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In this study, the authors consider the problem of cooperative spectrum sensing (CSS) based on linear combination of observations over correlated log-normal shadow-fading channels. To reduce the effects of imperfect reporting channels, a cluster-based CSS framework and a new cluster head selection algorithm are proposed. Using the received energies (as local observations) from different clusters, the fusion centre can make the final decision by linearly combining the noisy cluster observations. To calculate the combination weights, the authors come across the problem of joint distribution approximation of sum of the correlated log-normal random variables corresponding to different clusters. A joint moment generating function (MGF) matching algorithm is proposed in this study to estimate the summations by a single log-normal vector. Monte Carlo simulations confirm the accuracy of the proposed MGF-based approach in estimating the desired statistics and efficiency of the cluster-based spectrum-sensing algorithm in terms of primary signal detection.