## 3 Subchannel Selection Policies  
Suppose the wireless channel is a frequency-selective Additive White Gaussian Noise (AWGN), the bandwidth is \(B\) Hz, and the power spectral density is \(N_{0}\). In this paper, it is divided into \(N\) Rayleigh fading subchannels, and the subchannel coherence bandwidth is \(\Delta f\) Hz. Therefore, \(B=N\Delta f\). These subchannels are indexed by \(i=1\), \(2\),..., \(N\), and the gains of every subchannel are independent and identically distributed (i.i.d). Active PUs use DS-CDMA technology to access the spectrum band with spreading gain \(G\). According to the Central Limit Theorem, the interference process in the receiver of the SUs caused by a large number of PUs is considered a Gaussian approximation. Furthermore, according to the second-order statistics, the interference process is a white process [14]. Therefore, in each subchannel, the average interference introduced by the PUs at the receiver of the SUs is \((K-1)N_{0}\Delta f,K\geq 1\), where \(K\) is a system parameter related to the characteristics of PUs network [15]. As shown in Figure 4, the SUs utilize NC-OFDM to access the SUTCH which is indexed by \(j=1\), \(2\),..., \(M\), \(0\leq M\leq N\). The SUs spread their SUCCH power spectrum density over an ultra-wide bandwidth to exploit the underused (gray) spectral. \(Q\) is defined as the interference threshold of the PUs, which is the maximum allowable temporal interference in the receiver of the PUs caused by concurrent activity of the SUs in the same subchannel. As mentioned in Figure 4, the protocol to set up CR sensor networks is based on the time-slot structure. Therefore, in order to satisfy the interference threshold constraint, the power of the SUs accessing the SUTCH should be controlled in each time-slot. In this paper, the structure of the accessing system is depicted in Figure 5. For subchannel \(j\), the instantaneous gain between the transmitter and receiver of the SU is defined as \(g_{ss}^{j}\), and the instantaneous gain between the transmitter of the SU and the receiver of the PU is defined as \(g_{ps}^{j}\). Subscripts \(s\) and \(p\) refer to the secondary and the primary user, respectively. The \(g_{ss}^{j}\) and \(g_{ps}^{j}\) are assumed as the stationary and ergodic independent distributed random variables with unit-mean. Their Probability Density Functions (PDFs) are defined as \(f_{ss}^{j}(g_{ss}^{j})\) and \(f_{ps}^{j}(g_{ps}^{j})\), respectively. Channel gains \(g_{ss}^{j}\) and \(g_{ps}^{j}\) are i.i.d., \(j=1\), \(2\),..., \(M\). In this paper, we suppose the perfect Channel Side Information (CSI) pair (\(g_{ss}^{j}\), \(g_{ps}^{j}\)) can be available in the transmitters. Here, the CSI contains the probability distribution of the channel gain, as well as the actual value at a certain time-slot. Actually, the CSI pair can be estimated by a spectral coordinator or proper signaling. Note that, the result derived from this assumption is an upper-bound in the case without a perfect CSI pair. In this paper, we focus on the maximum achievable spectrum capacity of SUTCH, which is studied [16; 17]. Since more than one SUs will compete to access to the underused frequency band. The SUs' total available spectrum capacity is upper-bound by the case of only one SU, which is due to  
Figure 5: The structure of the accessing system for subchannel \(j\). the fact that SUs will impose interference on each other. Therefore, the discussion of the individual SU can also be used as the upper-bound of the total spectrum capacity of all SUs.