ics Co-Channel 1 Noise no,((t) Co-Channel 2 Co-Channel 3 Co-Channel C FIGURE 8.3 Architecture of the Beamforming-based Spectrum Sensing System Model. (8.2) At each node, the received signal xo,jt]is sampled with a rate of f==1/Th = over a time duration of T=NT S where Ts denotes the sampling period, and N denotes the number of samples that were utilized for the SS. The sampled version of the received signal is represented by xo,in at discrete time n which can be expressed as: whw(T) In the above, the average transmitted symbol energy of the licensed user (PU) signal and the energy of the co-channel interferer (CCI) signal are denoted as (E/E ) respectively. The square root of the symbol energy is multiplied with the Hermitian of the channel fading coefficient between a PU node and an SU node hoti Also, the square root of the energy of the co-channel interferer is multiplied with the Hermitian of the channel fading coefficient between the ith CCI node and SU node represents the sensing noise, assuming additive white Gaussian random variables with zero mean and variance o's experiencing independently and identically distributed (i.i.d.) fading effects. From the above two hypotheses, the distribution of the received signal at each antenna element is given below, where it is assumed to be an i.i.d. random process with complex normal distribution with zero mean and a correlation matrix defined below. Optimized Cognitive Radio Networks R1+n is the correlation matrix of Howhich represents the expectation of an interference term plus a sensing noise term. Because of the i.i.d. condition, we have E[In"]=E[n]"]=0 Using the trace properties, we can rewrite it as: The expectation of a CCI channel vector multiplied by its Hermitian E[hch produces a correlation matrix of that channel vector (R). Similarly, Rs+1+n is the correlation matrix of the received signal for H1 which represents the expectation of a PU transmitted signal term plus an interference term plus a sensing noise term, multiplied by their conjugate transpose. wh