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In this paper, we propose an efficient moment generating function (MGF)-based method to evaluate the performance of generalized selection combining (GSC) over different fading channels. Employing a recently proposed method which is, however, only applicable to GSC diversity with independent and identically distributed (i.i.d.) branches, we derive a MGF expression of the GSC output signal-to-noise ratio (SNR) for generalized fading channels, where the channel statistics in different diversity branches may be non-identical, or even distributed according to different distribution families. The resulting MGF expression is applicable to the analysis of the error probability, the outage probability, and the SNR statistics for GSC in a lot of wireless communications scenarios with generalized fading. Numerical examples are presented to illustrate the application of the new analysis.