Although a low FSR can benefit the overall system throughput, Figure 4.2(b) indi-cates that it results in a high missing ratio at low SNR. The primary users then suffer from larger MAI. To maintain the primary users’ transmissions under a tolerable inter-ference level, controlling the false ignorance rate (FIR) seems to be a feasible option and is presented in this chapter.
Define the FIR as the expected proportion of the number of falsely ignored channels over the total number of channels which are declared unoccupied. Based on this defini-tion, the new two hypotheses for a sub-band j scanned by a secondary user i are now given by
H0 : Yi,j = Xi,j+ Ni,j H1 : Yi,j = Ni,j
where H0 assumes that the sub-band is occupied and H1 assumes the sub-band is unoc-cupied.
Unlike the FSR case, the observed energy |Yi,j|2 of the i-th secondary user to the j-th sub-band have different exponential distributions depending on the different signal strength under the null hypothesis H0. An ideal operating mode which assumes that the fusion center has the SNR information of the cognitive users is proposed first. This assumption is made here to demonstrate the effect of controlling the FIR. We consider later a more practical case in which the fusion center is operated under a nominal SNR in the end of this chapter.
4.2.1 The Ideal SNR Mode
In this mode, the fusion center is assumed to have the perfect knowledge of SNRs from each cognitive user. Assume that Ni,j ∼ CN (0, 1) and Xi,j ∼ CN (0, q), with q > 1. Conditioned on H0, Y = |Yi,j|2is a random variable with exponential distribution denoted by Y ∼ E(q+11 ). Denote PM as the probability of success (missing probability) of the Bernoulli trial when each time a cognitive user makes his own decision, i.e.
PM = P r{|Yi,j|2 ≤ λ |H0, q} = P r{Y ≤ λ |H0, q}
At the fusion center, the p-values of each sub-band are calculated by the CCDF of the Binomial distribution.
where Nj is the total number of decision bits in sub-band j and xj is the number of 0’s in Nj.
Now, we apply the BH procedure at level α to detect the availability of the sub-bands as well. The steps are listed below.
1. Calculate each p-value pj, j = 1, 2, ..., M . 2. Sorting pj, j = 1, 2, ..., M in ascending order.
3. Find the largest index, imax, such that pimax ≤ imaxM α.
4. Declare sub-band j unoccupied for 1 ≤ j ≤ imax.
4.2.2 The Nominal SNR Mode
We consider here a more practical operating environment where the fusion center has no idea about the SNR information of each cognitive user. Instead of using the perfect SNR, a nominal SNR is set for the fusion center. We note that for each cognitive user, the decisions of the chosen sub-bands keep the same as the ideal SNR mode. Since the fusion center always operates at the nominal SNR, only the p-values of each sub-band are changed in this case.
Denote q∗ as the nominal SNR and PM∗ as the probability of success corresponding to q∗ at the fusion center. The expression of PM∗ is obtained straightforwardly by substi-tuting q∗ for q in (4.4). Besides, by substituting PM∗ for PM in (4.6), the nominal p-value for each sub-band denoted by p∗j, j = 1, . . . , M , is also obtained. The equations for PM∗ and p∗j, j = 1, . . . , M are given below respectively.
PM∗ = P r{|Yi,j|2 ≤ λ |H0, q∗} = P r{Y ≤ λ |H0, q∗}
= Z λ
0
1
q∗+ 1e−q∗+1y dy
= 1 − e−q∗+1λ (4.7)
p∗j =
Again, the BH procedure is applied to detect the availability of each sub-band based on the nominal p-values.
E(# false alarms / # unoccupied sub−bands)
False alarm ratio for controlling FIR
Ideal SNR S=30
E(# missings / # occupied sub−bands)
Missing ratio for controlling FIR
Ideal SNR S=30 Nominal SNR S=30 FSR DF with S=30
(a) (b)
Figure 4.3: False alarm ratio vs. missing ratio for Ideal SNR mode and Nominal SNR mode with M = 100, K = 20, λF IR = 1, α = 0.05 and Nominal SNR = 10 dB.
As can be seen in Figure 4.1(b) and Figure 4.3, the ideal SNR mode obviously keeps the FIR under the desired level α and achieves a really low missing ratio at the same time. While it also makes the false alarm ratio at low SNR quite high, which is similar to what the missing ratio behaves in Figure 4.2(b) for FSR. However, to protect the primary users from large interference with low missing ratio is our major concern here.
The nominal SNR mode is discussed below, and the fusion center is set to operate at
the nominal SNR = 10 (dB). Figure 4.3(b) and Figure 4.1(b) show respectively that the missing ratio increases in the SNR region lower than 10 (dB) and the FIR is also out of control in that region. In addition, the false alarm ratio becomes stable around 0.1 in Figure 4.3(a). Despite losing some control, the system seems to be operated under a good balance between the missing ratio and the false alarm ratio.
The reason why the nominal SNR mode behave in Figure 4.1(b) and Figure 4.3 is described here. When 0 ≤SNR≤ 10 (dB), the fusion center takes PM∗ which is greater than the real PM as the probability of success for calculating the p-values of each sub-band. Therefore a larger p-value is obtained than the ideal SNR mode which implies higher availability of the sub-band. The missing ratio then increases after applying the BH procedure. For SNR ≥ 10 (dB), since PM∗ is always smaller than the real PM, the missing ratio will only be better than the ideal SNR mode. Note that PM∗ is fixed once we set the nominal SNR at the fusion center. In order to compensate the effect caused by PM∗, we should adjust the energy threshold to keep the missing ratio in an acceptable level.
Chapter 5 Application
In this chapter some applications are demonstrated to show the valuable extensions of the proposed cooperative sensing method. The first application is called ”The com-bined FSR and FIR approach”. In order to enhance the sensing performance, the BH procedures for controlling the FSR and the FIR are combined at the fusion center. The second application is called ”Detection of the signal strength in each sub-band”. By setting multiple hypotheses of different levels, we can estimate the signal strength in each sub-band after adopting the BH procedure. Since the BH procedure is a simple and low complexity multiple comparison procedure, it will not add much loading to the fusion center even if the BH procedure is applied more than one time.