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Asynchronous impulsive noise and periodic impulsive noises limit communication performance in OFDM powerline communication systems. Conventional OFDM receivers that assume additive white Gaussian noise experience degradation in communication performance in impulsive noise. Alternate designs assume a statistical noise model and use the model parameters in mitigating impulsive noise. These receivers require training overhead for parameter estimation, and degrade due to model and parameter mismatch. To mitigate asynchronous impulsive noise, we exploit its sparsity in the time domain, and apply sparse Bayesian learning methods to estimate and subtract the noise impulses. We propose three iterative algorithms with different complexity vs. performance trade-offs: (1) we utilize the noise projection onto null and pilot tones; (2) we add the information in the date tones to perform joint noise estimation and symbol detection; (3) we use decision feedback from the decoder to further enhance the accuracy of noise estimation. These algorithms are also embedded in a time-domain block interleaving OFDM system to mitigate periodic impulsive noise. Compared to conventional OFDM receivers, the proposed methods achieve SNR gains of up to 9 dB in coded and 10 dB in uncoded systems in asynchronous impulsive noise, and up to 6 dB in coded systems in periodic impulsive noise.