ere the PU signal, interference, and noise are all independent. RS+I+n = Using trace properties, we can rewrite it as: The expectation of a PU channel multiplied by its Hermitian a correlation matrix of that channel (R). For the ED-based spectrum sensing in the CRN, the ith SU measures the average energy of the received signal as follows: (8.3) 5G Wireless Communication System in Healthcare Informatics 8.4 DERIVATION OF THE PROBABILITY OF DETECTION AND THE PROBABILITY OF FALSE ALARM In essence, the decision variable given in (8.3) can be reformulated in the indefinite quadratic form (IQF): (8.4) Without loss of generality, we assume that complex Gaussian with zero mean and correlation matrix R. Where is the identity matrix for uncorrelated Xi. Now let x ~ N (0,1) be the whitened version of X N (0,Rx) X = R - H 12x Thus, we rewrite yi in terms of matrix A as where 0 0 0 Here, o2 is the signal energy depending on the hypothesis H / H1. Thus, it can be evaluated as (8.5) The probability of detection(p): and the probability of false alarm (p) for the ith CR relay can be obtained by evaluating the following probabilities: (8.6) Optimized Cognitive Radio Networks (8.7) The probabilities pr (y<8H) and (y<eH1) can be found similarly as Equation (24) from [24]. ex(jw)B) Since all diagonal elements are the same in a whited matrix A, all eigenvalues are the = to solve the above integration, we need apply partial fractions for the inner fraction of the below expression, which can be expanded as: After simplifying the partial fraction, the evaluation of A0=1,A,=== which results in the following two integrals: Hence, applying the residue theory approach of [24], we can formulate the final CDF as below: (8.8) for k = 0, 1. 8.5 PROPOSED HEURISTIC METHODS FOR SPECTRUM SENSING OPTIMIZATION In the proposed solution, we develop heuristic algorithms to obtain the optimum solu- tion for beam vector w by maximizing the probability of detection while constraining the probability of false alarm to some threshold value, that is: min J (w)