## 6 Complexity and Volume of Data Sent to the FC  
In this section, we generalize the exemplifying analysis given in the previous subsection, concerning the tradeoff between the volume of data sent to the FC and the complexity related to the number of samples handled and to the computations of eigenvalues for all fusion methods under analysis. We consider a WiMAX system as a case study for providing numerical results as well. Assume that the WiMAX channel has \(2048\) subcarriers. For spectral roll-off reasons, only \(1,680\) subcarriers are utilized, leaving unused the subcarriers at the edge of the channel. The OFDMA subchannels are created by partitioning the \(K=1,680\) useful subcarriers in groups of \(K^{\prime}=24\) subcarriers, resulting in \(P=70\) subchannels. Let us consider first that detecting the _unused subcarriers_ of the OFDM signal is the objective of the spectrum sensing. For a sample fusion scheme, the number of samples must be much greater (say, 10 times or more) than the number of cooperating CRs, so that a good estimate of the sample covariance matrix is achieved. Considering a secondary network with \(m=10\) cooperating CRs and assuming that each CR collects \(N=100\) samples per subcarrier, a total of \(n=KN=1,680\times 100=168,000\) samples are collected per CR. Therefore, \(mKN=10\times 1,680\times 100=1,680,000\) samples are sent to the FC. Assuming that each sample undergoes a 3-bit quantization [25], then the total amount of data sent to the FC is \(5,040,000\) bits. Consider now an eigenvalue fusion scheme. For a fair comparison, assume again that received sample matrices of order \(J\times N/J=10\times 100\) are formed at each CR per subcarrier, also leading to covariance matrices of order \(J\times J=10\times 10\). The total number of samples taken by each CR is then \(n=KJ(N/J)=KN=1680\times 1000=1,680,000\), ten times more than in the case of sample fusion (notice that this increased number of samples can be obtained during the same sensing period by increasing the sampling rate \(J\) times compared with the sample fusion case). Each CR computes \(J=10\) eigenvalues per subcarrier to be transmitted to the FC. Also assuming that each eigenvalue undergoes a 3-bit quantization, the total amount of data sent to the FC is \(3 mKJ=3\times 10\times 1,680\times 10=504,000\) bits. The above analysis shows that the volume of data sent to the FC in the case of sample fusion (\(\propto mKN\) bits) is much greater than in the case of eigenvalue fusion (\(\propto mKJ\) bits), since \(N\gg J\). In the scenario of decision fusion, only the individual CR decisions are sent to the FC. Therefore, the FC receives \(mK\) binary decisions, which represents a volume of data much smaller than in the case of other fusion schemes. In terms of complexity, in the case of eigenvalue and decision fusions, each CR must be capable of processing \(n=KJ(N/J)=KN\) samples and estimate \(KJ\) eigenvalues. In the case of sample fusion, the complexity of the CR is reduced, since the eigenvalues are computed at the FC. However, the FC must be capable of processing \(m\) times more samples, _i.e._, \(mn=mKN\) samples and estimate \(mK\) eigenvalues (notice that \(m=J\) in the above fair scenario). Now, let us consider that detecting the unused subchannels of the OFDMA signal is the goal. Consider first the eigenvalue fusion scheme. The number of lines in the received sample matrix Equation (22) is equal to the number of subcarriers in a subchannel, which is \(K^{\prime}=24\) in the WiMAX example under analysis. Again, for a good estimation of the sample covariance matrix, the number of samples must be much greater (say, 10 times or more) than the number of subcarriers in a subchannel, which is the order of the covariance matrices. Then, let us consider that the number of samples per subcarrier is \(N=240\). Thus, the total number of samples collected by the each CR is \(n=KN=1,680\times 240=403,200\).