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In this paper, two narrowband-interference (NBI) rejection methods are formulated for power-line orthogonal frequency-division multiplexing (OFDM) communications. The inherent NBI suppression capability of the OFDM system is improved upon through the application of a spreading sequence (i.e., IS-OFDM) and further improved through the utilization of NBI filtering implemented at the OFDM receiver. A block least-mean-square (BLMS) algorithm, called cyclic prefix BLMS (CP-BLMS), is constructed for time-domain NBI filtering and a corresponding fast BLMS (FBLMS) algorithm, called linear FBLMS (LFBLMS), is also presented which performs the NBI filtering in the frequency domain. The fast implementation reduces the computational burden of the additional filtering process for a large number of subcarriers without impact on filter performance. Both simulation and theoretical analysis demonstrate the bit-error rate improvement of the proposed approaches compared to previous methods. Numerical results from theoretical analysis also illustrate the speed advantage of the LFBLMS algorithm.