## 6 Complexity and Volume of Data Sent to the FC
Each CR generates \(P=70\) sample matrices (one per subchannel) of order \(K^{\prime}\times N=24\times 240\), resulting in 70 covariance matrices of order \(K^{\prime}\times K^{\prime}=24\times 24\) and \(K^{\prime}=24\) eigenvalues per subchannel. Also assuming that each eigenvalue undergoes a 3-bit quantization, then the total amount of data sent to the FC is \(3 mK^{\prime}P=3 mK=3\times 10\times 1,680=50,400\) bits in the case of eigenvalue fusion. The scenario of decision fusion is similar, but only the individual CR decisions about a subchannel occupation are sent to the FC. Therefore, only \(mP=10\times 70=700\) binary decisions are sent to the FC, which represents a much smaller volume than in the case of the other fusion schemes. Let us now consider a sample fusion process, using the concatenation approach to form the received sample matrices, as determined from Equation (28). The choice for the concatenation approach is based on the fact that it produces better performance than MRC, also avoiding the need of knowing the channel gains used in the MRC. A number of \(K^{\prime}N=24\times 240=5,760\) samples are collected by each CR for each of the \(P=70\) subchannels. These samples are then sent to the FC, where a number of \(mP=10\times 70=700\) sample matrices of order \(K^{\prime}\times N=24\times 240\), as in Equation (22), are formed. The matrices corresponding to the subchannels from all CRs are concatenated according to Equation (28),leading to P = 70 sample matrices of order \(K^{\prime}\times mN=24\times 2400\). A number of \(PK^{\prime}=70\times 24=1,680\) eigenvalues are subsequently estimated to form the test statistics for the \(P=70\) subchannels. The analysis considering the detection of unused OFDMA subchannels shows that the volume of data sent to the FC in the case of decision fusion is \(mP\) bits. For the eigenvalue fusion this volume is \(bmK=bK^{\prime}mP\) bits, where \(b\) is the number of bits per eigenvalue. For the sample fusion, this volume grows to \(bmKN\) bits. Then, the volume of data sent to the FC in the case of eigenvalue fusion is \(bK^{\prime}\) times the volume of data in the case of decision fusion, and \(N\) times smaller than in the case of sample fusion. In terms of complexity, in the case of eigenvalue and decision fusions, each CR must be capable of processing \(n=KN\) samples and estimate \(K^{\prime}P=(K/P)P=K\) 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 \(mKN\) samples, which is \(m\) times greater than the number of samples processed by each CR. Nevertheless, the FC must estimate \(PK^{\prime}\) eigenvalues, the same number of eigenvalues estimated by each CR in the decision fusion and eigenvalue fusion cases. Let us make a last comparison between the eigenvalue fusion and the samples fusion by fixing the number of samples. Assume that each CR will take \(N\) samples per subcarrier. Therefore each CR will collect \(n=NK\) samples in the time domain. If the purpose is to sense each subcarrier or sub-band of a general wideband signal, in a sample fusion scenario the amount of samples sent to the FC will be \(mNK=m^{2}KN/m\). In the eigenvalue fusion scheme, if each CR forms covariance matrices of order \(J=m\) (as before) and computes \(J=m\) eigenvalues, the total amount of samples sent to the FC is \(m^{2}K\). Notice that the sample fusion scheme sends \(N/m\) times more data than the proposed eigenvalue scheme. Since the expected number of cooperating CRs is by far smaller than the number of collected samples in a real system, one can conclude that the eigenvalue fusion is capable of reducing considerably the amount of data in the control channel when compared with the sample fusion.