The OFDM peak-to-average power ratio reduction apparatus and method provides a compressive sensing algorithm that estimates the sparsity pattern of a sparse vector by a limited number of measurements. When the positions of the clipped peaks are known beforehand by, e.g., a genie-augmented receiver, then the algorithm optimally performs amplitude estimation utilizing a least squares estimation technique. When the cardinality of the peak-reducing signal is known at the receiver at initialization, the receiver optimizes the estimated peak-reducing signal by using least squares.