as a linear combination of two Gaussian components with different means and covariances. The PU transmit power is assumed to be one Watt and the noise is complex AWGN with power, ση2.
In Figure 4.2, the constellation plot of the K-means classifier’s input and output at SN Rof -13 dB, the sample number, Ns = 2000 and sensor num-
Section 4.3. Multivariate Gaussian Mixture Model Technique for Cooperative Spectrum Sensing91
ber,M = 2 is shown. Under the operating condition shown, the clusters can be seen to be overlapping and by examining the output of the algorithm, we notice how all data points are strictly assigned to one of the two classes as described. It is clear to see here that K-means algorithm made some cluster- ing error on both clusters which invariably affects the overall classification performance in terms of the spectrum hole detection.
18.5 19 19.5 20 20.5 21 21.5 22 22.5 18 20 22 24 Sensor 1 Sensor 2
K−means classifier input, SNR = −13dB, Ns = 2000, 1 PU
18.5 19 19.5 20 20.5 21 21.5 22 22.5 18 20 22 24 Sensor 2 Sensor 1
K−means classifier output, SNR = −13dB, Ns = 2000, 1 PU
Figure 4.2. Constellation plot showing clustering performance of K-means algorithm, SN R = -13dB, number of PU, P = 1, number of sensors, M = 2, number of samples,N s= 2000.
In Figure 4.3, the performance of the K-means algorithm in terms of
ROC curve is shown. Antenna number, M is set to 2 while SN R and Ns
are both varied. GivenP f aof 0.1, it can be seen that asNsis increased from
1000 to 2000, P d rises from 0.82 to 0.95 atSN R of -13 dB and from about 0.52 to 0.78 when SN R equals -15 dB, thus suggesting an improvement of about 13% and 26% respectively. Similarly, at the same P f a, when Ns is
fixed at 1000, P d rises from 0.52 to 0.81 (about 29 % gain) and 0.78 to 0.95 (about 17% gain) at Ns equals 2000 when SN R is raised from -15 dB
Section 4.3. Multivariate Gaussian Mixture Model Technique for Cooperative Spectrum Sensing92 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
Probability of false alarm
Probability of detection
K−means, Near LOS channel
Ns = 2000, SNR = −13dB, M = 2 Ns = 1000, SNR = −13dB, M = 2 Ns = 2000, SNR = −15dB, M = 2 Ns = 1000, SNR = −15dB, M = 2
Figure 4.3. ROC curves showing the sensing performance of theK-means algorithm, number of PU, P = 1, number of sensors, M = 2, number of samples, N s= 1000 and 2000, SN R = -13 dB and -15 dB.
to -13 dB. As expected, this also suggests that the scheme offers significant performance gain with increase in SN R. Using the same PU-SU operating scenario, the performance of the GMM scheme is investigated in Figure 4.4 at
SN Rof -13 dB and sample number, Ns= 2000 where the constellation plot
of Gaussian mixture with two components is shown as well as the contours of its corresponding probability density as obtained using the EM algorithm. Here, the capability of the EM algorithm to recognize and capture the user specified Gaussian components that are present in the mixture is also clearly seen. Although, the two bivariate normal components overlap, it is seen here that their peaks are reasonably distinguishable, thereby making clustering feasible. In Figure 4.5, the constellation plot of the training data is shown along with the estimated posterior probability for every data points which is used for deriving other underlying statistical properties of the clusters that are represented in the training data. In performing clustering, the data
Section 4.3. Multivariate Gaussian Mixture Model Technique for Cooperative Spectrum Sensing93
points are assigned to one of the two components in the mixture distribution corresponding to the highest posterior probability as shown in Figure 4.6.
18 19 20 21 22 23 18 18.5 19 19.5 20 20.5 21 21.5 22 22.5 23 Sensor 2 Sensor 1
EM−GMM classifier’s input
Figure 4.4. Constellation plot showing probability distribution of mixture components, SN R = -13dB, number of PU, P = 1, number of sensors, M
= 2, number of samples,N s= 2000.
Figure 4.7 shows therocof the EM based GMM spectrum sensing scheme where we investigated the performance of the scheme using Ns of 1000 and
2000 while theSN Ris set to -13 dB and -15 dB. It can be seen here that at the P f a of 0.1, detection probability increases from 0.4 to 0.7 (about 30% gain in P d) when M = 2, SN R = -15 dB and Ns is increased from 1000
to 2000. Similar trend in performance improvement can be observed when the SN R is adjusted from -15 dB to -13 dB, given M = 2 and Ns equals
1000 where P d rises rapidly from about 0.4 to 0.8, corresponding to a gain of about 40%.
These observable improvements indicate that in the lowSN Rregime, as the sample number is increased (more time is spent in sensing) or the receive
Section 4.3. Multivariate Gaussian Mixture Model Technique for Cooperative Spectrum Sensing94 18.5 19 19.5 20 20.5 21 21.5 22 22.5 18.5 19 19.5 20 20.5 21 21.5 22 22.5 23 Sensor 2 Sensor 1 Cluster 1 Cluster 2
Component 1 Posterior Probability
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 4.5. Constellation plot showing the mixture components’ posterior probability derived from the E-M algorithm, number of PU,P = 1, number of sensors, M = 2, number of samples,N s= 2000, SN R = -13 dB .
18.5 19 19.5 20 20.5 21 21.5 22 22.5 18.5 19 19.5 20 20.5 21 21.5 22 22.5 23 Sensor 2 Sensor 1
EM−GMM classifier’s output Cluster 1
Cluster 2
Figure 4.6. Constellation plot showing the clustering capability of the E-M algorithm, number of PU, P = 1, number of sensors, M = 2, number of samples,N s= 2000, SN R = -13 dB.
Section 4.4. Enhancing the Performance of Parametric Classifiers Using Kalman Filter 95 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
EMGMM, Near LOS channel
Probability of false alarm
Probability of detection
SNR = −13dB, Ns = 2000, M = 2 SNR = −15dB, Ns = 2000, M = 2 SNR = −13dB, Ns = 1000, M = 2 SNR = −15dB, Ns = 1000, M = 2
Figure 4.7. ROC curves showing the sensing performance of the E-M algorithm, number of PU, P = 1, number of sensors, M = 2, number of samples, N s= 1000 and 2000, SN R = -13 dB and -15 dB.
same time the representative components are more clearly separable, thus benefiting the GMM based sensing algorithm.
4.4 Enhancing the Performance of Parametric Classifiers Using