The basis of this paper is Wei and Tirkkonen, 2012, in which expressions for the key performance metrics ofthe sphericity test applied to the multiantenna cooperative spectrum sensing of multiple primarytransmitters in cognitive radio networks over nonfading channels are provided. The false alarm andthe detection probabilities were derived in Wei and Tirkkonen, 2012, based on approximations obtained by matching themoments of the test statistics to the Beta distribution. In this paper we show that the model adoptedin Wei and Tirkkonen, 2012, does not apply directly to fading channels, yet being considerably inaccurate for some systemparameters and channel conditions. Nevertheless, we show that the original expressions from Wei and Tirkkonen, 2012, canbe simply and accurately applied to a modified model that considers fixed or time-varying channelswith any fading statistic. We also analyze the performance of the sphericity test and other competingdetectors with a varying number of primary transmitters, considering different situations in termsof the channel gains and channel dynamics. Based on our results, we correct several interpretationsfrom Wei and Tirkkonen, 2012, in what concerns the performance of the detectors, both over a fixed-gain additive whiteGaussian noise channel and over a time-varying Rayleigh fading channel.
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