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A classifier for MFSK signals contaminated with Middleton's class-A impulsive noise and transmitted over a time-varying flat correlated fading channel is developed. The correlated fading process of the channel is expanded using the Karhunen-Loeve expansion. The classifier is derived for both synchronous and asynchronous waveforms. Asymptotic theoretical performance evaluation of the classifier is derived. Computer simulations are illustrated to validate the theoretical developments. The performance of the developed classifier is compared with the performance of the conventional classifier (designed for AWGN). It is shown that the conventional classifier has severe performance degradation if it is subject to impulsive noise. A greater performance improvement is obtained when using the developed classifier, especially for smaller impulsive indexes of the noise. It is also shown that the performance of the developed classifier is sensitive to the impulsive index (A) of the noise. Increasing A enhances the performance of the classifier.