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In this paper, we analyze the performance of different diversity combining techniques over fading channels with impulsive noise. We use Middleton's Class A model for the noise distribution and adopt two noise models, which assume dependent and independent noise components on each branch. We systematically analyze the performance of maximum ratio combing (MRC), equal gain combining (EGC), selection combining (SC), and post-detection combining (PDC) under these impulsive noise models, and derive insightful lower and upper bounds. We show that even under impulsive noise, the diversity order is retained for each combining scheme. However, we also show that under both models, there is a fundamental tradeoff between diversity gain and coding gain. Under the independent noise model, PDC is shown to combat impulsive noise more effectively than MRC, EGC, and SC. Our simulation results also corroborate our analysis.